Publisher: AGH University of Science and Technology Press (Total: 6 journals) [Sort by number of followers]

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Opuscula Mathematica
Journal Prestige (SJR): 0.378 Citation Impact (citeScore): 1 Number of Followers: 0 Open Access journal ISSN (Print) 12329274  ISSN (Online) 23006919 Published by AGH University of Science and Technology Press [6 journals] 
 Existence of positive continuous weak solutions for some semilinear
elliptic eigenvalue problems
Abstract:
Author(s): Noureddine Zeddini, Rehab Saeed Sari.
Abstract: Let \(D\) be a bounded \(C^{1,1}\)domain in \(\mathbb{R}^d\), \(d\geq 2\). The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions \(K(D)\) that was defined by N. Zeddini for \(d=2\) and by H. Mâagli and M. Zribi for \(d\geq 3\) and adapted to study some nonlinear elliptic problems in \(D\). The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants \(\lambda\) and \(\mu\) to the following system \(\Delta u=\lambda f(x,u,v)\), \(\Delta v=\mu g(x,u,v)\) in \(D\), \(u=\phi_1\) and \(v=\phi_2\) on \(\partial D\), where \(\phi_1\) and \(\phi_2\) are nontrivial nonnegative continuous functions on \(\partial D\). The functions \(f\) and \(g\) are nonnegative and belong to a class of functions containing in particular all functions of the type \(f(x,u,v)=p(x) u^{\alpha}h_1(v)\) and \(g(x,u,v)=q(x)h_2(u)v^{\beta}\) with \(\alpha\geq 1\), \(\beta \geq 1\), \(h_1\), \(h_2\) are continuous on \([0,\infty)\) and \(p\), \(q\) are nonnegative functions in \(K(D)\).
Keywords: Green function, Kato class, nonlinear elliptic systems, positive solution, maximum principle, Schauder fixed point theorem.
Mathematics Subject Classification: 31A35, 31B35, 31A16, 35B09, 35B50, 35J08, 35J57.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 489519, https://doi.org/10.7494/OpMath.2022.42.3.489.
PubDate: Fri, 29 Apr 2022 21:30:07 +020
 Spectral resolutions for nonselfadjoint block convolution operators
Abstract:
Author(s): Ewelina Zalot.
Abstract: The paper concerns the spectral theory for a class of nonselfadjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.
Keywords: spectral operators, chains, triangular decomposition, Laurent operators, Jacobi matrices.
Mathematics Subject Classification: 47B40, 47B28, 47B36, 47B35, 47B39.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 459487, https://doi.org/10.7494/OpMath.2022.42.3.459.
PubDate: Fri, 29 Apr 2022 21:30:06 +020
 Distance irregularity strength of graphs with pendant vertices
Abstract:
Author(s): Faisal Susanto, Kristiana Wijaya, Slamin, Andrea SemaničováFeňovčíková.
Abstract: A vertex \(k\)labeling \(\phi:V(G)\rightarrow\{1,2,\dots,k\}\) on a simple graph \(G\) is said to be a distance irregular vertex \(k\)labeling of \(G\) if the weights of all vertices of \(G\) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in \(G\). The least integer \(k\) for which \(G\) has a distance irregular vertex \(k\)labeling is called the distance irregularity strength of \(G\) and denoted by \(\mathrm{dis}(G)\). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.
Keywords: vertex \(k\)labeling, distance irregular vertex \(k\)labeling, distance irregularity strength, pendant vertices.
Mathematics Subject Classification: 05C78, 05C12.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 439458, https://doi.org/10.7494/OpMath.2022.42.3.439.
PubDate: Fri, 29 Apr 2022 21:30:05 +020
 On Ambarzumian type theorems for tree domains
Abstract:
Author(s): Vyacheslav Pivovarchik.
Abstract: It is known that the spectrum of the spectral SturmLiouville problem on an equilateral tree with (generalized) Neumann's conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian's theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral SturmLiouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian's theorem can't be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the NeumannDirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees' roots and the Dirichlet condition at the subtrees' roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.
