Opuscula MathematicaA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.
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AGH University of Science and Technology PressenCopyright AGH University of Science and Technology PressOpuscula Mathematica1232-92742300-6919Copyright AGH University of Science and Technology PressOpuscula Mathematicahttp://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
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Multiplicity results for perturbed fourth-order Kirchhoff-type problemsIn this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods.
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3740.pdf
Multiplicity results for perturbed fourth-order Kirchhoff-type problemsMohamad Reza Heidari TavaniGhasem Alizadeh AfrouziShapour Heidarkhanimultiplicity results; multiple solutions; fourth-order Kirchhoff-type equation; variational methods; critical point theorydoi:10.7494/OpMath.2017.37.5.755Opuscula Math. 37, no. 5 (2017), 755-772, http://dx.doi.org/10.7494/OpMath.2017.37.5.755Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.755http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3740.pdf375755772 Title: Multiplicity results for perturbed fourth-order Kirchhoff-type problems.

Abstract: In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods. Keywords: multiplicity results, multiple solutions, fourth-order Kirchhoff-type equation, variational methods, critical point theory. Mathematics Subject Classification: 34B15, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 755-772, http://dx.doi.org/10.7494/OpMath.2017.37.5.755.

]]>On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological modelThe paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3739.pdf
On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological modelPetro PukachVolodymyr Il'kivZinovii NytrebychMyroslava Vovkboundary value problem; beam vibrations; nonlinear evolution equation; Voigt-Kelvin model; memory; blowupdoi:10.7494/OpMath.2017.37.5.735Opuscula Math. 37, no. 5 (2017), 735-753, http://dx.doi.org/10.7494/OpMath.2017.37.5.735Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.735http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3739.pdf375735753 Title: On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model.

Author(s): Petro Pukach, Volodymyr Il'kiv, Zinovii Nytrebych, Myroslava Vovk.

Abstract: The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable. Keywords: boundary value problem, beam vibrations, nonlinear evolution equation, Voigt-Kelvin model, memory, blowup. Mathematics Subject Classification: 35G20, 35G31. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 735-753, http://dx.doi.org/10.7494/OpMath.2017.37.5.735.

]]>On 3-total edge product cordial connected graphsA \(k\)-total edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize connected graphs of order at least 15 admitting a 3-total edge product cordial labeling.
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3738.pdf
On 3-total edge product cordial connected graphsJaroslav Ivančo3-total edge product cordial labelings; 3-TEPC graphsdoi:10.7494/OpMath.2017.37.5.725Opuscula Math. 37, no. 5 (2017), 725-734, http://dx.doi.org/10.7494/OpMath.2017.37.5.725Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.725http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3738.pdf375725734 Title: On 3-total edge product cordial connected graphs.

Author(s): Jaroslav Ivančo.

Abstract: A \(k\)-total edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize connected graphs of order at least 15 admitting a 3-total edge product cordial labeling. Keywords: 3-total edge product cordial labelings, 3-TEPC graphs. Mathematics Subject Classification: 05C78. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 725-734, http://dx.doi.org/10.7494/OpMath.2017.37.5.725.

]]>Existence of minimal and maximal solutions to RL fractional integro-differential initial value problemsIn this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3737.pdf
Existence of minimal and maximal solutions to RL fractional integro-differential initial value problemsZ. DentonJ. D. RamírezRiemann Liouville derivative; integro-differential equation; monotone methoddoi:10.7494/OpMath.2017.37.5.705Opuscula Math. 37, no. 5 (2017), 705-724, http://dx.doi.org/10.7494/OpMath.2017.37.5.705Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.705http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3737.pdf375705724 Title: Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems.

Author(s): Z. Denton, J. D. Ramírez.

Abstract: In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example. Keywords: Riemann Liouville derivative, integro-differential equation, monotone method. Mathematics Subject Classification: 26A33, 34A08, 34A45, 45J05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 705-724, http://dx.doi.org/10.7494/OpMath.2017.37.5.705.

]]>Semicircular elements induced by p-adic number fieldsIn this paper, we study semicircular-like elements, and semicircular elements induced by \(p\)-adic analysis, for each prime \(p\). Starting from a \(p\)-adic number field \(\mathbb{Q}_{p}\), we construct a Banach \(*\)-algebra \(\mathfrak{LS}_{p}\), for a fixed prime \(p\), and show the generating elements \(Q_{p,j}\) of \(\mathfrak{LS}_{p}\) form weighted-semicircular elements, and the corresponding scalar-multiples \(\Theta_{p,j}\) of \(Q_{p,j}\) become semicircular elements, for all \(j\in\mathbb{Z}\). The main result of this paper is the very construction of suitable linear functionals \(\tau_{p,j}^{0}\) on \(\mathfrak{LS}_{p}\), making \(Q_{p,j}\) be weighted-semicircular, for all \(j\in\mathbb{Z}\).
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3736.pdf
Semicircular elements induced by p-adic number fieldsIlwoo ChoPalle E. T. Jorgensenfree probability; primes; \(p\)-adic number fields \(\mathbb{Q}_{p}\); Hilbert-space representations; \(C^{*}\)-algebras; wighted-semicircular elements; semicircular elementsdoi:10.7494/OpMath.2017.37.5.665Opuscula Math. 37, no. 5 (2017), 665-703, http://dx.doi.org/10.7494/OpMath.2017.37.5.665Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.665http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3736.pdf375665703 Title: Semicircular elements induced by p-adic number fields.

Author(s): Ilwoo Cho, Palle E. T. Jorgensen.

Abstract: In this paper, we study semicircular-like elements, and semicircular elements induced by \(p\)-adic analysis, for each prime \(p\). Starting from a \(p\)-adic number field \(\mathbb{Q}_{p}\), we construct a Banach \(*\)-algebra \(\mathfrak{LS}_{p}\), for a fixed prime \(p\), and show the generating elements \(Q_{p,j}\) of \(\mathfrak{LS}_{p}\) form weighted-semicircular elements, and the corresponding scalar-multiples \(\Theta_{p,j}\) of \(Q_{p,j}\) become semicircular elements, for all \(j\in\mathbb{Z}\). The main result of this paper is the very construction of suitable linear functionals \(\tau_{p,j}^{0}\) on \(\mathfrak{LS}_{p}\), making \(Q_{p,j}\) be weighted-semicircular, for all \(j\in\mathbb{Z}\). Keywords: free probability, primes, \(p\)-adic number fields \(\mathbb{Q}_{p}\), Hilbert-space representations, \(C^{*}\)-algebras, wighted-semicircular elements, semicircular elements. Mathematics Subject Classification: 05E15, 11R47, 11R56, 46L10, 46L40, 47L15, 47L30, 47L55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 665-703, http://dx.doi.org/10.7494/OpMath.2017.37.5.665.

]]>Block colourings of 6-cycle systemsLet \(\Sigma=(X,\mathcal{B})\) be a \(6\)-cycle system of order \(v\), so \(v\equiv 1,9\mod 12\). A \(c\)-colouring of type \(s\) is a map \(\phi\colon\mathcal {B}\rightarrow \mathcal{C}\), with \(C\) set of colours, such that exactly \(c\) colours are used and for every vertex \(x\) all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(\frac{v-1}{2}=qs+r\), with \(q, r\geq 0\). \(\phi\) is equitable if for every vertex \(x\) the set of the \(\frac{v-1}{2}\) blocks containing \(x\) is partitioned in \(r\) colour classes of cardinality \(q+1\) and \(s-r\) colour classes of cardinality \(q\). In this paper we study bicolourings and tricolourings, for which, respectively, \(s=2\) and \(s=3\), distinguishing the cases \(v=12k+1\) and \(v=12k+9\). In particular, we settle completely the case of \(s=2\), while for \(s=3\) we determine upper and lower bounds for \(c\).
http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3735.pdf
Block colourings of 6-cycle systemsPaola BonaciniMario GionfriddoLucia Marino6-cycles; block-colourings; G-decompositionsdoi:10.7494/OpMath.2017.37.5.647Opuscula Math. 37, no. 5 (2017), 647-664, http://dx.doi.org/10.7494/OpMath.2017.37.5.647Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.5.647http://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3735.pdf375647664 Title: Block colourings of 6-cycle systems.

