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The aim of this paper is to present existence and uniqueness results for the solution of a general system of Fredholm integral equations by applying the vectorial form of Maia's fixed‐point theorem. A Gronwall‐type lemma and comparison theorems are also obtained. [ABSTRACT FROM AUTHOR]

In this article, a study of the fixed point problem for Ćirić type multi-valued operators is presented. More precisely, some variants ofĆirić’s contraction principle for multi-valued operators, as well as a strict fixed point principle forĆirić type multi-valued will be given.

In this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:

In this paper, we introduce the idea of gH-weak subdifferential for interval-valued functions (IVFs) and show how to calculate gH-weak subgradients. It is observed that a nonempty gH-weak subdifferential set is convex and closed. In characterizing the class of functions for which the gH-weak subdifferential set is nonempty, it is identified that this class is the collection of gH-lower Lipschitz IVFs. In checking the validity of the sum rule of gH-weak subdifferential for a pair of IVFs, a counterexample is obtained, which reflects that the sum rule does not hold. However, under a mild restriction on one of the IVFs, one-sided inclusion for the sum rule holds. As applications, we employ gH-weak subdifferential to provide a few optimality conditions for nonsmooth IVFs. Further, a necessary optimality condition for interval optimization problems with a difference of two nonsmooth IVFs as the objective is established. Next, a necessary and sufficient condition via augmented normal cone and gH-weak subdifferential of IVFs for finding weak efficient points is presented. Lastly, in investigating a 'sup-relation' between gH-direction derivative and gH-weak subgradients, we approximately compute gH-weak subgradient at each iterative step. In the sequel, we propose W -gH-weak subgradient method to identify a weak efficient solution of an unconstrained nonsmooth IOP. We apply the proposed method to solve an interval optimization problem by taking a test example. We present a convergence analysis of the proposed method for constant and diminishing step sizes. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng–Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side. [ABSTRACT FROM AUTHOR]

In this paper, by combining a modified extragradient scheme with the viscosity approximation technique, an iterative scheme is developed for computing the common element of the set of fixed points of a sequence of asymptotically nonexpansive mappings and the set of solutions of the variational inequality problem for an α-inverse strongly monotone mapping. We prove a strong convergence theorem for the sequences generated by this scheme and give some applications of our convergence theorem.