Keywords: SturmLiouville equation, eigenvalue, equilateral tree, star graph, Dirichlet boundary condition, Neumann boundary condition.
Mathematics Subject Classification: 34B45, 34B24, 34L20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 427437, https://doi.org/10.7494/OpMath.2022.42.3.427.
PubDate: Fri, 29 Apr 2022 21:30:04 +020
 Growth of solutions of a class of linear fractional differential equations
with polynomial coefficients
Abstract:
Author(s): Saada Hamouda, Sofiane Mahmoudi.
Abstract: This paper is devoted to the study of the growth of solutions of certain class of linear fractional differential equations with polynomial coefficients involving the Caputo fractional derivatives by using the generalized WimanValiron theorem in the fractional calculus.
Keywords: linear fractional differential equations, growth of solutions, Caputo fractional derivative operator.
Mathematics Subject Classification: 34M10, 26A33.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 415426, https://doi.org/10.7494/OpMath.2022.42.3.415.
PubDate: Fri, 29 Apr 2022 21:30:03 +020
 New aspects for the oscillation of firstorder difference equations with
deviating arguments
Abstract:
Author(s): Emad R. Attia, Bassant M. ElMatary.
Abstract: We study the oscillation of firstorder linear difference equations with nonmonotone deviating arguments. Iterative oscillation criteria are obtained which essentially improve, extend, and simplify some known conditions. These results will be applied to some numerical examples.
Keywords: difference equations, oscillation, nonmonotone advanced arguments.
Mathematics Subject Classification: 39A10, 39A21.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 393413, https://doi.org/10.7494/OpMath.2022.42.3.393.
PubDate: Fri, 29 Apr 2022 21:30:02 +020
 Monodromy invariant Hermitian forms for second order Fuchsian differential
equations with four singularities
Abstract:
Author(s): Shunya Adachi.
Abstract: We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the nondegenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlevé differential equations.
Keywords: Fuchsian differential equations, monodromy representation, monodromy invariant Hermitian form.
Mathematics Subject Classification: 34M35, 34M15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 3 (2022), 361391, https://doi.org/10.7494/OpMath.2022.42.3.361.
PubDate: Fri, 29 Apr 2022 21:30:01 +020
 Ground states of coupled critical Choquard equations with weighted
potentials
Abstract:
Author(s): Gaili Zhu, Chunping Duan, Jianjun Zhang, Huixing Zhang.
Abstract: In this paper, we are concerned with the following coupled Choquard type system with weighted potentials \[\begin{cases} \Delta u+V_{1}(x)u=\mu_{1}(I_{\alpha}\!\ast\![Q(x) u ^{\frac{N+\alpha}{N}}])Q(x) u ^{\frac{\alpha}{N}1}u+\beta(I_{\alpha}\!\ast\![Q(x) v ^{\frac{N+\alpha}{N}}])Q(x) u ^{\frac{\alpha}{N}1}u,\\ \Delta v+V_{2}(x)v=\mu_{2}(I_{\alpha}\!\ast\![Q(x) v ^{\frac{N+\alpha}{N}}])Q(x) v ^{\frac{\alpha}{N}1}v+\beta(I_{\alpha}\!\ast\![Q(x) u ^{\frac{N+\alpha}{N}}])Q(x) v ^{\frac{\alpha}{N}1}v,\\ u,v\in H^{1}(\mathbb{R}^{N}),\end{cases}\] where \(N\geq3\), \(\mu_{1},\mu_{2},\beta\gt 0\) and \(V_{1}(x)\), \(V_{2}(x)\) are nonnegative functions. Via the variational approach, one positive ground state solution of this system is obtained under some certain assumptions on \(V_{1}(x)\), \(V_{2}(x)\) and \(Q(x)\). Moreover, by using Hardy's inequality and one Pohozǎev identity, a nonexistence result of nontrivial solutions is also considered.