Author(s): Paola Bonacini, Mario Gionfriddo, Lucia Marino.

Abstract: Let \(\Sigma=(X,\mathcal{B})\) be a \(6\)-cycle system of order \(v\), so \(v\equiv 1,9\mod 12\). A \(c\)-colouring of type \(s\) is a map \(\phi\colon\mathcal {B}\rightarrow \mathcal{C}\), with \(C\) set of colours, such that exactly \(c\) colours are used and for every vertex \(x\) all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(\frac{v-1}{2}=qs+r\), with \(q, r\geq 0\). \(\phi\) is equitable if for every vertex \(x\) the set of the \(\frac{v-1}{2}\) blocks containing \(x\) is partitioned in \(r\) colour classes of cardinality \(q+1\) and \(s-r\) colour classes of cardinality \(q\). In this paper we study bicolourings and tricolourings, for which, respectively, \(s=2\) and \(s=3\), distinguishing the cases \(v=12k+1\) and \(v=12k+9\). In particular, we settle completely the case of \(s=2\), while for \(s=3\) we determine upper and lower bounds for \(c\). Keywords: 6-cycles, block-colourings, G-decompositions. Mathematics Subject Classification: 05C15, 05B05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 5 (2017), 647-664, http://dx.doi.org/10.7494/OpMath.2017.37.5.647.

]]>Fan's condition on induced subgraphs for circumference and pancyclicityLet \(\mathcal{H}\) be a family of simple graphs and \(k\) be a positive integer. We say that a graph \(G\) of order \(n\geq k\) satisfies Fan's condition with respect to \(\mathcal{H}\) with constant \(k\), if for every induced subgraph \(H\) of \(G\) isomorphic to any of the graphs from \(\mathcal{H}\) the following holds: \[\forall u,v\in V(H)\colon d_H(u,v)=2\,\Rightarrow \max\{d_G(u),d_G(v)\}\geq k/2.\] If \(G\) satisfies the above condition, we write \(G\in\mathcal{F}(\mathcal{H},k)\). In this paper we show that if \(G\) is \(2\)-connected and \(G\in\mathcal{F}(\{K_{1,3},P_4\},k)\), then \(G\) contains a cycle of length at least \(k\), and that if \(G\in\mathcal{F}(\{K_{1,3},P_4\},n)\), then \(G\) is pancyclic with some exceptions. As corollaries we obtain the previous results by Fan, Benhocine and Wojda, and Ning.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3734.pdf
Fan's condition on induced subgraphs for circumference and pancyclicityWojciech WidełFan's condition; circumference; hamiltonian cycle; pancyclicitydoi:10.7494/OpMath.2017.37.4.617Opuscula Math. 37, no. 4 (2017), 617-639, http://dx.doi.org/10.7494/OpMath.2017.37.4.617Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.617http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3734.pdf374617639 Title: Fan's condition on induced subgraphs for circumference and pancyclicity.

Author(s): Wojciech Wideł.

Abstract: Let \(\mathcal{H}\) be a family of simple graphs and \(k\) be a positive integer. We say that a graph \(G\) of order \(n\geq k\) satisfies Fan's condition with respect to \(\mathcal{H}\) with constant \(k\), if for every induced subgraph \(H\) of \(G\) isomorphic to any of the graphs from \(\mathcal{H}\) the following holds: \[\forall u,v\in V(H)\colon d_H(u,v)=2\,\Rightarrow \max\{d_G(u),d_G(v)\}\geq k/2.\] If \(G\) satisfies the above condition, we write \(G\in\mathcal{F}(\mathcal{H},k)\). In this paper we show that if \(G\) is \(2\)-connected and \(G\in\mathcal{F}(\{K_{1,3},P_4\},k)\), then \(G\) contains a cycle of length at least \(k\), and that if \(G\in\mathcal{F}(\{K_{1,3},P_4\},n)\), then \(G\) is pancyclic with some exceptions. As corollaries we obtain the previous results by Fan, Benhocine and Wojda, and Ning. Keywords: Fan's condition, circumference, hamiltonian cycle, pancyclicity. Mathematics Subject Classification: 05C38, 05C45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 617-639, http://dx.doi.org/10.7494/OpMath.2017.37.4.617.

]]>On the chromatic number of (P_{5},windmill)-free graphsIn this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G\) admits a polynomial \(\chi\)-binding function. Moreover, we will present polynomial \(\chi\)-binding functions for several other subclasses of \(P_5\)-free graphs.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3733.pdf
On the chromatic number of (P_{5},windmill)-free graphsIngo Schiermeyervertex colouring; perfect graphs; \(\chi\)-binding function; forbidden induced subgraphdoi:10.7494/OpMath.2017.37.4.609Opuscula Math. 37, no. 4 (2017), 609-615, http://dx.doi.org/10.7494/OpMath.2017.37.4.609Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.609http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3733.pdf374609615 Title: On the chromatic number of (P_{5},windmill)-free graphs.

Author(s): Ingo Schiermeyer.

Abstract: In this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G\) admits a polynomial \(\chi\)-binding function. Moreover, we will present polynomial \(\chi\)-binding functions for several other subclasses of \(P_5\)-free graphs. Keywords: vertex colouring, perfect graphs, \(\chi\)-binding function, forbidden induced subgraph. Mathematics Subject Classification: 05C15, 05C17. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 609-615, http://dx.doi.org/10.7494/OpMath.2017.37.4.609.

]]>A hierarchy of maximal intersecting triple systemsWe reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each \(n\geq 7\) there are exactly 15 pairwise non-isomorphic such systems (and 13 for \(n=6\)). We present our result in terms of a hierarchy of Turán numbers \(\operatorname{ex}^{(s)}(n; M_2^{3})\), \(s\geq 1\), where \(M_2^{3}\) is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle \(C_3\) is defined as \(C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}\). Along the way we show that the largest intersecting triple system \(H\) on \(n\geq 6\) vertices, which is not a star and is triangle-free, consists of \(\max\{10,n\}\) triples. This facilitates our main proof's philosophy which is to assume that \(H\) contains a copy of the triangle and analyze how the remaining edges of \(H\) intersect that copy.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3732.pdf
A hierarchy of maximal intersecting triple systemsJoanna PolcynAndrzej Rucińskimaximal intersecting family; 3-uniform hypergraph; triple systemdoi:10.7494/OpMath.2017.37.4.597Opuscula Math. 37, no. 4 (2017), 597-608, http://dx.doi.org/10.7494/OpMath.2017.37.4.597Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.597http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3732.pdf374597608 Title: A hierarchy of maximal intersecting triple systems.

Author(s): Joanna Polcyn, Andrzej Ruciński.

Abstract: We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each \(n\geq 7\) there are exactly 15 pairwise non-isomorphic such systems (and 13 for \(n=6\)). We present our result in terms of a hierarchy of Turán numbers \(\operatorname{ex}^{(s)}(n; M_2^{3})\), \(s\geq 1\), where \(M_2^{3}\) is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle \(C_3\) is defined as \(C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}\). Along the way we show that the largest intersecting triple system \(H\) on \(n\geq 6\) vertices, which is not a star and is triangle-free, consists of \(\max\{10,n\}\) triples. This facilitates our main proof's philosophy which is to assume that \(H\) contains a copy of the triangle and analyze how the remaining edges of \(H\) intersect that copy. Keywords: maximal intersecting family, 3-uniform hypergraph, triple system. Mathematics Subject Classification: 05D05, 05C65. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 597-608, http://dx.doi.org/10.7494/OpMath.2017.37.4.597.

