Opuscula Mathematica
https://www.opuscula.agh.edu.pl
enA 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.Mon, 01 Jul 2024 23:50:00 +0200Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4435.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4435.pdfMon, 01 Jul 2024 23:40:07 +0200 Author(s): César E. Torres Ledesma, Hernán C. Gutierrez, Jesús A. Rodríguez, Manuel M. Bonilla.

Abstract: In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result. Keywords: harmonic potential, fractional Sobolev space, ground state solution, bifurcation result, variational method. Mathematics Subject Classification: 45G05, 35J60, 35B25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 749-765, https://doi.org/10.7494/OpMath.2024.44.5.749.

]]>Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4434.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4434.pdfMon, 01 Jul 2024 23:40:06 +0200 Author(s): Kazuki Ishibashi.

Abstract: In this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type. Keywords: nonoscillation, conformable differential equation, proportional-derivative controller, Riccati technique, Euler equation. Mathematics Subject Classification: 34C10, 26A24. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 727-748, https://doi.org/10.7494/OpMath.2024.44.5.727.

]]>Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4433.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4433.pdfMon, 01 Jul 2024 23:40:05 +0200 Author(s): Gonzalo Galiano.

Abstract: We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem. Keywords: nonlocal diffusion, Schwarz's symmetrization, Talenti's theorem, Riesz's inequality. Mathematics Subject Classification: 35B45, 35R09, 35J25, 35K20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 707-726, https://doi.org/10.7494/OpMath.2024.44.5.707.

]]>Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4432.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4432.pdfMon, 01 Jul 2024 23:40:04 +0200 Author(s): Anastasia Dudko, Oleksandr Lesechko, Vyacheslav Pivovarchik.

Abstract: We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs. Keywords: Sturm-Liouville equation, eigenvalue, spectrum, equilateral tree, caterpillar tree, snowflake graph, root, standard conditions, Dirichlet boundary condition, Neumann boundary condition. Mathematics Subject Classification: 34B45, 34B24, 34L20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 689-705, https://doi.org/10.7494/OpMath.2024.44.5.689.

]]>Seven largest trees pack
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4431.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4431.pdfMon, 01 Jul 2024 23:40:03 +0200 Author(s): Maciej Cisiński, Andrzej Żak.

Abstract: The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees \(T_2,\dots,T_{n-1}, T_n\) such that \(T_i\) has \(i\) vertices pack into \(K_n\). The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that \(k\) largest trees pack. The latter is true if none tree is a star, but in general, it is known only for \(k=5\). In this paper we prove, among other results, that seven largest trees pack. Keywords: tree, packing, tree packing conjecture. Mathematics Subject Classification: 05C35, 05C05, 05C70. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 673-688, https://doi.org/10.7494/OpMath.2024.44.5.673.

]]>Unitarily equivalent bilateral weighted shifts with operator weights
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4430.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4430.pdfMon, 01 Jul 2024 23:40:02 +0200 Author(s): Michał Buchała.

Abstract: The aim of this paper is to study unitarily equivalent bilateral weighted shifts with operator weights. Our purpose is to establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. Under further assumptions on weights it was proved that unitary equivalence of bilateral weigthed shifts with operator weights defined on \(\mathbb{C}^{2}\) can always be given by a unitary operator with at most two non-zero diagonals. The paper contains also examples of unitarily equivalent shifts with weights defined on \(\mathbb{C}^{k}\) such that every unitary operator, which intertwines them has at least \(k\) non-zero diagonals. Keywords: weighted shifts, operator weights, unitary equivalence. Mathematics Subject Classification: 47B37, 47B02. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 651-672, https://doi.org/10.7494/OpMath.2024.44.5.651.

]]>Analysis of a multiphase free boundary problem
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4429.pdf
https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4429.pdfMon, 01 Jul 2024 23:40:01 +0200 Author(s): Ahlem Abdelouahab, Sabri Bensid.

Abstract: In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution. Keywords: discontinuous nonlinearity, free boundary, perturbation, tumor growth. Mathematics Subject Classification: 34R35, 35J25, 92B05, 35R35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 5 (2024), 631-649, https://doi.org/10.7494/OpMath.2024.44.5.631.

