New convergence Techniques for Identifying with Fixed Point by Nonexapnsive Mappings

1. INTRODUCTION

We take a set E which is uniformly convex Banach space and E is the super set of C and C is closed convex set. In this paper, N indicates the set of all positive integers and G (T) = {x: Gx = x}. A mapping T: CC is called nonexpansive if N. For any arbitrary we can take,1CxGenerate a sequencenx, where xn is characterized by positive integer n1 as: ,1nnTxx (i) Hear (i) known as Picard sequence. ,)1(1nnnnnTxxx (ii) Here (ii) known as Mann [8] sequence. ,)1(1nnnnnTxxx nnnnnTxxy)1( (iii) Here (iii) known as Ishikawa [5] sequence. Ishikawa, Mann and other iteration methods have considered by few researchers for approximation fixed point of nonexpansive mapping [6, 11, 13-15]. Noor [9] defined iterative method by ,1Cxas follows: ,)1(1nnnnnTyxx nnnnnTzxy)1( nnnnnTxxz)1(,1n, (iv) Where nn,and nare sequence in (0, 1). Agrawal [2] in 2007 constructs accompanying strategy: ,)1(1nnnnnTyTxx ,)1(nnnnnTxxy (v) Where }{nand }{nare in (0, 1). They demonstrated that this scheme converges at equivalence comparatively as Picard schemes and it‘s comparatively finer than Mann iterative schemes for contration mappings. Abbas et. al. [1] constructs the other following schemes, where sequence }{nxis generated from any arbitrary Cx1 ,)1(1nnnnnTzTyx nnnnnTzTxy)1( nnnnnTxxz)1(, (vi)

than iterative techniques (v) [2]. Now, motivated by above all techniques we build another iterative process for determining the fixed point of nonexpansive mapping. Where sequence }{nxis generated by Cx1and written as follows: ,)1(1nnnnnTyTxx (vii) ,)1(nnnnnTxxy The primary goal of this article to satisfies convergent outcomes for nonexpansive mappings by iteration (vii). We also show that iteration (vii) converges faster than various other iterative schemes.

Consider K is a Banach space and SK = {1:xKx} is sphere on K having magnitude is identity. For all )1,0(and KSyx,with ,yxif ,1)1(yxthen is said to be convex i.e. strictly convex Banach space.yxyx)1( and  (0,1), then x= y. Then space K is called smooth if

xtyxt0lim. The space K is satisfies the Opial‘s condition [10] for every sequence }{nxin K, i.e yxxxnnnnsuplimsuplim Kywith.yx Definition 2.1 Assume that }{nrand }{nsare sequences that both sequences have limit point i.e both are converges to r and s. if

,0lim 

nnthen nrconverges faster thanns. Definition 2.2 Consider two fixed point iteration processes }{npand}{nq, both are conversing to nnrtp nnstqboth are defined for ,1n Here }{nrand }{nsare two real sequences of positive number tends to 0. If }{nrconverges finer than}{ns, then }{npis also finer than }{nuto t. Here we will characterize a few lemmas for further uses in this paper which are as follows: Lemma2.3 [4] Assume C is a nonempty closed convex subset of a uniformly convex Banach space K and T is a nonexpansive mapping on C. So I – T is demeclosed at 0. Lemma 2.4[12] Consider E is a uniformly comvex Banach space and 10qtpn. Now }{nxand }{nyis the two sequences of E such that ryrxnnnnsuplim,suplimand ,)1(suplimrytxtnnnnnhold for some .0r Then 0limnnnyx. Lemma 2.5 [2] Assume that E be a reflexive Banach space fulfilling the Opial condition and a function T: C X such that I- T demiclosed at 0 and F (T) where C is a convex subset of E. Let }{nxbe a sequence in C such that 0limnnnTxx and pxnnlim exist for all ).(TFp Then }{nxconverges to a fixed point of T.

In this part, we will prove that iteration method (vii) converges finer than the other methods. Theorem 3.1 Let we define a set E such that set E is normed linear space and there is the subset C of E, which is nonvoid closed convex set and let T be a constructive mapping included a factor )1,0(kand fixed point p. Let }{nube characterized by the iteration techniques (vi) and }{nxby (vii), where}{n and }{n are in

Proof: As demonstrated in Theorem 3 of M. Abbas and T. Nazir[1]. ,])1(1[11pukkpunnn.Nn Let pukkannn1})1(1{ pTxxpynnnnn)1( pTxpxnnnn)1( pxknn))1(1( Thus pTyTxpxnnnnn)1(1 pTypTxnnnn)1( pTxxpTxnnnnnnn)1()1(

pTxpxpTxnnnnnnnn)1()1(

pxkknnnn)})1(1(1{ .})1(1{pxkknnn Suppose pxkkbnnn1})1(1{ as .n Consequently }{nxconverges finer than }.{nu Theorem 3.2 Here we consider a self mapping T on set C and set C is nonvoid closed set. Now assume E as a normed linear space E where E is the super set of C, a sequence }{nx defined by (vii) and F (T). Then pxnnlim exist for each ).(TFp pTxxpynnnnn)1( pTxpxnnnn)1( pxpxnnnn)1( pxn (viii) so pTyTxpxnnnnn)1(1 pTypTxnnnn)1( pxpxnnnn)1( .pxn (ix) Thus pxnnlimexists ).(TFp Theorem 3.3 Let we have a set E which is uniformly Banach space and E is the super set of C i.e E C where C is nonempty closed convex set. Let be a nonexpansive self-mapping on C, and a sequence }{nxprovided by (vii) and F (T) then .0limnnnTxx Proof By theorem 3.2 we have pxnnlim is exists. Then assume .limcpxnn By from (viii) and (ix) we have ,suplimcpxnnand cpynnsuplim (x) But T is a nonexpansive map so we have pxpTxnnand pypTynn After getting limsup on both sides,

And cpTynnsuplim (xii) Since

)())(1(limlim1pTypTxpxnnnnnnn

Hence by the using of lemma 2.4 .0limnnnyx Now, pTyTxpxnnnnn)1(1 nnnnTyTxpTx Yield that cpTxnnsuplim (xiii) Now form (xi) and (xiii) ,limcpTxnn additionally we have pTxTxTypTynnnn pyTxTynnn and cpTynnsuplim (xiv) So from (xii) & (xiv) and .inflimcpynn Now by utilizing the lemma 2.4, from we have since pycnnlim = pTxxnnnnn)1(lim Hence by the lemma 2.4 .0limnnnTxx Hence the theorem is verified.

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Department of Mathematics/Computer Science, Govt. Science College, Rewa (M.P.) 486001, India