Ekeland's Variational Principle Research Articles

Optimal control of a multi-scale immuno-influenza A transmission model with viral load-dependent infection

Influenza A is a highly contagious respiratory illness that spreads globally and results in millions of cases worldwide. The transmission rate and mortality rate of influenza in the population are determined by the load of the influenza A virus. In this study, we have developed a multi-scale immuno-influenza model considering incomplete vaccine immunity. We have calculated the basic reproduction numbers for both immunological and epidemiological models. Additionally, we have examined the stability of the disease-free equilibrium and the existence of the endemic equilibrium at the population level. To assess the cost-effectiveness of antiviral therapy at an individual level and vaccination at a population level, we have formulated an optimal control problem. The objective of this problem is to minimize both the number of influenza cases and the cost of vaccination. To solve this optimal control problem, we have employed the Ekeland variational principle. Furthermore, we have designed a global forward and backward sweep algorithm to estimate the impact of antiviral treatment on reducing influenza transmission among humans. Our findings indicate that the use of antiviral treatment significantly reduces viral loads in individuals. However, it may also increase the potential risk at the population level. Therefore, vaccination against influenza is the most effective option from a public health perspective.

Optimal boundary control for a competitive population system with size structure and time delay in a polluted environment

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

The Strong Ekeland Variational Principle in Quasi-Pseudometric Spaces

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).

On almost regular mappings

Ioffe's criterion and various reformulations of it have become a standard tool in proving theorems guaranteeing various regularity properties such as metric regularity, i.e. the openness with a linear rate around the reference point, of a (set-valued) mapping. We derive an analogue of it guaranteeing the almost openness with a linear rate of mappings acting in incomplete spaces and having non-closed graphs, in general. The main tool used is an approximate version of Ekeland's variational principle for a function that is not necessarily lower semi-continuous and is defined on an abstract (possibly incomplete) space. Further, we focus on the stability of this property under additive single-valued and set-valued perturbations.

Ekeland's variational principle for the set-valued map with variable ordering structures and applications

Ekeland's variational principle for the set-valued map with variable ordering structures and applications

On solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory with Berestycki–Lions conditions on $ \mathbb{R}^3 $

<abstract><p>In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions.</p></abstract>

The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

<abstract><p>In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.</p></abstract>

Localization of Nash-type equilibria for systems with partial variational structure

In this paper, we aim to generalize an existing result by obtaining localized solutions within bounded convex sets, while also relaxing specific initial assumptions. To achieve this, we employ an iterative scheme that combines a fixed-point argument based on the Minty-Browder Theorem with a modified version of the Ekeland variational principle for bounded sets. An application to a system of second-order differential equations with Dirichlet boundary conditions is presented.

A type of minimal and maximal point theorem in locally convex product cones

Using the neighbourhoods in upper relative topologies, we introduce a quasi-metric on cones which leads to a version of Cantors's intersection theorem for upper and lower relative v-topologies. Then we prove a type of minimal and maximal point theorem for subsets of product cone topologies and obtain the corresponding Ekeland's variational principle (EVP) results for locally convex cone-valued functions.

Necessary and Sufficient Conditions for Pareto Optimal Solution of Backward Stochastic System With Application

This paper is interested in studying a new kind of Pareto cooperative differential game of backward stochastic differential equation. Based on characterizations of Pareto optimal solution, the game problem is transformed into a set of single objective optimal control problems with constraints of backward stochastic differential equations. In the first place, a necessary condition for Pareto optimal strategy is established by virtue of Ekeland's variational principle, and then it is proved that the necessary condition is also sufficient under certain convex assumption. To shed light on the application of the above theoretical results, a linear-quadratic game and a kind of optimal portfolio and consumption selection problem are also solved explicitly.

