Opuscula MathematicaA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.
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Quadratic inequalities for functionals in l^{∞}For a class of operators \(T\) on \(l^{\infty}\) and \(T\)-invariant functionals \(\varphi\) we prove inequalities between \(\varphi(x)\), \(\varphi(x^2)\) and the upper density of the sets \[P_r:=\{n \in \mathbb{N}_0: \varphi((T^{n}x)\cdot x) \gt r\}.\] Applications are given to Banach limits and integrals.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4121.pdf
Quadratic inequalities for functionals in l^{∞}Gerd HerzogPeer Chr. KunstmannBanach algebras of bounded functions, operator-invariant functionals, Banach limitsdoi:10.7494/OpMath.2021.41.3.437Opuscula Math. 41, no. 3 (2021), 437-446, https://doi.org/10.7494/OpMath.2021.41.3.437Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.437https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4121.pdf413437446 Title: Quadratic inequalities for functionals in l^{∞}.

Author(s): Gerd Herzog, Peer Chr. Kunstmann.

Abstract: For a class of operators \(T\) on \(l^{\infty}\) and \(T\)-invariant functionals \(\varphi\) we prove inequalities between \(\varphi(x)\), \(\varphi(x^2)\) and the upper density of the sets \[P_r:=\{n \in \mathbb{N}_0: \varphi((T^{n}x)\cdot x) \gt r\}.\] Applications are given to Banach limits and integrals. Keywords: Banach algebras of bounded functions, operator-invariant functionals, Banach limits. Mathematics Subject Classification: 47B37, 47B48, 47B60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 437-446, https://doi.org/10.7494/OpMath.2021.41.3.437.

]]>On the S-matrix of Schrödinger operator with nonlocal δ-interactionSchrödinger operators with nonlocal \(\delta\)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the \(S\)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The \(S\)-matrix \(S(z)\) is analytical in the lower half-plane \(\mathbb{C}_{−}\) when the Schrödinger operator with nonlocal \(\delta\)-interaction is positive self-adjoint. Otherwise, \(S(z)\) is a meromorphic matrix-valued function in \(\mathbb{C}_{−}\) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of \(S\)-matrices are given.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4120.pdf
On the S-matrix of Schrödinger operator with nonlocal δ-interactionAnna GłówczykSergiusz KużelLax-Phillips scattering scheme, scattering matrix, \(S\)-matrix, nonlocal \(\delta\)-interaction, non-cyclic functiondoi:10.7494/OpMath.2021.41.3.413Opuscula Math. 41, no. 3 (2021), 413-435, https://doi.org/10.7494/OpMath.2021.41.3.413Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.413https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4120.pdf413413435 Title: On the S-matrix of Schrödinger operator with nonlocal δ-interaction.

Author(s): Anna Główczyk, Sergiusz Kużel.

Abstract: Schrödinger operators with nonlocal \(\delta\)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the \(S\)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The \(S\)-matrix \(S(z)\) is analytical in the lower half-plane \(\mathbb{C}_{−}\) when the Schrödinger operator with nonlocal \(\delta\)-interaction is positive self-adjoint. Otherwise, \(S(z)\) is a meromorphic matrix-valued function in \(\mathbb{C}_{−}\) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of \(S\)-matrices are given. Keywords: Lax-Phillips scattering scheme, scattering matrix, \(S\)-matrix, nonlocal \(\delta\)-interaction, non-cyclic function. Mathematics Subject Classification: 47B25, 47A40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 413-435, https://doi.org/10.7494/OpMath.2021.41.3.413.

]]>Spectrum localization of a perturbed operator in a strip and applicationsLet \(A\) and \(\tilde{A}\) be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of \(A\) lie in some strip. In what strip the spectrum of \(\tilde{A}\) lies if \(A\) and \(\tilde{A}\) are "close"? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4119.pdf
Spectrum localization of a perturbed operator in a strip and applicationsMichael Gil'operator, spectrum, perturbation, approximation, integral operator, matrixdoi:10.7494/OpMath.2021.41.3.395Opuscula Math. 41, no. 3 (2021), 395-412, https://doi.org/10.7494/OpMath.2021.41.3.395Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.395https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4119.pdf413395412 Title: Spectrum localization of a perturbed operator in a strip and applications.