]]>On the solvability of some parabolic equations involving nonlinear boundary conditions with L^{1} data
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4428.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4428.pdfMon, 29 Apr 2024 19:00:06 +0200 Author(s): Laila Taourirte, Abderrahim Charkaoui, Nour Eddine Alaa.

Abstract: We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and \(L^1\) data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models. Keywords: quasilinear parabolic equation, nonlinear boundary conditions, weak solutions, Leray-Schauder topological degree, \(L^1\)-data. Mathematics Subject Classification: 35K59, 35K55, 35A01, 35B09, 35D30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 587-623, https://doi.org/10.7494/OpMath.2024.44.4.587.

]]>Geometric properties of the lattice of polynomials with integer coefficients
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4427.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4427.pdfMon, 29 Apr 2024 19:00:05 +0200 Author(s): Artur Lipnicki, Marek J. Śmietański.

Abstract: This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let \(r\), \(n\) be positive integers with \(n \ge 6r\). Let \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) be the space of polynomials of degree at most \(n\) on \([0,1]\) with integer coefficients such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\) and let \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) from \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) in \(L_2(0,1)\). We give rather precise quantitative estimations for successive minima of \(\boldsymbol{P}_n^\mathbb{Z}\) in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\). Keywords: approximation by polynomials with integer coefficients, lattice, covering radius, roots of polynomial. Mathematics Subject Classification: 41A10, 52C07, 26C10, 65H04. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 565-585, https://doi.org/10.7494/OpMath.2024.44.4.565.

]]>Graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4426.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4426.pdfMon, 29 Apr 2024 19:00:04 +0200 Author(s): Teresa W. Haynes, Michael A. Henning.

Abstract: A graph \(G\) whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. We give constructions that yield infinite families of graphs that are TI-graphs, as well as constructions that yield infinite families of graphs that are not TI-graphs. We study regular graphs that are TI-graphs. Among other results, we prove that all toroidal graphs are TI-graphs. Keywords: total domination, vertex partitions, independent domination. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 543-563, https://doi.org/10.7494/OpMath.2024.44.4.543.

]]>Asymptotic analysis for confluent hypergeometric function in two variables given by double integral
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4425.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4425.pdfMon, 29 Apr 2024 19:00:03 +0200 Author(s): Yoshishige Haraoka.

Abstract: We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along \(x=\infty\) and \(y=\infty\). Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around \((\infty,\infty)\). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new. Keywords: strong asymptotic expansion, Stokes phenomenon, middle convolution, twisted homology. Mathematics Subject Classification: 33C70, 34E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 505-541, https://doi.org/10.7494/OpMath.2024.44.4.505.

]]>Degenerate singular parabolic problems with natural growth
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4424.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4424.pdfMon, 29 Apr 2024 19:00:02 +0200 Author(s): Mounim El Ouardy, Youssef El Hadfi, Abdelaaziz Sbai.

Abstract: In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term \[\begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases}\] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^{N}\), \(N\gt 2\), \(Q\) is the cylinder \(\Omega \times (0,T)\), \(T\gt 0\), \(\Gamma\) the lateral surface \(\partial \Omega \times (0,T)\), \(2\leq p\lt N\), \(a(x,t)\) and \(b(x,t)\) are positive measurable bounded functions, \(q\geq 0\), \(0\leq\gamma\lt 1\), and \(f\) non-negative function belongs to the Lebesgue space \(L^{m}(Q)\) with \(m\gt 1\), and \(u_{0}\in L^{\infty}(\Omega)\) such that \[\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\] More precisely, we study the interaction between the term \(u^{q}\) (\(q>0\)) and the singular lower order term \(d(x,t)|\nabla u|^{p}u^{-\gamma}\) (\(0\lt\gamma\lt 1\)) in order to get a solution to the above problem. The regularizing effect of the term \(u^q\) on the regularity of the solution and its gradient is also analyzed. Keywords: degenerate parabolic equations, singular parabolic equations, natural growth term. Mathematics Subject Classification: 35A25, 35B45, 35B09, 35D30, 35K65, 35K67. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 471-503, https://doi.org/10.7494/OpMath.2024.44.4.471.

]]>Study of fractional semipositone problems on R^{N}
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4423.pdf
https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4423.pdfMon, 29 Apr 2024 19:00:01 +0200 Author(s): Nirjan Biswas.

