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

Abstraction

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

u­₁i?Z K, U_n+1= ( l-i??­_N) U_N+i??_NTu_N, n i‚? 1, ( 1.1 )

Ishikawa loop [ 2 ] , defined by

z­₁i?Z K, omega_n+1= ( l-i??­_N) omega_N+i??_NTy_N,

Y_N= ( l-i??­_N) omega_N+i??_NTz_N, n i‚? 1, ( 1.2 )

Noor loop [ 10 ] , defined by

v­₁i?Z K, V_n+1= ( l-i??­_N) V_N+i??_NTw_N,

tungsten_N= ( l-i??­_N) V_N+i??_NTerrestrial time_N,

T_N= ( l-i?§­_N) V_N+i?§_NTelevision_N, 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

ten_n+1= ( l-i??­_N) ten_N+i??_NTy ‘_N,

Y ‘_N= ( l-i??’­_N) ten_N+i?? ‘_NTy, one = 1, 2, … . , p-2 ( 1.4 )

Y= ( l-i??) ten_N+i??Texas_N,

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 ten_n+1= degree Fahrenheit ( T, ten_N) is an loop process, affecting T, which yields a sequence { ten_N} of point from X. Suppose { ten_N} 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 ten_n+l= degree Fahrenheit ( T, ten_N) 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 ten_n+1=f ( T, ten_N) . Let i??₀=x₀. Now x_{cubic decimeter}= degree Fahrenheit ( T, ten_O) . Because of rounding or discretization in the map T, a new value i??_{cubic decimeter}about equal to x_{cubic decimeter}might be obtained alternatively of the true value of degree Fahrenheit ( T, ten_O) . 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 { ten_N} .

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 { u_N} be the Mann loop, { omega_N} be the Ishikawa loop and { ten_N} be the Multistep loop. Let { x_N} , { omega_N} and { u_N} i?? X be such that ten_O=z_O=u_Oand allow ( i??_N)_Ni?? ( 0,1 ) , ( i??_N)_Ni?? [ 0,1 ) and ( i??)_Ni?? [ 0,1 ) satisfy ( 1.5 ) and ( 1.6 ) , and

ten_n+1= ( 1-i??_N) ten_N+ i??_NTy, Y= ( 1-i??) ten_N+ i??Ty, one = 1, 2, … .. , p – 2 ( 2.1 )

Y= ( 1 – i??) ten_N+ i??Texas_N.

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

i?Z_N= ||x_n+1– ( ( 1 – i??_N) ten_N+ i??_NTy) || , ( 2.2 )

i?¤_N= ||z_n+1– ( ( 1 – i??_N) omega_N+ i??_NTy_N) || , ( 2.3 )

i?§_N= ||u_n+1– ( ( 1 – i??_N) U_N+ i??_NTu_N) || ( 2.4 )

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

( I ) Ifi?Z_N= 0 implies thatten_N= x* , so the Multistep loop ( 1.4 ) , is said to be T -stable.

( two ) Ifi?¤_N= 0 implies thatomega_N= x* so the Ishikawa loop ( 1.2 ) is said to be T -stable.

( three ) Ifi?¬_N= 0 implies thatU_N=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? { ten_N} i??X:

i?Z_N=||x_n+1– ( ( 1 – i??_N) ten_N+ i??_NTy) || = 0 i?zten_N= 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? { omega_N} i??X:

i?¤_N=||z_n+1– ( ( 1 – i??_N) omega_N+ i??_NTy_N) || = 0 i?zomega_N= 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? { u_N} i??X:

i?§_N=||u_n+1– ( ( 1 – i??_N) U_N+ i??_NTu_N) || = 0 i?zU_N= 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.Letten_N= 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|| = ||z_n+1– ( ( 1 – i??_N) omega_N+ i??_NTy_N) ||

i‚? ||z_n+1-x*|| + ||- ( ( 1 – i??_N) omega_N+ i??_NTy_N) + x*||

i‚? ||z_n+1-x*|| + ( 1 – i??_N) ||z_N– x*|| + i??_N|| ( ( 1 – i??_N) ( omega_N– x* ) + i??_N( omega_N– x* ) ||

i‚? ||z_n+1-x*|| + ( 1 – i??_N) ||z_N– x*|| + i??_N|| ( ( 1 – i??_N) ( omega_N– x* ) + i??_Ni??_N||z_N– x*||

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

Therefore, for a { omega_N} fulfillingi?¤_N=||z_n+1– ( ( 1 – i??_N) omega_N+ i??_NTy_N) || = 0, we have shown thatomega_N= x* .

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

0 i‚? i?Z_N= ||x_n+1– ( ( 1 – i??_N) ten_N+ i??_NTy) ||

i‚? ||x_n+1– x*||+||x* – ( 1 – i??_N) ten_N+ i??_NTy) ||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f|| + i??_N|| ( 1 – i??) ten_N+ i??Ty-f||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f||

+ i??_N|| ( 1 – i??) ( ten_N– degree Fahrenheit ) + i??[ P ( 1 – i??) ten_N+ i??Texas_N) -f ] ||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f||

+ i??_N|| ( 1 – i??) ( ten_N– degree Fahrenheit ) + i??[ ( 1 – i??) ten_N+ i??Texas_N) -f ] ||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f||

+ i??_N( 1 – i??) || ten_N– f|| + i??_Ni??|| ( 1 – i??) ten_N+ i??Texas_N) -f||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f|| + i??_N( 1 – i??) || ten_N– f||

+ i??_Ni??|| ( 1 – i??) ( ten_N– degree Fahrenheit ) + i??( ten_N-f ) ||

i‚? ||x_n+1– x*||+ ( 1 – i??_N) ||x_N– f|| + i??_N( 1 – i??) || ten_N– f||

+ i??_Ni??|| ( 1 – i??) ||x_N– f|| + i??_Ni??i??||x_N-f||

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

Therefore, for a { ten_N} fulfillingi?Z_N=||x_n+1– ( ( 1 – i??_N) ten_N+ i??_NTy) || = 0, we have shown thatten_N= x* . Then this complete the cogent evidence of Theorem 2.5.

Set in ( 1.1 ) , ( 1.2 ) and ( 1.4 ) , T: = Thymine^Nto 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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