Author(s): Michael Gil'.

Abstract: Let \(A\) and \(\tilde{A}\) be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of \(A\) lie in some strip. In what strip the spectrum of \(\tilde{A}\) lies if \(A\) and \(\tilde{A}\) are "close"? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices. Keywords: operator, spectrum, perturbation, approximation, integral operator, matrix. Mathematics Subject Classification: 47A10, 47A55, 47B10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 395-412, https://doi.org/10.7494/OpMath.2021.41.3.395.

]]>Extensions of dissipative operators with closable imaginary partGiven a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4118.pdf
Extensions of dissipative operators with closable imaginary partChristoph Fischbacherextension theory, dissipative operators, ordinary differential operatorsdoi:10.7494/OpMath.2021.41.3.381Opuscula Math. 41, no. 3 (2021), 381-393, https://doi.org/10.7494/OpMath.2021.41.3.381Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.381https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4118.pdf413381393 Title: Extensions of dissipative operators with closable imaginary part.

Author(s): Christoph Fischbacher.

Abstract: Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval. Keywords: extension theory, dissipative operators, ordinary differential operators. Mathematics Subject Classification: 34L99, 47H06. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 381-393, https://doi.org/10.7494/OpMath.2021.41.3.381.

]]>Multi-variable quaternionic spectral analysisIn this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii).
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4117.pdf
Multi-variable quaternionic spectral analysisIlwoo ChoPalle E.T. Jorgensenthe quaternions \(\mathbb{H}\), vector spaces \(\mathbb{H}^n\) over \(\mathbb{H}\), \(q\)-spectral forms, \(q\)-spectral functionsdoi:10.7494/OpMath.2021.41.3.335Opuscula Math. 41, no. 3 (2021), 335-379, https://doi.org/10.7494/OpMath.2021.41.3.335Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.335https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4117.pdf413335379 Title: Multi-variable quaternionic spectral analysis.

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

Abstract: In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii). Keywords: the quaternions \(\mathbb{H}\), vector spaces \(\mathbb{H}^n\) over \(\mathbb{H}\), \(q\)-spectral forms, \(q\)-spectral functions. Mathematics Subject Classification: 20G20, 46S10, 47S10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 335-379, https://doi.org/10.7494/OpMath.2021.41.3.335.

]]>Perturbation series for Jacobi matrices and the quantum Rabi modelWe investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4116.pdf
Perturbation series for Jacobi matrices and the quantum Rabi modelMirna CharifLech ZielinskiJacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, quantum Rabi model, rotating wave approximationdoi:10.7494/OpMath.2021.41.3.301Opuscula Math. 41, no. 3 (2021), 301-333, https://doi.org/10.7494/OpMath.2021.41.3.301Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.301https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4116.pdf413301333 Title: Perturbation series for Jacobi matrices and the quantum Rabi model.

Author(s): Mirna Charif, Lech Zielinski.

Abstract: We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings. Keywords: Jacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, quantum Rabi model, rotating wave approximation. Mathematics Subject Classification: 81Q10, 47B36, 15A18. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 301-333, https://doi.org/10.7494/OpMath.2021.41.3.301.

]]>New characterizations of reproducing kernel Hilbert spaces and applications to metric geometryWe give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4115.pdf
New characterizations of reproducing kernel Hilbert spaces and applications to metric geometryDaniel AlpayPalle E.T. Jorgensenreproducing kernel, positive definite functions, approximation, algorithms, measures, stochastic processesdoi:10.7494/OpMath.2021.41.3.283Opuscula Math. 41, no. 3 (2021), 283-300, https://doi.org/10.7494/OpMath.2021.41.3.283Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.3.283https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4115.pdf413283300 Title: New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry.

