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Mathematics
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New Trends in Variational Methods in Nonlinear Analysis
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New Trends in Variational Methods in Nonlinear Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 9778

Special Issue Editor

Prof. Dr. Marek Galewski

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Guest Editor
Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland
Interests: variational methods; nonlinear analysis

Special Issue Information

Dear Colleagues,

Variational methods with their numerous applications in differential and difference equations are among the central topics in nonlinear analysis. Growing out of the Dirichlet principle, variational methods have followed many paths leading to solutions to nonlinear equations and their multiplicity.

This Special Issue, entitled “New Trends in Variational Methods in Nonlinear Analysis”, is aimed at the applications of various variational techniques showing their interplays with other fields of nonlinear analysis. Application-oriented papers as well as survey papers describing recent developments of variational methods and their applications to both ordinary and partial differential equations are welcomed.

As topics within the scope of this issue, let us mention (without being exhaustive) the following: existence and multiplicity of solutions to nonlinear equations, variational principles, localization results, links to topological and monotonicity methods, and diverse applications to boundary value problems.

Each paper should clearly indicate its focus and motivation for the setting and what kind of impact one may expect from the results presented in this paper.

Prof. Dr. Marek Galewski
Guest Editor

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Published Papers (5 papers)

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Research

8 pages, 229 KiB  
Article
Contractive Mappings on Metric Spaces with Graphs
by Simeon Reich and Alexander J. Zaslavski
Mathematics 2021, 9(21), 2774; https://doi.org/10.3390/math9212774 - 1 Nov 2021
Cited by 6 | Viewed by 1374
Abstract
We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs. Full article
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
9 pages, 264 KiB  
Article
Existence of Nontrivial Solutions for Sixth-Order Differential Equations
by Gabriele Bonanno, Pasquale Candito and Donal O’Regan
Mathematics 2021, 9(16), 1852; https://doi.org/10.3390/math9161852 - 5 Aug 2021
Cited by 5 | Viewed by 1488
Abstract
We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory. Full article
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
19 pages, 818 KiB  
Article
Multiple Solutions for Double Phase Problems with Hardy Type Potential
by Chun-Bo Lian, Bei-Lei Zhang and Bin Ge
Mathematics 2021, 9(4), 376; https://doi.org/10.3390/math9040376 - 13 Feb 2021
Cited by 2 | Viewed by 1567
Abstract
In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions [...] Read more.
In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness. Full article
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
13 pages, 280 KiB  
Article
Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz Conditions
by Bei-Lei Zhang, Bin Ge and Xiao-Feng Cao
Mathematics 2020, 8(11), 2068; https://doi.org/10.3390/math8112068 - 19 Nov 2020
Cited by 8 | Viewed by 2086
Abstract
In this paper, we consider a nonlocal p(x)-Kirchhoff problem with a p+-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. [...] Read more.
In this paper, we consider a nonlocal p(x)-Kirchhoff problem with a p+-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. Our results are an improvement and generalization of the corresponding results obtained by Hamdani et al. (2020). Full article
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
21 pages, 430 KiB  
Article
Resonant Anisotropic (p,q)-Equations
by Leszek Gasiński and Nikolaos S. Papageorgiou
Mathematics 2020, 8(8), 1332; https://doi.org/10.3390/math8081332 - 10 Aug 2020
Cited by 5 | Viewed by 2044
Abstract
We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction [...] Read more.
We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones. Full article
(This article belongs to the Special Issue New Trends in Variational Methods in Nonlinear Analysis)
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