Opuscula Mathematica A list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics. https://www.opuscula.agh.edu.pl AGH University of Science and Technology Press en Copyright AGH University of Science and Technology Press Opuscula Mathematica 1232-9274 2300-6919 Copyright AGH University of Science and Technology Press Opuscula Mathematica https://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg https://www.opuscula.agh.edu.pl The intersection graph of annihilator submodules of a module Let $$R$$ be a commutative ring and $$M$$ be a Noetherian $$R$$-module. The intersection graph of annihilator submodules of $$M$$, denoted by $$GA(M)$$ is an undirected simple graph whose vertices are the classes of elements of $$Z_R(M)\setminus \text{Ann}_R(M)$$, for $$a,b \in R$$ two distinct classes $$[a]$$ and $$[b]$$ are adjacent if and only if $$\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0$$. In this paper, we study diameter and girth of $$GA(M)$$ and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that $$GA(M)$$ is complete if and only if $$Z_R(M)$$ is an ideal of $$R$$. Also, we show that if $$M$$ is a finitely generated $$R$$-module with $$r(\text{Ann}_R(M))\neq \text{Ann}_R(M)$$ and $$|m-\text{Ass}_R(M)|=1$$ and $$GA(M)$$ is a star graph, then $$r(\text{Ann}_R(M))$$ is not a prime ideal of $$R$$ and $$|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1$$. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3933.pdf The intersection graph of annihilator submodules of a module S.B. Pejman Sh. Payrovi S. Babaei prime submodule; annihilator submodule; intersection annihilator graph doi:10.7494/OpMath.2019.39.4.577 Opuscula Math. 39, no. 4 (2019), 577-588, https://doi.org/10.7494/OpMath.2019.39.4.577 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.577 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3933.pdf 39 4 577 588
Title: The intersection graph of annihilator submodules of a module.

Author(s): S.B. Pejman, Sh. Payrovi, S. Babaei.

Abstract: Let $$R$$ be a commutative ring and $$M$$ be a Noetherian $$R$$-module. The intersection graph of annihilator submodules of $$M$$, denoted by $$GA(M)$$ is an undirected simple graph whose vertices are the classes of elements of $$Z_R(M)\setminus \text{Ann}_R(M)$$, for $$a,b \in R$$ two distinct classes $$[a]$$ and $$[b]$$ are adjacent if and only if $$\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0$$. In this paper, we study diameter and girth of $$GA(M)$$ and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that $$GA(M)$$ is complete if and only if $$Z_R(M)$$ is an ideal of $$R$$. Also, we show that if $$M$$ is a finitely generated $$R$$-module with $$r(\text{Ann}_R(M))\neq \text{Ann}_R(M)$$ and $$|m-\text{Ass}_R(M)|=1$$ and $$GA(M)$$ is a star graph, then $$r(\text{Ann}_R(M))$$ is not a prime ideal of $$R$$ and $$|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1$$.
Keywords: prime submodule, annihilator submodule, intersection annihilator graph.
Mathematics Subject Classification: 13C05, 13C99.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 577-588, https://doi.org/10.7494/OpMath.2019.39.4.577.

]]>
Description of the scattering data for Sturm-Liouville operators on the half-line We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to $$L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)$$, where $$\rho:\mathbb{R}_+\to\mathbb{R}_+$$ is a monotonically nondecreasing function from some family $$\mathscr{R}$$. In particular, $$\mathscr{R}$$ includes the functions $$\rho(x)=(1+x)^{\alpha}$$ with $$\alpha\geq 1$$. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3932.pdf Description of the scattering data for Sturm-Liouville operators on the half-line Yaroslav Mykytyuk Nataliia Sushchyk inverse scattering; Schrödinger operator; Banach algebra doi:10.7494/OpMath.2019.39.4.557 Opuscula Math. 39, no. 4 (2019), 557-576, https://doi.org/10.7494/OpMath.2019.39.4.557 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.557 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3932.pdf 39 4 557 576
Title: Description of the scattering data for Sturm-Liouville operators on the half-line.

Author(s): Yaroslav Mykytyuk, Nataliia Sushchyk.

Abstract: We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to $$L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)$$, where $$\rho:\mathbb{R}_+\to\mathbb{R}_+$$ is a monotonically nondecreasing function from some family $$\mathscr{R}$$. In particular, $$\mathscr{R}$$ includes the functions $$\rho(x)=(1+x)^{\alpha}$$ with $$\alpha\geq 1$$.
Keywords: inverse scattering, Schrödinger operator, Banach algebra.
Mathematics Subject Classification: 34L25, 34L40, 47L10, 81U40.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 557-576, https://doi.org/10.7494/OpMath.2019.39.4.557.

]]>
On unitary equivalence of bilateral operator valued weighted shifts We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We provide an example of unitary equivalence between shifts with weights defined on $$\mathbb{C}^2$$ which cannot be given by any unitary operator of diagonal form. The paper also contains some remarks regarding unitary operators that can give unitary equivalence of bilateral operator valued weighted shifts. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3931.pdf On unitary equivalence of bilateral operator valued weighted shifts Jakub Kośmider unitary equivalence; bilateral weighted shift; quasi-invertible weights; partial isometry doi:10.7494/OpMath.2019.39.4.543 Opuscula Math. 39, no. 4 (2019), 543-555, https://doi.org/10.7494/OpMath.2019.39.4.543 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.543 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3931.pdf 39 4 543 555
Title: On unitary equivalence of bilateral operator valued weighted shifts.

Author(s): Jakub Kośmider.

