An iterative process for a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems in reflexive Banach spaces

The equilibrium problem and its generalizations had a great influence in the development of some branches of pure and applied sciences. The equilibrium problems theory provides a natural and novel approach for some problems arising in nonlinear analysis, physics and engineering, image reconstruction, economics, finance, game theory and optimization. In recent times, there were many methods in order to solve the equilibrium problem and its generalizations. Some authors proposed many iterative methods and studied the convergence of such iterative methods for equilibrium problems and nonexpansive mappings in the setting of Hilbert spaces and Banach spaces. Note that a generalized mixed equilibrium problem is a generalization of an equilibrium problem and a Bregman totally quasi-asymptotically nonexpansive mapping is a generalization of a nonexpansive mapping in reflexive Banach spaces. The purpose of this paper is to combine the parallel method with the Bregman distance and the Bregman projection in order to introduce a new parallel hybrid iterative process which is to find common solutions of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems. After that, we prove that the proposed iteration strongly converges to the Bregman projection of initial element on the intersection of common fixed point set of a finite family of Bregman totally quasi-asymptotically nonexpansive mappings and the solution set of a system of generalized mixed equilibrium problems in reflexive Banach spaces. As application, we obatin some strong convergence results for a Bregman totally quasi-asymptotically nonexpansive mapping and a generalized mixed equilibrium problem in reflexive Banach spaces. These results are extensions and improvements to the main results in [7, 8]. In addition, a numerical example is provided to illustrate for the obtained result.

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved.

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u↦fl(t,x,u),l=1,2 and no compact property of C0-semigroups eAl(t),l=1,2.

New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main feature of our approach is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of it) does not appear in our formulas. Another aspect of our analysis is that it emphasizes the relationship with the subdifferential of the supremum of finite subfamilies, or equivalently, finite weighted sums. Some specific results are given in the setting of reflexive Banach spaces, showing that the subdifferential of the supremum can be reduced to the supremum of a countable family.

Topological concepts in partially ordered vector spaces

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.

The Solvability of Generalized Systems of Time-Dependent Hemivariational Inequalities Enjoying Symmetric Structure in Reflexive Banach Spaces

In real reflexive Banach spaces, let the GSTDHVI, SHVI, DVIP, VIT, and KKM represent a generalized system of time-dependent hemivariational inequalities, a system of hemivariational inequalities, a derived vector inclusion problem, Volterra integral term, and Knaster–Kuratowski–Mazurkiewicz, respectively, where the GSTDHVI consists of two parts which are of symmetric structure mutually. By virtue of the surjectivity theorem for pseudo-monotonicity mappings and the Banach contraction mapping principle, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, we consider and study a GSTDHVI with VITs. Under quite mild assumptions, it is shown that there exists only a solution to the investigated problem via demonstrating that a DVIP with VIT is solvable.