Keywords: ground states, Choquard equations, HardyLittlewoodSobolev inequality, lower critical exponent.
Mathematics Subject Classification: 35B25, 35B33, 35J61.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 337354, https://doi.org/10.7494/OpMath.2022.42.2.337.
PubDate: Fri, 25 Feb 2022 14:30:09 +010
 On some inverse problem for biparabolic equation with observed data in
L^{p} spaces
Abstract:
Author(s): Nguyen Huy Tuan.
Abstract: The biparabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for biparabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for biparabolic equation on this direction.
Keywords: biparabolic equations, Fourier truncation method, inverse source parabolic, inverse initial problem, Sobolev embeddings, Sobolev embeddings.
Mathematics Subject Classification: 35A05, 35A08.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 305335, https://doi.org/10.7494/OpMath.2022.42.2.305.
PubDate: Fri, 25 Feb 2022 14:30:08 +010
 Entire solutions for some critical equations in the Heisenberg group
Abstract:
Author(s): Patrizia Pucci, Letizia Temperini.
: We complete the study started in the paper [P. Pucci, L.Temperini, On the concentrationcompactness principle for FollandStein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal \(p\)Laplacian equations.
Keywords: Heisenberg group, entire solutions, critical exponents.
Mathematics Subject Classification: 35J62, 35J70, 35B08, 35J20, 35B09.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 279303, https://doi.org/10.7494/OpMath.2022.42.2.279.
PubDate: Fri, 25 Feb 2022 14:30:07 +010
 Double phase problems: a survey of some recent results
Abstract:
Author(s): Nikolaos S. Papageorgiou.
Abstract: We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.
Keywords: double phase integrand, generalized Orlicz spaces, regularity theory, maximum principle, Nehari manifold.
Mathematics Subject Classification: 35J20, 35J60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 257278, https://doi.org/10.7494/OpMath.2022.42.2.257.
PubDate: Fri, 25 Feb 2022 14:30:06 +010
 Exponential decay of solutions to a class of fourthorder nonlinear
hyperbolic equations modeling the oscillations of suspension bridges
Abstract:
Author(s): Yang Liu, Chao Yang.
Abstract: This paper is concerned with a class of fourthorder nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges.
Keywords: fourthorder nonlinear hyperbolic equations, weak solutions, exponential decay, a family of potential wells.
Mathematics Subject Classification: 35L35, 35D30, 35B40.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 239255, https://doi.org/10.7494/OpMath.2022.42.2.239.
PubDate: Fri, 25 Feb 2022 14:30:05 +010
 Blowup phenomena for some fourthorder strain wave equations at arbitrary
positive initial energy level
Abstract:
Author(s): Qiang Lin, Yongbing Luo.
Abstract: In this paper, we study a series of fourthorder strain wave equations involving dissipative structure, which appears in elastoplasticmicrostructure models. By some differential inequalities, we derive the finite time blow up results and the estimates of the upper bound blowup time with arbitrary positive initial energy. We also discuss the influence mechanism of the linear weak damping and strong damping on blowup time, respectively.
Keywords: fourthorder strain wave equation, arbitrary positive initial energy, blowup, blowup time.
Mathematics Subject Classification: 35L05, 35A01, 35L55.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 219238, https://doi.org/10.7494/OpMath.2022.42.2.219.
PubDate: Fri, 25 Feb 2022 14:30:04 +010
 The dbar formalism for the modified VeselovNovikov equation on the
halfplane
Abstract:
Author(s): Guenbo Hwang, Byungsoo Moon.
Abstract: We study the modified VeselovNovikov equation (mVN) posed on the halfplane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural \((2+1)\)dimensional generalization of the \((1+1)\)dimensional modified Kortewegde Vries equation in the sense as to how the NovikovVeselov equation is related to the Kortewegde Vries equation. In this paper, by means of the Fokas method, we present the socalled global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the \(d\)bar formalism, also known as Pompieu's formula. In addition, we characterize the \(d\)bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions.