Abstract: We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We provide an example of unitary equivalence between shifts with weights defined on $$\mathbb{C}^2$$ which cannot be given by any unitary operator of diagonal form. The paper also contains some remarks regarding unitary operators that can give unitary equivalence of bilateral operator valued weighted shifts.
Keywords: unitary equivalence, bilateral weighted shift, quasi-invertible weights, partial isometry.
Mathematics Subject Classification: 47B37, 47A62.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 543-555, https://doi.org/10.7494/OpMath.2019.39.4.543.

]]>
Decomposition of Gaussian processes, and factorization of positive definite kernels We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $$K$$, and the other for Gaussian processes, say $$V$$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $$K$$, presented as a covariance kernel for a Gaussian process $$V$$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $$K$$, vs for Gaussian process $$V$$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $$K$$ is the exact same as that which yield factorizations for $$V$$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3930.pdf Decomposition of Gaussian processes, and factorization of positive definite kernels Palle Jorgensen Feng Tian reproducing kernel Hilbert space; frames; generalized Ito-integration; the measurable category; analysis/synthesis; interpolation; Gaussian free fields; non-uniform sampling; optimization; transform; covariance; feature space doi:10.7494/OpMath.2019.39.4.497 Opuscula Math. 39, no. 4 (2019), 497-541, https://doi.org/10.7494/OpMath.2019.39.4.497 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.497 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3930.pdf 39 4 497 541
Title: Decomposition of Gaussian processes, and factorization of positive definite kernels.

Author(s): Palle Jorgensen, Feng Tian.

Abstract: We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels $$K$$, and the other for Gaussian processes, say $$V$$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $$K$$, presented as a covariance kernel for a Gaussian process $$V$$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $$K$$, vs for Gaussian process $$V$$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $$K$$ is the exact same as that which yield factorizations for $$V$$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.
Keywords: reproducing kernel Hilbert space, frames, generalized Ito-integration, the measurable category, analysis/synthesis, interpolation, Gaussian free fields, non-uniform sampling, optimization, transform, covariance, feature space.
Mathematics Subject Classification: 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 60J20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 497-541, https://doi.org/10.7494/OpMath.2019.39.4.497.

]]>
Oscillatory results for second-order noncanonical delay differential equations The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation $\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,$ under the condition $\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.$ Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3929.pdf Oscillatory results for second-order noncanonical delay differential equations Jozef Džurina Irena Jadlovská Ioannis P. Stavroulakis linear differential equation; delay; second-order; noncanonical; oscillation doi:10.7494/OpMath.2019.39.4.483 Opuscula Math. 39, no. 4 (2019), 483-495, https://doi.org/10.7494/OpMath.2019.39.4.483 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.483 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3929.pdf 39 4 483 495
Title: Oscillatory results for second-order noncanonical delay differential equations.

Author(s): Jozef Džurina, Irena Jadlovská, Ioannis P. Stavroulakis.

Abstract: The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation $\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,$ under the condition $\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.$ Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.
Keywords: linear differential equation, delay, second-order, noncanonical, oscillation.
Mathematics Subject Classification: 34C10, 34K11.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 483-495, https://doi.org/10.7494/OpMath.2019.39.4.483.

]]>
Applications of PDEs inpainting to magnetic particle imaging and corneal topography In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques. https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3928.pdf Applications of PDEs inpainting to magnetic particle imaging and corneal topography Andrea Andrisani Rosa Maria Mininni Francesca Mazzia Giuseppina Settanni Alessandro Iurino Sabina Tangaro Andrea Tateo Roberto Bellotti PDEs inpainting; medical imaging; magnetic particle imaging; radial curvature image; anterior surface of a cornea doi:10.7494/OpMath.2019.39.4.453 Opuscula Math. 39, no. 4 (2019), 453-482, https://doi.org/10.7494/OpMath.2019.39.4.453 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.4.453 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3928.pdf 39 4 453 482
Title: Applications of PDEs inpainting to magnetic particle imaging and corneal topography.

Author(s): Andrea Andrisani, Rosa Maria Mininni, Francesca Mazzia, Giuseppina Settanni, Alessandro Iurino, Sabina Tangaro, Andrea Tateo, Roberto Bellotti.

Abstract: In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques.
Keywords: PDEs inpainting, medical imaging, magnetic particle imaging, radial curvature image, anterior surface of a cornea.
Mathematics Subject Classification: 35Q68, 68U10, 82D80, 92C55, 94A08.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 4 (2019), 453-482, https://doi.org/10.7494/OpMath.2019.39.4.453.

]]>
General multiplicative Zagreb indices of graphs with given clique number We obtain lower and upper bounds on general multiplicative Zagreb indices for graphs of given clique number and order. Bounds on the basic multiplicative Zagreb indices and on the multiplicative sum Zagreb index follow from our results. We also determine graphs with the smallest and the largest indices. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3927.pdf General multiplicative Zagreb indices of graphs with given clique number Tomáš Vetrík Selvaraj Balachandran clique number; multiplicative Zagreb index; chromatic number doi:10.7494/OpMath.2019.39.3.433 Opuscula Math. 39, no. 3 (2019), 433-446, https://doi.org/10.7494/OpMath.2019.39.3.433 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.433 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3927.pdf 39 3 433 446
Title: General multiplicative Zagreb indices of graphs with given clique number.

Author(s): Tomáš Vetrík, Selvaraj Balachandran.

Abstract: We obtain lower and upper bounds on general multiplicative Zagreb indices for graphs of given clique number and order. Bounds on the basic multiplicative Zagreb indices and on the multiplicative sum Zagreb index follow from our results. We also determine graphs with the smallest and the largest indices.
Keywords: clique number, multiplicative Zagreb index, chromatic number.
Mathematics Subject Classification: 05C35, 05C07.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 433-446, https://doi.org/10.7494/OpMath.2019.39.3.433.

