The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1-3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms...

in this paper, we propose a new method for the numerical solution of two-dimensional linear and nonlinear volterra integral equations of the first and second kinds, which avoids from using starting values. an existence and uniqueness theorem is proved and convergence isverified by using an appropriate variety of the gronwall inequality. application of the method is demonstrated for solving the ...

We introduce an implicit method for nding a common element of the set of solutions of systems of equilibrium problems and the set of common xed points of a sequence of nonexpansive mappings and a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit schemes to the unique solution of a variational inequality, which is the optimality condition for ...

Correspondence: [email protected] Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand Abstract In this article, we introduce a new mapping generated by infinite family of nonexpansive mapping and infinite real numbers. By means of the new mapping, we prove a strong convergence theorem for finding a common element of the se...

In this paper, we first introduce a new Halpern-type iterative scheme to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. We then apply our method to approximate zeros of an infinite family of accretive operators and derive a st...

in this paper, we propose a new method for the numerical solution of two-dimensional linear and nonlinear volterra integral equations of the first and second kinds, which avoids from using starting values. an existence and uniqueness theorem is proved and convergence isverified by using an appropriate variety of the gronwall inequality. application of the method is demonstrated for solving the ...

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation: { un+1 − 2un + un−1 ∈ cnAun; n ≥ 1 u0 ∈ H, supn≥0 |un| < +∞, where A is a maximal monotone operator in a real Hilbert space H, and {cn} is a positive sequence of real numbers. We do not assume that A−1(0) ̸= ∅, and we prove among ...