Keywords: initialboundary value problem, integrable nonlinear PDE, spectral analysis, \(d\)bar.
Mathematics Subject Classification: 35G31, 35Q53, 37K15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 179217, https://doi.org/10.7494/OpMath.2022.42.2.179.
PubDate: Fri, 25 Feb 2022 14:30:03 +010
 Ground states for fractional nonlocal equations with logarithmic
nonlinearity
Abstract:
Author(s): Lifeng Guo, Yan Sun, Guannan Shi.
Abstract: In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by \[\begin{cases}\mathcal{L}_{K}u(x)+u\log u + u ^{q2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}\] where \(2\lt q\lt 2^{*}_s\), \(L_{K}\) is a nonlocal operator, \(\Omega\) is an open bounded set of \(\mathbb{R}^{n}\) with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.
Keywords: linking theorem, ground state, logarithmic nonlinearity, variational methods.
Mathematics Subject Classification: 35J20, 35B33, 58E05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 157178, https://doi.org/10.7494/OpMath.2022.42.2.157.
PubDate: Fri, 25 Feb 2022 14:30:02 +010
 Global existence and blowup phenomenon for a quasilinear viscoelastic
equation with strong damping and source terms
Abstract:
Author(s): Huafei Di, Zefang Song.
Abstract: Considered herein is the global existence and nonglobal existence of the initialboundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with \(t\). Thirdly, we obtain the finite time blowup result for certain solutions with the nonpositive initial energy and positive initial energy, and then give the upper bound for the blowup time \(T^\ast\). Especially, the threshold result between global existence and nonglobal existence is given under some certain conditions. Finally, a lower bound for the life span \(T^\ast\) is derived by the means of integrodifferential inequality techniques.
Keywords: viscoelastic equation, strong damping and source, blowup, upper and lower bounds, invariant set, potential well.
Mathematics Subject Classification: 35L35, 35L75, 35R15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 2 (2022), 119155, https://doi.org/10.7494/OpMath.2022.42.2.119.
PubDate: Fri, 25 Feb 2022 14:30:01 +010
 All metric bases and faulttolerant metric dimension for square of grid
Abstract:
Author(s): Laxman Saha, Mithun Basak, Kalishankar Tiwary.
Abstract: For a simple connected graph \(G=(V,E)\) and an ordered subset \(W = \{w_1,w_2,\ldots, w_k\}\) of \(V\), the code of a vertex \(v\in V\), denoted by \(\mathrm{code}(v)\), with respect to \(W\) is a \(k\)tuple \((d(v,w_1),\ldots, d(v, w_k))\), where \(d(v, w_t)\) represents the distance between \(v\) and \(w_t\). The set \(W\) is called a resolving set of \(G\) if \(\mathrm{code}(u)\neq \mathrm{code}(v)\) for every pair of distinct vertices \(u\) and \(v\). A metric basis of \(G\) is a resolving set with the minimum cardinality. The metric dimension of \(G\) is the cardinality of a metric basis and is denoted by \(\beta(G)\). A set \(F\subset V\) is called faulttolerant resolving set of \(G\) if \(F\setminus{\{v\}}\) is a resolving set of \(G\) for every \(v\in F\). The faulttolerant metric dimension of \(G\) is the cardinality of a minimal faulttolerant resolving set. In this article, a complete characterization of metric bases for \(G_{mn}^2\) has been given. In addition, we prove that the faulttolerant metric dimension of \(G_{mn}^2\) is 4 if \(m+n\) is even. We also show that the faulttolerant metric dimension of \(G_{mn}^2\) is at least 5 and at most 6 when \(m+n\) is odd.
Keywords: code, resolving set, metric dimension, faulttolerant resolving set, faulttolerant metric dimension.
Mathematics Subject Classification: 05C12, 05C05, 05C90, 05C76.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 93111, https://doi.org/10.7494/OpMath.2022.42.1.93.
