Modified three-step iteration process with errors for three Local strongly pseudo contractive operators

The purpose of this paper is to introduce more general modified two step and three step Ishikawa iterative with errors for local strongly pseudo contractive and local strongly accretive mappings which is much more general than the important class of strongly pseudo contractive and strongly accretive mappings.Also,we study the convergence of this iteration for locally mappings in the framework of Banach space. The results presented in this paper improve, generalize of the results of Mogbademn and OlaleauRafq, Yuguang and Fang and others. Introduction and preliminaries: Let X be a real Banach spaces, X* be the dual space on X The normalized duality mapping is defined by where denotes the generalized duality pairing. It is well know that if X is an uniformly smooth Banach space, then J is single-valued and it is uniformly continuous on any bonded sub set of . In the sequel we shall denote single-valued normalized duality mapping by j. By means of the normalized duality mapping by J. The symbole is J and F(T) the identity mapping on X and the set of all fixed points of T respectively. Let us recall the following three iteration processes due to Ishikawa[1],Mann[2]and Xu [3] Let k be a nonempty convex subset of an arbitrary normal linear space X and T : K K be an operator. i. For any given the sequence defined by (1.1) Is called the Ishikawa iteration sequence where are real sequences in [0,1] satisfying appropriate conditions. ii. In particular if for all then the sequence defined by: Is called the Mann iteration sequence. iii. For any the sequence defined by: (1.2) Where are arbitrary bounded sequences in K and , are real sequences in [0,1] such that is called the Ishikawa iteration sequences with errors. iv. If , with the same notations and definitions as in (iii), for all then the sequence now defined by: The Bulletin of Society for Mathematical Services and Standards Online: 2013-06-03 ISSN: 2277-8020, Vol. 6, pp 13-23 doi:10.18052/www.scipress.com/BSMaSS.6.13 © 2013 SciPress Ltd., Switzerland SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ Is called the Mann iteration sequences with errors. It is clear that the Ishikawa and Mann iteration sequences are all special cases of the Ishikawa and Mann iteration sequences with errors respectively. Now Let, be two mappings. For any given the more general modified two step iteration defined by: (1.3) Where the read sequence as in (1.2) It is that the iteration schemes (1.1) – (1.2) are special cases of (1.3). Noor[4] gave the following threestep iteration process for solving non-linear operator equations in real Banach space, let K be a nonempty closed convex subset of X and be mapping. For an arbitrary be mapping. For an arbitrary the sequence defined by: (1.4) Where are three sequences in[0,1], is called the three-step iteration or (Noor iteration). Rafiq[7] introduced the following new type of iteration, the modified three-step iteration is defined by:


Introduction and preliminaries:
Let X be a real Banach spaces, X* be the dual space on X The normalized duality mapping is defined by where denotes the generalized duality pairing. It is well know that if X is an uniformly smooth Banach space, then J is single-valued and it is uniformly continuous on any bonded sub set of . In the sequel we shall denote single-valued normalized duality mapping by j. By means of the normalized duality mapping by J. The symbole is J and F(T) the identity mapping on X and the set of all fixed points of T respectively. Let us recall the following three iteration processes due to Ishikawa [1],Mann [2]and Xu [3] Let k be a non-empty convex subset of an arbitrary normal linear space X and T : K K be an operator. i. For any given the sequence defined by  .2) It is that the iteration schemes (1.1) -(1.2) are special cases of (1.3). Noor [4] gave the following three-step iteration process for solving non-linear operator equations in real Banach space, let K be a nonempty closed convex subset of X and be mapping.
For an arbitrary be mapping. For an arbitrary the sequence defined by: Where are three sequences in[0,1], is called the three-step iteration or (Noor iteration). Rafiq [7] introduced the following new type of iteration, the modified three-step iteration is defined by: are real sequences in[0,1]. It is clear that the iteration schemes (1.4) is special case of (1.5), we define the more general modified three -step iteration process with errors by: Where are arbitrary bounded sequences in k and are real sequences in [0,1] satisfying some conditions. Now, we introduce local strongly pseudo -contractive (local strongly accretive) operators as follows. 1. Each strongly pseudo contractive operator is local strongly pseudo-contractive and each stronglyaccretive operator is local strongly accretive.