Abstract: Let \(s\in (0,1)\) and \(N\gt 2s\). In this paper, we consider the following class of nonlocal semipositone problems: \[(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,\] where the weight \(g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a\gt 0\) is a parameter, and \(f_a \in \mathcal{C}(\mathbb{R})\) is strictly negative on \((-\infty,0]\). For \(f_a\) having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution \(u_a\), provided \(a\) is near zero. To obtain the positivity of \(u_a\), we establish a Brezis-Kato type uniform estimate of \((u_a)\) in \(L^r(\mathbb{R}^N)\) for every \(r \in [\frac{2N}{N-2s}, \infty]\). Keywords: semipositone problems, fractional operator, uniform regularity estimates, positive solutions. Mathematics Subject Classification: 35R11, 35J50, 35B65, 35B09. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 4 (2024), 445-470, https://doi.org/10.7494/OpMath.2024.44.4.445.

]]>Reduction of positive self-adjoint extensions
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4422.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4422.pdfThu, 15 Feb 2024 20:00:08 +0100 Author(s): Zsigmond Tarcsay, Zoltán Sebestyén.

Abstract: We revise Krein's extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the "resolvent operator" \((I+T)^{-1}\) of \(T\). Our treatment is somewhat simpler and more natural than Krein's original method which was based on the Krein transform \((I-T)(I+T)^{-1}\). Apart from being positive and symmetric, we do not impose any further constraints on the operator \(T\): neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces. Keywords: positive selfadjoint contractive extension, nonnegative selfadjoint extension, Friedrichs and Krein-von Neumann extension. Mathematics Subject Classification: 47A57, 47A20, 47B25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 425-438, https://doi.org/10.7494/OpMath.2024.44.3.425.

]]>Positive solutions for nonparametric anisotropic singular solutions
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4421.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4421.pdfThu, 15 Feb 2024 20:00:07 +0100 Author(s): Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Xueying Sun.

Abstract: We consider an elliptic equation driven by a nonlinear, nonhomogeneous differential operator with nonstandard growth. The reaction has the combined effects of a singular term and of a "superlinear" perturbation. There is no parameter in the problem. Using variational tools and truncation and comparison techniques, we show the existence of at least two positive smooth solutions. Keywords: variable Lebesgue and Sobolev spaces, anisotropic regularity, anisotropic maximum principle, truncations and comparisons, Hardy inequality. Mathematics Subject Classification: 35B51, 35J60, 35B65, 35J75, 35J92, 46E35, 47J20, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 409-423, https://doi.org/10.7494/OpMath.2024.44.3.409.

]]>On the Möbius invariant principal functions of Pincus
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4420.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4420.pdfThu, 15 Feb 2024 20:00:06 +0100 Author(s): Sagar Ghosh, Gadadhar Misra.

Abstract: In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator \(T\) with \(\operatorname{rank} (T^*T-TT^*) =1\), modulo unitary equivalence, is the unilateral shift. Keywords: hyponormal operator, multiplicity, trace formula, homogeneous operators, principal function. Mathematics Subject Classification: 47B20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 391-407, https://doi.org/10.7494/OpMath.2024.44.3.391.

]]>Cesàro summability of Taylor series in higher order weighted Dirichlet-type spaces
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4419.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4419.pdfThu, 15 Feb 2024 20:00:05 +0100 Author(s): Soumitra Ghara, Rajeev Gupta, Md. Ramiz Reza.

Abstract: For a positive integer \(m\) and a finite non-negative Borel measure \(\mu\) on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces \(\mathcal H_{\mu, m}\). We show that if \(\alpha\gt\frac{1}{2}\), then for any \(f\) in \(\mathcal H_{\mu, m}\) the sequence of generalized Cesàro sums \(\{\sigma_n^{\alpha}[f]\}\) converges to \(f\). We further show that if \(\alpha=\frac{1}{2}\) then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer \(m\). Keywords: weighted Dirichlet-type integrals, Cesàro mean, summability, Hadamard multiplication. Mathematics Subject Classification: 41A10, 40G05, 46E20, 41A17. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 373-390, https://doi.org/10.7494/OpMath.2024.44.3.373.

]]>A note on the general moment problem
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4418.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4418.pdfThu, 15 Feb 2024 20:00:04 +0100 Author(s): Hamza El Azhar, Abdelouahab Hanine, El Hassan Zerouali.

