Ekeland's Variational Principle Research Articles

A concave–convex Kirchhoff type elliptic equation involving the fractional p-Laplacian and steep well potential

In this paper, we study the following Schrödinger–Kirchhoff type equation: , where , , N>ps, and , are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution by a truncated functional and discuss their asymptotical behavior for when p<r<2p and b lies in suitable range. Secondly, for all b>0, we obtain a positive energy solution and discuss their asymptotical behavior for when , where . Moreover, we obtain other asymptotical behavior of positive energy solution when . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.

Vector Optimization with Domination Structures: Variational Principles and Applications

This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by applications to some models in behavioral sciences, we establish new variational principles that can be viewed as far-going extensions of the Ekeland variational principle to cover domination vector settings. Our approach combines advantages of both primal and dual techniques in variational analysis with providing useful sufficient conditions for the existence of variational traps in behavioral science models with variable domination structures.

Three solutions for a new Kirchhoff-type problem

This article concerns on the existence of multiple solutions for a new Kirchhoff-type problem with negative modulus. We prove that there exist three nontrivial solutions when the parameter is enough small via the variational methods and algebraic analysis. Moreover, our fundamental technique is one of the Mountain Pass Lemma, Ekeland variational principle, and Minimax principle.

Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment.

This study examines an optimal harvesting problem for a periodic n-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the ith population, the concentration of toxicants in the ith population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting n species.

Stationary Kirchhoff equations and systems with reaction terms

&lt;abstract&gt;&lt;p&gt;In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland's variational principle and whose development simulates a noncooperative game.&lt;/p&gt;&lt;/abstract&gt;

Existence results of nontrivial solutions for a new $ p(x) $-biharmonic problem with weight function

&lt;abstract&gt;&lt;p&gt;In this paper, we study a class of $ p(x) $-biharmonic problems with negative nonlocal terms and weight function. Using the mountain pass theorem and the Ekeland's variational principle, at least three solutions are obtained. We also give some comments on the existence of infinite many solutions for our problem when the nonlinear term is a general function.&lt;/p&gt;&lt;/abstract&gt;

Ekeland's variational principle in fuzzy quasi-normed spaces

&lt;abstract&gt; &lt;p&gt;Fuzzy quasi-normed space provides an ideal mathematical framework for studying asymmetric phenomena. In this paper, we prove a version of the Ekeland variational principle in fuzzy quasi-normed spaces and apply it to Caristi's fixed point theorem and Takahashi minimization theorem. Moreover, we prove the equivalence relations among these theorems.&lt;/p&gt; &lt;/abstract&gt;

Optimal harvesting for a periodic competing system with size structure in a polluted environment

&lt;abstract&gt;&lt;p&gt;As a renewable resource, biological population not only has direct economic value to people's lives, but also has important ecological and environmental value. This study examines an optimal harvesting problem for a periodic, competing hybrid system of three species that is dependent on size structure in a polluted environment. The existence and uniqueness of the nonnegative solution are proved via an operator theory and fixed point theorem. The necessary optimality conditions are derived by constructing an adjoint system and using the tangent-normal cone technique. The existence of unique optimal control pair is verified by means of the Ekeland variational principle and a feedback form of the optimal policy is presented. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. The objective functional represents the total profit obtained from harvesting three species. The results obtained in this work can be extended to a wide variety of fields.&lt;/p&gt;&lt;/abstract&gt;

Ekeland variational principle in complete weakly symmetric (1, q 2)-quasimetric spaces and applications

In this paper, Ekeland variational principle for bifunctions is given in complete weakly symmetric -quasimetric spaces and it is shown to be equivalent to Oettli–Théra theorem, Caristi–Kirk fixed point theorem and Takahashi's nonconvex minimization principle. Moreover, Ekeland variational principle for the system of bifunctions is obtained. Finally, by applying obtained results, the existence of solutions of quasi-equilibrium problems (QEP) and uncountable system of QEP is provided under the compactness condition and the generalized Caristi-like condition, respectively.

Robust Ekeland variational principles. Application to the formation and stability of partnerships

This paper has two parts. The mathematical part provides generalized versions of the robust Ekeland variational principle in terms of set-valued EVP with variable preferences, uncertain parameters and changing weights given to vectorial perturbation functions. The behavioural part that motivates our findings models the formation and stability of a partnership in a changing, uncertain and complex environment in the context of the variational rationality approach of stop, continue and go human dynamics. Our generalizations allow us to consider two very important psychological effects relative to ego depletion and goal gradient hypothesis.

Multiple solutions for [formula omitted]-Kirchhoff type problem in [formula omitted

In the paper, we prove that the N-Kirchhoff type problem in RN has at least three solutions by using the Ekeland variational principle and fibering map.