T is local strongly pseudo-contractive if and only if (I-T) is local strongly accretive and
, where t x and k x are the constants oppearing in (1.8) and (1.9) respectively. 3. If T is local strongly accretive then accretive.

Lemma (1.3)[5]:-
Let X be a real Banach space and J: be the normalized duality mapping. Then , for any x,y  X

Let
be a non-negative sequence which satisfied the following inequality our purpose in this paper to prove that the modified three-step iteration process with error for three local strongly pseudo-contractive operators strongly convergence to fixed points. The results presented in this paper generalize the corresponding Main Results in [4], [5], [9], [10], [11], [12], and others Theorem (2.1):-Let X be a uniformly smooth Banach space and be a local strongly accretive mapping suppose that there exists a solution of the equation for some fX define suppose that R(X) is bounded. Let the two step iteration sequence with errors defined by: (2)  (2) and(iii) , we get (6) Now , we show by induction that For all for n=0 we have Therefore; the inequality (7) holds Substrituting (7) into (6) Therefore; Since are bounded and j is uniformly continuous on any bounded subset of X we have Where are sequences as in theorem (2.1). then <x n > converges strongly to the unique fixed point of T i . Proof:-Obviously <x n > and <y n > are both contained in K and therefore, bounded Since T i is local strongly pseudo-contractive, then (I -T i ) is local strongly accretive for all (i=1,2)put y=w and (T i = H i ), we get (7) the proof of theorem (2.1) follows. Now, we establish the convergence of more general modified three-step to the unique solution of uniformly continuous and locally operators in arbitrary Banach space.

Theorem (2.3):-
Let X be a uniformly smooth Basnch space, let k be a non empty bounded closed subset of X (I=1,2,3) and T i (i=1,2,3) is a local strongly pseudo-contractive mappings of k and Define sequence <x n > iteratively for by For all k x  (0,1) Then the sequence <x n > converges strongly to the unique a fixed point of T i The Bulletin of Society for Mathematical Services and Standards Vol. 6

Proof:-
Since it follows from(1.8) that is singleton sag q .The operators T i is local strongly pseudo contractive implies that (I -T i ) is local strongly accretive and therefore is accretive. Hence , for all r>0 and k x  (0,1) , we have: Which implies that the sequence <x n > converges strongly to q corollary (2.4).

Corollary(2.4):-
Let X be a real arbitrary Banach space and K be a nonempty closed bounded and convex subset of X . T i local strongly pseudo contractive self mapping of k and uniformly continuous such that and the sequence <x n > iteratively for defined by for all k x (0,1).  [4] and Imoru [10] , [11] via replace three-step iteration of strongly pseudo contractive (strongly accretive) by a more general modified three-step iterative of local strongly pseudo contractive (local strongly accretive) respectively. 2-Results of Xue and Fan [8] and Rafiq [5] via replace modified three-step iterative scheme by a more general modified three-step iterative scheme with errors of local strongly pseudo contractive (strongly accretive) operators. 3-Results of mohbademu and Olalean [9]. via replace strongly pseudo contractive (local strongly accretive) by local strongly pseudo contractive (local strongly accretive) operators respectively. For all k x (0,1). Then the sequence <x n > converges strongly to the unique solution of the equation T i x=f.
Proof:-It follows from definition of local strongly accretive mappings, that for given , there exists k x (0,1). such that We observe that R i , T i are uniformly continuous and for any given fk .

Which implies that
That is (I -R i ) is local strongly accretive. Thus R i is local strongly pseudo contractive. thus theorem(2.5) follows from theorem(2.3)