Continuous Pseudocontractions Research Articles

Strong convergence of viscosity iterations with error terms for cosine families in Banach spaces

In this paper viscosity iterations with error terms for approximating common fixed points of a nonexpansive cosine family and a continuous pseudocontractions are firstly considered. By using the theory of cosine families and constructing some control conditions, several new strong convergence results for this iterative process are presented in the bounded subsets of real uniformly convex and uniformly Gâteaux differentiable Banach spaces, and in the unnecessarily bounded subsets of real reflexive Banach spaces which admit a weakly sequentially continuous duality mapping, respectively. The results contain the corresponding ones without errors as their special cases.

Convergence of Ishikawa iterations on noncompact sets

Recall that Ishikawa's theorem [4] provides an iterative procedure that yields a sequence which converges to a fixed point of a Lipschitz pseudocontrative map T: C →C, where C is a compact convex subset of a Hilbert space X. The conditions on T and C, as well as the fact that X has to be a Hilbert space, are clearly very restrictive. Modifications of the Ishikawa's iterative scheme have been suggested to take care of, for example, the case where C is no longer compact or where T is only continuous. The purpose of this paper is to explore those cases where the unmodified Ishikawa iterative procedure still yields a sequence that converges to a fixed point of T, with C no longer compact. We show that, if T has a fixed point, then every Ishikawa iteration sequence converges in norm to a fixed point of T if C is boundedly compact or if the set of fixed points of T is “suitably large”. In the process, we also prove a convexity result for the fixed points of continuous pseudocontractions.

Some results on continuous pseudo-contractions in a reflexive Banach space

In this paper, we investigate fixed point problems of a continuous pseudo-contraction based on a viscosity iterative scheme. Strong convergence theorems are established in a reflexive Banach space which also enjoys a weakly continuous duality mapping.

Iterative algorithms with errors for zero points of m-accretive operators

Abstract In this paper, we study the convergence of paths for continuous pseudocontractions in a real Banach space. As an application, we consider the problem of finding zeros of m-accretive operators based on an iterative algorithm with errors. Strong convergence theorems for zeros of m-accretive operators are established in a real Banach space. MSC:47H05, 47H09, 47J25, 65J15.

Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space

In this paper, we introduce a new implicit iterative algorithm for finding a common element of a countable family of continuous pseudocontractions in a uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results extend the recent results announced by many others.

Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gâteaux differentiable and T : C → C be a continuous pseudo-contraction with a nonempty fixed point set F ( T ) . For arbitrary given element u ∈ C and for t ∈ ( 0 , 1 ) , let { y t } be the unique continuous path such that y t = ( 1 − t ) T y t + t u . Assume that y t → p ∈ F ( T ) as t → 0 . Let { α n } , { β n } and { γ n } be three real sequences in (0, 1) satisfying the following conditions: (i) α n + β n + γ n = 1 ; (ii) lim n → ∞ α n = lim n → ∞ β n = 0 ; (iii) lim n → ∞ β n 1 − γ n = 0 ; or (iii)′ ∑ n = 0 ∞ α n 1 − γ n = ∞ . Let { ϵ n } be a summable sequence of positive numbers. For arbitrary initial datum x 0 = x 0 0 ∈ C and a fixed n ≥ 0 , construct elements { x n m } as follows: x n m + 1 = α n u + β n x n + γ n T x n m , m = 0 , 1 , 2 , … . Suppose that there exists a least positive integer N ( n ) satisfying the following condition: ‖ T x n N ( n ) + 1 − T x n N ( n ) ‖ ≤ γ n − 1 ( 1 − γ n ) ϵ n . Define iteratively a sequence { x n } in an explicit manner as follows: x n + 1 = x n + 1 0 = x n N ( n ) + 1 = α n u + β n x n + γ n T x n N ( n ) , n ≥ 0 . Then { x n } converges strongly to a fixed point of T . For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x n , this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311–3317].

Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces

We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

In this paper, we continue to study weak convergence problems for the implicit iteration process for a finite family of Lipschitzian continuous pseudocontractions in general Banach spaces. The results presented in this paper improve and extend the corresponding ones of Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Chen et al. [R. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709] and others.

Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E , and let T : K ⟶ E be a continuous pseudocontraction which satisfies the weakly inward condition. For f : K ⟶ K any contraction map on K , and every nonempty closed convex and bounded subset of K having the fixed point property for nonexpansive self-mappings, it is shown that the path x → x t , t ∈ [ 0 , 1 ) , in K , defined by x t = t T x t + ( 1 − t ) f ( x t ) is continuous and strongly converges to the fixed point of T , which is the unique solution of some co-variational inequality. If, in particular, T is a Lipschitz pseudocontractive self-mapping of K , it is also shown, under appropriate conditions on the sequences of real numbers { α n } , { μ n } , that the iteration process: z 1 ∈ K , z n + 1 = μ n ( α n T z n + ( 1 − α n ) z n ) + ( 1 − μ n ) f ( z n ) , n ∈ N , strongly converges to the fixed point of T , which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.

Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let be a nonempty closed convex subset of , and let be a uniformly continuous pseudocontraction. If is any contraction map on and if every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers , , that the iteration process , , , strongly converges to the fixed point of , which is the unique solution of some variational inequality, provided that is bounded.

Strong convergence theorems for uniformly continuous pseudocontractive maps

Let K be a nonempty closed convex subset of a real Banach space E and let T : K → K be a uniformly continuous pseudocontraction. Fix any u ∈ K . Let { x n } be defined by the iterative process: x 0 ∈ K , x n + 1 : = μ n ( α n T x n + ( 1 − α n ) x n ) + ( 1 − μ n ) u . Let δ ( ϵ ) denote the modulus of continuity of T with pseudo-inverse ϕ. If { ϕ ( t ) t : 0 < t < 1 } and { x n } are bounded then, under some mild conditions on the sequences { α n } n and { μ n } n , the strong convergence of { x n } to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on { ϕ ( t ) t : 0 < t < 1 } and { x n } can be dispensed with.

A double-sequence iteration process for fixed points of continuous pseudocontractions

Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H. The idea of a double-sequence iteration is introduced, and it is proved that a Mann-type double-sequence iteration process converges strongly to a fixed point of a continuous pseudocontractive map T which maps C into C. Related results deal with the strong convergence of the iteration process to fixed points of nonexpansive maps.

Iterative solutions of nonlinear accretive operator equations in arbitrary Banach spaces

Suppose E is an arbitrary real Banach space and K is a nonempty closed convex and bounded subset of E. Suppose T : K —>• K is a uniformly continuous strong pseudocontraction. It is proved that the Mann and the Ishikawa iteration methods converge strongly to the unique fixed point of T. Furthermore, our results also hold for the slightly more general class of strictly hemicontractive maps. Related results deal with the iterative approximation of solutions of accretive operator equations in arbitrary real Banach spaces. MIRAMARE TRIESTE April 1996 Permanent address: Department of Mathematics, University of Nigeria, Nsukka, Nigeria.