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We consider a new system of generalized variational inequalities (SGVI) defined on two closed convex subsets of a real Hilbert space. To find the solution of considered SGVI, a parallel Mann iteration process and a parallel S-iteration process have been proposed and the strong convergence of the sequences generated by these parallel iteration processes is discussed. Numerical example illustrates that the proposed parallel S-iteration process has an advantage over parallel Mann iteration process in computing altering points of some mappings. [ABSTRACT FROM AUTHOR]

The Mann iterates behave well for nonexpansive mappings for any initial guess in the domain. Our aim in this article is to extend this method to a broad class of inexact fixed point algorithms generated by nearly nonexpansive sequences in Banach spaces and to locate the weak limit of the iterates by its initial guesses. Due to the inexactness, our algorithms become efficiently applicable for a wider class of problems. As applications, we give convergence theorems for finding solutions of variational inclusion problems and constrained multiple-sets split feasibility problems. Our results are significant refinements and improvements of the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the semilocal convergence analysis of a Newton-like method discussed by Bartle (Amer Math Soc 6: 827-831, ) to solve the generalized operator equations containing nondifferentiatble term in Banach spaces. This method has also been studied by Rheinboldt (SIAM J Numer Anal 5: 42-63, ). The aim of the paper is to discuss the convergence analysis under local Lipschitz condition $\|F'_{x}-F'_{x_{0}}\|\le \omega (\|x-x_{0}\|)$ for a given point $x_{0}$. Our results extend and improve the previous ones in the sense of local Lipschitz conditions. We apply our results to solve the Fredholm-type operator equations. [ABSTRACT FROM AUTHOR]

It is well known that the rate of convergence of 5-iteration process introduced by Agarwal et al. [J. Nonlinear Convex Anal., 8 (1) (2007), 61-79.] is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce a parallel S-iteration process for finding altering points of nonlinear operators. We apply our algorithms to solve a system of operator equations in Banach space setting. This work also includes convergence analysis of hybrid steepest-descent-like method and hybrid Newton-like method in the context of altering points. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss some fundamental properties of Bregman distance, generalized projection operators, firmly nonexpansive mappings, and resolvent operators of set-valued monotone operators corresponding to a functional Φ(|| · ||). We further study some proximal point algorithms for finding zeros of monotone operators and solving generalized mixed equilibrium problems in Banach spaces. Our results improve and extend some recent results concerning generalized projection operators corresponding to Bregman distance. [ABSTRACT FROM AUTHOR]

Let X be a real reflexive Banach space with a weakly continuous duality mapping Jφ. Let C be a nonemptyweakly closed star-shaped (with respect to u) subset of X. Let F= {T(t) : t ∊ [0, + ∞)} be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of C, which is uniformly continuous at zero. We will showthat the implicit iteration scheme: yn = αnu+(1-αn)T(tn)yn, for all n ∊ N, converges strongly to a common fixed point of the semigroup F for some suitably chosen parameters {αn} and {tn}. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009). [ABSTRACT FROM AUTHOR]

We study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in Banach spaces. The convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the "viscosity approximation technique" in reflexive Banach spaces with uniform Gâteaux differentiable norms. This technique should make existing or new results in solving variational inequalities more applicable. [ABSTRACT FROM AUTHOR]

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theorem extends and improves several corresponding results in the context of nearly nonexpansive mappings. [ABSTRACT FROM AUTHOR]

The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature. [ABSTRACT FROM AUTHOR]