]]>
The complexity of open k-monopolies in graphs for negative k Let $$G$$ be a graph with vertex set $$V(G)$$, $$\delta(G)$$ minimum degree of $$G$$ and $$k\in\left\{1-\left\lceil\frac{\delta(G)}{2}\right\rceil,\ldots ,\left\lfloor \frac{\delta(G)}{2}\right\rfloor\right\}$$. Given a nonempty set $$M\subseteq V(G)$$ a vertex $$v$$ of $$G$$ is said to be $$k$$-controlled by $$M$$ if $$\delta_M(v)\ge\frac{\delta_{V(G)}(v)}{2}+k$$ where $$\delta_M(v)$$ represents the number of neighbors of $$v$$ in $$M$$. The set $$M$$ is called an open $$k$$-monopoly for $$G$$ if it $$k$$-controls every vertex $$v$$ of $$G$$. In this short note we prove that the problem of computing the minimum cardinality of an open $$k$$-monopoly in a graph for a negative integer $$k$$ is NP-complete even restricted to chordal graphs. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3926.pdf The complexity of open k-monopolies in graphs for negative k Iztok Peterin open $$k$$-monopolies; complexity; total domination doi:10.7494/OpMath.2019.39.3.425 Opuscula Math. 39, no. 3 (2019), 425-431, https://doi.org/10.7494/OpMath.2019.39.3.425 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.425 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3926.pdf 39 3 425 431
Title: The complexity of open k-monopolies in graphs for negative k.

Author(s): Iztok Peterin.

Abstract: Let $$G$$ be a graph with vertex set $$V(G)$$, $$\delta(G)$$ minimum degree of $$G$$ and $$k\in\left\{1-\left\lceil\frac{\delta(G)}{2}\right\rceil,\ldots ,\left\lfloor \frac{\delta(G)}{2}\right\rfloor\right\}$$. Given a nonempty set $$M\subseteq V(G)$$ a vertex $$v$$ of $$G$$ is said to be $$k$$-controlled by $$M$$ if $$\delta_M(v)\ge\frac{\delta_{V(G)}(v)}{2}+k$$ where $$\delta_M(v)$$ represents the number of neighbors of $$v$$ in $$M$$. The set $$M$$ is called an open $$k$$-monopoly for $$G$$ if it $$k$$-controls every vertex $$v$$ of $$G$$. In this short note we prove that the problem of computing the minimum cardinality of an open $$k$$-monopoly in a graph for a negative integer $$k$$ is NP-complete even restricted to chordal graphs.
Keywords: open $$k$$-monopolies, complexity, total domination.
Mathematics Subject Classification: 05C85, 05C69, 05C07.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 425-431, https://doi.org/10.7494/OpMath.2019.39.3.425.

]]>
Metric dimension of Andrásfai graphs A set $$W\subseteq V(G)$$ is called a resolving set, if for each pair of distinct vertices $$u,v\in V(G)$$ there exists $$t\in W$$ such that $$d(u,t)\neq d(v,t)$$, where $$d(x,y)$$ is the distance between vertices $$x$$ and $$y$$. The cardinality of a minimum resolving set for $$G$$ is called the metric dimension of $$G$$ and is denoted by $$\dim_M(G)$$. This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and $$And(k)\square P_n$$. Also, we provide upper and lower bounds for $$dim_M(And(k)\square C_n)$$. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3925.pdf Metric dimension of Andrásfai graphs S. Batool Pejman Shiroyeh Payrovi Ali Behtoei resolving set; metric dimension; Andrásfai graph; Cayley graph; Cartesian product doi:10.7494/OpMath.2019.39.3.415 Opuscula Math. 39, no. 3 (2019), 415-423, https://doi.org/10.7494/OpMath.2019.39.3.415 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.415 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3925.pdf 39 3 415 423
Title: Metric dimension of Andrásfai graphs.

Author(s): S. Batool Pejman, Shiroyeh Payrovi, Ali Behtoei.

Abstract: A set $$W\subseteq V(G)$$ is called a resolving set, if for each pair of distinct vertices $$u,v\in V(G)$$ there exists $$t\in W$$ such that $$d(u,t)\neq d(v,t)$$, where $$d(x,y)$$ is the distance between vertices $$x$$ and $$y$$. The cardinality of a minimum resolving set for $$G$$ is called the metric dimension of $$G$$ and is denoted by $$\dim_M(G)$$. This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and $$And(k)\square P_n$$. Also, we provide upper and lower bounds for $$dim_M(And(k)\square C_n)$$.
Keywords: resolving set, metric dimension, Andrásfai graph, Cayley graph, Cartesian product.
Mathematics Subject Classification: 05C12, 05C25.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 415-423, https://doi.org/10.7494/OpMath.2019.39.3.415.

]]>
Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term We give an existence theorem of global solution to the initial-boundary value problem for $$u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)$$ under some smallness conditions on the initial data, where $$\sigma (v^2)$$ is a positive function of $$v^2\ne 0$$ admitting the degeneracy property $$\sigma(0)=0$$. We are interested in the case where $$\sigma(v^2)$$ has no exponent $$m \geq 0$$ such that $$\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0$$. A typical example is $$\sigma(v^2)=\operatorname{log}(1+v^2)$$. $$f(u)$$ is a function like $$f=|u|^{\alpha} u$$. A decay estimate for $$\|\nabla u(t)\|_{\infty}$$ is also given. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3924.pdf Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term Mitsuhiro Nakao degenerate quasilinear parabolic equation; nonlinear source term; Moser's method doi:10.7494/OpMath.2019.39.3.395 Opuscula Math. 39, no. 3 (2019), 395-414, https://doi.org/10.7494/OpMath.2019.39.3.395 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.395 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3924.pdf 39 3 395 414
Title: Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term.

Author(s): Mitsuhiro Nakao.