Abstract: In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in \(\mathbb{R}_+\). Keywords: general moment problem, charge sequences, atomic measure. Mathematics Subject Classification: 44A60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 359-372, https://doi.org/10.7494/OpMath.2024.44.3.359.

]]>Shifted model spaces and their orthogonal decompositions
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4417.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4417.pdfThu, 15 Feb 2024 20:00:03 +0100 Author(s): M. Cristina Câmara, Kamila Kliś-Garlicka, Marek Ptak.

Abstract: We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift \(S^*\). In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator \(T\), with respect to another operator, follow from certain commutation relations between the two operators involved. Keywords: model space, Toeplitz operator, Toeplitz kernel, truncated Toeplitz operator, nearly invariant, shift invariant. Mathematics Subject Classification: 47B32, 47B35, 30H10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 341-357, https://doi.org/10.7494/OpMath.2024.44.3.341.

]]>Finitely additive functions in measure theory and applications
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4416.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4416.pdfThu, 15 Feb 2024 20:00:02 +0100 Author(s): Daniel Alpay, Palle Jorgensen.

Abstract: In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures \(\mu\), and to adjoints of composition operators. Keywords: Hilbert space, reproducing kernels, probability space, Gaussian fields, transforms, covariance, Itô integration, Itô calculus, generalized Brownian motion. Mathematics Subject Classification: 47B32, 60G20, 60G15, 60H05, 60J60, 46E22. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 323-339, https://doi.org/10.7494/OpMath.2024.44.3.323.

]]>Jan Stochel, a stellar mathematician
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4415.pdf
https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4415.pdfThu, 15 Feb 2024 20:00:01 +0100 Author(s): Sameer Chavan, Raúl Curto, Zenon Jan Jabłoński, Il Bong Jung, Mihai Putinar.

Abstract: The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics. Keywords: unbounded subnormal operator, moment problem, composition operator, Cauchy dual. Mathematics Subject Classification: 47B20, 47B25, 30E05, 47B33. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 3 (2024), 303-321, https://doi.org/10.7494/OpMath.2024.44.3.303.

]]>Weak signed Roman k-domination in digraphs
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdfMon, 15 Jan 2024 18:00:07 +0100 Author(s): Lutz Volkmann.

Abstract: Let \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). The weight of a WSRkDF \(f\) is \(w(f)=\sum_{v\in V(D)}f(v)\). The weak signed Roman \(k\)-domination number \(\gamma_{wsR}^k(D)\) is the minimum weight of a WSRkDF on \(D\). In this paper we initiate the study of the weak signed Roman \(k\)-domination number of digraphs, and we present different bounds on \(\gamma_{wsR}^k(D)\). In addition, we determine the weak signed Roman \(k\)-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number \(\gamma_{wsR}(D)=\gamma_{wsR}^1(D)\) and the signed Roman \(k\)-domination number \(\gamma_{sR}^k(D).\) Keywords: digraph, weak signed Roman \(k\)-dominating function, weak signed Roman \(k\)-domination number, signed Roman \(k\)-dominating function, signed Roman \(k\)-domination number. Mathematics Subject Classification: 05C20, 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 285-296, https://doi.org/10.7494/OpMath.2024.44.2.285.

]]>Positive solutions to a third order nonlocal boundary value problem with a parameter
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4413.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4413.pdfMon, 15 Jan 2024 18:00:06 +0100 Author(s): Gabriela Szajnowska, Mirosława Zima.

Abstract: We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel'skiĭ-Guo fixed point theorem in cones and the properties of the Green's function corresponding to the BVP under study. The main results are illustrated by suitable examples. Keywords: boundary value problem, nonlocal boundary conditions, positive solution, cone. Mathematics Subject Classification: 34B10, 34B15, 34B18, 34B27. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 267-283, https://doi.org/10.7494/OpMath.2024.44.2.267.

]]>Anisotropic p-Laplace Equations on long cylindrical domain
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4412.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4412.pdfMon, 15 Jan 2024 18:00:05 +0100 Author(s): Purbita Jana.

Abstract: The main aim of this article is to study the Poisson type problem for anisotropic \(p\)-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo \(p\)-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension. Keywords: pseudo \(p\)-Laplace equation, cylindrical domains, asymptotic analysis. Mathematics Subject Classification: 35P15, 35P30, 35B38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 249-265, https://doi.org/10.7494/OpMath.2024.44.2.249.