]]>Toward Wojda's conjecture on digraph packingGiven a positive integer \(m\leq n/2\), Wojda conjectured in 1985 that if \(D_1\) and \(D_2\) are digraphs of order \(n\) such that \(|A(D_1)|\leq n-m\) and \(|A(D_2)|\leq 2n-\lfloor n/m\rfloor-1\) then \(D_1\) and \(D_2\) pack. The cases when \(m=1\) or \(m = n/2\) follow from known results. Here we prove the conjecture for \(m\geq\sqrt{8n}+418275\).
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3731.pdf
Toward Wojda's conjecture on digraph packingJerzy KonarskiAndrzej Żakpacking; digraph; sizedoi:10.7494/OpMath.2017.37.4.589Opuscula Math. 37, no. 4 (2017), 589-595, http://dx.doi.org/10.7494/OpMath.2017.37.4.589Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.589http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3731.pdf374589595 Title: Toward Wojda's conjecture on digraph packing.

Author(s): Jerzy Konarski, Andrzej Żak.

Abstract: Given a positive integer \(m\leq n/2\), Wojda conjectured in 1985 that if \(D_1\) and \(D_2\) are digraphs of order \(n\) such that \(|A(D_1)|\leq n-m\) and \(|A(D_2)|\leq 2n-\lfloor n/m\rfloor-1\) then \(D_1\) and \(D_2\) pack. The cases when \(m=1\) or \(m = n/2\) follow from known results. Here we prove the conjecture for \(m\geq\sqrt{8n}+418275\). Keywords: packing, digraph, size. Mathematics Subject Classification: 05C35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 589-595, http://dx.doi.org/10.7494/OpMath.2017.37.4.589.

]]>A general 2-part Erdȍs-Ko-Rado theoremA two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into \(X_1\) and \(X_2\). Some positive integers \(k_i\), \(\ell_i\) (\(1\leq i\leq m\)) are given. We prove that if \(\mathcal{F}\)) is an intersecting family containing members \(F\) such that \(|F\cap X_1|=k_i\), \(|F\cap X_2|=\ell_i\) holds for one of the values \(i\) (\(1\leq i\leq m\)) then \(|\mathcal{F}|\) cannot exceed the size of the largest subfamily containing one element.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3730.pdf
A general 2-part Erdȍs-Ko-Rado theoremGyula O. H. Katonaextremal set theory; two-part problem; intersecting familydoi:10.7494/OpMath.2017.37.4.577Opuscula Math. 37, no. 4 (2017), 577-588, http://dx.doi.org/10.7494/OpMath.2017.37.4.577Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.577http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3730.pdf374577588 Title: A general 2-part Erdȍs-Ko-Rado theorem.

Author(s): Gyula O. H. Katona.

Abstract: A two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into \(X_1\) and \(X_2\). Some positive integers \(k_i\), \(\ell_i\) (\(1\leq i\leq m\)) are given. We prove that if \(\mathcal{F}\)) is an intersecting family containing members \(F\) such that \(|F\cap X_1|=k_i\), \(|F\cap X_2|=\ell_i\) holds for one of the values \(i\) (\(1\leq i\leq m\)) then \(|\mathcal{F}|\) cannot exceed the size of the largest subfamily containing one element. Keywords: extremal set theory, two-part problem, intersecting family. Mathematics Subject Classification: 05D05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 577-588, http://dx.doi.org/10.7494/OpMath.2017.37.4.577.

]]>Anti-Ramsey numbers for disjoint copies of graphsA subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by Erdȍs, Simonovits and Sós and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(ar(n,G)\). In this paper we show the lower bound for \(ar(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3729.pdf
Anti-Ramsey numbers for disjoint copies of graphsIzolda GorgolAgnieszka Görlichanti-Ramsey number; rainbow number; disjoint copiesdoi:10.7494/OpMath.2017.37.4.567Opuscula Math. 37, no. 4 (2017), 567-575, http://dx.doi.org/10.7494/OpMath.2017.37.4.567Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.567http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3729.pdf374567575 Title: Anti-Ramsey numbers for disjoint copies of graphs.

Author(s): Izolda Gorgol, Agnieszka Görlich.

Abstract: A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by Erdȍs, Simonovits and Sós and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(ar(n,G)\). In this paper we show the lower bound for \(ar(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp. Keywords: anti-Ramsey number, rainbow number, disjoint copies. Mathematics Subject Classification: 05C55, 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 567-575, http://dx.doi.org/10.7494/OpMath.2017.37.4.567.

]]>A note on incomplete regular tournaments with handicap two of order n≡8(mod 16) A \(d\)-handicap distance antimagic labeling of a graph \(G=(V,E)\) with \(n\) vertices is a bijection \(f:V\to \{1,2,\ldots ,n\}\) with the property that \(f(x_i)=i\) and the sequence of weights \(w(x_1),w(x_2),\ldots,w(x_n)\) (where \(w(x_i)=\sum_{x_i x_j\in E}f(x_j)\)) forms an increasing arithmetic progression with common difference \(d\). A graph \(G\) is a \(d\)-handicap distance antimagic graph if it allows a \(d\)-handicap distance antimagic labeling. We construct a class of \(k\)-regular \(2\)-handicap distance antimagic graphs for every order \(n\equiv8\pmod{16}\), \(n\geq56\) and \(6\leq k\leq n-50\).
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3728.pdf
A note on incomplete regular tournaments with handicap two of order n≡8(mod 16)Dalibor Froncekincomplete tournaments; handicap tournaments; distance magic labeling; handicap labelingdoi:10.7494/OpMath.2017.37.4.557Opuscula Math. 37, no. 4 (2017), 557-566, http://dx.doi.org/10.7494/OpMath.2017.37.4.557Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.557http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3728.pdf374557566 Title: A note on incomplete regular tournaments with handicap two of order n≡8(mod 16).

Author(s): Dalibor Froncek.

Abstract: A \(d\)-handicap distance antimagic labeling of a graph \(G=(V,E)\) with \(n\) vertices is a bijection \(f:V\to \{1,2,\ldots ,n\}\) with the property that \(f(x_i)=i\) and the sequence of weights \(w(x_1),w(x_2),\ldots,w(x_n)\) (where \(w(x_i)=\sum_{x_i x_j\in E}f(x_j)\)) forms an increasing arithmetic progression with common difference \(d\). A graph \(G\) is a \(d\)-handicap distance antimagic graph if it allows a \(d\)-handicap distance antimagic labeling. We construct a class of \(k\)-regular \(2\)-handicap distance antimagic graphs for every order \(n\equiv8\pmod{16}\), \(n\geq56\) and \(6\leq k\leq n-50\). Keywords: incomplete tournaments, handicap tournaments, distance magic labeling, handicap labeling. Mathematics Subject Classification: 05C78. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 557-566, http://dx.doi.org/10.7494/OpMath.2017.37.4.557.

]]>Acyclic sum-list-colouring of grids and other classes of graphsIn this paper we consider list colouring of a graph \(G\) in which the sizes of lists assigned to different vertices can be different. We colour \(G\) from the lists in such a way that each colour class induces an acyclic graph. The aim is to find the smallest possible sum of all the list sizes, such that, according to the rules, \(G\) is colourable for any particular assignment of the lists of these sizes. This invariant is called the \(D_1\)-sum-choice-number of \(G\). In the paper we investigate the \(D_1\)-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the \(D_1\)-sum-choice-number for each grid \(P_n\square P_m\), when at least one of the numbers \(n\), \(m\) is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the \(D_1\)-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3727.pdf
Acyclic sum-list-colouring of grids and other classes of graphsEwa Drgas-BurchardtAgata Drzysteksum-list colouring; acyclic colouring; grids; generalized Petersen graphsdoi:10.7494/OpMath.2017.37.4.535Opuscula Math. 37, no. 4 (2017), 535-556, http://dx.doi.org/10.7494/OpMath.2017.37.4.535Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.535http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3727.pdf374535556 Title: Acyclic sum-list-colouring of grids and other classes of graphs.

