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.
https://www.opuscula.agh.edu.pl
AGH University of Science and Technology PressenOpuscula Mathematica1232-92742300-6919Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
Oscillatory behavior of second-order damped differential equations with a superlinear neutral termThis article concerns the oscillatory behavior of solutions to second-order damped nonlinear differential equations with a superlinear neutral term. The results are obtained by a Riccati type transformation as well as by an integral criterion. Examples illustrating the results are provided and some suggestions for further research are indicated.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4035.pdf
Oscillatory behavior of second-order damped differential equations with a superlinear neutral termErcan TunçOsman Özdemiroscillation, second-order, neutral differential equation, damping termdoi:10.7494/OpMath.2020.40.5.629Opuscula Math. 40, no. 5 (2020), 629-639, https://doi.org/10.7494/OpMath.2020.40.5.629Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.629https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4035.pdf405629639 Title: Oscillatory behavior of second-order damped differential equations with a superlinear neutral term.

Author(s): Ercan Tunç, Osman Özdemir.

Abstract: This article concerns the oscillatory behavior of solutions to second-order damped nonlinear differential equations with a superlinear neutral term. The results are obtained by a Riccati type transformation as well as by an integral criterion. Examples illustrating the results are provided and some suggestions for further research are indicated. Keywords: oscillation, second-order, neutral differential equation, damping term. Mathematics Subject Classification: 34C10, 34K11, 34K40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 629-639, https://doi.org/10.7494/OpMath.2020.40.5.629.

]]>On 2-rainbow domination number of functigraph and its complementLet \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4034.pdf
On 2-rainbow domination number of functigraph and its complementAthena ShaminezhadEbrahim Vatandoost2-rainbow domination number, functigraph, complement, cubic graphdoi:10.7494/OpMath.2020.40.5.617Opuscula Math. 40, no. 5 (2020), 617-627, https://doi.org/10.7494/OpMath.2020.40.5.617Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.617https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4034.pdf405617627 Title: On 2-rainbow domination number of functigraph and its complement.

Abstract: Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp. Keywords: 2-rainbow domination number, functigraph, complement, cubic graph. Mathematics Subject Classification: 05C69, 05C75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 617-627, https://doi.org/10.7494/OpMath.2020.40.5.617.

]]>Outer independent rainbow dominating functions in graphs A 2-rainbow dominating function (2-rD function) of a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}\) having the property that if \(f(x)=\emptyset\), then \(f(N(x))=\{1,2\}\). The 2-rainbow domination number \(\gamma_{r2}(G)\) is the minimum weight of \(\sum_{v\in V(G)}|f(v)|\) taken over all 2-rainbow dominating functions \(f\). An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph \(G\) is a 2-rD function \(f\) for which the set of all \(v\in V(G)\) with \(f(v)=\emptyset\) is independent. The outer independent 2-rainbow domination number \(\gamma_{oir2}(G)\) is the minimum weight of an OI2-rD function of \(G\). In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on \(\gamma_{oir2}(G)\). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair \((a,b)\) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if \(a+1\leq b\leq 2a\).
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4033.pdf
Outer independent rainbow dominating functions in graphsZhila MansouriDoost Ali Mojdehouter-independent rainbow domination, \(K_{1,r}\)-free graphs, treesdoi:10.7494/OpMath.2020.40.5.599Opuscula Math. 40, no. 5 (2020), 599-615, https://doi.org/10.7494/OpMath.2020.40.5.599Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.599https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4033.pdf405599615 Title: Outer independent rainbow dominating functions in graphs.

Author(s): Zhila Mansouri, Doost Ali Mojdeh.

Abstract: A 2-rainbow dominating function (2-rD function) of a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}\) having the property that if \(f(x)=\emptyset\), then \(f(N(x))=\{1,2\}\). The 2-rainbow domination number \(\gamma_{r2}(G)\) is the minimum weight of \(\sum_{v\in V(G)}|f(v)|\) taken over all 2-rainbow dominating functions \(f\). An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph \(G\) is a 2-rD function \(f\) for which the set of all \(v\in V(G)\) with \(f(v)=\emptyset\) is independent. The outer independent 2-rainbow domination number \(\gamma_{oir2}(G)\) is the minimum weight of an OI2-rD function of \(G\). In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on \(\gamma_{oir2}(G)\). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair \((a,b)\) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if \(a+1\leq b\leq 2a\). Keywords: outer-independent rainbow domination, \(K_{1,r}\)-free graphs, trees. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 599-615, https://doi.org/10.7494/OpMath.2020.40.5.599.

]]>On some extensions of the A-modelThe A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4032.pdf
On some extensions of the A-modelRytis Juršėnasfinite rank higher order singular perturbation, cascade (A) model, peak model, Hilbert space, scale of Hilbert spaces, Pontryagin space, ordinary boundary triple, Krein \(Q\)-function, Weyl function, gamma field, symmetric operator, proper extension, resolventdoi:10.7494/OpMath.2020.40.5.569Opuscula Math. 40, no. 5 (2020), 569-597, https://doi.org/10.7494/OpMath.2020.40.5.569Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.569https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4032.pdf405569597 Title: On some extensions of the A-model.

Author(s): Rytis Juršėnas.

Abstract: The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces. Keywords: finite rank higher order singular perturbation, cascade (A) model, peak model, Hilbert space, scale of Hilbert spaces, Pontryagin space, ordinary boundary triple, Krein \(Q\)-function, Weyl function, gamma field, symmetric operator, proper extension, resolvent. Mathematics Subject Classification: 47A56, 47B25, 47B50, 35P05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 569-597, https://doi.org/10.7494/OpMath.2020.40.5.569.

]]>On the nonoscillatory behavior of solutions of three classes of fractional difference equationsIn this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4031.pdf
On the nonoscillatory behavior of solutions of three classes of fractional difference equationsSaid Rezk GraceJehad AlzabutSakthivel PunithaVelu MuthulakshmiHakan AdıgüzelCaputo difference operator, nonoscillation criteria, fractional difference equation, mathematical inequalitiesdoi:10.7494/OpMath.2020.40.5.549Opuscula Math. 40, no. 5 (2020), 549-568, https://doi.org/10.7494/OpMath.2020.40.5.549Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.549https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4031.pdf405549568 Title: On the nonoscillatory behavior of solutions of three classes of fractional difference equations.

Abstract: In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark. Keywords: Caputo difference operator, nonoscillation criteria, fractional difference equation, mathematical inequalities. Mathematics Subject Classification: 34A08, 39A21. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 549-568, https://doi.org/10.7494/OpMath.2020.40.5.549.

]]>Existence results for a sublinear second order Dirichlet boundary value problem on the half-lineIn this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by
using Ekeland's variational principle and critical point theory.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4030.pdf
Existence results for a sublinear second order Dirichlet boundary value problem on the half-lineDahmane BouafiaToufik MoussaouiEkeland's variational principle, critical pointdoi:10.7494/OpMath.2020.40.5.537Opuscula Math. 40, no. 5 (2020), 537-548, https://doi.org/10.7494/OpMath.2020.40.5.537Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.537https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4030.pdf405537548 Title: Existence results for a sublinear second order Dirichlet boundary value problem on the half-line.

Author(s): Dahmane Bouafia, Toufik Moussaoui.

Abstract: In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by
using Ekeland's variational principle and critical point theory. Keywords: Ekeland's variational principle, critical point. Mathematics Subject Classification: 34B40, 35A15, 35B38, 45C05, 34B24, 46T20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 537-548, https://doi.org/10.7494/OpMath.2020.40.5.537.

]]>Oscillatory criteria via linearization of half-linear second order delay differential equationsIn the paper, we study oscillation of the half-linear second order delay differential equations of the form \[\left(r(t)(y'(t))^{\alpha}\right)'+p(t)y^{\alpha}(\tau(t))=0.\] We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones.
https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4029.pdf
Oscillatory criteria via linearization of half-linear second order delay differential equationsBlanka BaculíkováJozef Džurinasecond order differential equations, delay, monotonic properties, linearization, oscillationdoi:10.7494/OpMath.2020.40.5.523Opuscula Math. 40, no. 5 (2020), 523-536, https://doi.org/10.7494/OpMath.2020.40.5.523Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.5.523https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4029.pdf405523536 Title: Oscillatory criteria via linearization of half-linear second order delay differential equations.

Author(s): Blanka Baculíková, Jozef Džurina.

Abstract: In the paper, we study oscillation of the half-linear second order delay differential equations of the form \[\left(r(t)(y'(t))^{\alpha}\right)'+p(t)y^{\alpha}(\tau(t))=0.\] We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones. Keywords: second order differential equations, delay, monotonic properties, linearization, oscillation. Mathematics Subject Classification: 34K11, 34C10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 5 (2020), 523-536, https://doi.org/10.7494/OpMath.2020.40.5.523.

]]>Decompositions of complete 3-uniform hypergraphs into cycles of constant prime lengthA complete \(3\)-uniform hypergraph of order \(n\) has vertex set \(V\) with \(|V|=n\) and the set of all \(3\)-subsets of \(V\) as its edge set. A \(t\)-cycle in this hypergraph is \(v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1\) where \(v_1, v_2,\dots, v_t\) are distinct vertices and \(e_1, e_2,\dots, e_t\) are distinct edges such that \(v_i, v_{i+1}\in e_i\) for \(i \in \{1, 2,\dots, t-1\}\) and \(v_t, v_1 \in e_t\). A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete \(3\)-uniform hypergraph of order \(n\) into \(p\)-cycles, whenever \(p\) is prime.
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4028.pdf
Decompositions of complete 3-uniform hypergraphs into cycles of constant prime lengthR. LakshmiT. Poovaragavanuniform hypergraph, cycle decompositiondoi:10.7494/OpMath.2020.40.4.509Opuscula Math. 40, no. 4 (2020), 509-516, https://doi.org/10.7494/OpMath.2020.40.4.509Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.509https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4028.pdf404509516 Title: Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length.

