Eric Uwadiegwu Ofoedu

Strong and weak convergence theorems for asymptotically nonexpansive mappings

Abstract Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {kn}n⩾1⊂[1,∞), limkn=1, F(T):={x∈K: Tx=x}≠∅ . Suppose {xn}n⩾1 is generated iteratively by x 1 ∈K, x n+1 =P (1−α n )x n +α n T(PT) n−1 x n , n⩾1, where {αn}n⩾1⊂(0,1) is such that ϵ 0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑n⩾1(kn2−1) x ∗ ∈F(T) is proved. If T is not assumed to be completely continuous but E also has a Frechet differentiable norm, then weak convergence of {xn} to some x ∗ ∈F(T) is obtained.

Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings

In this paper, we introduce an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasi- nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Our theorems improve, generalize, unify and extend several results recently announced.

A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

Let be a real Banach space, and a closed convex nonempty subset of . Let be total asymptotically nonexpansive mappings. A simple iterative sequence is constructed in and necessary and sufficient conditions for this sequence to converge to a common fixed point of are given. Furthermore, in the case that is a uniformly convex real Banach space, strong convergence of the sequence to a common fixed point of the family is proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.

Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.

Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, a new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of finite family of relatively quasi-nonexpansive mappings and set of common solutions of a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space is proved using the properties of generalized f-projection operator. Our results extend important recent results.

Iterative approximation of a common zero of a countably infinite family of -accretive operators in Banach spaces.

Let E be a real reflexive and strictly convex Banach space which has a uniformly Gâteaux differentiable norm and let C be a closed convex nonempty subset of E. Strong convergence theorems for approximation of a common zero of a countably infinite family of m-accretive mappings from C to E are proved. Consequently, we obtained strong convergence theorems for a countably infinite family of pseudocontractive mappings.

New Implicit and explicit approximation methods for solutions of integral equations of Hammerstein type

Let H be a real Hilbert spae and F , K : H ? H be mappings such that D ( K ) = D ( F ) = H . Suppose that Hammerstein equation of the type u + KFu = 0 has a solution in H, then we studied in this paper methods that contain an auxiliary mapping (defined on an appropriate real Hilbert space in terms of the mappings K and F) which is pseudocontractive whenever K and F are monotone; and approximation of a fixed point of this pseudocontractive mapping induces approximation of a solution of the equation u + KFu = 0 . Moreover, the mappings K and F need not be defined on compact subset of H or angle bounded on H. Furthermore, our methods which do not involve K - 1 provide an implicit algorithm for approximation of solutions of the equation u + KFu = 0 whenever K and F are assumed to be bounded and continuous; if K and F are assumed to be Lipschitz continuous, then an explicit iterative algorithm for computation of solutions of the equation u + KFu = 0 is provided, still without involving K - 1 .

An algorithm for finding common solutions of various problems in nonlinear operator theory

AbstractIn this paper, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets. MSC:47H06, 47H09, 47J05, 47J25.

Convergence of Path and Approximation of Common Element of Null Spaces of Countably Infinite Family of -Accretive Mappings in Uniformly Convex Banach Spaces

We prove path convergence theorems and introduce a new iterative sequence for a countably infinite family of -accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors.

Approximation of common fixed point of families of nonlinear mappings with applications

Abstract It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gâteaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly Gâteaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.