]]>An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4411.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4411.pdfMon, 15 Jan 2024 18:00:04 +0100 Author(s): Michael Gil'.

Abstract: Let \(A\) be a bounded linear operator in a complex separable Hilbert space, \(A^*\) be its adjoint one and \(A_I:=(A-A^*)/(2i)\). Assuming that \(A_I\) is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of \(A\). Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality \(\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)\), where \(\lambda_k(A)\) \((k=1,2, \ldots )\) are the eigenvalues of \(A\) and \(N_2(\cdot)\) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators. Keywords: Hilbert space, linear operators, eigenvalues. Mathematics Subject Classification: 47A10, 47A55, 47B10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 241-248, https://doi.org/10.7494/OpMath.2024.44.2.241.

]]>Parabolic turbulence k-epsilon model with applications in fluid flows through permeable media
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4410.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4410.pdfMon, 15 Jan 2024 18:00:03 +0100 Author(s): Hermenegildo Borges de Oliveira.

Abstract: In this work, we study a one-equation turbulence \(k\)-epsilon model that governs fluid flows through permeable media. The model problem under consideration here is derived from the incompressible Navier-Stokes equations by the application of a time-averaging operator used in the \(k\)-epsilon modeling and a volume-averaging operator that is characteristic of modeling unsteady porous media flows. For the associated initial- and boundary-value problem, we prove the existence of suitable weak solutions (average velocity field and turbulent kinetic energy) in the space dimensions of physics interest. Keywords: turbulence, \(k\)-epsilon modelling, permeable media, existence. Mathematics Subject Classification: 76F60, 76S05, 35Q35, 35K55, 35A01, 76D03. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 197-240, https://doi.org/10.7494/OpMath.2024.44.2.197.

]]>Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4409.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4409.pdfMon, 15 Jan 2024 18:00:02 +0100 Author(s): Alberto Cabada, Nikolay D. Dimitrov, Jagan Mohan Jonnalagadda.

Abstract: This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems. Keywords: fractional difference, Dirichlet conditions, Green's function, existence of solutions. Mathematics Subject Classification: 26A33, 39A12, 39A27. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 167-195, https://doi.org/10.7494/OpMath.2024.44.2.167.

]]>On the structure of the diffusion distance induced by the fractional dyadic Laplacian
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4408.pdf
https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4408.pdfMon, 15 Jan 2024 18:00:01 +0100 Author(s): María Florencia Acosta, Hugo Aimar, Ivana Gómez, Federico Morana.

Abstract: In this note we explore the structure of the diffusion metric of Coifman-Lafon determined by fractional dyadic Laplacians. The main result is that, for each \(t\gt 0\), the diffusion metric is a function of the dyadic distance, given in \(\mathbb{R}^+\) by \(\delta(x,y) = \inf\{|I|\colon I \text{ is a dyadic interval containing } x \text{ and } y\}\). Even if these functions of \(\delta\) are not equivalent to \(\delta\), the families of balls are the same, to wit, the dyadic intervals. Keywords: diffusion metrics, dyadic diffusion. Mathematics Subject Classification: 54E35, 35K08. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 2 (2024), 157-165, https://doi.org/10.7494/OpMath.2024.44.2.157.

]]>The extensive 1-median problem with radius on networks
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4407.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4407.pdfFri, 27 Oct 2023 09:00:07 +0200 Author(s): Tran Hoai Ngoc Nhan, Nguyen Thanh Hung, Kien Trung Nguyen.

Abstract: The median location problem concerns finding locations of one or several new facilities that minimize the overall weighted distances from the existing to the new facilities. We address the problem of locating one new facility with a radius \(r\) on networks. Furthermore, the radius \(r\) is flexible and the objective function is the conic combination of the traditional 1-median function and the value \(r\). We call this problem an extensive 1-median problem with radius on networks. To solve the problem, we first induce the so-called finite dominating set, that contains all points on the underlying network and radius values which are candidate for the optimal solution of the problem. This helps to develop a combinatorial algorithm that solves the problem on a general network \(G=(V,E)\) in \(O(|E||V|^3)\) time. We also consider the underlying problem with improved algorithm on trees. Based the convexity of the objective function with variable radius, we develop a linear time algorithm to find an extensive 1-median with radius on the underlying tree. Keywords: extensive facility, median problem, tree, convex. Mathematics Subject Classification: 90B10, 90C35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 135-149, https://doi.org/10.7494/OpMath.2024.44.1.135.