Author(s): R. Lakshmi, T. Poovaragavan.

Abstract: A complete \(3\)-uniform hypergraph of order \(n\) has vertex set \(V\) with \(|V|=n\) and the set of all \(3\)-subsets of \(V\) as its edge set. A \(t\)-cycle in this hypergraph is \(v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1\) where \(v_1, v_2,\dots, v_t\) are distinct vertices and \(e_1, e_2,\dots, e_t\) are distinct edges such that \(v_i, v_{i+1}\in e_i\) for \(i \in \{1, 2,\dots, t-1\}\) and \(v_t, v_1 \in e_t\). A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete \(3\)-uniform hypergraph of order \(n\) into \(p\)-cycles, whenever \(p\) is prime. Keywords: uniform hypergraph, cycle decomposition. Mathematics Subject Classification: 05C65, 05C85. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 509-516, https://doi.org/10.7494/OpMath.2020.40.4.509.

]]>Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4027.pdf
Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spacesChing-on LoAnthony Wai-keung Lohweighted composition operators, Hardy spaces, compact operators, Hilbert-Schmidt operatorsdoi:10.7494/OpMath.2020.40.4.495Opuscula Math. 40, no. 4 (2020), 495-507, https://doi.org/10.7494/OpMath.2020.40.4.495Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.495https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4027.pdf404495507 Title: Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces.

Author(s): Ching-on Lo, Anthony Wai-keung Loh.

Abstract: Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\). Keywords: weighted composition operators, Hardy spaces, compact operators, Hilbert-Schmidt operators. Mathematics Subject Classification: 47B33, 30H10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 495-507, https://doi.org/10.7494/OpMath.2020.40.4.495.

]]>Properties of solutions to some weighted p-Laplacian equation In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\).
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4026.pdf
Properties of solutions to some weighted p-Laplacian equationPrashanta Garain\(p\)-Laplacian, degenerate elliptic equations, weighted Sobolev spacedoi:10.7494/OpMath.2020.40.4.483Opuscula Math. 40, no. 4 (2020), 483-494, https://doi.org/10.7494/OpMath.2020.40.4.483Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.483https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4026.pdf404483494 Title: Properties of solutions to some weighted p-Laplacian equation.

Author(s): Prashanta Garain.

Abstract: In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\). Keywords: \(p\)-Laplacian, degenerate elliptic equations, weighted Sobolev space. Mathematics Subject Classification: 35A01, 35J62, 35J70. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 483-494, https://doi.org/10.7494/OpMath.2020.40.4.483.

]]>Facial rainbow edge-coloring of simple 3-connected plane graphsA facial rainbow edge-coloring of a plane graph \(G\) is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of \(G\). The minimum number of colors used in such a coloring is denoted by \(\text{erb}(G)\). Trivially, \(\text{erb}(G) \geq \text{L}(G)+1\) holds for every plane graph without cut-vertices, where \(\text{L}(G)\) denotes the length of a longest facial path in \(G\). Jendroľ in 2018 proved that every simple \(3\)-connected plane graph admits a facial rainbow edge-coloring with at most \(\text{L}(G)+2\) colors, moreover, this bound is tight for \(\text{L}(G)=3\). He also proved that \(\text{erb}(G) = \text{L}(G)+1\) for \(\text{L}(G)\not\in\{3,4,5\}\). He posed the following conjecture: There is a simple \(3\)-connected plane graph \(G\) with \(\text{L}(G)=4\) and \(\text{erb}(G)=\text{L}(G)+2\). In this note we answer the conjecture in the affirmative.
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4025.pdf
Facial rainbow edge-coloring of simple 3-connected plane graphsJúlius Czapplane graph, facial path, edge-coloringdoi:10.7494/OpMath.2020.40.4.475Opuscula Math. 40, no. 4 (2020), 475-482, https://doi.org/10.7494/OpMath.2020.40.4.475Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.475https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4025.pdf404475482 Title: Facial rainbow edge-coloring of simple 3-connected plane graphs.

Author(s): Július Czap.

Abstract: A facial rainbow edge-coloring of a plane graph \(G\) is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of \(G\). The minimum number of colors used in such a coloring is denoted by \(\text{erb}(G)\). Trivially, \(\text{erb}(G) \geq \text{L}(G)+1\) holds for every plane graph without cut-vertices, where \(\text{L}(G)\) denotes the length of a longest facial path in \(G\). Jendroľ in 2018 proved that every simple \(3\)-connected plane graph admits a facial rainbow edge-coloring with at most \(\text{L}(G)+2\) colors, moreover, this bound is tight for \(\text{L}(G)=3\). He also proved that \(\text{erb}(G) = \text{L}(G)+1\) for \(\text{L}(G)\not\in\{3,4,5\}\). He posed the following conjecture: There is a simple \(3\)-connected plane graph \(G\) with \(\text{L}(G)=4\) and \(\text{erb}(G)=\text{L}(G)+2\). In this note we answer the conjecture in the affirmative. Keywords: plane graph, facial path, edge-coloring. Mathematics Subject Classification: 05C10, 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 475-482, https://doi.org/10.7494/OpMath.2020.40.4.475.

