Stability of iterative algorithm for a countable family of quasi-contractive operators

Adv. Inequal. Appl. 2017, 2017:14 ISSN: 2050-7461

STABILITY OF ITERATIVE ALGORITHM FOR A COUNTABLE FAMILY OF QUASI-CONTRACTIVE OPERATORS

Q. YUAN

Department of Mathematics, Linyi University, Linyi 276000, China

Copyright c2017 Qing Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper we establish the stability of an iterative algorithm for a countable family of quasi-contractive operators in an arbitrary Banach space in a more general form.

Keywords:fixed point; quasi-contraction; iteration procedure, stability.

2010 AMS Subject Classification:47H10.

1.

Preliminaries

LetX be a Banach space. T a selfmap ofX. Lety_n+1= f(T,y_n)be some iteration procedure. Suppose that F(T), the fixed point set of T, is nonempty and that x_n converges to a point p∈F(T). Let {xn} ⊂X be bounded, and define εn=kyn+1,f(T,yn)k. If limεn=0 implies that limyn=p, then the iteration procedureyn+1= f(T,yn)is said to beT-stable or stable with respect toT.

Harder and Hicks [4] showed how such sequences{xn}could arise in practice and the impor-tance of investigating the stability of various iteration procedures for certain classes of nonlinear mappings. It was remarked by Massa (Math. Reviews 90a(1990), no. 54109a, 54H25) that the

E-mail address: [email protected] Received May 23, 2017

discussion about stability is very rich in examples. In [5], some applications of stability results to first order differential equations are discussed. Stability results for several iteration proce-dures for certain classes of nonlinear mappings have been established in recent papers by many authors (see, for example, A. M. Harder [4] [5], A. M. Harder and T. L. Hicks[6], M. O. Osilike [7] [8] [9] [10] [11] [12], and B.E.Rhoades [13]).

Ravi P. Agarwal, Yeol Je Cho, Jun Li, and Nan Jing Huang [3] prove a stability result of an Ishikawa type iteration procedure for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces.

We shall prove a stability result of an iteration procedure for a countable family of quasi-contractions in an arbitrary Banach space.

LetX be a Banach space and{T_k}∞

k=1 a countable family of selfmap of X, satisfying there exist 0≤h<1 such that

kTix−Tjyk ≤hmax{kx−yk,kx−Tixk,kx−Tjyk,ky−Tjyk,ky−Tixk}

for alli,j. Then we call{Tk}∞_k=1a countable family of quasi-contractions. Define the iteration procedure for{T_k}∞

k=1as follows:

y_n= 1

n−1 n−1

∑

T_ky_n−1

LetF({T_k}∞

k=1)denote the common fixed point set of{Tk}∞k=1. Suppose{Tk}∞k=1 is nonempty and y_n converges to a point p ∈F({Tk}_k∞=1). Let {xn} ⊂X be bounded, and define εn = kxn−_n−1₁∑n_k−₌₁1Tkxn−1k. If limεn =0 implies that limxn = p, then the iteration procedure

y_n= _n−1₁∑n_k−₌₁1Tkyn−1is said to be stable with respect to{Tk}∞_k=1.

We shall proveyn=_n−1₁∑n_k=−1₁Tkyn−1is stable with respect to{Tk}∞k=1in a more general form. We need the following lemmas in order to prove our main theorem.

Lemma 1. [2] Let{xn},{εn} be nonnegative sequences satisfying xn+1≤hxn+εn for all n∈

N,0≤h<1,limεn=0. Thenlimxn=0.

Lemma 2. Let{T_k}∞

k=1be a countable family of quasi-contractions, then

kTix−pk ≤ h

Proof.

kTix−pk ≤hmax{kx−pk,kx−Tixk,kx−Tipk,kp−Tixk,kp−Tipk}

≤hmax{kx−pk,kx−T_ixk,kx−pk,kp−T_ixk,0}

=hmax{kx−pk,kx−Tixk,kx−pk,kp−Tixk}

HencekTix−pk ≤hkx−pkorkTix−pk ≤hkx−TixkorkTix−pk ≤hkp−Tixk. IfkTix−pk ≤hkx−pk, it is clear

kT_ix−pk ≤hkx−pk ≤ h

1−hkx−pk.

IfkTix−pk ≤hkp−Tixk, then

kTix−pk=0≤ h

1−hkx−pk

IfkTix−pk ≤hkx−Tixk, then

kx−T_ixk ≤ kT_ix−pk+kx−pk ≤hkx−T_ixk+kx−pk.

Hence,kTix−pk ≤ ₁₋h_hkx−pk.

Lemma 3. Let{T_k}∞

k=1be a countable family of quasi-contractions, then{Tk}∞k=1have a unique

common fixed point.

