Ishikawa Iterative Scheme Research Articles

STABLE AND ALMOST STABLE ITERATION SCHEMES FOR STRICTLY HEMI-CONTRACTIVE OPERATORS IN ARBITRARY BANACH SPACES

ABSTRACT Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a Lipschitz strictly hemi-contractive operator. We establish that a certain Ishikawa iteration scheme with errors both converges strongly to a unique fixed point of T and is almost T-stable on K or T-stable on K. Our results extend, improve and unify the corresponding results in Refs. 1-14, [21], 23-24, [27], [30] and others.

Convergence of an Ishikawa iteration scheme for nonlinear quasi-contractive mappings in convex metric spaces

In this paper, we show that the Ishikawa iteration scheme for a nonlinear quasicontractive selfmap of a nonempty closed convex subset of a complete convex metric space, converges to the unique fixed point. Our theorem generalizes Ciric's result (1999) and partially solves an open problem of Ciric [5].

Convergence of a random iteration scheme to a random fixed point

This paper discusses the convergence of random Ishikawa iteration scheme to a random fixed point for a certain class of random operators.

Approximation of fixed points of strongly pseudocontractive mappings

Let E E be a real Banach space with a uniformly convex dual, and let K K be a nonempty closed convex and bounded subset of E E . Let T : K → K T:K \to K be a continuous strongly pseudocontractive mapping of K K into itself. Let { c n } n = 1 ∞ \{ {c_n}\} _{n = 1}^\infty be a real sequence satisfying: (i) 0 > c n > 1 0 > {c_n} > 1 for all n ⩾ 1 n \geqslant 1 ; (ii) ∑ n = 1 ∞ c n = ∞ \sum \nolimits _{n = 1}^\infty {{c_n} = \infty } ; and (iii) ∑ n = 1 ∞ c n b ( c n ) > ∞ \sum \nolimits _{n = 1}^\infty {{c_n}b({c_n}) > \infty } , where b : [ 0 , ∞ ) → [ 0 , ∞ ) b:[0,\infty ) \to [0,\infty ) is some continuous nondecreasing function satisfying b ( 0 ) = 0 , b ( c t ) ⩽ c b ( t ) b(0) = 0,\,b(ct) \leqslant cb(t) for all c ⩾ 1 c \geqslant 1 . Then the sequence { x n } n = 1 ∞ \{ {x_n}\} _{n = 1}^\infty generated by x 1 ∈ K {x_1} \in K , \[ x n + 1 = ( 1 − c n ) x n + c n T x n , n ⩾ 1 , {x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1, \] converges strongly to the unique fixed point of T T . A related result deals with the Ishikawa iteration scheme when T T is Lipschitzian and strongly pseudocontractive.

Fixed Point Iterations For Strictly Hemi-Contractive Maps In Uniformly Smooth Banach Spaces

It is proved that the Mann iteration process converges strongly to the fixed point of a strictly hemi-contractive map in real uniformly smooth Banach spaces. The class of strictly hemi-contractive maps includes all strictly pseudo-contractive maps with nonempty fixed point sets. A related result deals with the Ishikawa iteration scheme when the mapping is Lipschitzian and strictly hemi-contractive. Our theorems generalize important known results.

Ishikawa's iterations of real Lipschitz functions

In this paper, we consider Ishikawa's iteration scheme to compute fixed points of real Lipschitz functions. Two general convergence theorems are obtained. Our results generalise the result of Hillam.

A note on the Ishikawa iteration scheme

A note on the Ishikawa iteration scheme