### T-stabilities of Mann­ Ishikawa Iterations and Multistep Iteration for Nonexpansive Mapping

T-stabilities of Mann­Ishikawa Iterations and Multistep Iteration for Nonexpansive Mapping

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We show that the T-stability of Mann, Ishikawa loops are tantamount to the T-stability of a multistep loop.

Mathematical Subject Classification:47H10

Keywords:Mann, Ishikawa, Noor and Multistep loops ; T-stability

1. Introduction and preliminaries

Let E be a Banach infinite, K a nonempty, bulging subset of E, and T a self map of K. Three most popular loop processs for obtaining fixed points of T, if they exist, are Mann loop [ 1 ] , defined by

1i?Z K, Un+1= ( l-i??­N) UN+i??NTuN, n i‚? 1, ( 1.1 )

Ishikawa loop [ 2 ] , defined by

1i?Z K, omegan+1= ( l-i??­N) omegaN+i??NTyN,

YN= ( l-i??­N) omegaN+i??NTzN, n i‚? 1, ( 1.2 )

Noor loop [ 10 ] , defined by

1i?Z K, Vn+1= ( l-i??­N) VN+i??NTwN,

tungstenN= ( l-i??­N) VN+i??NTerrestrial timeN,

TN= ( l-i?§­N) VN+i?§NTelevisionN, n i‚? 1, ( 1.3 )

for certain picks of { i??­N} , { i??N} and { i?§N} i?? [ 0, cubic decimeter ] .

The multistep loop [ 9 ] , arbitrary fixed order P i‚? 2, defined by

tenn+1= ( l-i??­N) tenN+i??NTy ‘N,

Y ‘N= ( l-i??’­N) tenN+i?? ‘NTy , one = 1, 2, … . , p-2 ( 1.4 )

Y = ( l-i?? ) tenN+i?? TexasN,

where the sequence { i??N} is such that for all n i?Z N

{ i??N} i?? ( 0, 1 ) ( 1.5 )

and for all n i?Z N

{ i?? } i?? [ 0, 1 ) , 1 i‚? I i‚? p – 1, ( 1.6 )

In the above taking p = 3 in ( 1.4 ) we obtain loop ( 1.3 ) . Taking p = 2 in ( 1.4 ) we obtain loop ( 1.2 ) .

Let K be a closed convex bounded subset of normed additive infinite E = ( E, ) and T self-mappings of E. Then T is called nonexpansive on K if ( 1.7 )

for all x, y i?Z K. Let F ( T ) : = { ten i?Z K: Tx = x } be denoted as the set of fixed points of a function T.

Let X be a normed infinite and T a nonexpansive selfmap of X. Let ten0be a point of X, and presume that tenn+1= degree Fahrenheit ( T, tenN) is an loop process, affecting T, which yields a sequence { tenN} of point from X. Suppose { tenN} converges to some ten*i?Z F ( T ) : = { ten i?Z K: Tx = x } i‚? i?¦ . Let { i??N} be an arbitrary sequence in X, and we consider i?-N= ||i??n+l– degree Fahrenheit ( T, i??N) || for n = 1,2,3, …

Definition 1.1.[ 3 ] If ( ( i?-N= 0 ) i?z ( i??N= P ) ) so the loop process tenn+l= degree Fahrenheit ( T, tenN) is said to be T -stable with regard to T.

Remark 1.2.[ 3 ] In pattern, such a sequence { i??N} could originate in the undermentioned manner. Let ten0be a point in X. Set tenn+1=f ( T, tenN) . Let i??0=x0. Now xcubic decimeter= degree Fahrenheit ( T, tenO) . Because of rounding or discretization in the map T, a new value i??cubic decimeterabout equal to xcubic decimetermight be obtained alternatively of the true value of degree Fahrenheit ( T, tenO) . Then to come close i??2, the value degree Fahrenheit ( T, i??­1) is computed to outputs i??2, an estimate of degree Fahrenheit ( T, i??1) . This calculation is continued to obtain { i??N} an approximative sequence of { tenN} .

A sensible speculation is that the Ishikawa loop and the corresponding Mann loop are tantamount for all maps for which either method provides convergence to a fixed point. In an effort to verify this speculation the writers, in a series of documents [ 4-9 ] have shown the equality for several categories of maps.

In [ 11 ] , Rhoades and Soltuz considered the equality between T -stabilities of ( 1.1 ) and ( 1.2 ) . More exactly, they proved the undermentioned theorem.

Theorem ( Rhoades and Soltuz [ 11 ] ) :Let X be a normed infinite and T: Xi‚®X a map. Then the following are tantamount:

( I ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the Ishikawa loop ( 1.2 ) is T -stable,

( two ) for all { i??N} i?? ( 0,1 ) , fulfilling ( 1.5 ) , the Mann loop ( 1.1 ) is T-stable.

In this paper, we shall turn out the equality between T -stabilities of ( 1.4 ) and ( 1.2 ) . Consequently, we shall turn out the equality between T -stabilities of ( 1.4 ) and ( 1.1 ) . Throughout this paper, we shall presume that both Mann, Ishikawa and multistep loops converge to a fixed point of T.