Author(s): Daniel Alpay, Palle E.T. Jorgensen.

Abstract: We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples. Keywords: reproducing kernel, positive definite functions, approximation, algorithms, measures, stochastic processes. Mathematics Subject Classification: 46E22, 43A35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 3 (2021), 283-300, https://doi.org/10.7494/OpMath.2021.41.3.283.

]]>Uniqueness of series in the Franklin system and the Gevorkyan problemsIn 1870 G. Cantor proved that if \(\lim_{N \rightarrow \infty}\sum_{n=-N}^N c_{n}e^{inx} = 0\), \(\bar{c}_{n}=c_{n}\), then \(c_{n}=0\) for \(n\in\mathbb{Z}\). In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. He solved this conjecture in 2015. In 2014 Z. Wronicz proved that there exists a Franklin series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In the present paper we show that to the uniqueness of the Franklin system \(\lim_{n\rightarrow \infty}\sum_{n=0}^\infty a_{n}f_{n}\) it suffices to prove the convergence its subsequence \(s_{2^{n}}\) to zero by the condition \(a_{n}=o(\sqrt{n})\). It is a solution of the Gevorkyan problem formulated in 2016.
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4114.pdf
Uniqueness of series in the Franklin system and the Gevorkyan problemsZygmunt WroniczFranklin system, orthonormal spline system, uniqueness of seriesdoi:10.7494/OpMath.2021.41.2.269Opuscula Math. 41, no. 2 (2021), 269-276, https://doi.org/10.7494/OpMath.2021.41.2.269Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.269https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4114.pdf412269276 Title: Uniqueness of series in the Franklin system and the Gevorkyan problems.

Author(s): Zygmunt Wronicz.

Abstract: In 1870 G. Cantor proved that if \(\lim_{N \rightarrow \infty}\sum_{n=-N}^N c_{n}e^{inx} = 0\), \(\bar{c}_{n}=c_{n}\), then \(c_{n}=0\) for \(n\in\mathbb{Z}\). In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. He solved this conjecture in 2015. In 2014 Z. Wronicz proved that there exists a Franklin series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In the present paper we show that to the uniqueness of the Franklin system \(\lim_{n\rightarrow \infty}\sum_{n=0}^\infty a_{n}f_{n}\) it suffices to prove the convergence its subsequence \(s_{2^{n}}\) to zero by the condition \(a_{n}=o(\sqrt{n})\). It is a solution of the Gevorkyan problem formulated in 2016. Keywords: Franklin system, orthonormal spline system, uniqueness of series. Mathematics Subject Classification: 42C10, 42C25, 41A15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 269-276, https://doi.org/10.7494/OpMath.2021.41.2.269.

]]>Remarks on the outer-independent double Italian domination numberLet \(G\) be a graph with vertex set \(V(G)\). If \(u\in V(G)\), then \(N[u]\) is the closed neighborhood of \(u\). An outer-independent double Italian dominating function (OIDIDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) such that if \(f(v)\in\{0,1\}\) for a vertex \(v\in V(G)\), then \(\sum_{x\in N[v]}f(x)\ge 3\), and the set \(\{u\in V(G):f(u)=0\}\) is independent. The weight of an OIDIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The outer-independent double Italian domination number \(\gamma_{oidI}(G)\) equals the minimum weight of an OIDIDF on \(G\). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4113.pdf
Remarks on the outer-independent double Italian domination numberLutz Volkmanndouble Italian domination number, outer-independent double Italian domination number, Nordhaus-Gaddum bounddoi:10.7494/OpMath.2021.41.2.259Opuscula Math. 41, no. 2 (2021), 259-268, https://doi.org/10.7494/OpMath.2021.41.2.259Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.259https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4113.pdf412259268 Title: Remarks on the outer-independent double Italian domination number.

Author(s): Lutz Volkmann.