Author(s): Ewa Drgas-Burchardt, Agata Drzystek.

Abstract: In this paper we consider list colouring of a graph \(G\) in which the sizes of lists assigned to different vertices can be different. We colour \(G\) from the lists in such a way that each colour class induces an acyclic graph. The aim is to find the smallest possible sum of all the list sizes, such that, according to the rules, \(G\) is colourable for any particular assignment of the lists of these sizes. This invariant is called the \(D_1\)-sum-choice-number of \(G\). In the paper we investigate the \(D_1\)-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the \(D_1\)-sum-choice-number for each grid \(P_n\square P_m\), when at least one of the numbers \(n\), \(m\) is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the \(D_1\)-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs. Keywords: sum-list colouring, acyclic colouring, grids, generalized Petersen graphs. Mathematics Subject Classification: 05C30, 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 535-556, http://dx.doi.org/10.7494/OpMath.2017.37.4.535.

]]>The metric dimension of circulant graphs and their Cartesian productsLet \(G=(V,E)\) be a connected graph (or hypergraph) and let \(d(x,y)\) denote the distance between vertices \(x,y\in V(G)\). A subset \(W\subseteq V(G)\) is called a resolving set for \(G\) if for every pair of distinct vertices \(x,y\in V(G)\), there is \(w\in W\) such that \(d(x,w)\neq d(y,w)\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\beta(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) has vertex set \(\{v_0,v_1,\ldots,v_{n-1}\}\) and edges \(v_iv_{i+j}\) where \(0\leq i\leq n-1\) and \(1\leq j\leq t\) and the indices are taken modulo \(n\) (\(2\leq t\leq\left\lfloor\frac{n}{2}\right\rfloor\)). In this paper we determine the exact metric dimension of the circulant graphs \(C_n(1,2,\ldots,t)\), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrík (2016). In particular, we show that \(\beta(C_n(1,2,\ldots,t))=\beta(C_{n+2t}(1,2,\ldots,t))\) for large enough \(n\), which implies that the metric dimension of these circulants is completely determined by the congruence class of \(n\) modulo \(2t\). We determine the exact value of \(\beta(C_n(1,2,\ldots,t))\) for \(n\equiv 2\bmod 2t\) and \(n\equiv (t+1)\bmod 2t\) and we give better bounds on the metric dimension of these circulants for \(n\equiv 0\bmod 2t\) and \(n\equiv 1 \bmod 2t\). In addition, we bound the metric dimension of Cartesian products of circulant graphs.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3726.pdf
The metric dimension of circulant graphs and their Cartesian productsKevin ChauShonda Gosselinmetric dimension; circulant graph; cartesian productdoi:10.7494/OpMath.2017.37.4.509Opuscula Math. 37, no. 4 (2017), 509-534, http://dx.doi.org/10.7494/OpMath.2017.37.4.509Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.509http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3726.pdf374509534 Title: The metric dimension of circulant graphs and their Cartesian products.

Author(s): Kevin Chau, Shonda Gosselin.

Abstract: Let \(G=(V,E)\) be a connected graph (or hypergraph) and let \(d(x,y)\) denote the distance between vertices \(x,y\in V(G)\). A subset \(W\subseteq V(G)\) is called a resolving set for \(G\) if for every pair of distinct vertices \(x,y\in V(G)\), there is \(w\in W\) such that \(d(x,w)\neq d(y,w)\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\beta(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) has vertex set \(\{v_0,v_1,\ldots,v_{n-1}\}\) and edges \(v_iv_{i+j}\) where \(0\leq i\leq n-1\) and \(1\leq j\leq t\) and the indices are taken modulo \(n\) (\(2\leq t\leq\left\lfloor\frac{n}{2}\right\rfloor\)). In this paper we determine the exact metric dimension of the circulant graphs \(C_n(1,2,\ldots,t)\), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrík (2016). In particular, we show that \(\beta(C_n(1,2,\ldots,t))=\beta(C_{n+2t}(1,2,\ldots,t))\) for large enough \(n\), which implies that the metric dimension of these circulants is completely determined by the congruence class of \(n\) modulo \(2t\). We determine the exact value of \(\beta(C_n(1,2,\ldots,t))\) for \(n\equiv 2\bmod 2t\) and \(n\equiv (t+1)\bmod 2t\) and we give better bounds on the metric dimension of these circulants for \(n\equiv 0\bmod 2t\) and \(n\equiv 1 \bmod 2t\). In addition, we bound the metric dimension of Cartesian products of circulant graphs. Keywords: metric dimension, circulant graph, cartesian product. Mathematics Subject Classification: 05C12, 05C65, 05C25, 05E18. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 509-534, http://dx.doi.org/10.7494/OpMath.2017.37.4.509.

]]>Spanning trees with a bounded number of leavesIn 1998, H. Broersma and H. Tuinstra proved that: Given a connected graph \(G\) with \(n\geq 3\) vertices, if \(d(u)+d(v)\geq n-k+1\) for all non-adjacent vertices \(u\) and \(v\) of \(G\) (\(k\geq 1\)), then \(G\) has a spanning tree with at most \(k\) leaves. In this paper, we generalize this result by using implicit degree sum condition of \(t\) (\(2\leq t\leq k\)) independent vertices and we prove what follows: Let \(G\) be a connected graph on \(n\geq 3\) vertices and \(k\geq 2\) be an integer. If the implicit degree sum of any \(t\) independent vertices is at least \(\frac{t(n-k)}{2}+1\) for (\(k\geq t\geq 2\)), then \(G\) has a spanning tree with at most \(k\) leaves.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3725.pdf
Spanning trees with a bounded number of leavesJunqing CaiEvelyne FlandrinHao LiQiang Sunspanning tree; implicit degree; leavesdoi:10.7494/OpMath.2017.37.4.501Opuscula Math. 37, no. 4 (2017), 501-508, http://dx.doi.org/10.7494/OpMath.2017.37.4.501Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.501http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3725.pdf374501508 Title: Spanning trees with a bounded number of leaves.

Abstract: In 1998, H. Broersma and H. Tuinstra proved that: Given a connected graph \(G\) with \(n\geq 3\) vertices, if \(d(u)+d(v)\geq n-k+1\) for all non-adjacent vertices \(u\) and \(v\) of \(G\) (\(k\geq 1\)), then \(G\) has a spanning tree with at most \(k\) leaves. In this paper, we generalize this result by using implicit degree sum condition of \(t\) (\(2\leq t\leq k\)) independent vertices and we prove what follows: Let \(G\) be a connected graph on \(n\geq 3\) vertices and \(k\geq 2\) be an integer. If the implicit degree sum of any \(t\) independent vertices is at least \(\frac{t(n-k)}{2}+1\) for (\(k\geq t\geq 2\)), then \(G\) has a spanning tree with at most \(k\) leaves. Keywords: spanning tree, implicit degree, leaves. Mathematics Subject Classification: 05C07. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 501-508, http://dx.doi.org/10.7494/OpMath.2017.37.4.501.

]]>Colourings of (k-r,k)-treesTrees are generalized to a special kind of higher dimensional complexes known as \((j,k)\)-trees ([L. W. Beineke, R. E. Pippert, On the structure of \((m,n)\)-trees, Proc. 8th S-E Conf. Combinatorics, Graph Theory and Computing, 1977, 75-80]), and which are a natural extension of \(k\)-trees for \(j=k-1\). The aim of this paper is to study\((k-r,k)\)-trees ([H. P. Patil, Studies on \(k\)-trees and some related topics, PhD Thesis, University of Warsaw, Poland, 1984]), which are a generalization of \(k\)-trees (or usual trees when \(k=1\)). We obtain the chromatic polynomial of \((k-r,k)\)-trees and show that any two \((k-r,k)\)-trees of the same order are chromatically equivalent. However, if \(r\neq 1\) in any \((k-r,k)\)-tree \(G\), then it is shown that there exists another chromatically equivalent graph \(H\), which is not a \((k-r,k)\)-tree. Further, the vertex-partition number and generalized total colourings of \((k-r,k)\)-trees are obtained. We formulate a conjecture about the chromatic index of \((k-r,k)\)-trees, and verify this conjecture in a number of cases. Finally, we obtain a result of [M. Borowiecki, W. Chojnacki, Chromatic index of \(k\)-trees, Discuss. Math. 9 (1988), 55-58] as a corollary in which \(k\)-trees of Class 2 are characterized.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3724.pdf
Colourings of (k-r,k)-treesM. BorowieckiH. P. Patilchromatic polynomial; partition number; colouring; treedoi:10.7494/OpMath.2017.37.4.491Opuscula Math. 37, no. 4 (2017), 491-500, http://dx.doi.org/10.7494/OpMath.2017.37.4.491Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.491http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3724.pdf374491500 Title: Colourings of (k-r,k)-trees.

Author(s): M. Borowiecki, H. P. Patil.

Abstract: Trees are generalized to a special kind of higher dimensional complexes known as \((j,k)\)-trees ([L. W. Beineke, R. E. Pippert, On the structure of \((m,n)\)-trees, Proc. 8th S-E Conf. Combinatorics, Graph Theory and Computing, 1977, 75-80]), and which are a natural extension of \(k\)-trees for \(j=k-1\). The aim of this paper is to study\((k-r,k)\)-trees ([H. P. Patil, Studies on \(k\)-trees and some related topics, PhD Thesis, University of Warsaw, Poland, 1984]), which are a generalization of \(k\)-trees (or usual trees when \(k=1\)). We obtain the chromatic polynomial of \((k-r,k)\)-trees and show that any two \((k-r,k)\)-trees of the same order are chromatically equivalent. However, if \(r\neq 1\) in any \((k-r,k)\)-tree \(G\), then it is shown that there exists another chromatically equivalent graph \(H\), which is not a \((k-r,k)\)-tree. Further, the vertex-partition number and generalized total colourings of \((k-r,k)\)-trees are obtained. We formulate a conjecture about the chromatic index of \((k-r,k)\)-trees, and verify this conjecture in a number of cases. Finally, we obtain a result of [M. Borowiecki, W. Chojnacki, Chromatic index of \(k\)-trees, Discuss. Math. 9 (1988), 55-58] as a corollary in which \(k\)-trees of Class 2 are characterized. Keywords: chromatic polynomial, partition number, colouring, tree. Mathematics Subject Classification: 05C75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 491-500, http://dx.doi.org/10.7494/OpMath.2017.37.4.491.

]]>On Fibonacci numbers in edge coloured treesIn this paper we show the applications of the Fibonacci numbers in edge coloured trees. We determine the second smallest number of all \((A,2B)\)-edge colourings in trees. We characterize the minimum tree achieving this second smallest value.
http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3723.pdf
On Fibonacci numbers in edge coloured treesUrszula BednarzDorota BródAnetta Szynal-LianaIwona WłochMałgorzata Wołowiec-Musiałedge colouring; tree; tripod; Fibonacci numbersdoi:10.7494/OpMath.2017.37.4.479Opuscula Math. 37, no. 4 (2017), 479-490, http://dx.doi.org/10.7494/OpMath.2017.37.4.479Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.4.479http://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3723.pdf374479490 Title: On Fibonacci numbers in edge coloured trees.

Abstract: In this paper we show the applications of the Fibonacci numbers in edge coloured trees. We determine the second smallest number of all \((A,2B)\)-edge colourings in trees. We characterize the minimum tree achieving this second smallest value. Keywords: edge colouring, tree, tripod, Fibonacci numbers. Mathematics Subject Classification: 11B37, 11C20, 15B36, 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 4 (2017), 479-490, http://dx.doi.org/10.7494/OpMath.2017.37.4.479.

]]>On criteria for algebraic independence of collections of functions satisfying algebraic difference relationsThis paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vignéras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, \(q\)-exponential functions and \(q\)-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3722.pdf
On criteria for algebraic independence of collections of functions satisfying algebraic difference relationsHiroshi Ogawaradifference algebra; systems of algebraic difference equations; algebraic independence; Vignéras' multiple gamma functions; \(q\)-polylogarithm functionsdoi:10.7494/OpMath.2017.37.3.457Opuscula Math. 37, no. 3 (2017), 457-472, http://dx.doi.org/10.7494/OpMath.2017.37.3.457Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.457http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3722.pdf373457472 Title: On criteria for algebraic independence of collections of functions satisfying algebraic difference relations.

Author(s): Hiroshi Ogawara.

Abstract: This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vignéras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, \(q\)-exponential functions and \(q\)-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem. Keywords: difference algebra, systems of algebraic difference equations, algebraic independence, Vignéras' multiple gamma functions, \(q\)-polylogarithm functions. Mathematics Subject Classification: 12H10, 39A10, 39A13. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 457-472, http://dx.doi.org/10.7494/OpMath.2017.37.3.457.

]]>On the inverse signed total domination number in graphsIn this paper, we study the inverse signed total domination number in graphs and present new sharp lower and upper bounds on this parameter. For example by making use of the classic theorem of Turán (1941), we present a sharp upper bound on \(K_{r+1}\)-free graphs for \(r\geq 2\). Also, we bound this parameter for a tree from below in terms of its order and the number of leaves and characterize all trees attaining this bound.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3721.pdf
On the inverse signed total domination number in graphsD. A. MojdehB. Samadiinverse signed total dominating function; inverse signed total domination number; \(k\)-tuple total domination numberdoi:10.7494/OpMath.2017.37.3.447Opuscula Math. 37, no. 3 (2017), 447-456, http://dx.doi.org/10.7494/OpMath.2017.37.3.447Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.447http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3721.pdf373447456 Title: On the inverse signed total domination number in graphs.

Author(s): D. A. Mojdeh, B. Samadi.

Abstract: In this paper, we study the inverse signed total domination number in graphs and present new sharp lower and upper bounds on this parameter. For example by making use of the classic theorem of Turán (1941), we present a sharp upper bound on \(K_{r+1}\)-free graphs for \(r\geq 2\). Also, we bound this parameter for a tree from below in terms of its order and the number of leaves and characterize all trees attaining this bound. Keywords: inverse signed total dominating function, inverse signed total domination number, \(k\)-tuple total domination number. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 447-456, http://dx.doi.org/10.7494/OpMath.2017.37.3.447.

]]>On strongly spanning k-edge-colorable subgraphsA subgraph \(H\) of a multigraph \(G\) is called strongly spanning, if any vertex of \(G\) is not isolated in \(H\). \(H\) is called maximum \(k\)-edge-colorable, if \(H\) is proper \(k\)-edge-colorable and has the largest size. We introduce a graph-parameter \(sp(G)\), that coincides with the smallest \(k\) for which a multigraph \(G\) has a maximum \(k\)-edge-colorable subgraph that is strongly spanning. Our first result offers some alternative definitions of \(sp(G)\). Next, we show that \(\Delta(G)\) is an upper bound for \(sp(G)\), and then we characterize the class of multigraphs \(G\) that satisfy \(sp(G)=\Delta(G)\). Finally, we prove some bounds for \(sp(G)\) that involve well-known graph-theoretic parameters.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3720.pdf
On strongly spanning k-edge-colorable subgraphsVahan V. MkrtchyanGagik N. Vardanyan\(k\)-edge-colorable subgraph; maximum \(k\)-edge-colorable subgraph; strongly spanning \(k\)-edge-colorable subgraph; \([1,k]\)-factordoi:10.7494/OpMath.2017.37.3.435Opuscula Math. 37, no. 3 (2017), 435-446, http://dx.doi.org/10.7494/OpMath.2017.37.3.435Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.435http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3720.pdf373435446 Title: On strongly spanning k-edge-colorable subgraphs.