Abstract: We give an existence theorem of global solution to the initial-boundary value problem for $$u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)$$ under some smallness conditions on the initial data, where $$\sigma (v^2)$$ is a positive function of $$v^2\ne 0$$ admitting the degeneracy property $$\sigma(0)=0$$. We are interested in the case where $$\sigma(v^2)$$ has no exponent $$m \geq 0$$ such that $$\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0$$. A typical example is $$\sigma(v^2)=\operatorname{log}(1+v^2)$$. $$f(u)$$ is a function like $$f=|u|^{\alpha} u$$. A decay estimate for $$\|\nabla u(t)\|_{\infty}$$ is also given.
Keywords: degenerate quasilinear parabolic equation, nonlinear source term, Moser's method.
Mathematics Subject Classification: 35B40, 35D35, 58J35, 58K30.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 395-414, https://doi.org/10.7494/OpMath.2019.39.3.395.

]]>
Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete $$k$$-uniform hypergraph $$K^{(k)}_{n}$$ into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For $$n\equiv 2,4,5\pmod 6$$, we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of $$K^{(3)}_{n}$$ into 5-cycles has been presented for all admissible $$n\leq17$$, and for all $$n=4^{m}+1$$ when $$m$$ is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if $$42~|~(n-1)(n-2)$$ and if there exist $$\lambda=\frac{(n-1)(n-2)}{42}$$ sequences $$(k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})$$ on $$D_{all}(n)$$, then $$K^{(3)}_{n}$$ can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of $$K^{(3)}_{37}$$ and $$K^{(3)}_{43}$$ into 7-cycles. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles Meihua Meiling Guan Jirimutu uniform hypergraph; 7-cycle; cycle decomposition doi:10.7494/OpMath.2019.39.3.383 Opuscula Math. 39, no. 3 (2019), 383-393, https://doi.org/10.7494/OpMath.2019.39.3.383 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.383 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf 39 3 383 393
Title: Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles.

Author(s): Meihua, Meiling Guan, Jirimutu.

Abstract: We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete $$k$$-uniform hypergraph $$K^{(k)}_{n}$$ into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For $$n\equiv 2,4,5\pmod 6$$, we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of $$K^{(3)}_{n}$$ into 5-cycles has been presented for all admissible $$n\leq17$$, and for all $$n=4^{m}+1$$ when $$m$$ is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if $$42~|~(n-1)(n-2)$$ and if there exist $$\lambda=\frac{(n-1)(n-2)}{42}$$ sequences $$(k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})$$ on $$D_{all}(n)$$, then $$K^{(3)}_{n}$$ can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of $$K^{(3)}_{37}$$ and $$K^{(3)}_{43}$$ into 7-cycles.
Keywords: uniform hypergraph, 7-cycle, cycle decomposition.
Mathematics Subject Classification: 05C65, 05C85.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 383-393, https://doi.org/10.7494/OpMath.2019.39.3.383.

]]>
On the zeros of the Macdonald functions We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3922.pdf On the zeros of the Macdonald functions Yuji Hamana Hiroyuki Matsumoto Tomoyuki Shirai zeros; Macdonald functions; Bessel functions doi:10.7494/OpMath.2019.39.3.361 Opuscula Math. 39, no. 3 (2019), 361-382, https://doi.org/10.7494/OpMath.2019.39.3.361 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.361 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3922.pdf 39 3 361 382
Title: On the zeros of the Macdonald functions.

Author(s): Yuji Hamana, Hiroyuki Matsumoto, Tomoyuki Shirai.

Abstract: We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented.
Keywords: zeros, Macdonald functions, Bessel functions.
Mathematics Subject Classification: 33C10, 30C15, 32A60, 33F05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 361-382, https://doi.org/10.7494/OpMath.2019.39.3.361.

]]>
A partial refining of the Erdős-Kelly regulation The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs &amp; Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple $$n$$-vertex graph $$G$$ with maximum vertex degree $$\Delta$$, the exact minimum number, say $$\theta =\theta(G)$$, of new vertices in a $$\Delta$$-regular graph $$H$$ which includes $$G$$ as an induced subgraph. The number $$\theta(G)$$, which we call the cost of regulation of $$G$$, has been upper-bounded by the order of $$G$$, the bound being attained for each $$n\ge4$$, e.g. then the edge-deleted complete graph $$K_n-e$$ has $$\theta=n$$. For $$n\ge 4$$, we present all factors of $$K_n$$ with $$\theta=n$$ and next $$\theta=n-1$$. Therein in case $$\theta=n-1$$ and $$n$$ odd only, we show that a specific extra structure, non-matching, is required. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3921.pdf A partial refining of the Erdős-Kelly regulation Joanna Górska Zdzisław Skupień inducing $$\Delta$$-regulation; cost of regulation doi:10.7494/OpMath.2019.39.3.355 Opuscula Math. 39, no. 3 (2019), 355-360, https://doi.org/10.7494/OpMath.2019.39.3.355 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.355 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3921.pdf 39 3 355 360
Title: A partial refining of the Erdős-Kelly regulation.

Author(s): Joanna Górska, Zdzisław Skupień.

Abstract: The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs &amp; Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple $$n$$-vertex graph $$G$$ with maximum vertex degree $$\Delta$$, the exact minimum number, say $$\theta =\theta(G)$$, of new vertices in a $$\Delta$$-regular graph $$H$$ which includes $$G$$ as an induced subgraph. The number $$\theta(G)$$, which we call the cost of regulation of $$G$$, has been upper-bounded by the order of $$G$$, the bound being attained for each $$n\ge4$$, e.g. then the edge-deleted complete graph $$K_n-e$$ has $$\theta=n$$. For $$n\ge 4$$, we present all factors of $$K_n$$ with $$\theta=n$$ and next $$\theta=n-1$$. Therein in case $$\theta=n-1$$ and $$n$$ odd only, we show that a specific extra structure, non-matching, is required.
Keywords: inducing $$\Delta$$-regulation, cost of regulation.
Mathematics Subject Classification: 05C35, 05C75.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 355-360, https://doi.org/10.7494/OpMath.2019.39.3.355.

]]>
Oscillations of equations caused by several deviating arguments Linear delay or advanced differential equations with variable coefficients and several not necessarily monotone arguments are considered, and some new oscillation criteria are given. More precisely, sufficient conditions, involving $$\lim\sup$$ and $$\lim\inf$$, are obtained, which essentially improve several known criteria existing in the literature. Examples illustrating the results are also given, numerically solved in MATLAB. https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3920.pdf Oscillations of equations caused by several deviating arguments George E. Chatzarakis differential equation; non-monotone argument; oscillatory solution; nonoscillatory solution doi:10.7494/OpMath.2019.39.3.321 Opuscula Math. 39, no. 3 (2019), 321-353, https://doi.org/10.7494/OpMath.2019.39.3.321 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.3.321 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3920.pdf 39 3 321 353
Title: Oscillations of equations caused by several deviating arguments.

Author(s): George E. Chatzarakis.

Abstract: Linear delay or advanced differential equations with variable coefficients and several not necessarily monotone arguments are considered, and some new oscillation criteria are given. More precisely, sufficient conditions, involving $$\lim\sup$$ and $$\lim\inf$$, are obtained, which essentially improve several known criteria existing in the literature. Examples illustrating the results are also given, numerically solved in MATLAB.
Keywords: differential equation, non-monotone argument, oscillatory solution, nonoscillatory solution.
Mathematics Subject Classification: 34K11, 34K06.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 3 (2019), 321-353, https://doi.org/10.7494/OpMath.2019.39.3.321.

]]>
Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3919.pdf Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations Yang Yanbing Md Salik Ahmed Qin Lanlan Xu Runzhang fourth-order nonlinear wave equation; strong damping; blow up; global existence doi:10.7494/OpMath.2019.39.2.297 Opuscula Math. 39, no. 2 (2019), 297-313, https://doi.org/10.7494/OpMath.2019.39.2.297 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.297 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3919.pdf 39 2 297 313
Title: Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations.

Author(s): Yang Yanbing, Md Salik Ahmed, Qin Lanlan, Xu Runzhang.

Abstract: Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time.
Keywords: fourth-order nonlinear wave equation, strong damping, blow up, global existence.
Mathematics Subject Classification: 35B44, 35L35, 35L05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 297-313, https://doi.org/10.7494/OpMath.2019.39.2.297.

]]>
Extremal length and Dirichlet problem on Klein surfaces The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3918.pdf Extremal length and Dirichlet problem on Klein surfaces Monica Roşiu Klein surface; extremal length; extremal distance doi:10.7494/OpMath.2019.39.2.281 Opuscula Math. 39, no. 2 (2019), 281-296, https://doi.org/10.7494/OpMath.2019.39.2.281 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.281 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3918.pdf 39 2 281 296
Title: Extremal length and Dirichlet problem on Klein surfaces.

Author(s): Monica Roşiu.

Abstract: The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.
Keywords: Klein surface, extremal length, extremal distance.
Mathematics Subject Classification: 30F50, 35J05, 31A15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 281-296, https://doi.org/10.7494/OpMath.2019.39.2.281.

]]>
Isotropic and anisotropic double-phase problems: old and new We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the $$(p,q)$$- and $$(p(x),q(x))$$-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcellini and G. Mingione. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3917.pdf Isotropic and anisotropic double-phase problems: old and new Vicenţiu D. Rădulescu differential operator with unbalanced growth; double-phase energy; variable exponent doi:10.7494/OpMath.2019.39.2.259 Opuscula Math. 39, no. 2 (2019), 259-279, https://doi.org/10.7494/OpMath.2019.39.2.259 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.259 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3917.pdf 39 2 259 279
Title: Isotropic and anisotropic double-phase problems: old and new.

Author(s): Vicenţiu D. Rădulescu.

Abstract: We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the $$(p,q)$$- and $$(p(x),q(x))$$-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcellini and G. Mingione.
Keywords: differential operator with unbalanced growth, double-phase energy, variable exponent.
Mathematics Subject Classification: 35J60, 35J65, 58E05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 259-279, https://doi.org/10.7494/OpMath.2019.39.2.259.

]]>
Existence and multiplicity results for quasilinear equations in the Heisenberg group In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation $$(\mathcal{E}_{\lambda})$$ in $$\mathbb{H}^{n}$$, depending on a real parameter $$\lambda$$, which involves a general elliptic operator $$\mathbf{A}$$ in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all $$\lambda\gt 0$$ and, for special elliptic operators $$\mathbf{A}$$, existence of infinitely many solutions $$(u_k)_k$$. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3916.pdf Existence and multiplicity results for quasilinear equations in the Heisenberg group Patrizia Pucci Heisenberg group; entire solutions; critical exponents doi:10.7494/OpMath.2019.39.2.247 Opuscula Math. 39, no. 2 (2019), 247-257, https://doi.org/10.7494/OpMath.2019.39.2.247 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.247 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3916.pdf 39 2 247 257
Title: Existence and multiplicity results for quasilinear equations in the Heisenberg group.

Author(s): Patrizia Pucci.