Abstract: Let \(G\) be a graph with vertex set \(V(G)\). If \(u\in V(G)\), then \(N[u]\) is the closed neighborhood of \(u\). An outer-independent double Italian dominating function (OIDIDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) such that if \(f(v)\in\{0,1\}\) for a vertex \(v\in V(G)\), then \(\sum_{x\in N[v]}f(x)\ge 3\), and the set \(\{u\in V(G):f(u)=0\}\) is independent. The weight of an OIDIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The outer-independent double Italian domination number \(\gamma_{oidI}(G)\) equals the minimum weight of an OIDIDF on \(G\). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs. Keywords: double Italian domination number, outer-independent double Italian domination number, Nordhaus-Gaddum bound. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 259-268, https://doi.org/10.7494/OpMath.2021.41.2.259.

]]>Introduction to dominated edge chromatic number of a graphWe introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs.
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4112.pdf
Introduction to dominated edge chromatic number of a graphMohammad R. PiriSaeid Alikhanidominated edge chromatic number, subdivision, operation, coronadoi:10.7494/OpMath.2021.41.2.245Opuscula Math. 41, no. 2 (2021), 245-257, https://doi.org/10.7494/OpMath.2021.41.2.245Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.245https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4112.pdf412245257 Title: Introduction to dominated edge chromatic number of a graph.

Author(s): Mohammad R. Piri, Saeid Alikhani.

Abstract: We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs. Keywords: dominated edge chromatic number, subdivision, operation, corona. Mathematics Subject Classification: 05C25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 245-257, https://doi.org/10.7494/OpMath.2021.41.2.245.

]]>Dimension of the intersection of certain Cantor sets in the planeIn this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf
Dimension of the intersection of certain Cantor sets in the planeSteen PedersenVincent T. ShawCantor set, fractal, self-similar, translation, intersection, dimension, Minkowski dimensiondoi:10.7494/OpMath.2021.41.2.227Opuscula Math. 41, no. 2 (2021), 227-244, https://doi.org/10.7494/OpMath.2021.41.2.227Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.227https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf412227244 Title: Dimension of the intersection of certain Cantor sets in the plane.

Author(s): Steen Pedersen, Vincent T. Shaw.

Abstract: In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane. Keywords: Cantor set, fractal, self-similar, translation, intersection, dimension, Minkowski dimension. Mathematics Subject Classification: 28A80, 51F99. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 227-244, https://doi.org/10.7494/OpMath.2021.41.2.227.

]]>On the gauge-natural operators similar to the twisted Dorfman-Courant bracketAll \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdf
On the gauge-natural operators similar to the twisted Dorfman-Courant bracketWłodzimierz M. Mikulskinatural operator, linear vector field, linear form, twisted Dorfman-Courant bracket, the Jacobi identity in Leibniz formdoi:10.7494/OpMath.2021.41.2.205Opuscula Math. 41, no. 2 (2021), 205-226, https://doi.org/10.7494/OpMath.2021.41.2.205Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.205https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdf412205226 Title: On the gauge-natural operators similar to the twisted Dorfman-Courant bracket.

Author(s): Włodzimierz M. Mikulski.

Abstract: All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented. Keywords: natural operator, linear vector field, linear form, twisted Dorfman-Courant bracket, the Jacobi identity in Leibniz form. Mathematics Subject Classification: 53A55, 53A45, 53A99. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 205-226, https://doi.org/10.7494/OpMath.2021.41.2.205.

]]>Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curveWe consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) nor on the potential \(h\). For \(N=3\), the existence of ground-state solution may depends on the trace of the regular part of the Green function of \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\).
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf
Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curveIdowu Esther IjaodoroEl Hadji Abdoulaye ThiamHardy-Sobolev inequality, positive minimizers, parametrized curve, mass, Green functiondoi:10.7494/OpMath.2021.41.2.187Opuscula Math. 41, no. 2 (2021), 187-204, https://doi.org/10.7494/OpMath.2021.41.2.187Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.187https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf412187204 Title: Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve.