Proof. For eachi, it follows from Ciric [1] thatT_ihas a unique fixed point p_i. It is sufficient to prove p_i=p_j, for alli,j.

kpi−pjk=kTipi−Tjpjk

≤hmax{kpi−pjk,0,kpi−pjk,0,kpj−pik}=hkpi−pjk

2.

Main results

Theorem 1. Let X be a Banach space, {T_k}∞

k=1a countable family of quasi-contractions. Let

{xn} ⊂X be bounded, and defineεn=kxn−_n−1₁∑n_k−₌₁1Tkxn−1k. Assumelimεn=0. Define pn

to be the diameter of{x_k:k≥n}S

{T_lx_k :k≥n,l ≥1}; i.e., pn=δ({xk:k≥n} S

{T_lx_k :k≥

n,l≥1}). Thenlimp_n=0.

Proof. First, we show p_n is bounded. Since {xn} is bounded, there exist M >0 such that kxnk ≤M. It follows from Lemma 2 that|Tlxk−pk ≤ ₁₋h_hkxk−pk, for alll,k. Thus, |Tlxkk ≤

h

1−h(kxkk+kpk) +kpk ≤ h

1−h(M+kpk) +kpk, for alll,k. So it is easy to see pnis bounded and p_n≤p_n−1.

Defineδn=sup{εk}∞_k=n, then limδn=0 since limεn=0. Next, we show pn≤hpn−2+2δn. ∀i,j≥n,

kT_kx_i−T_lx_jk ≤hmax{kx_i−y_jk,kx_i−T_lx_jk,kx_i−T_kx_ik,

kxj−Tkxik,kxj−Tlxjk} ≤hpn

kxi−Tlxjk ≤ kxi− 1 i−1

i−1

∑

T_kx_i−1k+k 1 i−1

i−1

∑

T_kx_i−1−T_lx_jk

≤εi+k 1 i−1

i−1

∑

(T_kx_i−1−T_lx_j)k

≤εi+ 1 i−1

i−1

∑

kT_kx_i−1−T_lx_jk

≤εi+ 1 i−1

i−1

∑

hp_n−1

=εi+hpn−1

≤δn+hpn−1

kxi−xjk ≤ kxi− 1 i−1

i−1

∑

T_kx_i−1k+k 1 i−1

i−1

∑

≤εi+k 1 i−1

i−1

∑

(T_kx_i−1−x_j)k

≤εi+ 1 i−1

i−1

∑

kT_kx_i−1−x_jk

≤εi+ 1 i−1

i−1

∑

(δn+hpn−2)

=εi+δn+hpn−2

≤2δn+hpn−2

So,δ({x_k:k≥n}S{T_lx_k:k≥n,l≥1})≤2δn+hpn−2, i.e., pn≤hpn−2+2δn.

Leta_k=p_2k, then p_2k≤hp_2k−2+2δ2k impliesak≤hak−1+2δk. It follows from Lemma 1 that lima_k=0, i.e., limp_2k=0.

Let b_k = p_2k+1, then p_2k+1≤hp_2k−1+2δ2k+1 implies bk≤hbk−1+2δk. It follows from Lemma 1 that limb_k=0, i.e., limp_2k+1=0.

Hence limp_n=0.

As a corollary, we establish the stability of the iteration procedurey_n=_n−1₁∑n_k−₌₁1Tkyn−1with respect to a countable family of quasi-contractions{T_k}∞

k=1.

Theorem 2. Let X be a Banach space,{Tk}∞_k=1a countable family of quasi-contractions. Then

the iteration procedure for{T_k}∞

k=1defined as

y_n= 1

n−1 n−1

∑

T_ky_n−1

is stable with respect to{T_k}∞

k=1.

Proof.

kxn−pk ≤ kxn− 1 n−1

n−1

∑

T_kxn−1k+k 1 n−1

n−1

∑

T_kxn−1−pk

≤εn+ 1 n−1

n−1

∑

But

kT_kxn−1−Tkpk ≤hmax{kxn−1−pk,kxn−1−Tkxn−1k,kTkxn−1−pk}

So,

kT_kx_n−1−T_kpk ≤hmax{kx_n−1−pk,kx_n−1−T_kx_n−1k}

≤h(kx_n−1−pk+kxn−1−Tkxn−1k)

≤hkxn−1−pk+hpn−1)

substituting it into the first inequality, we obtain

kxn−pk ≤εn+ 1 n−1

n−1

∑

kT_kxn−1−Tkpk

≤εn+ 1 n−1

n−1

∑

(hkxn−1−pk+hpn−1)

=εn+hkxn−1−pk+hpn−1

Noting that 0 ≤h<1, limεn=0 with addition limpn =0, by Theorem 1. It follows from

Lemma 1 that limkxn−pk=0, i.e., limxn=p.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgement

The author was supported by Shandong Provincial Natural Science Foundation, China (ZR2017LA001) and the Youth Foundation of Linyi University (LYDX2016BS023).

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