On the nonlocal Schrödinger‐poisson type system in the Heisenberg group

This paper is concerned with a class of nonlocal Schrödinger‐Poisson type system with sublinear term. With the aid of the Ekeland's variational principle and the mountain pass theorem, the existence of negative energy solution, positive energy solution, and positive ground state solution is obtained, respectively. Moreover, we also obtain the multiplicity of solutions by using the Clark theorem. The novelty of this paper is that the equation has not only nonlocal term but also nonlocal coefficient.

On Ekeland's variational principle for interval-valued functions with applications

In this paper, we obtain a version of Ekeland's variational principle for interval-valued functions by means of the Dancs-Hegedüs-Medvegyev theorem (Dancs et al. (1983) [14]). We also derive two versions of Ekeland's variational principle involving the generalized Hukuhara Gâteaux differentiability of interval-valued functions as well as a version of Ekeland's variational principle for interval-valued bifunctions. Finally, we apply these new versions of Ekeland's variational principle to fixed point theorems, to interval-valued optimization problems, to the interval-valued Mountain Pass Theorem, to noncooperative interval-valued games, and to interval-valued optimal control problems described by interval-valued differential equations.

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

Abstract In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &amp;\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &amp;\mbox{in}\, \Omega,\\ u=\phi=0,\quad &amp;\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b &gt; 0, 1 &lt; q &lt; 2 or 2 &lt; q &lt; 4, λ &gt; 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Variational rationality, variational principles and the existence of traps in a changing environment

This paper has two aspects. Mathematically, in the context of global optimization, it provides the existence of an optimum of a perturbed optimization problem that generalizes the celebrated Ekeland variational principle and equivalent formulations (Caristi, Takahashi), whenever the perturbations need not satisfy the triangle inequality. Behaviorally, it is a continuation of the recent variational rationality approach of stay (stop) and change (go) human dynamics. It gives sufficient conditions for the existence of traps in a changing environment. In this way it emphasizes even more the striking correspondence between variational analysis in mathematics and variational rationality in psychology and behavioral sciences.

Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Let \(\Omega\subset\mathbb{R}^{N}\) be a bounded domain with smooth boundary and \(0\in\Omega\). For \(0&lt;s&lt;1\), \(1\le r&lt;q&lt;p\), \(0\le\alpha&lt;ps&lt;N\) and a positive parameter \(\lambda\), we consider the fractional \((p,q)\)-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where \(a,b&gt;0\), \(c&lt;sr+N(1-r/p)\), \(\theta\in(1,p_{s}^{*}(\alpha)/p)\) and \(p_{s}^{*}(\alpha)\) is critical Sobolev-Hardy exponent. For a given suitable \(f(x)\), we prove that there are least two nontrivial solutions for small \(\lambda\), by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as \(a\to 0^{+}\). For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when \(\lambda\) belongs to a suitable range. For more information see https://ejde.math.txstate.edu/Volumes/2021/66/abstr.html

New contributions for tripled fixed point methodologies via a generalized variational principle with applications

This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications.

Existence of solution for a class of quasilinear Schrödinger equation in [formula omitted] with zero-mass

In this paper we investigate the existence of solutions for the following quasilinear Schrödinger problem with zero-mass:{−Δu−kΔ(u2)u=h(x)uqinRN,u≥0,u∈D1,2(RN)∩L∞(RN), where k>0, 0≤q<2.2⁎−1 and 2⁎=2N/(N−2);N≥3 is the critical Sobolev exponent and h∈Lloc∞(RN) is a function that can change sign. To establish existence results we used the variational method. More precisely, we used a change of variables combined with the Ekeland Variational Principle, Mountain Pass Theorem, careful estimates on energy functionals and an argument of passing to the limit.

Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem in fuzzy quasi-metric spaces

In this paper, a partial order and its basic properties are exploited in a fuzzy quasi-metric space in the sense of George and Veeramani (1994) [17]. Based on this partial order, Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem are extended to fuzzy quasi-metric spaces by using the Brézis and Browder principle on ordered sets. Moreover, an equivalence chain among these theorems is provided.

Legendre’s Necessary Condition for Fractional Bolza Functionals with Mixed Initial/Final Constraints

The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore, we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely, we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order $$0 < \alpha \le 1$$ and on a Riemann–Liouville fractional integral of order $$\beta > 0$$ , the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore, we underline that some subsidiary contributions are provided all along the paper. In particular, we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order $$0< \alpha <1$$ , compactly supported. Moreover, we also discuss some evidences claiming that Riemann–Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions.