MDPI - Publisher of Open Access Journals

14 pages, 292 KB

Open AccessArticle

Oettli-Théra Theorem and Ekeland Variational Principle in Fuzzy b-Metric Spaces

by Xuan Liu, Fei He and Ning Lu

Axioms 2025, 14(9), 679; https://doi.org/10.3390/axioms14090679 - 3 Sep 2025

Viewed by 311

The purpose of this paper is to establish the Oettli–Th

\overset{´}{e}

ra theorem in the setting of KM-type fuzzy b-metric spaces. To achieve this, we first prove a lemma that removes the constraints on the space coefficients, which significantly simplifies the [...] Read more.

The purpose of this paper is to establish the Oettli–Th

\overset{´}{e}

\overset{´}{e}

ra theorem, we further demonstrate the equivalence of Ekeland variational principle, Caristi’s fixed point theorem, and Takahashi’s nonconvex minimization theorem in fuzzy b-metric spaces. Notably, the results obtained in this paper are consistent with the conditions of the corresponding theorems in classical fuzzy metric spaces, thereby extending the existing theories to the broader framework of fuzzy b-metric spaces. Full article

(This article belongs to the Section Mathematical Analysis)

31 pages, 459 KB

Open AccessArticle

Multiple Solutions to the Fractional p-Laplacian Equations of Schrödinger–Hardy-Type Involving Concave–Convex Nonlinearities

by Yun-Ho Kim

Fractal Fract. 2024, 8(7), 426; https://doi.org/10.3390/fractalfract8070426 - 20 Jul 2024

Cited by 1 | Viewed by 987

Abstract

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the

L^{\infty}

-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for [...] Read more.

This paper is concerned with nonlocal fractional p-Laplacian Schrödinger–Hardy-type equations involving concave–convex nonlinearities. The first aim is to demonstrate the

L^{\infty}

-bound for any possible weak solution to our problem. As far as we know, the global a priori bound for weak solutions to nonlinear elliptic problems involving a singular nonlinear term such as Hardy potentials has not been studied extensively. To overcome this, we utilize a truncated energy technique and the De Giorgi iteration method. As its application, we demonstrate that the problem above has at least two distinct nontrivial solutions by exploiting a variant of Ekeland’s variational principle and the classical mountain pass theorem as the key tools. Furthermore, we prove the existence of a sequence of infinitely many weak solutions that converges to zero in the

L^{\infty}

-norm. To derive this result, we employ the modified functional method and the dual fountain theorem. Full article

(This article belongs to the Special Issue Fractional Elliptic and Parabolic Equations: Analysis and Related Topics)

66 pages, 6454 KB

Open AccessReview

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

by Irina Meghea

Axioms 2023, 12(6), 532; https://doi.org/10.3390/axioms12060532 - 29 May 2023

Cited by 3 | Viewed by 1748

Abstract

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics. Full article

(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)

8 pages, 235 KB

Open AccessArticle

A New Equilibrium Version of Ekeland’s Variational Principle and Its Applications

by Yuqiang Feng, Juntao Xie and Bo Wu

Axioms 2022, 11(2), 68; https://doi.org/10.3390/axioms11020068 - 9 Feb 2022

Viewed by 2325

Abstract

In this note, a new equilibrium version of Ekeland’s variational principle is presented. It is a modification and promotion of previous results. Subsequently, the principle is applied to discuss the equilibrium points for binary functions and the fixed points for nonlinear mappings. Full article

(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)

13 pages, 300 KB

Open AccessArticle

Remarks on Surjectivity of Gradient Operators

by Raffaele Chiappinelli and David E. Edmunds

Mathematics 2020, 8(9), 1538; https://doi.org/10.3390/math8091538 - 8 Sep 2020

Cited by 8 | Viewed by 2419

Abstract

Let X be a real Banach space with dual

X^{*}

and suppose that

F : X \to X^{*}

. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent [...] Read more.

Let X be a real Banach space with dual

X^{*}

and suppose that

F : X \to X^{*}

. We give a characterisation of the property that F is locally proper and establish its stability under compact perturbation. Modifying an recent result of ours, we prove that any gradient map that has this property and is additionally bounded, coercive and continuous is surjective. As before, the main tool for the proof is the Ekeland Variational Principle. Comparison with known surjectivity results is made; finally, as an application, we discuss a Dirichlet boundary-value problem for the p-Laplacian

(1 < p < \infty)

, completing our previous result which was limited to the case

p \geq 2

. Full article

(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (5)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI