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33 pages, 603 KB

Open AccessReview

Remarks on Multivalued Variants of Ekeland Principle with Applications

by Irina Meghea

Symmetry 2025, 17(11), 1848; https://doi.org/10.3390/sym17111848 - 3 Nov 2025

Viewed by 542

This review provides an extended discussion on multivalued variants of Ekeland’s variational principle, with the aim of highlighting the importance of this special type of result. After a presentation of the background containing general problems which appear in relation to this extended and [...] Read more.

This review provides an extended discussion on multivalued variants of Ekeland’s variational principle, with the aim of highlighting the importance of this special type of result. After a presentation of the background containing general problems which appear in relation to this extended and current subject, three series of results are developed to describe three multiple-valued versions of this important principle. In this part of the work, some different or improved proofs have been given, along with several special remarks from the author, together with their demonstrations. The discussion section contains an overview of many applications of such multivalued variants in solving problems issuing from real phenomena modeling. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis: 3rd Edition)

16 pages, 330 KB

Open AccessArticle

New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

by Yuan Gao, Lishan Liu, Na Wei, Haibo Gu and Yonghong Wu

Fractal Fract. 2025, 9(10), 646; https://doi.org/10.3390/fractalfract9100646 - 4 Oct 2025

Viewed by 551

Abstract

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: [...] Read more.

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations:

M ({[u]}_{s, p}^{p} + \int_{R^{N}} V (x) {| u |}^{p} d x) [{(- Δ)}_{p}^{s} u +

{V (x) | u |}^{p - 2} u] = {λ | u |}^{q - 2} u \ln | u |, x \in R^{N}, (P)

. In these equations,

λ \in R ∖ {0},

p \in (1, + \infty)

,

s \in (0, 1), s p < N,

p_{s}^{*} = \frac{N p}{N - s p}

,

M (τ) = a + b τ^{θ - 1}

,

a, b \in R^{+},

1 < θ < \frac{p_{s}^{*}}{p}

,

V (x) \in C (R^{N}, R)

is a potential function and

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of

λ

. More precisely, (i) Equation (P) has one ground state solution for

λ > 0

and

p θ < q < p_{s}^{*}

, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for

λ < 0

and

p < q < p_{s}^{*}

, with positive and negative corresponding energy values, respectively. Full article

(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)

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 538

Abstract

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}

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 proof process. Based on the Oettli–Th

\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)

19 pages, 292 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fixed-Point Results and the Ekeland Variational Principle in Vector B-Metric Spaces

by Radu Precup and Andrei Stan

Axioms 2025, 14(4), 250; https://doi.org/10.3390/axioms14040250 - 26 Mar 2025

Cited by 1 | Viewed by 982

Abstract

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in [...] Read more.

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland’s variational principle. As a consequence, we also derive a variant of Caristi’s fixed-point theorem. Full article

(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)

15 pages, 294 KB

Open AccessFeature PaperReview

Approximate Solutions of Variational Inequalities and the Ekeland Principle

by Raffaele Chiappinelli and David E. Edmunds

Mathematics 2025, 13(6), 1016; https://doi.org/10.3390/math13061016 - 20 Mar 2025

Viewed by 665

Abstract

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual

X^{*}

. We study the so-called variational inequality problem: given

y \in X^{*,},

does [...] Read more.

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual

X^{*}

. We study the so-called variational inequality problem: given

y \in X^{*,},

does there exist

x_{0} \in K

such that (in standard notation)

⟨F (x_{0}) - y, x - x_{0}⟩ \geq 0

for all

x \in K ?

After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 3rd Edition)

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 2 | Viewed by 1180

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)

13 pages, 277 KB

Open AccessArticle

The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces

by Ştefan Cobzaş

Mathematics 2024, 12(3), 471; https://doi.org/10.3390/math12030471 - 1 Feb 2024

Cited by 1 | Viewed by 1439

Abstract

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. [...] Read more.

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results). Full article

(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures, 2nd Edition)

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 1966

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 [...] Read more.

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)

15 pages, 302 KB

Open AccessArticle

Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property

by Basit Ali, Ştefan Cobzaş and Mokhwetha Daniel Mabula

Axioms 2023, 12(3), 247; https://doi.org/10.3390/axioms12030247 - 28 Feb 2023

Cited by 2 | Viewed by 2766

Abstract

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in [...] Read more.

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph G. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of weighted graphs for which one of these principles holds, are also considered. Full article

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

16 pages, 323 KB

Open AccessArticle

The Existence and Multiplicity of Homoclinic Solutions for a Fractional Discrete p−Laplacian Equation

by Yong Wu, Bouali Tahar, Guefaifia Rafik, Abita Rahmoune and Libo Yang

Mathematics 2022, 10(9), 1400; https://doi.org/10.3390/math10091400 - 22 Apr 2022

Cited by 4 | Viewed by 1842

Abstract

In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on

Z

. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain [...] Read more.

In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on

Z

. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter

λ > 0

is large enough. Full article

(This article belongs to the Special Issue Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications)

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 2527

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)

10 pages, 282 KB

Open AccessFeature PaperArticle

New Generalized Ekeland’s Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du’s Abstract Maximal Element Principle

by Junjian Zhao and Wei-Shih Du

Axioms 2021, 10(1), 11; https://doi.org/10.3390/axioms10010011 - 20 Jan 2021

Cited by 3 | Viewed by 2293

Abstract

In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland’s variational principle and common (fuzzy) fixed point theorem for essential distances. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)

17 pages, 341 KB

Open AccessArticle

Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝ^N

by Yun-Ho Kim

Mathematics 2020, 8(10), 1792; https://doi.org/10.3390/math8101792 - 15 Oct 2020

Cited by 6 | Viewed by 2458

Abstract

We are concerned with the following elliptic equations: [...] Read more.

We are concerned with the following elliptic equations:

{(- Δ)}_{p}^{s} v + {V (x) | v |}^{p - 2} v = λ a (x) {| v |}^{r - 2} v + g (x, v) i n R^{N},

where

{(- Δ)}_{p}^{s}

is the fractional p-Laplacian operator with

0 < s < 1 < r < p < + \infty

,

s p < N

, the potential function

V : R^{N} \to (0, \infty)

is a continuous potential function, and

g : R^{N} \times R \to R

satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem. Full article

(This article belongs to the Special Issue Mathematical Analysis and Boundary Value Problems)

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 2568

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)

6 pages, 262 KB

Open AccessArticle

Ekeland Variational Principle in the Variable Exponent Sequence Spaces ℓ_p(·)

by Monther R. Alfuraidan and Mohamed A. Khamsi

Mathematics 2020, 8(3), 375; https://doi.org/10.3390/math8030375 - 7 Mar 2020

Viewed by 2743

Abstract

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces

ℓ_{p (\cdot)}

. The core obstacle in the development of a modular version of the EVP is the [...] Read more.

In this work, we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces

ℓ_{p (\cdot)}

. The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the establishment of the classical EVP, that has hitherto prevented a successful treatment of the modular case. As an application, we establish a modular version of Caristi’s fixed point theorem in

ℓ_{p (\cdot)}

. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)

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