Abstract: In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation $$(\mathcal{E}_{\lambda})$$ in $$\mathbb{H}^{n}$$, depending on a real parameter $$\lambda$$, which involves a general elliptic operator $$\mathbf{A}$$ in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all $$\lambda\gt 0$$ and, for special elliptic operators $$\mathbf{A}$$, existence of infinitely many solutions $$(u_k)_k$$.
Keywords: Heisenberg group, entire solutions, critical exponents.
Mathematics Subject Classification: 35J62, 35J70, 35B08, 35J20, 35B09.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 247-257, https://doi.org/10.7494/OpMath.2019.39.2.247.

]]>
On a Robin (p,q)-equation with a logistic reaction We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a $$p$$-Laplacian and of a $$q$$-Laplacian ($$(p,q)$$-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $$\lambda \gt 0$$ varies. Also, we show that for every admissible parameter $$\lambda \gt 0$$, the problem admits a smallest positive solution. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3915.pdf On a Robin (p,q)-equation with a logistic reaction Nikolaos S. Papageorgiou Calogero Vetro Francesca Vetro positive solutions; superdiffusive reaction; local minimizers; maximum principle; minimal positive solutions; Robin boundary condition; indefinite potential doi:10.7494/OpMath.2019.39.2.227 Opuscula Math. 39, no. 2 (2019), 227-245, https://doi.org/10.7494/OpMath.2019.39.2.227 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.227 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3915.pdf 39 2 227 245
Title: On a Robin (p,q)-equation with a logistic reaction.

Author(s): Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro.

Abstract: We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a $$p$$-Laplacian and of a $$q$$-Laplacian ($$(p,q)$$-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $$\lambda \gt 0$$ varies. Also, we show that for every admissible parameter $$\lambda \gt 0$$, the problem admits a smallest positive solution.
Keywords: positive solutions, superdiffusive reaction, local minimizers, maximum principle, minimal positive solutions, Robin boundary condition, indefinite potential.
Mathematics Subject Classification: 35J20, 35J60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 227-245, https://doi.org/10.7494/OpMath.2019.39.2.227.

]]>
Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3914.pdf Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions Genni Fragnelli Dimitri Mugnai strongly singular/degenerate equations; non smooth coefficients; null controllability doi:10.7494/OpMath.2019.39.2.207 Opuscula Math. 39, no. 2 (2019), 207-225, https://doi.org/10.7494/OpMath.2019.39.2.207 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.207 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3914.pdf 39 2 207 225
Title: Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions.

Author(s): Genni Fragnelli, Dimitri Mugnai.

Abstract: We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case.
Keywords: strongly singular/degenerate equations, non smooth coefficients, null controllability.
Mathematics Subject Classification: 35Q93, 93B05, 34H05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 207-225, https://doi.org/10.7494/OpMath.2019.39.2.207.

]]>
Existence results and a priori estimates for solutions of quasilinear problems with gradient terms In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $$\mathbb{R}^N$$ with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of $$u$$ and by removing the growth condition for the nonlinearity $$f$$ at $$u=0$$. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3913.pdf Existence results and a priori estimates for solutions of quasilinear problems with gradient terms Roberta Filippucci Chiara Lini existence result; quasilinear problems; a priori estimates doi:10.7494/OpMath.2019.39.2.195 Opuscula Math. 39, no. 2 (2019), 195-206, https://doi.org/10.7494/OpMath.2019.39.2.195 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.195 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3913.pdf 39 2 195 206
Title: Existence results and a priori estimates for solutions of quasilinear problems with gradient terms.

Author(s): Roberta Filippucci, Chiara Lini.

Abstract: In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $$\mathbb{R}^N$$ with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of $$u$$ and by removing the growth condition for the nonlinearity $$f$$ at $$u=0$$.
Keywords: existence result, quasilinear problems, a priori estimates.
Mathematics Subject Classification: 35J92, 35J70.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 195-206, https://doi.org/10.7494/OpMath.2019.39.2.195.

]]>
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}$ where $$\Omega \subset \mathbb{R}^N$$ is an open bounded domain, $$N\geq 3$$, and $$A(x,t,\xi)$$, $$g(x,t)$$, $$h(x)$$ are given functions, with $$A_t = \frac{\partial A}{\partial t}$$, $$a = \nabla_{\xi} A$$, such that $$A(x,\cdot,\cdot)$$ is even and $$g(x,\cdot)$$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if $$A(x,t,\xi)$$ grows fast enough with respect to $$t$$, then the nonlinear term related to $$g(x,t)$$ may have also a supercritical growth. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3912.pdf Infinitely many solutions for some nonlinear supercritical problems with break of symmetry Anna Maria Candela Addolorata Salvatore quasilinear elliptic equation; weak Cerami-Palais-Smale condition; Ambrosetti-Rabinowitz condition; break of symmetry; perturbation method; supercritical growth doi:10.7494/OpMath.2019.39.2.175 Opuscula Math. 39, no. 2 (2019), 175-194, https://doi.org/10.7494/OpMath.2019.39.2.175 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.175 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3912.pdf 39 2 175 194
Title: Infinitely many solutions for some nonlinear supercritical problems with break of symmetry.

Author(s): Anna Maria Candela, Addolorata Salvatore.

Abstract: In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}$ where $$\Omega \subset \mathbb{R}^N$$ is an open bounded domain, $$N\geq 3$$, and $$A(x,t,\xi)$$, $$g(x,t)$$, $$h(x)$$ are given functions, with $$A_t = \frac{\partial A}{\partial t}$$, $$a = \nabla_{\xi} A$$, such that $$A(x,\cdot,\cdot)$$ is even and $$g(x,\cdot)$$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if $$A(x,t,\xi)$$ grows fast enough with respect to $$t$$, then the nonlinear term related to $$g(x,t)$$ may have also a supercritical growth.
Keywords: quasilinear elliptic equation, weak Cerami-Palais-Smale condition, Ambrosetti-Rabinowitz condition, break of symmetry, perturbation method, supercritical growth.
Mathematics Subject Classification: 35J20, 35J62, 35J66, 58E05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 175-194, https://doi.org/10.7494/OpMath.2019.39.2.175.

