On a new approach of enriched operators

<abstract><p>The goal of this manuscript is to introduce the JK iterative scheme for the numerical reckoning of fixed points in generalized contraction mappings. Also, weak and strong convergence results are investigated under this scheme in the setting of Banach spaces. Moreover, two numerical examples are given to illustrate that the JK iterative scheme is more effective than some other iterative schemes in the literature. Ultimately, as an application, the JK iterative scheme is applied to solve a discrete composite functional differential equation of the Volterra-Stieljes type.</p></abstract>

The purpose of this paper is to study the method of approximation for zeros of the sum of a finite family of maximal monotone mappings in the setting of Banach spaces. Under some mild conditions, we establish strong convergence results of the proposed approximation method. The assumptions that one of the mappings is single valued and α-inverse strongly monotone are dispensed with. In addition, we give some applications to the minimization problems. Finally, we provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Fixed points for multivalued mappings defined on unbounded sets in Banach spaces

In this paper, we give a necessary and sufficient condition under which a variational inequality defined on unbounded sets in a Banach space has a solution. Furthermore, we establish a necessary and sufficient condition under which the minimax equality on unbounded sets is true.

A "purely geometric" proof of the Lindenstrauss-Troyanski result ([2], [6]) on strongly exposed points of weakly compact sets in Banach spaces is given.I would like to thank the referee for his suggestions.We consider a Banach-space X, \\ ||.If G X and if e > 0, then B(,e) = {x G X; \\x -|| < e}.The closed convex hull of a set A G X, is denoted c(A).Let C be a convex subset of X, then Ce is the set of the extremal points of C. If F is a convex, weakly compact, F a convex, closed, bounded set of X and J a closed subinterval of [0,1 ], (K, B,J) denotes the closed, convex setWe have to introduce a few geometrical definitions, referring to [4].Suppose that C is a nonempty, bounded, closed and convex subset of X.Let"strongly exposed" if there exists / G X* (\\f\\ = 1) such that Ve > 0, 3a > 0 with G S(fa, C) G B(e).Let S be the set of all/ G X* such that 11/11 = 1 and/strongly exposes some point of C. Proposition I.If K is a nonempty, convex, weakly-compact subset of X and if B is convex, closed and bounded with K fl B = 0, then the set D = c(K U F) = (K,B, [0,1]) has the following property: Ve > 0, 3 G K such that c(D\B(,e)).First we observe that it is sufficient to prove the proposition for X separable.Indeed, suppose D does not have the required property.Then there exists e > 0 with Vx G K: x G c(D\B(x,e)) and hence, Vx G K, 3A(x) G D with the properties: x G c(A(x)), A(x) n B(x, e) = 0 and A(x) is countable.By induction we construct a sequence (Kn,Bn)n where Kn is countable in K and Bn is countable in F. Let (K0,B0) = ({x},0), where x is some element of K. Suppose we already found (Kn,Bn).Consider Vy G UxeKnA(x) an element ky in K and by in B, with y G c({ky,by)).Let

A new class of mappings, called relatively continuous, is introduced and incorporated to elicit best proximity pair theorems for a non-self-mapping in the setting of reflexive Banach space. As a consequence we obtain a generalization of Carathéodory extension theorem for an initial value problem with L 1 functions on the right hand side.

Motivated by Dotson's example we consider a certain class of mappings which includes the classes of mappings studied by Zamfirescu, Ćirić, Berinde and others. We prove several new results about convergence of distinct iterative processes in convex metric spaces. Furthermore, we study the stability for this class of mappings in the setting of metric spaces.

Iterative approximation of fixed points of nonexpansive mapping is a very active theme in many aspects of mathematical and engineering areas, in particular, in image recovery and signal processing. Because the errors usually occur in few places, it is necessary to show that whether the iterative algorithm is robust or not. In the present work, we prove that Krasnoselski-Mann's algorithm is robust for asymptotically nonexpansive mapping in a Banach space setting. Our results generalize the corresponding results existing in the literature.

We introduce a new iterative scheme for nding a common elementof the solutions set of a generalized mixed equilibrium problem and the xedpoints set of an in nitely countable family of nonexpansive mappings in a Banachspace setting. Strong convergence theorems of the proposed iterative scheme arealso established by the generalized projection method. Our results generalize thecorresponding results in the literature.

In this paper, we generalizes the notion of fundamentally nonexpansive mappings introduced in [Fixed Point Theory and Applications , 2014 (2014), Paper No. 76] to k-nonexpansive mappings. We investigate various properties associated with these mappings and demonstrate some existence and convergence results for k-nonexpansive mappings in the setting of Banach spaces.

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R n . In this context, we establish a subdifferential characterization and show that qualified convexly C 1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C 1 in the context of certain uniformly convex spaces.