Point Theorem For Set-valued Mappings Research Articles

Stabilization of reaction–diffusion fractional-order memristive neural networks

This paper investigates the stabilization control of fractional-order memristive neural networks with reaction–diffusion terms. With regard to the reaction–diffusion model, a novel processing method based on Hardy–Poincarè inequality is introduced, as a result, the diffusion terms are estimated associated with the information of the reaction–diffusion coefficients and the regional feature, which may be beneficial to obtain conditions with less conservatism. Then, based on Kakutani’s fixed point theorem of set-valued maps, new testable algebraic conclusion for ensuring the existence of the system’s equilibrium point is obtained. Subsequently, by means of Lyapunov stability theory, it is concluded that the resulting stabilization error system is global asymptotic/Mittag-Leffler stable with a prescribed controller. Finally, an illustrative example about is provided to show the effectiveness of the established results.

Weak solutions for contact models involving a class of generalized materials

We consider a contact model for a class of generalized materials whose behaviour is described by means of a bipotential. Using the theory of the bipotentials we deliver a weak formulation consisting of a system of three variational inequalities. The unknown is a triple having as components the displacement field, the Cauchy stress tensor and a Lagrange multiplier related to the friction force on the frictional contact zone. We investigate the existence of the weak solutions by using a fixed point theorem for set-valued mappings and a minimization technique.

Set-valued equilibrium problems based on the concept of null set with applications

In this paper, by means of the null set defined by Wu (J Math Anal Appl. 2019;472:1741–1761), we introduce a set-valued equilibrium problem based on the null set, where the objective mapping takes values in a hyperspace equipped with a convex cone. Moreover, we obtain new existence results for set-valued equilibrium problems defined on compact or noncompact sets. Some applications are given to set optimization problems, to a set-valued variational inequality, to saddle point theorems for set-valued mappings, and to a generalized noncooperative game.

On Caristi fixed point theorem for set-valued mappings

On Caristi fixed point theorem for set-valued mappings

Existence of solution for nonlinear integral inclusions

In this paper, we prove the existence of solution of two nonlinear integral inclusions by using generalization of Krasnoselskii fixed point theorem for set-valued mappings. As an application we prove the existence of solution of the boundary valued problem of ordinary differential inclusion.

Fixed point results for single valued and set valued P-contractions and application to second order boundary value problems

In this paper, by considering the concept of set-valued nonlinear P-contraction which is newly introduced, we present some new fixed point theorems for set-valued mappings on complete metric space. Then by considering a single-valued case we provide an existence and uniqueness result for a kind of second order two point boundary value problem.

Fixed point theorem for set-valued mappings with new type of inequalities

In this paper, we present a fixed point theorem for multivalued mappings satisfying new inequalities without using the Pompeiu–Hausdorff distance. The results generalize the famous Feng and Liu’s result [Feng and Liu, J. Math. Anal. Appl. 317 (2006) 103–112] and some known results in the literature. Some examples are given to illustrate our results.

Fixed point of set-valued mappings

We provide a new definitions of multivalued metric as well as we developing some fixed point theorems for set-valued maps inpartially D* - space by making use of implicit-relation with example.

Vectorial Ekeland variational principle for cyclically antimonotone vector equilibrium problems

ABSTRACTIn this paper, we first obtain some results for the vectorial Ekeland variational principle with a W-distance, in which the final space is a real linear space not necessarily endowed with a topology. We introduce a cyclically antimonotone bifunction taking values in a real linear space, show its properties, and then apply the results to deduce a general vector equilibrium version of Ekeland variational principle. Moreover, we apply the properties of the cyclically antimonotone bifunction to study the existence of solutions for vector equilibrium problems defined on compact or noncompact sets. We also prove the existence of solutions for uncountable systems of vector equilibrium problems. Finally, our main results are applied to establish two new general versions of Caristi-Kirk fixed point theorem for set-valued mappings and some new existence theorems for vector variational inequalities under mild conditions.

Nadler and Kannan Type Set Valued Mappings in M-Metric Spaces and an Application

This article intends to initiate the study of Pompeiu–Hausdorff distance induced by an M-metric. The Nadler and Kannan type fixed point theorems for set-valued mappings are also established in the said spaces. Moreover, the discussion is supported with the aid of competent examples and a result on homotopy. This approach improves the current state of art in fixed point theory.

Takahashi’s minimization theorem and some related results in quasi-metric spaces

In this paper, we establish Takahashi’s minimization theorem in the setting of quasi-metric spaces and provide its equivalence with Ekeland’s variational principle given in Cobzas (Topol Appl 158:1073–1084, 2011). We present an equilibrium version of Ekeland’s variational principle and extended Takahashi’s minimization theorem in the setting of quasi-metric spaces but without using the triangle inequality of the involved bifunction. We establish an equivalent chain of theorems containing Takahashi’s minimization theorem, Ekeland’s variational principle, the equilibrium version of Ekeland’s variational principle and Caristi–Kirk’s fixed point theorem for set-valued maps in the setting of quasi-metric spaces. As applications, we give an error bound for the solution set of the equilibrium problems and provide sufficient conditions for the existence of weak sharp solutions of equilibrium problems.

Common fixed point theorems for set-valued mappings in normed spaces

Let \(\Phi \) be the class of all real functions \(\varphi : [0, \infty [ \times [0, \infty [ \rightarrow [0, \infty [\) that satisfy the following condition: there exists \(\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) 0\). In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings \(T, S: X \rightrightarrows X\), with nonempty and convex values, have a common fixed point whenver there exists a function \(\varphi \in \Phi \) such that $$\begin{aligned} \Vert y - u\Vert \le \varphi (\Vert y - x\Vert , \Vert u - x\Vert ), \; \text { for all } x\in X, y\in T(x), u\in S(x). \end{aligned}$$ Next, we prove that the same conclusion holds when at least one of the set-valued mappings is lower semicontinuous with nonempty closed and convex values. Our common fixed point theorems turn out to be useful for a unitary treatment of several problems from optimization and nonlinear analysis (quasi-equilibrium problems, quasi-optimization problems, constrained fixed point problems, quasi-variational inequalities).

Stability and synchronization of fractional-order memristive neural networks with multiple delays

The paper presents theoretical results on the global asymptotic stability and synchronization of a class of fractional-order memristor-based neural networks (FMNN) with multiple delays. First, the asymptotic stability of fractional-order (FO) linear systems with single or multiple delays is discussed. Delay-independent stability criteria for the two types of systems are established by using the maximum modulus principle and the spectral radii of matrices. Second, new testable algebraic criteria for ensuring the existence and global asymptotic stability of the system equilibrium point are obtained by employing the Kakutani’s fixed point theorem of set-valued maps, the comparison theorem, and the stability criterion for FO linear systems with multiple delays. Third, the synchronization criterion for FMNN is presented based on the linear error feedback control. Finally, numerical examples are given demonstrating the effectiveness of the proposed results.

Fixed point theorems for set-valued mappings and variational principles in uniform spaces with w-distances

Fixed point theorems for set-valued mappings and variational principles in uniform spaces with w-distances

Fixed point theorems for set-valued mappings in <i>b</i>-metric spaces

Fixed point theorems for set-valued mappings in <i>b</i>-metric spaces

Caristi-type fixed point theorem of set-valued maps in metric spaces

Caristi-type fixed point theorem of set-valued maps in metric spaces

Ekeland Variational Principles in Vector Equilibrium Problems

In this paper, we focus on vector equilibrium problems whose final space is a real linear space not necessarily endowed with a topology. In this framework, some exact and approximate Ekeland variational principles are obtained. The main mathematical tool is a kind of strict fixed point theorem for set-valued mappings, from which the Ekeland variational principles are derived by means of algebraic notions and a concept of approximate solution for vector equilibrium problems based on free-disposal sets. The obtained results improve other recent ones of the literature, and clarify the role of some assumptions usually required on the vector bifunction to state Ekeland variational principles in vector equilibrium problems.

A local fixed point theorem for set-valued mappings on partial metric spaces

The purpose of this paper is to study the existence and location of fixed points for pseudo-contractive-type set-valued mappings in the setting of partial metric spaces by using Bianchini-Grundolfi gauge functions.

Finite-time stabilization control of memristor-based neural networks

This paper investigates the finite-time stabilization problem for a general class of memristor-based neural networks (MNNs). Firstly, based on set-valued analysis and Kakutani’s fixed point theorem of set-valued maps, the existence of equilibrium point can be guaranteed for MNNs. Then, by designing novel discontinuous controller, some sufficient conditions are proposed to stabilize the states of such MNNs in finite time. Moreover, we give the upper bound of the settling time for stabilization which depends on the system parameters and control gains. The main tools to be used involve the framework of Filippov differential inclusions, non-smooth analysis, matrix theory and the famous finite-time stability theorem of nonlinear system. Finally, the theoretical results are verified by concrete examples with computer simulations.

Fixed point theorems for set-valued mappings on TVS-cone metric spaces

In the context of TVS-cone metric spaces, we prove a Bishop-Phelps and a Caristi type theorem. These results allow us to prove a fixed point theorem for $(\delta, L)$ -weak contraction according to a pseudo Hausdorff metric defined by means of a cone metric.