PubDate: Thu, 20 Jan 2022 18:00:06 +010
 Solution of the boundary value problem of heat conduction in a cone
Abstract:
Author(s): Murat Ramazanov, Muvasharkhan Jenaliyev, Nurtay Gulmanov.
Abstract: In the paper we consider the boundary value problem of heat conduction in a noncylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We use the CarlemanVekua method of equivalent regularization to solve the obtained singular Volterra integral equation.
Keywords: noncylindrical domain, cone, boundary value problem of heat conduction, singular Volterra integral equation, CarlemanVekua regularization method.
Mathematics Subject Classification: 35K05, 45D99.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 7591, https://doi.org/10.7494/OpMath.2022.42.1.75.
PubDate: Thu, 20 Jan 2022 18:00:05 +010
 Edge homogeneous colorings
Abstract:
Author(s): Tomáš Madaras, Alfréd Onderko, Thomas Schweser.
Abstract: We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) \(q\) colors (resp. one end sees \(q\) colors and the color sets for both ends are the same), and a sufficient condition for 2coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether \(q\) colors. The relations of these colorings to \(M_q\)colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have \(q\) colors.
Keywords: homogeneous coloring, \(M_q\)coloring, line graph, role coloring.
Mathematics Subject Classification: 05C15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 6573, https://doi.org/10.7494/OpMath.2022.42.1.65.
PubDate: Thu, 20 Jan 2022 18:00:04 +010
 Knesertype oscillation criteria for secondorder halflinear advanced
difference equations
Abstract:
Author(s): N. Indrajith, John R. Graef, E. Thandapani.
Abstract: The authors present Knesertype oscillation criteria for a class of advanced type secondorder difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results.
Keywords: secondorder difference equations, advanced argument, halflinear, oscillation.
Mathematics Subject Classification: 39A10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 5564, https://doi.org/10.7494/OpMath.2022.42.1.55.
PubDate: Thu, 20 Jan 2022 18:00:03 +010
 γpaired dominating graphs of cycles
Abstract:
Author(s): Pannawat Eakawinrujee, Nantapath Trakultraipruk.
Abstract: A paired dominating set of a graph \(G\) is a dominating set whose induced subgraph contains a perfect matching. The paired domination number, denoted by \(\gamma_{pr}(G)\), is the minimum cardinality of a paired dominating set of \(G\). A \(\gamma_{pr}(G)\)set is a paired dominating set of cardinality \(\gamma_{pr}(G)\). The \(\gamma\)paired dominating graph of \(G\), denoted by \(PD_{\gamma}(G)\), as the graph whose vertices are \(\gamma_{pr}(G)\)sets. Two \(\gamma_{pr}(G)\)sets \(D_1\) and \(D_2\) are adjacent in \(PD_{\gamma}(G)\) if there exists a vertex \(u\in D_1\) and a vertex \(v\notin D_1\) such that \(D_2=(D_1\setminus \{u\})\cup \{v\}\). In this paper, we present the \(\gamma\)paired dominating graphs of cycles.
Keywords: paired dominating graph, paired dominating set, paired domination number.
Mathematics Subject Classification: 05C69, 05C38.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 3154, https://doi.org/10.7494/OpMath.2022.42.1.31.
PubDate: Thu, 20 Jan 2022 18:00:02 +010
 Uniqueness of solution of a nonlinear evolution dam problem in a
heterogeneous porous medium
Abstract:
Author(s): Messaouda Ben Attia, Elmehdi Zaouche, Mahmoud Bousselsal.
Abstract: By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of \(\mathbb{R}^n\) (\(n\in \{2,3\}\)) with an impermeable horizontal bottom.
Keywords: test function, method of doubling variables, nonlinear evolution dam problem, heterogeneous porous medium, uniqueness.
Mathematics Subject Classification: 35A02, 76S05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 42, no. 1 (2022), 529, https://doi.org/10.7494/OpMath.2022.42.1.5.
PubDate: Thu, 20 Jan 2022 18:00:01 +010