]]>Option pricing formulas under a change of numèraireWe present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4024.pdf
Option pricing formulas under a change of numèraireAntonio AttalientiMichele BufaloBlack-Scholes formula, binomial model, martingale measures, numèrairedoi:10.7494/OpMath.2020.40.4.451Opuscula Math. 40, no. 4 (2020), 451-473, https://doi.org/10.7494/OpMath.2020.40.4.451Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.451https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4024.pdf404451473 Title: Option pricing formulas under a change of numèraire.

Author(s): Antonio Attalienti, Michele Bufalo.

Abstract: We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper. Keywords: Black-Scholes formula, binomial model, martingale measures, numèraire. Mathematics Subject Classification: 91B25, 60G46. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 451-473, https://doi.org/10.7494/OpMath.2020.40.4.451.

]]>An inverse backward problem for degenerate two-dimensional parabolic equationThis paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4023.pdf
An inverse backward problem for degenerate two-dimensional parabolic equationKhalid AtifiEl-Hassan EssoufiBouchra Khouitidata assimilation, adjoint method, regularization, heat equation, inverse problem, degenerate equations, optimizationdoi:10.7494/OpMath.2020.40.4.427Opuscula Math. 40, no. 4 (2020), 427-449, https://doi.org/10.7494/OpMath.2020.40.4.427Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.427https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4023.pdf404427449 Title: An inverse backward problem for degenerate two-dimensional parabolic equation.

Abstract: This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach. Keywords: data assimilation, adjoint method, regularization, heat equation, inverse problem, degenerate equations, optimization. Mathematics Subject Classification: 15A29, 47A52, 34A38, 93C20, 60J70, 35K05, 35K65. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 427-449, https://doi.org/10.7494/OpMath.2020.40.4.427.

]]>Multiple solutions of boundary value problems on time scales for a φ-Laplacian operatorWe establish the existence and multiplicity of solutions for some boundary value problems on time scales with a \(\varphi\)-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete \(p\)-Laplacian as well as those for boundary value problems on time scales.
https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4022.pdf
Multiple solutions of boundary value problems on time scales for a φ-Laplacian operatorPablo AmsterMariel Paula KunaDionicio Pastor Santosdynamic equations on time scales, nonlinear boundary value problems, upper and lower solutions, Leray-Schauder degree, multiple solutionsdoi:10.7494/OpMath.2020.40.4.405Opuscula Math. 40, no. 4 (2020), 405-425, https://doi.org/10.7494/OpMath.2020.40.4.405Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.4.405https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4022.pdf404405425 Title: Multiple solutions of boundary value problems on time scales for a φ-Laplacian operator.

Author(s): Pablo Amster, Mariel Paula Kuna, Dionicio Pastor Santos.

Abstract: We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a \(\varphi\)-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete \(p\)-Laplacian as well as those for boundary value problems on time scales. Keywords: dynamic equations on time scales, nonlinear boundary value problems, upper and lower solutions, Leray-Schauder degree, multiple solutions. Mathematics Subject Classification: 34N05, 34K10, 47H11. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 4 (2020), 405-425, https://doi.org/10.7494/OpMath.2020.40.4.405.

]]>On the crossing numbers of join products of five graphs of order six with the discrete graphThe main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product \(G^{\ast} + D_n\), where the disconnected graph \(G^{\ast}\) of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the \(5\)-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph \(G^{\ast}\), the crossing numbers of \(G_i+D_n\) for four other graphs \(G_i\) of order six will be also established.
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4021.pdf
On the crossing numbers of join products of five graphs of order six with the discrete graphMichal Stašgraph, drawing, crossing number, join product, cyclic permutationdoi:10.7494/OpMath.2020.40.3.383Opuscula Math. 40, no. 3 (2020), 383-397, https://doi.org/10.7494/OpMath.2020.40.3.383Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.383https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4021.pdf403383397 Title: On the crossing numbers of join products of five graphs of order six with the discrete graph.

Author(s): Michal Staš.

Abstract: The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product \(G^{\ast} + D_n\), where the disconnected graph \(G^{\ast}\) of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the \(5\)-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph \(G^{\ast}\), the crossing numbers of \(G_i+D_n\) for four other graphs \(G_i\) of order six will be also established. Keywords: graph, drawing, crossing number, join product, cyclic permutation. Mathematics Subject Classification: 05C10, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 383-397, https://doi.org/10.7494/OpMath.2020.40.3.383.

]]>A note on bipartite graphs whose [1,k]-domination number equal to their number of verticesA subset \(D\) of the vertex set \(V\) of a graph \(G\) is called an \([1,k]\)-dominating set if every vertex from \(V-D\) is adjacent to at least one vertex and at most \(k\) vertices of \(D\). A \([1,k]\)-dominating set with the minimum number of vertices is called a \(\gamma_{[1,k]}\)-set and the number of its vertices is the \([1,k]\)-domination number \(\gamma_{[1,k]}(G)\) of \(G\). In this short note we show that the decision problem whether \(\gamma_{[1,k]}(G)=n\) is an \(NP\)-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph \(G\) of order \(n\) satisfying \(\gamma_{[1,k]}(G)=n\) is given for every integer \(n \geq (k+1)(2k+3)\).
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4020.pdf
A note on bipartite graphs whose [1,k]-domination number equal to their number of verticesNarges GhareghaniIztok PeterinPouyeh Sharifanidomination, \([1,k]\)-domination number, \([1,k]\)-total domination number, bipartite graphsdoi:10.7494/OpMath.2020.40.3.375Opuscula Math. 40, no. 3 (2020), 375-382, https://doi.org/10.7494/OpMath.2020.40.3.375Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.375https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4020.pdf403375382 Title: A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices.

Abstract: A subset \(D\) of the vertex set \(V\) of a graph \(G\) is called an \([1,k]\)-dominating set if every vertex from \(V-D\) is adjacent to at least one vertex and at most \(k\) vertices of \(D\). A \([1,k]\)-dominating set with the minimum number of vertices is called a \(\gamma_{[1,k]}\)-set and the number of its vertices is the \([1,k]\)-domination number \(\gamma_{[1,k]}(G)\) of \(G\). In this short note we show that the decision problem whether \(\gamma_{[1,k]}(G)=n\) is an \(NP\)-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph \(G\) of order \(n\) satisfying \(\gamma_{[1,k]}(G)=n\) is given for every integer \(n \geq (k+1)(2k+3)\). Keywords: domination, \([1,k]\)-domination number, \([1,k]\)-total domination number, bipartite graphs. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 375-382, https://doi.org/10.7494/OpMath.2020.40.3.375.

]]>A note on confidence intervals for deblurred imagesWe consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented.
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4019.pdf
A note on confidence intervals for deblurred imagesMichał BielZbigniew Szkutnikinverse problems, confidence intervals, convolution, deblurringdoi:10.7494/OpMath.2020.40.3.361Opuscula Math. 40, no. 3 (2020), 361-373, https://doi.org/10.7494/OpMath.2020.40.3.361Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.361https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4019.pdf403361373 Title: A note on confidence intervals for deblurred images.

Author(s): Michał Biel, Zbigniew Szkutnik.

Abstract: We consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented. Keywords: inverse problems, confidence intervals, convolution, deblurring. Mathematics Subject Classification: 62G08, 62G15, 62G20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 361-373, https://doi.org/10.7494/OpMath.2020.40.3.361.

]]>Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systemsWe investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature.
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4018.pdf
Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systemsMimia BenhadriTomás CaraballoHalim ZeghdoudiKrasnoselskii's fixed point theorem, positive periodic solutions, Lotka-Volterra competition systems, variable delaysdoi:10.7494/OpMath.2020.40.3.341Opuscula Math. 40, no. 3 (2020), 341-360, https://doi.org/10.7494/OpMath.2020.40.3.341Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.341https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4018.pdf403341360 Title: Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems.

Abstract: We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature. Keywords: Krasnoselskii's fixed point theorem, positive periodic solutions, Lotka-Volterra competition systems, variable delays. Mathematics Subject Classification: 34K20, 34K13, 92B20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 341-360, https://doi.org/10.7494/OpMath.2020.40.3.341.

]]>Stochastic Wiener filter in the white noise spaceIn this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4017.pdf
Stochastic Wiener filter in the white noise spaceDaniel AlpayAriel PinhasWiener filter, white noise space, Wick product, stochastic distributiondoi:10.7494/OpMath.2020.40.3.323Opuscula Math. 40, no. 3 (2020), 323-339, https://doi.org/10.7494/OpMath.2020.40.3.323Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.323https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4017.pdf403323339 Title: Stochastic Wiener filter in the white noise space.

Author(s): Daniel Alpay, Ariel Pinhas.

Abstract: In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions. Keywords: Wiener filter, white noise space, Wick product, stochastic distribution. Mathematics Subject Classification: 93E11, 60H40, 46F25, 13J99. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 323-339, https://doi.org/10.7494/OpMath.2020.40.3.323.

]]>A unique weak solution for a kind of coupled system of fractional Schrödinger equations
In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrödinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables.
https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4016.pdf
A unique weak solution for a kind of coupled system of fractional Schrödinger equationsFatemeh AbdolrazaghiAbdolrahman Razanifractional Laplacian, uniqueness, weak solution, nonlinear systemsdoi:10.7494/OpMath.2020.40.3.313Opuscula Math. 40, no. 3 (2020), 313-322, https://doi.org/10.7494/OpMath.2020.40.3.313Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.3.313https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4016.pdf403313322 Title: A unique weak solution for a kind of coupled system of fractional Schrödinger equations.

Abstract:
In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrödinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables. Keywords: fractional Laplacian, uniqueness, weak solution, nonlinear systems. Mathematics Subject Classification: 34A08, 35J10, 35D30, 93C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 3 (2020), 313-322, https://doi.org/10.7494/OpMath.2020.40.3.313.