]]>
Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions In this paper, a nonlinear differential problem involving the $$p$$-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3911.pdf Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions Gabriele Bonanno Giuseppina D'Aguì Angela Sciammetta mixed problem; critical points doi:10.7494/OpMath.2019.39.2.159 Opuscula Math. 39, no. 2 (2019), 159-174, https://doi.org/10.7494/OpMath.2019.39.2.159 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.159 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3911.pdf 39 2 159 174
Title: Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions.

Author(s): Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta.

Abstract: In this paper, a nonlinear differential problem involving the $$p$$-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.
Keywords: mixed problem, critical points.
Mathematics Subject Classification: 35J25, 35J20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 159-174, https://doi.org/10.7494/OpMath.2019.39.2.159.

]]>
Some remarks on the coincidence set for the Signorini problem We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3910.pdf Some remarks on the coincidence set for the Signorini problem Miguel de Benito Delgado Jesus Ildefonso Díaz Signorini problem; coincidence set; location estimates; free boundary problem; contact problems doi:10.7494/OpMath.2019.39.2.145 Opuscula Math. 39, no. 2 (2019), 145-157, https://doi.org/10.7494/OpMath.2019.39.2.145 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.145 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3910.pdf 39 2 145 157
Title: Some remarks on the coincidence set for the Signorini problem.

Author(s): Miguel de Benito Delgado, Jesus Ildefonso Díaz.

Abstract: We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data.
Keywords: Signorini problem, coincidence set, location estimates, free boundary problem, contact problems.
Mathematics Subject Classification: 35J86, 35R35, 35R70, 35B60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 145-157, https://doi.org/10.7494/OpMath.2019.39.2.145.

]]>
On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems. https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3909.pdf On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative Michał Bełdziński Marek Galewski diffeomorphism; uniqueness; non-potential problems; variational methods; monotone methods; Palais-Smale condition doi:10.7494/OpMath.2019.39.2.131 Opuscula Math. 39, no. 2 (2019), 131-144, https://doi.org/10.7494/OpMath.2019.39.2.131 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.2.131 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3909.pdf 39 2 131 144
Title: On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative.

Author(s): Michał Bełdziński, Marek Galewski.

Abstract: In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.
Keywords: diffeomorphism, uniqueness, non-potential problems, variational methods, monotone methods, Palais-Smale condition.
Mathematics Subject Classification: 34A12, 47H30, 47J07.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 2 (2019), 131-144, https://doi.org/10.7494/OpMath.2019.39.2.131.

]]>
Pseudo-differential equations and conical potentials: 2-dimensional case We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3908.pdf Pseudo-differential equations and conical potentials: 2-dimensional case Vladimir B. Vasilyev pseudo-differential equation; wave factorization; Dirichlet problem; system of linear integral equations doi:10.7494/OpMath.2019.39.1.109 Opuscula Math. 39, no. 1 (2019), 109-124, https://doi.org/10.7494/OpMath.2019.39.1.109 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.109 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3908.pdf 39 1 109 124
Title: Pseudo-differential equations and conical potentials: 2-dimensional case.

Abstract: We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.
Keywords: pseudo-differential equation, wave factorization, Dirichlet problem, system of linear integral equations.
Mathematics Subject Classification: 35S15, 45A05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 109-124, https://doi.org/10.7494/OpMath.2019.39.1.109.

]]>
Oscillation criteria for even order neutral difference equations In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form $\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,$ where $$m\geq 2$$ is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3907.pdf Oscillation criteria for even order neutral difference equations S. Selvarangam S. A. Rupadevi E. Thandapani S. Pinelas even order; neutral difference equation; oscillation; asymptotic behavior; mixed type doi:10.7494/OpMath.2019.39.1.91 Opuscula Math. 39, no. 1 (2019), 91-108, https://doi.org/10.7494/OpMath.2019.39.1.91 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.91 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3907.pdf 39 1 91 108
Title: Oscillation criteria for even order neutral difference equations.

Author(s): S. Selvarangam, S. A. Rupadevi, E. Thandapani, S. Pinelas.

Abstract: In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form $\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,$ where $$m\geq 2$$ is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.
Keywords: even order, neutral difference equation, oscillation, asymptotic behavior, mixed type.
Mathematics Subject Classification: 39A10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 91-108, https://doi.org/10.7494/OpMath.2019.39.1.91.

]]>
The existence of consensus of a leader-following problem with Caputo fractional derivative In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3906.pdf The existence of consensus of a leader-following problem with Caputo fractional derivative Ewa Schmeidel leader-following problem; Caputo fractional differential equation; consensus; nonlinear system; Schauder fixed point theorem doi:10.7494/OpMath.2019.39.1.77 Opuscula Math. 39, no. 1 (2019), 77-89, https://doi.org/10.7494/OpMath.2019.39.1.77 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.77 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3906.pdf 39 1 77 89
Title: The existence of consensus of a leader-following problem with Caputo fractional derivative.

Author(s): Ewa Schmeidel.

Abstract: In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.
Keywords: leader-following problem, Caputo fractional differential equation, consensus, nonlinear system, Schauder fixed point theorem.
Mathematics Subject Classification: 26A33, 34K20, 45D05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 77-89, https://doi.org/10.7494/OpMath.2019.39.1.77.