2.The T -stabilities

Let { uN} be the Mann loop, { omegaN} be the Ishikawa loop and { tenN} be the Multistep loop. Let { xN} , { omegaN} and { uN} i?? X be such that tenO=zO=uOand allow ( i??N)Ni?? ( 0,1 ) , ( i??N)Ni?? [ 0,1 ) and ( i?? )Ni?? [ 0,1 ) satisfy ( 1.5 ) and ( 1.6 ) , and

tenn+1= ( 1-i??N) tenN+ i??NTy , Y = ( 1-i?? ) tenN+ i?? Ty , one = 1, 2, … .. , p – 2 ( 2.1 )

Y = ( 1 – i?? ) tenN+ i?? TexasN.

We consider the following nonnegative sequences, for all n i?ZN:

i?ZN= ||xn+1– ( ( 1 – i??N) tenN+ i??NTy ) || , ( 2.2 )

i?¤N= ||zn+1– ( ( 1 – i??N) omegaN+ i??NTyN) || , ( 2.3 )

i?§N= ||un+1– ( ( 1 – i??N) UN+ i??NTuN) || ( 2.4 )

Definition 2.1.Definition 1.1 for ( 2.2 ) , ( 2.3 ) and ( 2.4 ) gives:

( I ) If i?ZN= 0 implies that tenN= x* , so the Multistep loop ( 1.4 ) , is said to be T -stable.

( two ) If i?¤N= 0 implies that omegaN= x* so the Ishikawa loop ( 1.2 ) is said to be T -stable.

( three ) If i?¬N= 0 implies that UN=x* so the Mann loop ( 1.1 ) is said to be T -stable.

Remark 2.2.Let X be a normed infinite and T: Ten i‚® X a nonexpansive map. The following are tantamount:

( I ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) and { i?? } i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the Multistep loop is T -stable,

( i* ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) and { i?? } i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , iˆ? { tenN} i??X: i?ZN= ||xn+1– ( ( 1 – i??N) tenN+ i??NTy ) || = 0 i?z tenN= x* , ( 2.5 )

Remark 2.3.Let X be a normed infinite and T: Ten i‚® X a nonexpanslve map. The following are tantamount:

( two ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the Ishikawa loop is T -stable,

( ii* ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , iˆ? { omegaN} i??X: i?¤N= ||zn+1– ( ( 1 – i??N) omegaN+ i??NTyN) || = 0 i?z omegaN= x* , ( 2.6 )

Remark 2.4. Let X be a normed infinite and T: Xi‚® X a nonexpansive map. The following are tantamount:

( three ) for all { i??N} i?? ( 0,1 ) satisfying ( 1.5 ) , the Mann loop is T -stable,

( iii* ) for all { i??N} i?? ( 0, cubic decimeter ) satisfying ( 1.5 ) , iˆ? { uN} i??X: i?§N= ||un+1– ( ( 1 – i??N) UN+ i??NTuN) || = 0 i?z UN= x* , ( 2.7 )

Theorem 2.5.Let X be a normed infinite and T: Ten i‚® X a nonexpanslve map. Then the following are tantamount:

( I ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) and { i?? } i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the Multistep loop is T -stable,

( two ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0, cubic decimeter ) satisfying ( 1.5 ) and ( 1.6 ) , the Ishikawa loop is T -stable,

( three ) for all { i??N} i?? ( 0,1 ) satisfying ( 1.5 ) , the Mali1 loop is T -stable.

Proof.Let tenN= x* . Since the Mann, Ishikawa and multistep loops converge, M & A ; lt ; i‚? . Remarks 2.2, 2.3 and 2.4 assure that ( I ) i?› ( two ) i?› ( three ) is tantamount to ( i* ) i?› ( ii* ) i?› ( iii* ) . ( ii* ) i?z ( iii* ) proved in [ 11 ] . We shall turn out that ( i* ) i?› ( ii* ) . Therefore, we shall turn out that ( i* ) i?› ( ii* ) i?› ( iii* ) .

We prove ( i* ) i?z ( ii* ) . The cogent evidence is complete if we consider

0 i‚? ||i?¤N|| = ||zn+1– ( ( 1 – i??N) omegaN+ i??NTyN) ||

i‚? ||zn+1-x*|| + ||- ( ( 1 – i??N) omegaN+ i??NTyN) + x*||

i‚? ||zn+1-x*|| + ( 1 – i??N) ||zN– x*|| + i??N|| ( ( 1 – i??N) ( omegaN– x* ) + i??N( omegaN– x* ) ||

i‚? ||zn+1-x*|| + ( 1 – i??N) ||zN– x*|| + i??N|| ( ( 1 – i??N) ( omegaN– x* ) + i??Ni??N||zN– x*||

= ||zn+1-x*|| + ||zN-x*|| i‚® 0 as n i‚® i‚? . ( 2.8 )

Therefore, for a { omegaN} fulfilling i?¤N= ||zn+1– ( ( 1 – i??N) omegaN+ i??NTyN) || = 0, we have shown that omegaN= x* .