]]>Oscillation of time fractional vector diffusion-wave equation with fractional dampingIn this paper, sufficient conditions for \(H\)-oscillation of solutions of a time fractional vector diffusion-wave equation with forced and fractional damping terms subject to the Neumann boundary condition are established by employing certain fractional differential inequality, where \(H\) is a unit vector in \(\mathbb{R}^n\). The examples are given to illustrate the main results.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4015.pdf
Oscillation of time fractional vector diffusion-wave equation with fractional dampingR. RameshS. HarikrishnanJ. J. NietoP. Prakashfractional diffusion-wave equation, \(H\)-oscillation, vector differential equationdoi:10.7494/OpMath.2020.40.2.291Opuscula Math. 40, no. 2 (2020), 291-305, https://doi.org/10.7494/OpMath.2020.40.2.291Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.291https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4015.pdf402291305 Title: Oscillation of time fractional vector diffusion-wave equation with fractional damping.

Author(s): R. Ramesh, S. Harikrishnan, J. J. Nieto, P. Prakash.

Abstract: In this paper, sufficient conditions for \(H\)-oscillation of solutions of a time fractional vector diffusion-wave equation with forced and fractional damping terms subject to the Neumann boundary condition are established by employing certain fractional differential inequality, where \(H\) is a unit vector in \(\mathbb{R}^n\). The examples are given to illustrate the main results. Keywords: fractional diffusion-wave equation, \(H\)-oscillation, vector differential equation. Mathematics Subject Classification: 35B05, 35R11, 34K37. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 291-305, https://doi.org/10.7494/OpMath.2020.40.2.291.

]]>Subdivision of hypergraphs and their coloringsIn this paper we introduce the subdivision of hypergraphs, study their properties and parameters and investigate their weak and strong chromatic numbers in various cases.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4014.pdf
Subdivision of hypergraphs and their coloringsMoharram N. Iradmusahypergraph, uniform hypergraph, subdivision of hypergraphdoi:10.7494/OpMath.2020.40.2.271Opuscula Math. 40, no. 2 (2020), 271-290, https://doi.org/10.7494/OpMath.2020.40.2.271Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.271https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4014.pdf402271290 Title: Subdivision of hypergraphs and their colorings.

Author(s): Moharram N. Iradmusa.

Abstract: In this paper we introduce the subdivision of hypergraphs, study their properties and parameters and investigate their weak and strong chromatic numbers in various cases. Keywords: hypergraph, uniform hypergraph, subdivision of hypergraph. Mathematics Subject Classification: 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 271-290, https://doi.org/10.7494/OpMath.2020.40.2.271.

]]>Asymptotic expansion of large eigenvalues for a class of unbounded Jacobi matricesWe investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4013.pdf
Asymptotic expansion of large eigenvalues for a class of unbounded Jacobi matricesAyoub HarratEl Hassan ZeroualiLech Zielinskitridiagonal matrix, band matrix, unbounded self-adjoint operator, discrete spectrum, large eigenvalues, asymptoticsdoi:10.7494/OpMath.2020.40.2.241Opuscula Math. 40, no. 2 (2020), 241-270, https://doi.org/10.7494/OpMath.2020.40.2.241Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.241https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4013.pdf402241270 Title: Asymptotic expansion of large eigenvalues for a class of unbounded Jacobi matrices.

Author(s): Ayoub Harrat, El Hassan Zerouali, Lech Zielinski.

Abstract: We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly. Keywords: tridiagonal matrix, band matrix, unbounded self-adjoint operator, discrete spectrum, large eigenvalues, asymptotics. Mathematics Subject Classification: 47B25, 47B36, 15A18. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 241-270, https://doi.org/10.7494/OpMath.2020.40.2.241.

]]>On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative termsThis paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdf
On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative termsJohn R. GraefSaid R. GraceErcan Tunçintegro-differential equations, fractional differential equations, nonoscillatory solutions, boundedness, Caputo derivativedoi:10.7494/OpMath.2020.40.2.227Opuscula Math. 40, no. 2 (2020), 227-239, https://doi.org/10.7494/OpMath.2020.40.2.227Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.227https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdf402227239 Title: On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms.

Author(s): John R. Graef, Said R. Grace, Ercan Tunç.

Abstract: This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained. Keywords: integro-differential equations, fractional differential equations, nonoscillatory solutions, boundedness, Caputo derivative. Mathematics Subject Classification: 34E10, 34A34. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 227-239, https://doi.org/10.7494/OpMath.2020.40.2.227.

]]>Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cyclesIt is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2\ (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4011.pdf
Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cyclesRyan C. BungeDontez CollinsDaryl Conko-CamelSaad I. El-ZanatiRachel LiebrechtAlexander Vasquezmaximum packing, \(\lambda\)-fold complete 3-uniform hypergraph, loose 3-cycledoi:10.7494/OpMath.2020.40.2.209Opuscula Math. 40, no. 2 (2020), 209-225, https://doi.org/10.7494/OpMath.2020.40.2.209Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.209https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4011.pdf402209225 Title: Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles.

