Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators

Abstract

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature.

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

“…which was considered and investigated by Nandal, Chugh and Postolache [28]. (6) If N = 2, C 1 = C 2 = C, then the problem in Equation ( 39) reduces to the following system of variational inequalities of finding (x…”

“…In a very recent paper [28], Nandal, Chugh and Postolache introduced an iterative algorithm to study fixed point problem of nonexpansive mappings and zero point problem of maximal monotone mappings. Then, applicability of the algorithm was shown by discussing and solving different kinds of problems; for instance, general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mapping, split feasibility and its connected problems were solved under weaker control conditions on the parameters.…”

“…Therefore, Alghamdi et al [40] established the implicit midpoint rule for nonexpansive mappings in Hilbert space and also proved that their iterative method can be applied to find periodic solution of a nonlinear time dependent evolution equation. In [41], Xu [28] iterative method for approximating a common element of the fixed point sets of nonexpansive mappings and the sets of zeros of maximal monotone mappings. Then, we introduce and analyze a new general system of generalized equilibrium problems and apply our main result to solve this problem.…”

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

“…Recently, various split type problems have been introduced by many authors (see [4,[7][8][9][10][11][12][13]). For important results in this direction or in similar subjects, see [14][15][16][17][18][19][20][21][22][23][24].…”

Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications

“…Further, several fixed point results were obtained by generalizing the concept of metric space [31]. For other new fixed point results and their applications, see [32][33][34].…”