On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379

Abstract Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”

Singular p-biharmonic problems involving the Hardy-Sobolev exponent

This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods combined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and example.
 For mote information see https://ejde.math.txstate.edu/Volumes/2023/61/abstr.html

Some existence results for critical nonlocal Choquard equation on the Heisenberg group

This paper deals with the following critical nonlocal Choquard equation on the Heisenberg group: −(a−b∫Ω|∇Hu|2dξ)ΔHu=μ|u|q−2u+∫Ω|u(η)|Qλ∗|η−1ξ|λdη|u|Qλ∗−2uinΩ,u=0on∂Ω,where Ω⊂HN is a smooth bounded domain, ΔH is the Kohn-Laplacian on the Heisenberg group HN, 1<q<2 or 2<q<Qλ∗, a,b>0, μ>0, 0<λ<4, and Qλ∗=2Q−λQ−2 is the critical exponent. Existence results are obtained by using the Ekeland variational principle, Clark critical point theorem, mountain pass theorem, and Krasnoselskii genus theorem, respectively. Due to critical nonlinearities as well as the presence of the double non-local teams, there are some difficulties on the Heisenberg group’s framework. Our results are new even in the Euclidean case.

Variable s(·)$$ s\left(\cdotp \right) $$‐order Kirchhoff‐type problem with a p(·)$$ p\left(\cdotp \right) $$‐fractional Laplace operator

This paper deals with the existence and multiplicity of weak solutions for the following Kirchhoff‐type problem where models a Kirchhoff coefficient and is a variable ‐order ‐fractional Laplace operator, with and . Here, is a bounded smooth domain with for any . By using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function having special structure, we establish conditions ensuring the multiplicity results.

Normalized solutions to the Sobolev critical Kirchhoff-type equation with non-trapping potential

The paper is concerned about the existence of solutions with prescribed L 2 -norm to the following Kirchhoff-type equation − ( a + b ∫ R 3 | ∇u | 2 ) Δu + ( V + λ ) u = | u | p − 2 u + μ | u | q − 2 u in R 3 , where a , b > 0 , 2 < q < 14 / 3 < p ⩽ 6 or 14 / 3 < q < p ⩽ 6 , μ > 0 . Noting that 14/3 is the mass critical exponent, a Pohozaev constraint method is adopted in two cases. In the mass mixed critical case, i.e., 2 < q < 10 / 3 , 14 / 3 < p ⩽ 6 , we get a normalized solution to above equation with small enough μ by Ekeland's variational principle. In the mass supercritical case, i.e., 14 / 3 < q < p ⩽ 6 , we obtain a positive ground state normalized solution, and energy comparison argument is used in the Sobolev critical case.

Multiple positive solutions for a Schrödinger-Poisson-Slater equation with critical growth

We consider the multiplicity of positive solutions for a Schrödinger-Poisson-Slater equation of the type−△u+(u2⁎1|4πx|)u=μ|u|p−1u+g(x)|u|4u,inR3, where μ>0, 3<p<5, and g∈C(R3,R+). Using Ekeland's variational principle and the well-known arguments of the concentration-compactness principle of Lions (1984) [24,25], when g has one local maximum point, we obtain a positive ground-state solution for all μ>0, while for g with k strict local maximum points, we prove that the equation has at least k distinct positive solutions for μ>0 small. The proof uses a minimization procedure on the Nehari manifold.

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

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.

Existence and multiplicity results for a critical superlinear fractional Ambrosetti–Prodi type problem

We consider the following class of non-local superlinear parametric problem (−Δ)su=λu+u+2s∗−1+f(x),inΩ,u=0,inRN∖Ω,where 0<s<1, Ω is a bounded domain in RN with N>2s and 2s∗=2N/(N−2s) is the fractional critical Sobolev exponent, u+(x)≔max{u(x),0} and λ>0 is a parameter.When λ is not an eigenvalue of (−Δ)s and N>6s, we apply variational methods (especially Linking Theorem) to show that the above problem has at least two non-trivial solutions. We also discuss the existence results of resonant problem (that is, λ=λ1,s with λ1,s is the principal eigenvalue of (−Δ)s) via Ekeland variational principle.