Author(s): Vahan V. Mkrtchyan, Gagik N. Vardanyan.

Abstract: A subgraph \(H\) of a multigraph \(G\) is called strongly spanning, if any vertex of \(G\) is not isolated in \(H\). \(H\) is called maximum \(k\)-edge-colorable, if \(H\) is proper \(k\)-edge-colorable and has the largest size. We introduce a graph-parameter \(sp(G)\), that coincides with the smallest \(k\) for which a multigraph \(G\) has a maximum \(k\)-edge-colorable subgraph that is strongly spanning. Our first result offers some alternative definitions of \(sp(G)\). Next, we show that \(\Delta(G)\) is an upper bound for \(sp(G)\), and then we characterize the class of multigraphs \(G\) that satisfy \(sp(G)=\Delta(G)\). Finally, we prove some bounds for \(sp(G)\) that involve well-known graph-theoretic parameters. Keywords: \(k\)-edge-colorable subgraph, maximum \(k\)-edge-colorable subgraph, strongly spanning \(k\)-edge-colorable subgraph, \([1,k]\)-factor. Mathematics Subject Classification: 05C70, 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 435-446, http://dx.doi.org/10.7494/OpMath.2017.37.3.435.

]]>Positive solutions of a singular fractional boundary value problem with a fractional boundary conditionFor \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Carathéodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3719.pdf
Positive solutions of a singular fractional boundary value problem with a fractional boundary conditionJeffrey W. LyonsJeffrey T. Neugebauerfractional differential equation; singular problem; fixed pointdoi:10.7494/OpMath.2017.37.3.421Opuscula Math. 37, no. 3 (2017), 421-434, http://dx.doi.org/10.7494/OpMath.2017.37.3.421Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.421http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3719.pdf373421434 Title: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition.

Author(s): Jeffrey W. Lyons, Jeffrey T. Neugebauer.

Abstract: For \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Carathéodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given. Keywords: fractional differential equation, singular problem, fixed point. Mathematics Subject Classification: 26A33, 34A08, 34B16. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 421-434, http://dx.doi.org/10.7494/OpMath.2017.37.3.421.

]]>Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamicsWe consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3718.pdf
Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamicsMaciej LeszczyńskiElżbieta RatajczykUrszula LedzewiczHeinz Schättleroptimal control; sufficient conditions for optimality; method of characteristics; pharmacodynamic modeldoi:10.7494/OpMath.2017.37.3.403Opuscula Math. 37, no. 3 (2017), 403-419, http://dx.doi.org/10.7494/OpMath.2017.37.3.403Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.403http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3718.pdf373403419 Title: Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics.

Author(s): Maciej Leszczyński, Elżbieta Ratajczyk, Urszula Ledzewicz, Heinz Schättler.

Abstract: We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given. Keywords: optimal control, sufficient conditions for optimality, method of characteristics, pharmacodynamic model. Mathematics Subject Classification: 49K15, 93C15, 92C45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 403-419, http://dx.doi.org/10.7494/OpMath.2017.37.3.403.

]]>General solutions of second-order linear difference equations of Euler typeThe purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation \(y^{\prime\prime}+(\lambda/t^2)y=0\) or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3717.pdf
General solutions of second-order linear difference equations of Euler typeAkane HongyoNaoto YamaokaEuler-Cauchy equations; oscillation; conditionally oscillatorydoi:10.7494/OpMath.2017.37.3.389Opuscula Math. 37, no. 3 (2017), 389-402, http://dx.doi.org/10.7494/OpMath.2017.37.3.389Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.389http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3717.pdf373389402 Title: General solutions of second-order linear difference equations of Euler type.

Author(s): Akane Hongyo, Naoto Yamaoka.

Abstract: The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation \(y^{\prime\prime}+(\lambda/t^2)y=0\) or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations. Keywords: Euler-Cauchy equations, oscillation, conditionally oscillatory. Mathematics Subject Classification: 39A06, 39A12, 39A21. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 389-402, http://dx.doi.org/10.7494/OpMath.2017.37.3.389.

]]>On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52-54] that the identity component \(\text{Diff}\,^r_{G,c}(M)_0\) of the group of equivariant \(C^r\)-diffeomorphisms of a principal \(G\) bundle \(M\) over a manifold \(B\) is perfect for a compact connected Lie group \(G\) and \(1 \leq r \leq \infty\) (\(r \neq \dim B + 1\)). In this paper, we study the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for a principal \(G\) bundle \(M\) over a manifold \(B\) by relating it to the uniform perfectness of the group of \(C^r\)-diffeomorphisms of \(B\) and show that under a certain condition, \(\text{Diff}\,^r_{G,c}(M)_0\) is uniformly perfect if \(B\) belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for principal \(G\) bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and \(r\neq 4\).
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3716.pdf
On the uniform perfectness of equivariant diffeomorphism groups for principal G manifoldsKazuhiko Fukuiuniform perfectness; principal \(G\) manifold; equivariant diffeomorphismdoi:10.7494/OpMath.2017.37.3.381Opuscula Math. 37, no. 3 (2017), 381-388, http://dx.doi.org/10.7494/OpMath.2017.37.3.381Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.381http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3716.pdf373381388 Title: On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds.

Author(s): Kazuhiko Fukui.

Abstract: We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52-54] that the identity component \(\text{Diff}\,^r_{G,c}(M)_0\) of the group of equivariant \(C^r\)-diffeomorphisms of a principal \(G\) bundle \(M\) over a manifold \(B\) is perfect for a compact connected Lie group \(G\) and \(1 \leq r \leq \infty\) (\(r \neq \dim B + 1\)). In this paper, we study the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for a principal \(G\) bundle \(M\) over a manifold \(B\) by relating it to the uniform perfectness of the group of \(C^r\)-diffeomorphisms of \(B\) and show that under a certain condition, \(\text{Diff}\,^r_{G,c}(M)_0\) is uniformly perfect if \(B\) belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for principal \(G\) bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and \(r\neq 4\). Keywords: uniform perfectness, principal \(G\) manifold, equivariant diffeomorphism. Mathematics Subject Classification: 58D05, 57R30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 381-388, http://dx.doi.org/10.7494/OpMath.2017.37.3.381.

]]>Existence of three solutions for impulsive multi-point boundary value problemsThis paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results.
http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3715.pdf
Existence of three solutions for impulsive multi-point boundary value problemsMartin BohnerShapour HeidarkhaniAmjad SalariGiuseppe Caristimulti-point boundary value problem; impulsive condition; classical solution; variational method; three critical points theoremdoi:10.7494/OpMath.2017.37.3.353Opuscula Math. 37, no. 3 (2017), 353-379, http://dx.doi.org/10.7494/OpMath.2017.37.3.353Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.3.353http://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3715.pdf373353379 Title: Existence of three solutions for impulsive multi-point boundary value problems.

Author(s): Martin Bohner, Shapour Heidarkhani, Amjad Salari, Giuseppe Caristi.

Abstract: This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results. Keywords: multi-point boundary value problem, impulsive condition, classical solution, variational method, three critical points theorem. Mathematics Subject Classification: 34B10, 34B15, 34A37. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 3 (2017), 353-379, http://dx.doi.org/10.7494/OpMath.2017.37.3.353.