Author(s): Idowu Esther Ijaodoro, El Hadji Abdoulaye Thiam.

Abstract: We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) nor on the potential \(h\). For \(N=3\), the existence of ground-state solution may depends on the trace of the regular part of the Green function of \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\). Keywords: Hardy-Sobolev inequality, positive minimizers, parametrized curve, mass, Green function. Mathematics Subject Classification: 35J91, 35J20, 35J75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 187-204, https://doi.org/10.7494/OpMath.2021.41.2.187.

]]>The achromatic number of K_{6} □ K_{7} is 18A vertex colouring \(f:V(G)\to C\) of a graph \(G\) is complete if for any two distinct colours \(c_1, c_2 \in C\) there is an edge \(\{v_1,v_2\}\in E(G)\) such that \(f(v_i)=c_i\), \(i=1,2\). The achromatic number of \(G\) is the maximum number \(\text{achr}(G)\) of colours in a proper complete vertex colouring of \(G\). In the paper it is proved that \(\text{achr}(K_6 \square K_7)=18\). This result finalises the determination of \(\text{achr}(K_6 \square K_q)\).
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4108.pdf
The achromatic number of K_{6} □ K_{7} is 18Mirko Horňákcomplete vertex colouring, achromatic number, Cartesian productdoi:10.7494/OpMath.2021.41.2.163Opuscula Math. 41, no. 2 (2021), 163-185, https://doi.org/10.7494/OpMath.2021.41.2.163Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.2.163https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4108.pdf412163185 Title: The achromatic number of K_{6} □ K_{7} is 18.

Author(s): Mirko Horňák.

Abstract: A vertex colouring \(f:V(G)\to C\) of a graph \(G\) is complete if for any two distinct colours \(c_1, c_2 \in C\) there is an edge \(\{v_1,v_2\}\in E(G)\) such that \(f(v_i)=c_i\), \(i=1,2\). The achromatic number of \(G\) is the maximum number \(\text{achr}(G)\) of colours in a proper complete vertex colouring of \(G\). In the paper it is proved that \(\text{achr}(K_6 \square K_7)=18\). This result finalises the determination of \(\text{achr}(K_6 \square K_q)\). Keywords: complete vertex colouring, achromatic number, Cartesian product. Mathematics Subject Classification: 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 2 (2021), 163-185, https://doi.org/10.7494/OpMath.2021.41.2.163.

]]>Exponential stability results for variable delay difference equationsSufficient conditions that guarantee exponential decay to zero of the variable delay difference equation \[x(n+1)=a(n)x(n)+b(n)x(n-g(n))\] are obtained. These sufficient conditions are deduced via inequalities by employing Lyapunov functionals. In addition, a criterion for the instability of the zero solution is established. The results in the paper generalizes some results in the literature.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf
Exponential stability results for variable delay difference equationsErnest Yanksonexponential stability, Lyapunov functional, instabilitydoi:10.7494/OpMath.2021.41.1.145Opuscula Math. 41, no. 1 (2021), 145-155, https://doi.org/10.7494/OpMath.2021.41.1.145Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.145https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf411145155 Title: Exponential stability results for variable delay difference equations.

Author(s): Ernest Yankson.

Abstract: Sufficient conditions that guarantee exponential decay to zero of the variable delay difference equation \[x(n+1)=a(n)x(n)+b(n)x(n-g(n))\] are obtained. These sufficient conditions are deduced via inequalities by employing Lyapunov functionals. In addition, a criterion for the instability of the zero solution is established. The results in the paper generalizes some results in the literature. Keywords: exponential stability, Lyapunov functional, instability. Mathematics Subject Classification: 34D20, 34D40, 34K20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 145-155, https://doi.org/10.7494/OpMath.2021.41.1.145.

