Opuscula Mathematica
https://www.opuscula.agh.edu.pl
enA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.Mon, 29 Nov 2021 23:30:00 +0100Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
μ-Hankel operators on Hilbert spaces
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4142.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4142.pdfMon, 29 Nov 2021 23:00:07 +0100 Author(s): Adolf Mirotin, Ekaterina Kuzmenkova.

Abstract: A class of operators is introduced (\(\mu\)-Hankel operators, \(\mu\) is a complex parameter), which generalizes the class of Hankel operators. Criteria for boundedness, compactness, nuclearity, and finite dimensionality are obtained for operators of this class, and for the case \(|\mu| = 1\) their description in the Hardy space is given. Integral representations of \(\mu\)-Hankel operators on the unit disk and on the Semi-Axis are also considered. Keywords: Hankel operator, \(\mu\)-Hankel operator, Hardy space, integral representation, nuclear operator, integral operator. Mathematics Subject Classification: 47B25, 47B35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 881-898, https://doi.org/10.7494/OpMath.2021.41.6.881.

]]>Discrete spectra for some complex infinite band matrices
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4141.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4141.pdfMon, 29 Nov 2021 23:00:06 +0100 Author(s): Maria Malejki.

Abstract: Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\). Keywords: unbounded operator, band-type matrix, complex tridiagonal matrix, discrete spectrum, eigenvalue, limit points of eigenvalues. Mathematics Subject Classification: 47B36, 47B37, 47A25, 47A75, 15A18. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 861-879, https://doi.org/10.7494/OpMath.2021.41.6.861.

]]>Spontaneous decay of level from spectral theory point of view
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4140.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4140.pdfMon, 29 Nov 2021 23:00:05 +0100 Author(s): Eduard Ianovich.

Abstract: In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent. Keywords: spectral theory, quantum field theory, self-adjoint operators, absolutely continuous spectrum, spontaneous decay. Mathematics Subject Classification: 47A10, 81Q10, 81T10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 849-859, https://doi.org/10.7494/OpMath.2021.41.6.849.

]]>Corona theorem for strictly pseudoconvex domains
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4139.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4139.pdfMon, 29 Nov 2021 23:00:04 +0100 Author(s): Sebastian Gwizdek.

Abstract: Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved "abstract" Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries. Keywords: Corona theorem, Banach algebra, uniform algebra, Arens product, Gleason part, band of measures, representing measure. Mathematics Subject Classification: 30H80, 32A38, 32A65, 32A70, 46J10, 46J15, 46J20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 843-848, https://doi.org/10.7494/OpMath.2021.41.6.843.

]]>The Krein-von Neumann extension of a regular even order quasi-differential operator
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4138.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4138.pdfMon, 29 Nov 2021 23:00:03 +0100 Author(s): Minsung Cho, Seth Hoisington, Roger Nichols, Brian Udall.

Abstract: We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl. Keywords: Krein-von Neumann extension, regular quasi-differential operator. Mathematics Subject Classification: 47B25, 47E05, 34B24. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 805-841, https://doi.org/10.7494/OpMath.2021.41.6.805.

]]>Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4137.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4137.pdfMon, 29 Nov 2021 23:00:02 +0100 Author(s): Ilwoo Cho.

Abstract: In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\). Keywords: separable Hilbert spaces, free Hilbert spaces, jump operators, shift operators, jump-shift operators, semicircular elements. Mathematics Subject Classification: 46L10, 46L54, 47L30, 47L55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 755-803, https://doi.org/10.7494/OpMath.2021.41.6.755.

]]>Generalized powers and measures
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4136.pdf
https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4136.pdfMon, 29 Nov 2021 23:00:01 +0100 Author(s): Zbigniew Burdak, Marek Kosiek, Patryk Pagacz, Krzysztof Rudol, Marek Słociński.

Abstract: Using the winding of measures on torus in "rational directions" special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces. Keywords: representing measures, Szegö-singular measure, compatible pair of isometries, spectral measure. Mathematics Subject Classification: 47B20, 47A13, 47B91. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 6 (2021), 747-754, https://doi.org/10.7494/OpMath.2021.41.6.747.

]]>Nonparametric bootstrap confidence bands for unfolding sphere size distributions
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4135.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4135.pdfThu, 30 Sep 2021 00:10:07 +0200 Author(s): Jakub Wojdyła.

Abstract: The stereological inverse problem of unfolding the distribution of spheres radii from measured planar sections radii, known as the Wicksell's corpuscle problem, is considered. The construction of uniform confidence bands based on the smoothed bootstrap in the Wicksell's problem is presented. Theoretical results on the consistency of the proposed bootstrap procedure are given, where the consistency of the bands means that the coverage probability converges to the nominal level. The finite-sample performance of the proposed method is studied via Monte Carlo simulations and compared with the asymptotic (non-bootstrap) solution described in literature. Keywords: bootstrap, confidence bands, inverse problem, nonparametric density estimation, Wicksell's problem. Mathematics Subject Classification: 45Q05, 62G05, 62G15, 62G20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 725-740, https://doi.org/10.7494/OpMath.2021.41.5.725.

]]>Certain properties of continuous fractional wavelet transform on Hardy space and Morrey space
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4134.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4134.pdfThu, 30 Sep 2021 00:10:06 +0200 Author(s): Amit K. Verma, Bivek Gupta.

Abstract: In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results. Keywords: fractional Fourier transform, continuous fractional wavelet transform, Hardy space, Morrey space. Mathematics Subject Classification: 42B10, 42C40, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 701-723, https://doi.org/10.7494/OpMath.2021.41.5.701.

]]>On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4133.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4133.pdfThu, 30 Sep 2021 00:10:05 +0200 Author(s): Ivan Tsyfra.

Abstract: We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation. Keywords: ordinary differential equation, partial differential equation, integrability, symmetry, quadrature, Lie transformation group. Mathematics Subject Classification: 34A30, 34C14. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 685-699, https://doi.org/10.7494/OpMath.2021.41.5.685.

Abstract: We show that if \(S\), \(T\) are two commuting automorphisms of a standard Borel space such that they generate a free Borel \(\mathbb{Z}^2\)-action then \(S\) and \(T\) do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel \(\mathbb{Z}^d\)-actions. Keywords: coboundries, Rokhlin Lemma, Borel \(\mathbb{Z}^d\)-action. Mathematics Subject Classification: 37A40, 37A99, 37B99. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 667-683, https://doi.org/10.7494/OpMath.2021.41.5.667.

]]>Closed range weighted composition operators between L^{p}-spaces
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4131.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4131.pdfThu, 30 Sep 2021 00:10:03 +0200 Author(s): Ching-on Lo, Anthony Wai-keung Loh.

Abstract: We characterize the closedness of ranges of weighted composition operators between \(L^p\)-spaces, where \(1 \leq p \leq \infty\). When the \(L^p\)-spaces are weighted sequence spaces, several corollaries about this class of operators are also deduced. Keywords: weighted composition operator, Lebesgue space, closed range. Mathematics Subject Classification: 47B33, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 649-665, https://doi.org/10.7494/OpMath.2021.41.5.649.

]]>Extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4130.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4130.pdfThu, 30 Sep 2021 00:10:02 +0200 Author(s): Fatiha Boulahia, Slimane Hassaine.

Abstract: In the present paper, we give criteria for the existence of extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm. Some properties of the set of attainable points of the Amemiya norm in this space are also discussed. Keywords: extreme points, strict convexity, almost periodic functions, Besicovitch-Orlicz spaces of almost periodic functions. Mathematics Subject Classification: 46B20, 46B25, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 629-648, https://doi.org/10.7494/OpMath.2021.41.5.629.

]]>Oscillation criteria for linear difference equations with several variable delays
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4129.pdf
https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4129.pdfThu, 30 Sep 2021 00:10:01 +0200 Author(s): Vasileios Benekas, Ábel Garab, Ardak Kashkynbayev, Ioannis P. Stavroulakis.

Abstract: We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay. Keywords: oscillation, difference equations, several delays, non-monotone argument, slowly varying function. Mathematics Subject Classification: 39A21, 39A10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 5 (2021), 613-627, https://doi.org/10.7494/OpMath.2021.41.5.613.

]]>A note on possible density and diameter of counterexamples to the Seymour's second neighborhood conjecture
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4128.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4128.pdfFri, 09 Jul 2021 22:00:07 +0200 Author(s): Oleksiy Zelenskiy, Valentyna Darmosiuk, Illia Nalivayko.

Abstract: Seymour's second neighborhood conjecture states that every simple digraph without loops or 2-cycles contains a vertex whose second neighborhood is at least as large as its first. In this paper we show, that from falsity of Seymour's second neighborhood conjecture it follows that there exist strongly-connected counterexamples with both low and high density (dense and sparse graph). Moreover, we show that if there is a counterexample to conjecture, then it is possible to construct counterexample with any diameter \(k\geq 3\). Keywords: graph theory, Seymour's second neighborhood conjecture, density of graph, diameter of graph. Mathematics Subject Classification: 05C12, 05C20, 05C42. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 601-605, https://doi.org/10.7494/OpMath.2021.41.4.601.

]]>Region of existence of multiple solutions for a class of Robin type four-point BVPs
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4127.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4127.pdfFri, 09 Jul 2021 22:00:06 +0200 Author(s): Amit K. Verma, Nazia Urus, Ravi P. Agarwal.

Abstract: This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent. Keywords: Green's function, monotone iterative technique, maximum principle, multi-point problem. Mathematics Subject Classification: 34B05, 34B15, 34B10, 65L10, 47J25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 571-600, https://doi.org/10.7494/OpMath.2021.41.4.571.

