Ishikawa Iterative Sequence Research Articles

Variational Inclusions Involving Generalized Set-Valued Quasi-Contractive Mappings in Banach Spaces

In this paper, we give a characterization of strong convergence theorem of the Ishikawa iterative sequence with errors for the generalized set-valued variational inclusion problem involving a generalized set-valued quasi-contractive mapping in a real Banach space. First, the existence of the solution of the generalized set-valued variational inclusion problem is proved. Then, using the existence result, the Ishikawa iterative algorithm with errors is constructed and the convergence of the iterative sequences is proved. Finally, numerical experiments are proposed to support the main result. The results improve and extend some known results.

Convergence theorems of a scheme for I-asymptotically quasi-nonexpansive type mapping in Banach space

Let X be a Banach space. Let K be a nonempty subset of X. Let T : K ? K be an I-asymptotically quasi-nonexpansive type mapping and I : K ? K be an asymptotically quasi-nonexpansive type mappings in the Banach space. Our aim is to establish the necessary and sufficient conditions for the convergence of the Ishikawa iterative sequences with errors of an I-asymptotically quasi-nonexpansive type mappping in Banach spaces to a common fixed point of T and I. Also, we study the convergence of the Ishikawa iterative sequences to common fixed point for nonself I-asymptotically quasinonexpansive type mapping in Banach spaces. The results presented in this paper extend and generalize some recent work of Chang and Zhou [1], Wang [19], Yao and Wang [20] and many others.

Convergence and stability of modified Ishikawa iteration sequence with errors

We show some stability and convergence theorems of the modified Ishikawa iterative sequence with errors for a strongly successively pseudocontractive and strictly asymptotically pseudocontractive mapping in a real Banach space. Additionally, we prove that if T is a uniformly Lipschitzian strongly accretive mapping, the modified Ishikawa iteration sequence with errors converges strongly to the unique solution of the equation . The main results of this paper improve and extend the known results in the current literature. MSC:47H09, 47H10, 47J25.

Strong Convergence of a Modified Ishikawa Iterative Sequence for Asymptotically Quasi-Pseudo-Contractive-Type Mappings

The purpose of this paper is to investigate the strong convergence problem of a modified mixed Ishikawa iterative sequence with errors for approximating the fixed points of an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi-pseudo-contractive-type mapping in an arbitrary real Banach space. The results here improve and extend the corresponding results reported by some other authors recently.

A Modified Mixed Ishikawa Iteration for Common Fixed Points of Two Asymptotically Quasi Pseudocontractive Type Non-Self-Mappings

A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontractive type non-self-mappings is introduced. By the flexible use of the iterative scheme and a new lemma, some strong convergence theorems are proved under suitable conditions. The results in this paper improve and generalize some existing results.

The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and be a generalized Lipschitz Φ-hemi-contractive mapping with . Let , , , be four real sequences in and satisfy the conditions (i) as and ; (ii) . For some , let , be any bounded sequences in D, and be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping. MSC:47H10.

A necessary and sufficient condition for the strong convergence of Lipschitzian pseudocontractive mapping in Banach spaces

The purpose of this paper is to establish some necessary and sufficient conditions for the strong convergence of the Ishikawa iterative sequence and the Mann iterative sequence to a fixed point of pseudocontractive mapping in Banach spaces. Our results, to some extent, improve and extend the well-known result of [S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1) (1974) 147–150.] to Banach spaces.

Equivalence problem of Mann and Ishikawa Iterations in Banach Spaces

In this paper,we prove some equivalence theorems between convergence and stability of modified Mann and modified Ishikawa iterative sequence with errors for uniformly Lipschitz strongly successively pseudo-contractive mappings in Banach spaces by using different methods,under the condition on the iteration para-meters limsupβ_nk/L(L+1).Our results prove and extend the results that need limβ_n=0 and initial points of two different iteration are the same one.

Strong Convergence Theorems of the Ishikawa Process with Errors for Strictly Pseudocontractive Mapping of Browder-Petryshyn Type in Banach Spaces

We prove several strong convergence theorems for the Ishikawa iterative sequence with errors to a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces and give sufficient and necessary conditions for the convergence of the scheme to a fixed point of the mapping. The results presented in this work give an affirmative answer to the open question raised by Zeng et al. 2006, and generalize the corresponding result of Zeng et al. 2006, Osilike and Udomene 2001, and others.