Author(s): Ryan C. Bunge, Dontez Collins, Daryl Conko-Camel, Saad I. El-Zanati, Rachel Liebrecht, Alexander Vasquez.

Abstract: It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2\ (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges. Keywords: maximum packing, \(\lambda\)-fold complete 3-uniform hypergraph, loose 3-cycle. Mathematics Subject Classification: 05C65, 05C85. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 209-225, https://doi.org/10.7494/OpMath.2020.40.2.209.

]]>On the deformed Besov-Hankel spacesIn this paper we introduce function spaces denoted by \(BH_{\kappa,\beta}^{p,r}\) (\(0\lt\beta\lt 1\), \(1\leq p, r \leq +\infty\)) as subspaces of \(L^p\) that we call deformed Besov-Hankel spaces. We provide characterizations of these spaces in terms of Bochner-Riesz means in the case \(1\leq p\leq +\infty\) and in terms of partial Hankel integrals in the case \(1\lt p\lt +\infty\) associated to the deformed Hankel operator by a parameter \(\kappa\gt 0\). For \(p=r=+\infty\), we obtain an approximation result involving partial Hankel integrals.
https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4010.pdf
On the deformed Besov-Hankel spacesSalem Ben SaïdMohamed Amine BoubatraMohamed Sifideformed Hankel kernel, Besov spaces, Bochner-Riesz means, partial Hankel integralsdoi:10.7494/OpMath.2020.40.2.171Opuscula Math. 40, no. 2 (2020), 171-207, https://doi.org/10.7494/OpMath.2020.40.2.171Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.2.171https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4010.pdf402171207 Title: On the deformed Besov-Hankel spaces.

Author(s): Salem Ben Saïd, Mohamed Amine Boubatra, Mohamed Sifi.

Abstract: In this paper we introduce function spaces denoted by \(BH_{\kappa,\beta}^{p,r}\) (\(0\lt\beta\lt 1\), \(1\leq p, r \leq +\infty\)) as subspaces of \(L^p\) that we call deformed Besov-Hankel spaces. We provide characterizations of these spaces in terms of Bochner-Riesz means in the case \(1\leq p\leq +\infty\) and in terms of partial Hankel integrals in the case \(1\lt p\lt +\infty\) associated to the deformed Hankel operator by a parameter \(\kappa\gt 0\). For \(p=r=+\infty\), we obtain an approximation result involving partial Hankel integrals. Keywords: deformed Hankel kernel, Besov spaces, Bochner-Riesz means, partial Hankel integrals. Mathematics Subject Classification: 44A15, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 2 (2020), 171-207, https://doi.org/10.7494/OpMath.2020.40.2.171.

]]>Concentration-compactness results for systems in the Heisenberg groupIn this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4009.pdf
Concentration-compactness results for systems in the Heisenberg groupPatrizia PucciLetizia TemperiniHeisenberg group, concentration-compactness, critical exponents, Hardy termsdoi:10.7494/OpMath.2020.40.1.151Opuscula Math. 40, no. 1 (2020), 151-163, https://doi.org/10.7494/OpMath.2020.40.1.151Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.151https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4009.pdf401151163 Title: Concentration-compactness results for systems in the Heisenberg group.

Author(s): Patrizia Pucci, Letizia Temperini.

Abstract: In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms. Keywords: Heisenberg group, concentration-compactness, critical exponents, Hardy terms. Mathematics Subject Classification: 22E30, 35B33, 35J50, 58E30, 35H05, 35A23. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 151-163, https://doi.org/10.7494/OpMath.2020.40.1.151.

]]>A multiplicity theorem for parametric superlinear (p,q)-equationsWe consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4008.pdf
A multiplicity theorem for parametric superlinear (p,q)-equationsFlorin-Iulian OneteNikolaos S. PapageorgiouCalogero Vetrosuperlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groupsdoi:10.7494/OpMath.2020.40.1.131Opuscula Math. 40, no. 1 (2020), 131-149, https://doi.org/10.7494/OpMath.2020.40.1.131Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.131https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4008.pdf401131149 Title: A multiplicity theorem for parametric superlinear (p,q)-equations.

Author(s): Florin-Iulian Onete, Nikolaos S. Papageorgiou, Calogero Vetro.

Abstract: We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information. Keywords: superlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groups. Mathematics Subject Classification: 35J20, 35J60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 131-149, https://doi.org/10.7494/OpMath.2020.40.1.131.

]]>Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearityIn this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4007.pdf
Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearityWei LianMd Salik AhmedRunzhang Xuglobal existence, blow-up, logarithmic and polynomial nonlinearity, potential welldoi:10.7494/OpMath.2020.40.1.111Opuscula Math. 40, no. 1 (2020), 111-130, https://doi.org/10.7494/OpMath.2020.40.1.111Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.111https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4007.pdf401111130 Title: Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity.

Author(s): Wei Lian, Md Salik Ahmed, Runzhang Xu.

Abstract: In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity. Keywords: global existence, blow-up, logarithmic and polynomial nonlinearity, potential well. Mathematics Subject Classification: 35L71, 35L20, 35L05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 111-130, https://doi.org/10.7494/OpMath.2020.40.1.111.