]]>The interaction between PDE and graphs in multiscale modelingIn this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\). The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3714.pdf
The interaction between PDE and graphs in multiscale modelingFernando A. MoralesSebastián Naranjo Álvarezcoupled PDE systems; mixed formulations; porous media; analytic graph theory; complex networksdoi:10.7494/OpMath.2017.37.2.327Opuscula Math. 37, no. 2 (2017), 327-345, http://dx.doi.org/10.7494/OpMath.2017.37.2.327Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.327http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3714.pdf372327345 Title: The interaction between PDE and graphs in multiscale modeling.

Author(s): Fernando A. Morales, Sebastián Naranjo Álvarez.

Abstract: In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\). The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation. Keywords: coupled PDE systems, mixed formulations, porous media, analytic graph theory, complex networks. Mathematics Subject Classification: 05C82, 05C10, 35R02, 35J50. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 327-345, http://dx.doi.org/10.7494/OpMath.2017.37.2.327.

]]>Control system defined by some integral operatorIn the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control \(u\) there exists a uniquely defined trajectory \(x_{u}\) which continuously depends on control \(u\) and the operator \(u\mapsto x_{u}\) is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3713.pdf
Control system defined by some integral operatorMarek MajewskiVolterra equation; implicit function theorem; sensitivitydoi:10.7494/OpMath.2017.37.2.313Opuscula Math. 37, no. 2 (2017), 313-325, http://dx.doi.org/10.7494/OpMath.2017.37.2.313Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.313http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3713.pdf372313325 Title: Control system defined by some integral operator.

Author(s): Marek Majewski.

Abstract: In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control \(u\) there exists a uniquely defined trajectory \(x_{u}\) which continuously depends on control \(u\) and the operator \(u\mapsto x_{u}\) is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions. Keywords: Volterra equation, implicit function theorem, sensitivity. Mathematics Subject Classification: 45D05, 34A12, 47J07, 46T20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 313-325, http://dx.doi.org/10.7494/OpMath.2017.37.2.313.

]]>Compact generalized weighted composition operators on the Bergman spaceWe characterize the compactness of the generalized weighted composition operators acting on the Bergman space.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3712.pdf
Compact generalized weighted composition operators on the Bergman spaceQinghua HuXiangling ZhuBergman space; generalized weighted composition operator; compactnessdoi:10.7494/OpMath.2017.37.2.303Opuscula Math. 37, no. 2 (2017), 303-312, http://dx.doi.org/10.7494/OpMath.2017.37.2.303Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.303http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3712.pdf372303312 Title: Compact generalized weighted composition operators on the Bergman space.

Author(s): Qinghua Hu, Xiangling Zhu.

Abstract: We characterize the compactness of the generalized weighted composition operators acting on the Bergman space. Keywords: Bergman space, generalized weighted composition operator, compactness. Mathematics Subject Classification: 30H30, 47B33. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 303-312, http://dx.doi.org/10.7494/OpMath.2017.37.2.303.

]]>Existence of three solutions for impulsive nonlinear fractional boundary value problemsIn this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3711.pdf
Existence of three solutions for impulsive nonlinear fractional boundary value problemsShapour HeidarkhaniMassimiliano FerraraGiuseppe CaristiAmjad Salarifractional differential equation; impulsive condition; classical solution; variational methods; critical point theorydoi:10.7494/OpMath.2017.37.2.281Opuscula Math. 37, no. 2 (2017), 281-301, http://dx.doi.org/10.7494/OpMath.2017.37.2.281Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.281http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3711.pdf372281301 Title: Existence of three solutions for impulsive nonlinear fractional boundary value problems.

Author(s): Shapour Heidarkhani, Massimiliano Ferrara, Giuseppe Caristi, Amjad Salari.

Abstract: In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Keywords: fractional differential equation, impulsive condition, classical solution, variational methods, critical point theory. Mathematics Subject Classification: 34A08, 34B37, 58E05, 58E30, 26A33. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 281-301, http://dx.doi.org/10.7494/OpMath.2017.37.2.281.

]]>Fractional boundary value problems on the half lineIn this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3710.pdf
Fractional boundary value problems on the half lineAssia FriouiAssia Guezane-LakoudRabah Khaldiboundary value problem at resonance; existence of solution; unbounded interval; coincidence degree of Mawhin; fractional differential equationdoi:10.7494/OpMath.2017.37.2.265Opuscula Math. 37, no. 2 (2017), 265-280, http://dx.doi.org/10.7494/OpMath.2017.37.2.265Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.265http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3710.pdf372265280 Title: Fractional boundary value problems on the half line.

Abstract: In this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example. Keywords: boundary value problem at resonance, existence of solution, unbounded interval, coincidence degree of Mawhin, fractional differential equation. Mathematics Subject Classification: 34B40, 34B15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 265-280, http://dx.doi.org/10.7494/OpMath.2017.37.2.265.

]]>Non-factorizable C-valued functions induced by finite connected graphsIn this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.
http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3709.pdf
Non-factorizable C-valued functions induced by finite connected graphsIlwoo Chodirected graphs; graph groupoids; Redei zeta functions; graph zeta functions; non-factorizable graphs; gluing on graphsdoi:10.7494/OpMath.2017.37.2.225Opuscula Math. 37, no. 2 (2017), 225-263, http://dx.doi.org/10.7494/OpMath.2017.37.2.225Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.2.225http://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3709.pdf372225263 Title: Non-factorizable C-valued functions induced by finite connected graphs.

Author(s): Ilwoo Cho.

Abstract: In this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function. Keywords: directed graphs, graph groupoids, Redei zeta functions, graph zeta functions, non-factorizable graphs, gluing on graphs. Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L40, 46L54. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 2 (2017), 225-263, http://dx.doi.org/10.7494/OpMath.2017.37.2.225.

]]>Hankel and Toeplitz operators: continuous and discrete representationsWe find a relation guaranteeing that Hankel operators realized in the space of sequences \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and in the space of functions \(L^2 (\mathbb{R}_{+})\) are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space \(\mathcal{l}^2 (\mathbb{Z}_{+})\) generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and \(L^2 (\mathbb{R}_{+})\).
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3708.pdf
Hankel and Toeplitz operators: continuous and discrete representationsDmitri R. Yafaevunbounded Hankel and Toeplitz operators; various representations; moment problems; generalized Hilbert matricesdoi:10.7494/OpMath.2017.37.1.189Opuscula Math. 37, no. 1 (2017), 189-218, http://dx.doi.org/10.7494/OpMath.2017.37.1.189Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.189http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3708.pdf371189218 Title: Hankel and Toeplitz operators: continuous and discrete representations.

Author(s): Dmitri R. Yafaev.

Abstract: We find a relation guaranteeing that Hankel operators realized in the space of sequences \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and in the space of functions \(L^2 (\mathbb{R}_{+})\) are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space \(\mathcal{l}^2 (\mathbb{Z}_{+})\) generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and \(L^2 (\mathbb{R}_{+})\). Keywords: unbounded Hankel and Toeplitz operators, various representations, moment problems, generalized Hilbert matrices. Mathematics Subject Classification: 47B25, 47B35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 189-218, http://dx.doi.org/10.7494/OpMath.2017.37.1.189.

]]>The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equationThe Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3707.pdf
The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equationDmitry ShepelskyLech ZielinskiDullin-Gottwald-Holm equation; Camassa-Holm equation; inverse scattering transform; Riemann-Hilbert problemdoi:10.7494/OpMath.2017.37.1.167Opuscula Math. 37, no. 1 (2017), 167-187, http://dx.doi.org/10.7494/OpMath.2017.37.1.167Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.167http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3707.pdf371167187 Title: The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation.

Author(s): Dmitry Shepelsky, Lech Zielinski.

Abstract: The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented. Keywords: Dullin-Gottwald-Holm equation, Camassa-Holm equation, inverse scattering transform, Riemann-Hilbert problem. Mathematics Subject Classification: 35Q53, 37K15, 35Q15, 35B40, 35Q51, 37K40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 167-187, http://dx.doi.org/10.7494/OpMath.2017.37.1.167.