]]>Singular quasilinear convective systems involving variable exponents
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4406.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4406.pdfFri, 27 Oct 2023 09:00:06 +0200 Author(s): Abdelkrim Moussaoui, Dany Nabab, Jean Vélin.

Abstract: The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder's fixed point theorem. Keywords: \(p(x)\)-Laplacian, variable exponents, fixed point, singular system, gradient estimate, regularity. Mathematics Subject Classification: 35J75, 35J48, 35J92. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 105-134, https://doi.org/10.7494/OpMath.2024.44.1.105.

]]>Conditional mean embedding and optimal feature selection via positive definite kernels
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4405.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4405.pdfFri, 27 Oct 2023 09:00:05 +0200 Author(s): Palle E.T. Jorgensen, Myung-Sin Song, James Tian.

Abstract: Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of kernels in a construction o foptimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel \(K\) in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of "optimal" feature representations. Keywords: positive-definite kernels, reproducing kernel Hilbert space, stochastic processes, frames, machine learning, embedding problems, optimization. Mathematics Subject Classification: 47N10, 47A52, 47B32, 42A82, 42C15, 62H12, 62J07, 65J20, 68T07, 90C20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 79-103, https://doi.org/10.7494/OpMath.2024.44.1.79.

]]>Two-weight norm inequalities for rough fractional integral operators on Morrey spaces
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4404.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4404.pdfFri, 27 Oct 2023 09:00:04 +0200 Author(s): Kwok-Pun Ho.

Abstract: We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces. Keywords: two-weight norm inequalities, rough fractional integral operators, Morrey spaces. Mathematics Subject Classification: 42B20, 42B25, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 67-77, https://doi.org/10.7494/OpMath.2024.44.1.67.

]]>Local irregularity conjecture for 2-multigraphs versus cacti
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4403.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4403.pdfFri, 27 Oct 2023 09:00:03 +0200 Author(s): Igor Grzelec, Mariusz Woźniak.

Abstract: A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph \(G\) such that every color induces a locally irregular submultigraph of \(G\). A locally irregular colorable multigraph \(G\) is any multigraph which admits a locally irregular coloring. We denote by \(\textrm{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph \(G\). In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph \(G\), which is not isomorphic to \(K_2\), multigraph \(^2G\) obtained from \(G\) by doubling each edge satisfies \(\textrm{lir}(^2G)\leq 2\). We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph \(G\) satisfies \(\textrm{lir}(G)\leq 3\). At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved. Keywords: locally irregular coloring, decomposable, cactus graphs, 2-multigraphs. Mathematics Subject Classification: 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 49-65, https://doi.org/10.7494/OpMath.2024.44.1.49.

]]>Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4402.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4402.pdfFri, 27 Oct 2023 09:00:02 +0200 Author(s): Sebastião Cordeiro, Carlos Raposo, Jorge Ferreira, Daniel Rocha, Mohammad Shahrouzi.

Abstract: This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma. Keywords: viscoelastic equation, dispersive term, logarithmic nonlinearity, local existence. Mathematics Subject Classification: 35A01, 35L20, 35L70. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 19-47, https://doi.org/10.7494/OpMath.2024.44.1.19.

]]>Uniqueness for a class p-Laplacian problems when a parameter is large
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4401.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4401.pdfFri, 27 Oct 2023 09:00:01 +0200 Author(s): B. Alreshidi, D.D. Hai.

Abstract: We prove uniqueness of positive solutions for the problem \[-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,\] where \(1\lt p\lt 2\) and \(p\) is close to 2, \(\Omega\) is bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f:[0,\infty)\rightarrow [0,\infty )\) with \(f(z)\sim z^{\beta }\) at \(\infty\) for some \(\beta \in (0,1)\), and \(\lambda\) is a large parameter. The monotonicity assumption on \(f\) is not required even for \(u\) large. Keywords: singular \(p\)-Laplacian, uniqueness, positive solutions. Mathematics Subject Classification: 35J92, 35J75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 5-17, https://doi.org/10.7494/OpMath.2024.44.1.5.