Necessary and Sufficient Condions for Strong Convergence of Ishikawa Iterative Sequence of Variational Inclusions with <I>φ</I>-Strongly Accretive Type Mappings

Necessary and Sufficient Condions for Strong Convergence of Ishikawa Iterative Sequence of Variational Inclusions with <I>φ</I>-Strongly Accretive Type Mappings

THE CONVERGENCE THEOREMS FOR COMMON FIXED POINTS OF UNIFORMLY L-LIPSCHITZIAN ASYMPTOTICALLY Φ-PSEUDOCONTRACTIVE MAPPINGS

In this paper, we show that the modified Mann iteration with errors converges strongly to fixed point for uniformly L-Lipschitzian asymptotically <TEX>$\Phi$</TEX>-pseudocontractive mappings in real Banach spaces. Meanwhile, it is proved that the convergence of Mann and Ishikawa iterations is equivalent for uniformly L-Lipschitzian asymptotically <TEX>$\Phi$</TEX>-pseudocontractive mappings in real Banach spaces. Finally, we obtain the convergence theorems of Ishikawa iterative sequence and the modified Ishikawa iterative process with errors.

Convergence theorems of a family of uniformly (L, α)-Lipschitz asymptotically quasi-nonexpansive type mappings

AbstractIn this paper, we study the convergence of the Ishikawa iterative sequence of rank-

Convergence Theorems of Modified Ishikawa Iterative Scheme for Two Nonexpansive Semigroups

We prove convergence theorems of modified Ishikawa iterative sequence for two nonexpansive semigroups in Hilbert spaces by the two hybrid methods. Our results improve and extend the corresponding results announced by Saejung (2008) and some others.

Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space

We introduce a modified Ishikawa iterative process for approximating a fixed point of two infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two infinitely nonexpansive self-mappings.

RANDOM ISHIKAWA ITERATIVE SEQUENCE WITH ERRORS FOR APPROXIMATING RANDOM FIXED POINTS

The purpose of this paper is to construct a random Ishikawa iterative sequence with errors for random strongly pseudo-contractive operator T in separable Banach spaces and to study that under suitable conditions this random iterative sequence with errors converges to a random fixed point of T. As applications, we utilize our results to study the existence problems of solutions for some kinds of nonlinear random operator equations in uniformly smooth separable Banach spaces.

RANDOM ISHIKAWA ITERATIVE SEQUENCE WITH ERRORS FOR APPROXIMATING RANDOM FIXED POINTS

The purpose of this paper is to construct a random Ishikawa iterative sequence with errors for random strongly pseudo-contractive operator T in separable Banach spaces and to study that under suitable conditions this random iterative sequence with errors converges to a random fixed point of T. As applications, we utilize our results to study the existence problems of solutions for some kinds of nonlinear random operator equations in uniformly smooth separable Banach spaces.

NEW ITERATIVE PROCESS FOR THE EQUATION INVOLVING STRONGLY ACCRETIVE OPERATORS IN BANACH SPACES

In this paper, under suitable conditions, we show that the new class of iterative process with errors introduced by Li et al converges strongly to the unique solution of the equation involving strongly accretive operators in real Banach spaces. Furthermore, we prove that it is equivalent to the classical Ishikawa iterative sequence with errors.

Weak stability of Mann and Ishikawa iterations with errors for [formula omitted]-hemicontractive operators

Let E be an arbitrary Banach space and T : E → E be a Lipschitzian and ϕ -hemicontractive mapping. This paper proves that, without the property lim inf n → ∞ ϕ ( t ) / t > 0 , the Mann and Ishikawa iterative sequences with errors are weakly T -stable. The related result deals with the weak T -stability of these sequences with errors to the unique solution of the equation f = T x when T : E → E is Lipschitzian and ϕ -strongly accretive operator. Our results improve and generalize the recent results of Zhou et al. [H.Y. Zhou, S.S. Chang, Y.J. Cho, Weak stability of Ishikawa iteration procedures for ϕ -hemicontractive and accretive operators, Appl. Math. Lett. 14 (2001) 949–954] without the strict requirement lim inf n → ∞ ϕ ( t ) / t > 0 and extend the results to the Mann and Ishikawa iterations with errors.

Stability and convergence of modified Ishikawa iterative sequences with errors in Banach spaces

Some stability and convergence theorems of the modified Ishikawa iterative sequences with errors for asymptotically nonexpansive mapping in the intermediate sense and asymptotically pseudo contractive and uniformly Lipschitzian mappings in Banach spaces are obtained.

CONVERGENCE OF ISHIKAWA ITERATIVE SEQUENCES FOR ACCRETIVE LIPSCHITZIAN MAPPINGS IN BANACH SPACES

In this paper, we study the convergence theorems of the Ishikawa iterative sequences with mixed errors for the accretive Lipschitzian mappings in Banach spaces.