]]>Reaction-diffusion coupled inclusions with variable exponents and large diffusion
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4126.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4126.pdfFri, 09 Jul 2021 22:00:05 +0200 Author(s): Jacson Simsen, Mariza Stefanello Simsen, Petra Wittbold.

Abstract: This work concerns the study of asymptotic behavior of coupled systems of \(p(x)\)-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity. Keywords: reaction-diffusion coupled systems, variable exponents, attractors, upper semicontinuity, large diffusion. Mathematics Subject Classification: 35K55, 35K92, 35A16, 35B40, 35B41. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 539-570, https://doi.org/10.7494/OpMath.2021.41.4.539.

]]>Asymptotic expansions for the first hitting times of Bessel processes
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4125.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4125.pdfFri, 09 Jul 2021 22:00:04 +0200 Author(s): Yuji Hamana, Ryo Kaikura, Kosuke Shinozaki.

Abstract: We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient. Keywords: Bessel process, hitting time, tail probability, modified Bessel function, asymptotic expansion, Laplace transform. Mathematics Subject Classification: 60G40, 60J60, 41A60, 30C10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 509-537, https://doi.org/10.7494/OpMath.2021.41.4.509.

]]>Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4124.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4124.pdfFri, 09 Jul 2021 22:00:03 +0200 Author(s): Abdelrachid El Amrouss, Omar Hammouti.

Abstract: Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory. Keywords: discrete boundary value problems, 2n-th order, variational methods, critical point theory. Mathematics Subject Classification: 39A10, 34B08, 34B15, 58E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 489-507, https://doi.org/10.7494/OpMath.2021.41.4.489.

]]>Remarks on damped Schrödinger equation of Choquard type
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4123.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4123.pdfFri, 09 Jul 2021 22:00:02 +0200 Author(s): Lassaad Chergui.

Abstract: This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient. Keywords: damped Choquard equation, global existence, scattering, invariant sets. Mathematics Subject Classification: 35Q55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 465-488, https://doi.org/10.7494/OpMath.2021.41.4.465.

]]>Total connected domination game
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdf
https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdfFri, 09 Jul 2021 22:00:01 +0200 Author(s): Csilla Bujtás, Michael A. Henning, Vesna Iršič, Sandi Klavžar.

Abstract: The (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number (\(\gamma_{\rm tcg}(G)\)) \(\gamma_{\rm cg}(G)\) of \(G\). We show that \(\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}\), and consequently define \(G\) as Class \(i\) if \(\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i\) for \(i \in \{0,1,2\}\). A large family of Class \(0\) graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least \(2\). We show that no tree is Class \(2\) and characterize Class \(1\) trees. We provide an infinite family of Class \(2\) bipartite graphs. Keywords: connected domination game, total connected domination game, graph product, tree. Mathematics Subject Classification: 05C57, 05C69, 91A43. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 4 (2021), 453-464, https://doi.org/10.7494/OpMath.2021.41.4.453.

]]>Quadratic inequalities for functionals in l^{∞}
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4121.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4121.pdfMon, 19 Apr 2021 14:00:07 +0200 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-interaction
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4120.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4120.pdfMon, 19 Apr 2021 14:00:06 +0200 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 applications
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4119.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4119.pdfMon, 19 Apr 2021 14:00:05 +0200 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 part
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4118.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4118.pdfMon, 19 Apr 2021 14:00:04 +0200 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.

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 model
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4116.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4116.pdfMon, 19 Apr 2021 14:00:02 +0200 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 geometry
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4115.pdf
https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4115.pdfMon, 19 Apr 2021 14:00:01 +0200 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 problems
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4114.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4114.pdfWed, 17 Mar 2021 18:00:07 +0100 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 number
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4113.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4113.pdfWed, 17 Mar 2021 18:00:06 +0100 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 graph
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4112.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4112.pdfWed, 17 Mar 2021 18:00:05 +0100 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 plane
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdfWed, 17 Mar 2021 18:00:04 +0100 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 bracket
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdfWed, 17 Mar 2021 18:00:03 +0100 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 curve
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdfWed, 17 Mar 2021 18:00:02 +0100 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 18
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4108.pdf
https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4108.pdfWed, 17 Mar 2021 18:00:01 +0100 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 equations
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdfMon, 08 Feb 2021 18:00:07 +0100 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 integrands
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdfMon, 08 Feb 2021 18:00:06 +0100 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}
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdfMon, 08 Feb 2021 18:00:05 +0100 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, I
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdfMon, 08 Feb 2021 18:00:04 +0100 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 maps
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdfMon, 08 Feb 2021 18:00:03 +0100 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 exponent
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdfMon, 08 Feb 2021 18:00:02 +0100 Author(s): Abderrahim Charkaoui, Houda Fahim, Nour Eddine Alaa.

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 problem
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdfMon, 08 Feb 2021 18:00:01 +0100 Author(s): Rachid Bentifour, Sofiane El-Hadi Miri.

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.