]]>The basis property of eigenfunctions in the problem of a nonhomogeneous damped stringThe equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator \(i A\). We prove that the set of root vectors of the operator \(A\) forms a basis of subspaces in a certain Hilbert space \(H\). Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator \(A\) is a Riesz basis for \(H\).
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3706.pdf
The basis property of eigenfunctions in the problem of a nonhomogeneous damped stringŁukasz Rzepnickinonhomogeneous damped string; Hilbert space; Riesz basis; modulus of continuity; basis with parentheses; basis of subspaces; string equationdoi:10.7494/OpMath.2017.37.1.141Opuscula Math. 37, no. 1 (2017), 141-165, http://dx.doi.org/10.7494/OpMath.2017.37.1.141Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.141http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3706.pdf371141165 Title: The basis property of eigenfunctions in the problem of a nonhomogeneous damped string.

Author(s): Łukasz Rzepnicki.

Abstract: The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator \(i A\). We prove that the set of root vectors of the operator \(A\) forms a basis of subspaces in a certain Hilbert space \(H\). Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator \(A\) is a Riesz basis for \(H\). Keywords: nonhomogeneous damped string, Hilbert space, Riesz basis, modulus of continuity, basis with parentheses, basis of subspaces, string equation. Mathematics Subject Classification: 34L10, 34B08. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 141-165, http://dx.doi.org/10.7494/OpMath.2017.37.1.141.

]]>Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularityWe find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3705.pdf
Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularityMedet NursultanovGrigori RozenblumSturm-Liouville operator; singular potential; asymptotics of eigenvaluesdoi:10.7494/OpMath.2017.37.1.109Opuscula Math. 37, no. 1 (2017), 109-139, http://dx.doi.org/10.7494/OpMath.2017.37.1.109Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.109http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3705.pdf371109139 Title: Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity.

Author(s): Medet Nursultanov, Grigori Rozenblum.

Abstract: We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously. Keywords: Sturm-Liouville operator, singular potential, asymptotics of eigenvalues. Mathematics Subject Classification: 34L20, 34L40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 109-139, http://dx.doi.org/10.7494/OpMath.2017.37.1.109.

]]>Seminormal systems of operators in Clifford environmentsThe primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex \(C^*\)-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a \(C^*\)-algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3704.pdf
Seminormal systems of operators in Clifford environmentsMircea Martinmultidimensional operator theory; joint seminormality; Riesz transforms; Putnam inequalitydoi:10.7494/OpMath.2017.37.1.81Opuscula Math. 37, no. 1 (2017), 81-107, http://dx.doi.org/10.7494/OpMath.2017.37.1.81Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.81http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3704.pdf37181107 Title: Seminormal systems of operators in Clifford environments.

Author(s): Mircea Martin.

Abstract: The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex \(C^*\)-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a \(C^*\)-algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities. Keywords: multidimensional operator theory, joint seminormality, Riesz transforms, Putnam inequality. Mathematics Subject Classification: 47B20, 47A13, 47A63, 44A15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 81-107, http://dx.doi.org/10.7494/OpMath.2017.37.1.81.

]]>Towards theory of C-symmetriesThe concept of \(\mathcal{C}\)-symmetry originally appeared in \(\mathcal{PT}\)-symmetric quantum mechanics is studied within the Krein spaces framework.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3703.pdf
Towards theory of C-symmetriesS. KuzhelV. SudilovskayaKrein space; \(J\)-self-adjoint operator; \(J\)-symmetric operator; Friedrichs extension; \(\mathcal{C}\)-symmetrydoi:10.7494/OpMath.2017.37.1.65Opuscula Math. 37, no. 1 (2017), 65-80, http://dx.doi.org/10.7494/OpMath.2017.37.1.65Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.65http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3703.pdf3716580 Title: Towards theory of C-symmetries.

Author(s): S. Kuzhel, V. Sudilovskaya.

Abstract: The concept of \(\mathcal{C}\)-symmetry originally appeared in \(\mathcal{PT}\)-symmetric quantum mechanics is studied within the Krein spaces framework. Keywords: Krein space, \(J\)-self-adjoint operator, \(J\)-symmetric operator, Friedrichs extension, \(\mathcal{C}\)-symmetry. Mathematics Subject Classification: 47A55, 47B25, 47A57, 81Q15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 65-80, http://dx.doi.org/10.7494/OpMath.2017.37.1.65.

]]>The LQ/KYP problem for infinite-dimensional systemsOur aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3702.pdf
The LQ/KYP problem for infinite-dimensional systemsPiotr Grabowskicontrol of infinite-dimensional systems; semigroups; infinite-time LQ-control problem; Lur'e feedback systemsdoi:10.7494/OpMath.2017.37.1.21Opuscula Math. 37, no. 1 (2017), 21-64, http://dx.doi.org/10.7494/OpMath.2017.37.1.21Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.21http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3702.pdf3712164 Title: The LQ/KYP problem for infinite-dimensional systems.

Author(s): Piotr Grabowski.

Abstract: Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system. Keywords: control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems. Mathematics Subject Classification: 49N10, 93B05, 93C25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 21-64, http://dx.doi.org/10.7494/OpMath.2017.37.1.21.

]]>Limit-point criteria for the matrix Sturm-Liouville operator and its powersWe consider matrix Sturm-Liouville operators generated by the formal expression \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] in the space \(L^2_n(I)\), \(I:=[0, \infty)\). Let the matrix functions \(P:=P(x)\), \(Q:=Q(x)\) and \(R:=R(x)\) of order \(n\) (\(n \in \mathbb{N}\)) be defined on \(I\), \(P\) is a nondegenerate matrix, \(P\) and \(Q\) are Hermitian matrices for \(x \in I\) and the entries of the matrix functions \(P^{-1}\), \(Q\) and \(R\) are measurable on \(I\) and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices \(P\), \(Q\) and \(R\) that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by \(l^k[y]\) (\(k \in \mathbb{N}\)). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.
http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3701.pdf
Limit-point criteria for the matrix Sturm-Liouville operator and its powersIrina N. Braeutigamquasi-derivative; quasi-differential operator; matrix Sturm-Liouville operator; deficiency numbers; distributionsdoi:10.7494/OpMath.2017.37.1.5Opuscula Math. 37, no. 1 (2017), 5-19, http://dx.doi.org/10.7494/OpMath.2017.37.1.5Copyright AGH University of Science and Technology Press, Krakow 2017Opuscula Mathematica20172017http://dx.doi.org/10.7494/OpMath.2017.37.1.5http://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3701.pdf371519 Title: Limit-point criteria for the matrix Sturm-Liouville operator and its powers.

Author(s): Irina N. Braeutigam.

Abstract: We consider matrix Sturm-Liouville operators generated by the formal expression \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] in the space \(L^2_n(I)\), \(I:=[0, \infty)\). Let the matrix functions \(P:=P(x)\), \(Q:=Q(x)\) and \(R:=R(x)\) of order \(n\) (\(n \in \mathbb{N}\)) be defined on \(I\), \(P\) is a nondegenerate matrix, \(P\) and \(Q\) are Hermitian matrices for \(x \in I\) and the entries of the matrix functions \(P^{-1}\), \(Q\) and \(R\) are measurable on \(I\) and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices \(P\), \(Q\) and \(R\) that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by \(l^k[y]\) (\(k \in \mathbb{N}\)). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients. Keywords: quasi-derivative, quasi-differential operator, matrix Sturm-Liouville operator, deficiency numbers, distributions. Mathematics Subject Classification: 34L05, 34B24, 47E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 37, no. 1 (2017), 5-19, http://dx.doi.org/10.7494/OpMath.2017.37.1.5.