]]>Fractional p&q-Laplacian problems with potentials vanishing at infinityIn this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4006.pdf
Fractional p&q-Laplacian problems with potentials vanishing at infinityTeresa Iserniafractional \(p\&q\)-Laplacian, vanishing potentials, ground state solutiondoi:10.7494/OpMath.2020.40.1.93Opuscula Math. 40, no. 1 (2020), 93-110, https://doi.org/10.7494/OpMath.2020.40.1.93Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.93https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4006.pdf40193110 Title: Fractional p&q-Laplacian problems with potentials vanishing at infinity.

Author(s): Teresa Isernia.

Abstract: In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator. Keywords: fractional \(p\&q\)-Laplacian, vanishing potentials, ground state solution. Mathematics Subject Classification: 35A15, 35J60, 35R11, 45G05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 93-110, https://doi.org/10.7494/OpMath.2020.40.1.93.

]]>Nonhomogeneous equations with critical exponential growth and lack of compactnessWe study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4005.pdf
Nonhomogeneous equations with critical exponential growth and lack of compactnessGiovany M. FigueiredoVicenţiu D. Rădulescuexponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iterationdoi:10.7494/OpMath.2020.40.1.71Opuscula Math. 40, no. 1 (2020), 71-92, https://doi.org/10.7494/OpMath.2020.40.1.71Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.71https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4005.pdf4017192 Title: Nonhomogeneous equations with critical exponential growth and lack of compactness.

Author(s): Giovany M. Figueiredo, Vicenţiu D. Rădulescu.

Abstract: We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth. Keywords: exponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iteration. Mathematics Subject Classification: 35J62, 35A15, 35B30, 35B33, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 71-92, https://doi.org/10.7494/OpMath.2020.40.1.71.

]]>On the regularity of solution to the time-dependent p-Stokes systemIn this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4004.pdf
On the regularity of solution to the time-dependent p-Stokes systemLuigi C. BerselliMichael Růžičkaregularity, evolution problem, \(p\)-Stokesdoi:10.7494/OpMath.2020.40.1.49Opuscula Math. 40, no. 1 (2020), 49-69, https://doi.org/10.7494/OpMath.2020.40.1.49Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.49https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4004.pdf4014969 Title: On the regularity of solution to the time-dependent p-Stokes system.

Author(s): Luigi C. Berselli, Michael Růžička.

Abstract: In this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces. Keywords: regularity, evolution problem, \(p\)-Stokes. Mathematics Subject Classification: 76D03, 35Q35, 76A05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 49-69, https://doi.org/10.7494/OpMath.2020.40.1.49.

]]>On solvability of elliptic boundary value problems via global invertibilityIn this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4003.pdf
On solvability of elliptic boundary value problems via global invertibilityMichał BełdzińskiMarek Galewskidiffeomorphism, Dirichlet conditions, Laplace operator, Neumann conditions, uniquenessdoi:10.7494/OpMath.2020.40.1.37Opuscula Math. 40, no. 1 (2020), 37-47, https://doi.org/10.7494/OpMath.2020.40.1.37Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.37https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4003.pdf4013747 Title: On solvability of elliptic boundary value problems via global invertibility.

Author(s): Michał Bełdziński, Marek Galewski.

Abstract: In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. Keywords: diffeomorphism, Dirichlet conditions, Laplace operator, Neumann conditions, uniqueness. Mathematics Subject Classification: 35J60, 46T20, 47H30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 37-47, https://doi.org/10.7494/OpMath.2020.40.1.37.

]]>Some multiplicity results of homoclinic solutions for second order Hamiltonian systems We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4002.pdf
Some multiplicity results of homoclinic solutions for second order Hamiltonian systemsSara BarileAddolorata Salvatoresecond order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddingsdoi:10.7494/OpMath.2020.40.1.21Opuscula Math. 40, no. 1 (2020), 21-36, https://doi.org/10.7494/OpMath.2020.40.1.21Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.21https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4002.pdf4012136 Title: Some multiplicity results of homoclinic solutions for second order Hamiltonian systems.

Author(s): Sara Barile, Addolorata Salvatore.

Abstract: We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough. Keywords: second order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddings. Mathematics Subject Classification: 34C37, 58E05, 70H05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 21-36, https://doi.org/10.7494/OpMath.2020.40.1.21.

]]>On some convergence results for fractional periodic Sobolev spacesIn this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4001.pdf
On some convergence results for fractional periodic Sobolev spacesVincenzo Ambrosiofractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergencedoi:10.7494/OpMath.2020.40.1.5Opuscula Math. 40, no. 1 (2020), 5-20, https://doi.org/10.7494/OpMath.2020.40.1.5Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.5https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4001.pdf401520 Title: On some convergence results for fractional periodic Sobolev spaces.

Author(s): Vincenzo Ambrosio.

Abstract: In this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series. Keywords: fractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergence. Mathematics Subject Classification: 42B05, 46E35, 49J45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 5-20, https://doi.org/10.7494/OpMath.2020.40.1.5.