]]>Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrandsMultiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf
Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrandsJoel Fotso TachagoHubert NnangElvira Zappaleconvex function, reiterated two-scale convergence, relaxation, Orlicz-Sobolev spacesdoi:10.7494/OpMath.2021.41.1.113Opuscula Math. 41, no. 1 (2021), 113-143, https://doi.org/10.7494/OpMath.2021.41.1.113Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.113https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf411113143 Title: Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands.

Author(s): Joel Fotso Tachago, Hubert Nnang, Elvira Zappale.

Abstract: Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function. Keywords: convex function, reiterated two-scale convergence, relaxation, Orlicz-Sobolev spaces. Mathematics Subject Classification: 35B27, 35B40, 35J25, 46J10, 49J45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 113-143, https://doi.org/10.7494/OpMath.2021.41.1.113.

]]>On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf
On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}Michal StašJuraj Valiskagraph, crossing number, join product, cyclic permutation, path, cycledoi:10.7494/OpMath.2021.41.1.95Opuscula Math. 41, no. 1 (2021), 95-112, https://doi.org/10.7494/OpMath.2021.41.1.95Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.95https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf41195112 Title: On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}.

Author(s): Michal Staš, Juraj Valiska.

Abstract: The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\). Keywords: graph, crossing number, join product, cyclic permutation, path, cycle. Mathematics Subject Classification: 05C10, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 95-112, https://doi.org/10.7494/OpMath.2021.41.1.95.

]]>Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, IWe consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf
Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, IManabu Naitoasymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequalitydoi:10.7494/OpMath.2021.41.1.71Opuscula Math. 41, no. 1 (2021), 71-94, https://doi.org/10.7494/OpMath.2021.41.1.71Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.71https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf4117194 Title: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I.

Author(s): Manabu Naito.

Abstract: We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\). Keywords: asymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequality. Mathematics Subject Classification: 34C11, 34C10, 26D10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 71-94, https://doi.org/10.7494/OpMath.2021.41.1.71.

]]>More on linear and metric tree mapsWe consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf
More on linear and metric tree mapsSergiy Kozerenkotree, Markov graph, metric map, non-expanding map, linear map, graph homomorphismdoi:10.7494/OpMath.2021.41.1.55Opuscula Math. 41, no. 1 (2021), 55-70, https://doi.org/10.7494/OpMath.2021.41.1.55Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.55https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf4115570 Title: More on linear and metric tree maps.

Author(s): Sergiy Kozerenko.

Abstract: We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree. Keywords: tree, Markov graph, metric map, non-expanding map, linear map, graph homomorphism. Mathematics Subject Classification: 05C05, 05C12, 05C20, 54E40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 55-70, https://doi.org/10.7494/OpMath.2021.41.1.55.

]]>Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponentWe are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf
Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponentAbderrahim CharkaouiHouda FahimNour Eddine Alaavariable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solutiondoi:10.7494/OpMath.2021.41.1.25Opuscula Math. 41, no. 1 (2021), 25-53, https://doi.org/10.7494/OpMath.2021.41.1.25Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.25https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf4112553 Title: Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent.

Abstract: We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems. Keywords: variable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solution. Mathematics Subject Classification: 35D30, 35K59, 35A01, 35K93, 35A16, 47H10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 25-53, https://doi.org/10.7494/OpMath.2021.41.1.25.

]]>Some existence results for a nonlocal non-isotropic problemIn this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf
Some existence results for a nonlocal non-isotropic problemRachid BentifourSofiane El-Hadi Mirianisotropic operator, integro-differential problem, variational methodsdoi:10.7494/OpMath.2021.41.1.5Opuscula Math. 41, no. 1 (2021), 5-23, https://doi.org/10.7494/OpMath.2021.41.1.5Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.5https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf411523 Title: Some existence results for a nonlocal non-isotropic problem.

Abstract: In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\) Keywords: anisotropic operator, integro-differential problem, variational methods. Mathematics Subject Classification: 35A15, 35B09, 35E15, 35J20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 5-23, https://doi.org/10.7494/OpMath.2021.41.1.5.