Conversely, we prove ( ii* ) i?z ( i* ) . The cogent evidence is complete if we consider

0 i‚? i?ZN= ||xn+1– ( ( 1 – i??N) tenN+ i??NTy ) ||

i‚? ||xn+1– x*||+||x* – ( 1 – i??N) tenN+ i??NTy ) ||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f|| + i??N|| ( 1 – i?? ) tenN+ i?? Ty -f||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f||

+ i??N|| ( 1 – i?? ) ( tenN– degree Fahrenheit ) + i?? [ P ( 1 – i?? ) tenN+ i?? TexasN) -f ] ||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f||

+ i??N|| ( 1 – i?? ) ( tenN– degree Fahrenheit ) + i?? [ ( 1 – i?? ) tenN+ i?? TexasN) -f ] ||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f||

+ i??N( 1 – i?? ) || tenN– f|| + i??Ni?? || ( 1 – i?? ) tenN+ i?? TexasN) -f||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f|| + i??N( 1 – i?? ) || tenN– f||

+ i??Ni?? || ( 1 – i?? ) ( tenN– degree Fahrenheit ) + i?? ( tenN-f ) ||

i‚? ||xn+1– x*||+ ( 1 – i??N) ||xN– f|| + i??N( 1 – i?? ) || tenN– f||

+ i??Ni?? || ( 1 – i?? ) ||xN– f|| + i??Ni?? i?? ||xN-f||

= ||xn+1– x*||+||xN– f|| i‚® 0 as n i‚® i‚? . ( 2.9 )

Therefore, for a { tenN} fulfilling i?ZN= ||xn+1– ( ( 1 – i??N) tenN+ i??NTy ) || = 0, we have shown that tenN= x* . Then this complete the cogent evidence of Theorem 2.5.

Set in ( 1.1 ) , ( 1.2 ) and ( 1.4 ) , T: = ThymineNto obtain the modified Mann, modified Ishikawa and modified multistep loops. We suppose that both modified Mann and modified Ishikawa besides modified multistep loops converge to a fixed point of T. Note that Definition 2.1, Remarks 2.2 and 2.3, and Theorem 2.5 clasp in this instance excessively.

Corollary 2.6.Let X be a normed infinite and T: Ten i‚® X a nonexpansive map. The following are tantamount:

( I ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) and { i?? } i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the modified Multistep loop is T -stable,

( two ) for all { i??N} i?? ( 0,1 ) , { i??N} i?? [ 0,1 ) satisfying ( 1.5 ) and ( 1.6 ) , the modified Ishikawa loop is T -stable,

( three ) all { i??N} i?? ( 0,1 ) satisfying ( 1.5 ) , the Mann modified loop is T -stable.

Mentions

[ 1 ] W.R. Mann, “ Mean value in Iteration ” , Proc. Amer. Mat. Soc. 4 ( 1953 ) 506-510.

[ 2 ] S. Ishikawa, : Fixed points by a new loop method ” , Proc. Amer. Math. Soc. 44 ( 1974 ) 147-150.

[ 3 ] A.M. Harder, T. Hicks, “ Stability consequences for fixed point loop processs ” , Math. Japonica 33 ( 1988 ) 693­-706.

[ 4 ] B.E. Rhoades, S.M. Soltuz, “ On the equality of Mann and Ishikawa loop methods ” , Internal. J. Math. Math. Sci. 2003 ( 2003 ) 451-459.

[ 5 ] B.E. Rhoades, S.M. Soltuz, “ The equality of Mann loop and Ishikawa loop for non-Lipschitzian operators ” , Internat. J. Math. Math. Sci. 2003 ( 2003 ) 2645-2652.

[ 6 ] B.E. Rhoades, S.M. Soltuz, “ The equality of Mann and Ishikawa loop for i??-uniformly pseudocontractive or i??-uniformly accretive maps ” , Internal. J. Math. Math. Sci. 46 ( 2004 ) 2443-2452.

[ 7 ] B.E. Rhoades, S.M. Soltuz, “ The equality of Mann and Ishikawa loop for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps ” , Tamkang J. Math. 35 ( 2004 ) 235-245.

[ 8 ] B.E. Rhoades, S.M. Soltuz, “ The equality between the convergences of Ishikawa and Mann loops for asymptotically nonexpansive in the intermediate sense and strongly in turn pseudocontractive maps ” , J. Math. Anal. Appl. 289 ( 2004 ) 266-278.

[ 9 ] B.E. Rhoades, S.M. Soltuz, “ The equality between Mann-Ishikawa loop and the multistep loop ” , Nonlinear Anal. 58 ( 2004 ) 219-228.

[ 10 ] M. A. Noor, “ New estimate strategies for general variational inequalities ” , J. Math. Anal. Appl. 251 ( 2000 ) 217-229

[ 11 ] R.E. Rhoades, S.M. Soltuz, “ The equality between the T-stabilities of Mann and Ishikawa loops ” , J. Math. Anal. Appl. 318 ( 2006 ) 472-475.

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