]]>
On the convergence of solutions to second-order neutral difference equations A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3905.pdf On the convergence of solutions to second-order neutral difference equations Małgorzata Migda Janusz Migda Małgorzata Zdanowicz second-order difference equation; asymptotic behavior; quasi-differences; Krasnoselskii's fixed point theorem doi:10.7494/OpMath.2019.39.1.61 Opuscula Math. 39, no. 1 (2019), 61-75, https://doi.org/10.7494/OpMath.2019.39.1.61 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.61 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3905.pdf 39 1 61 75
Title: On the convergence of solutions to second-order neutral difference equations.

Author(s): Małgorzata Migda, Janusz Migda, Małgorzata Zdanowicz.

Abstract: A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant.
Keywords: second-order difference equation, asymptotic behavior, quasi-differences, Krasnoselskii's fixed point theorem.
Mathematics Subject Classification: 39A10, 39A22, 39A30.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 61-75, https://doi.org/10.7494/OpMath.2019.39.1.61.

]]>
Boundary value problems with solutions in convex sets By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of concave solutions of certain boundary value problems in ordered Banach spaces. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3904.pdf Boundary value problems with solutions in convex sets Gerd Herzog Peer Chr. Kunstmann Dirichlet boundary value problems; solutions in convex sets; continuation method; ordered Banach spaces; concave solutions doi:10.7494/OpMath.2019.39.1.49 Opuscula Math. 39, no. 1 (2019), 49-60, https://doi.org/10.7494/OpMath.2019.39.1.49 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.49 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3904.pdf 39 1 49 60
Title: Boundary value problems with solutions in convex sets.

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

Abstract: By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of concave solutions of certain boundary value problems in ordered Banach spaces.
Keywords: Dirichlet boundary value problems, solutions in convex sets, continuation method, ordered Banach spaces, concave solutions.
Mathematics Subject Classification: 34B15, 47H10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 49-60, https://doi.org/10.7494/OpMath.2019.39.1.49.

]]>
Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3903.pdf Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term John R. Graef Said R. Grace Ercan Tunç oscillatory behavior; neutral differential equation; even-order doi:10.7494/OpMath.2019.39.1.39 Opuscula Math. 39, no. 1 (2019), 39-47, https://doi.org/10.7494/OpMath.2019.39.1.39 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.39 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3903.pdf 39 1 39 47
Title: Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term.

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

Abstract: The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.
Keywords: oscillatory behavior, neutral differential equation, even-order.
Mathematics Subject Classification: 34C10, 34K11.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 39-47, https://doi.org/10.7494/OpMath.2019.39.1.39.

]]>
Dynamic system with random structure for modeling security and risk management in cyberspace We deal with the investigation of $$L_{2}$$-stability of the trivial solution to the system of difference equations with coefficients depending on a semi-Markov chain. In our considerations, random transformations of solutions are assumed. Necessary and sufficient conditions for $$L_{2}$$-stability of the trivial solution to such systems are obtained. A method is proposed for constructing Lyapunov functions and the conditions for its existence are justified. The dynamic system and methods discussed in the paper are very well suited for use as models for protecting information in cyberspace. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3902.pdf Dynamic system with random structure for modeling security and risk management in cyberspace Irada Dzhalladova Miroslava Růžičková semi-Markov chain; random transformation of solutions; the Lyapunov function; $$L_{2}$$-stability; systems of difference equations; jumps of solutions; cybersecurity doi:10.7494/OpMath.2019.39.1.23 Opuscula Math. 39, no. 1 (2019), 23-37, https://doi.org/10.7494/OpMath.2019.39.1.23 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.23 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3902.pdf 39 1 23 37
Title: Dynamic system with random structure for modeling security and risk management in cyberspace.

Abstract: We deal with the investigation of $$L_{2}$$-stability of the trivial solution to the system of difference equations with coefficients depending on a semi-Markov chain. In our considerations, random transformations of solutions are assumed. Necessary and sufficient conditions for $$L_{2}$$-stability of the trivial solution to such systems are obtained. A method is proposed for constructing Lyapunov functions and the conditions for its existence are justified. The dynamic system and methods discussed in the paper are very well suited for use as models for protecting information in cyberspace.
Keywords: semi-Markov chain, random transformation of solutions, the Lyapunov function, $$L_{2}$$-stability, systems of difference equations, jumps of solutions, cybersecurity.
Mathematics Subject Classification: 34F05, 60J28.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 23-37, https://doi.org/10.7494/OpMath.2019.39.1.23.

]]>
Difference equations with impulses Difference equations with impulses are studied focussing on the existence of periodic or bounded orbits, asymptotic behavior and chaos. So impulses are used to control the dynamics of the autonomous difference equations. A model of supply and demand is also considered when Li-Yorke chaos is shown among others. https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3901.pdf Difference equations with impulses Marius Danca Michal Fečkan Michal Pospíšil difference equations; impulses; stability; fixed points; Li-Yorke chaos doi:10.7494/OpMath.2019.39.1.5 Opuscula Math. 39, no. 1 (2019), 5-22, https://doi.org/10.7494/OpMath.2019.39.1.5 Copyright AGH University of Science and Technology Press, Krakow 2019 Opuscula Mathematica 2019 2019 https://doi.org/10.7494/OpMath.2019.39.1.5 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3901.pdf 39 1 5 22
Title: Difference equations with impulses.

Author(s): Marius Danca, Michal Fečkan, Michal Pospíšil.

Abstract: Difference equations with impulses are studied focussing on the existence of periodic or bounded orbits, asymptotic behavior and chaos. So impulses are used to control the dynamics of the autonomous difference equations. A model of supply and demand is also considered when Li-Yorke chaos is shown among others.
Keywords: difference equations, impulses, stability, fixed points, Li-Yorke chaos.
Mathematics Subject Classification: 37B55, 39A23, 39A33, 39A60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 39, no. 1 (2019), 5-22, https://doi.org/10.7494/OpMath.2019.39.1.5.

]]>