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WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS
WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Jul, 21(3): 195-206
Copyright © 2014, Korean Society of Mathematical Education
  • Received : April 10, 2014
  • Accepted : June 19, 2014
  • Published : July 28, 2014
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About the Authors
SEUNG-HYUN KIM
aDEPARTMENT OF MATHEMATICS, KYUNGSUNG UNIVESRITY, BUSAN 608-736, KOREAEmail address:jiny0610@hotmail.com
BYUNG-SOO LEE
bDEPARTMENT OF MATHEMATICS, KYUNGSUNG UNIVESRITY, BUSAN 608-736, KOREAEmail address:bslee@ks.ac.kr

Abstract
In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al. [1] .
Keywords
1. Introduction and Preliminaries
The equilibrium problems were introduced by Blum and Oettli [2] in 1994. Numerous problems in applied sciences, for example optimization problems, saddle point problems, variational inequality problems and Nash equilibria in noncoopera- tive games, are reduced to find a solution of the following equilibrium problem [2 , 3] ; finding x C such that
PPT Slide
Lager Image
where ϕ : C × C → ℝ is a bifunction.
On the other hand, the problem of finding a common fixed point of a family of mappings is a classical problem in nonlinear analysis. Finding an optimal point in the set of common fixed points of a family of mappings is a task that occurs frequently in various areas of mathematical sciences and engineering. For example, the convex feasibility problem reduces to finding a point of the set of common fixed points of a family of nonexpansive mappings [4] .
In 2009, Qin et al. considered the following weak convergence theorem to a common fixed point of a finite family of asymptotically k -strictly pseudo-contractive mappings under a hybrid iterative scheme.
Theorem 1.1 ( [5] ). Assume the following conditions;
  • (1)C is a closed convex subset of a Hilbert spaceH,
  • (2)Ti:C→Cis an asymptotically ki-strictly pseudo-contractive mapping,
  •       where1 ≤i≤N for some natural number N and0 ≤ki< 1,
  • (3) {kn,i}is a sequence in[1, ∞)such that< ∞
  • (4)k= max{ki: 1 ≤i≤N}and
  • (5) {kn}is a sequence defined by kn= max{kn,i: 1 ≤i≤N}for n∈ ℕ.
Assume that F := (
PPT Slide
Lager Image
F ( Ti )) ≠
PPT Slide
Lager Image
. For any x0 C , let { xn } be a sequence generated by
  • xn=an−1xn−1+ (1 −an−1)∀n≥ 0,
where { an } is a sequence in (0, 1) such that k + Ɛ an ≤ 1 − Ɛ for some Ɛ ∈ (0, 1) and n = ( h − 1) N + i ( n ≥ 1), where i = i ( n ) ∈ {1, 2, ⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n → ∞. Then { xn } converges weakly to an element of F .
The existence of solutions of equilibrium problems and common fixed points of finite mappings are very important in nonlinear analysis with applications. Moreover, to find the intersection of solution sets of equilibrium problems and common fixed points of finite mappings and to apply the intersection are also important. Recently, there have been a few works for the intersection of the two sets to be the set of weakly convergent points of two given sequences in Hilbert spaces.
In 2000, Kumam et al. considered the following weak convergence theorem to a given common element of the set of common xed points of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction under a hybrid iterative scheme.
Theorem 1.2 ( [1] ). Assume the conditions (1)-(5) in Theorem 1.1. Let ϕ : C × C → ℝ be a bifunction satisfying the followings;
  • (A1)ϕ(x,x) = 0, ∀x∈C;
  • (A2)ϕis monotone, i.e.,ϕ(x,y) +ϕ(y,x) ≤ 0 for anyx,y∈C;
  • (A3)ϕis upper-hemicontinuous, i.e., for eachx,y,z∈C,
  •                  ϕ(tz+ (1 −t)x,y) ≤ϕ(x,y);
  • (A4)ϕ(x, · ) is convex and lower semicontinuous for eachx∈C.
Assume that F := (
PPT Slide
Lager Image
F ( Ti )) ∩ EP ( ϕ ) ≠
PPT Slide
Lager Image
, where EP ( ϕ ) is a set of solutions of equilibrium problem (1.1). For any x 0 C , let { xn } and { υn } be sequences generated by
PPT Slide
Lager Image
where n = ( h − 1) N + i ( n ≥ 1), i = i ( n ) ∈ {1, 2,⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n →∞. Let { an } and { rn } satisfy the following conditions:
  • (1) {an} ⊂ [α,β], for someα,β∈ (k; 1) and
  • (2) {rn} ⊂ (0, ∞) and
Then { xn } and { υn } converges weakly to an element of F .
On the other hand, the fixed point iterative scheme with errors was introduced by Liu [6] . The idea of considering fixed point iterative scheme problems with errors which comes from practical numerical computation usually concerns the approximation fixed point and is related to the stability of fixed point iterative schemes. The idea of considering iterative scheme procedures with errors leads to finding the approximate solution to equilibrium problems. In 2005, Combettes and Hirstoga [3] introduced an iterative scheme for a problem of finding best approximate solutions to equilibrium problem and proved the strong convergence result.
In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al. [1] .
The following results will be needed in the main result.
Lemma 1.1 ( [7 , 8] ). Let H be a real Hilbert space. There hold the following identities
  • (i) ∥x−y∥2=∥x∥2−∥y∥2− 2 ⟨x−y,y⟩, ∀x,y∈H
  • (ii) ∥ax+by+cz∥2=a∥x∥2+b∥y∥2+c∥z∥2−ab∥x−y∥2−bc∥y−z∥2−ca∥z−x∥2, ∀x,y∈H,where a,b,c∈ [0, 1]with a+b+c= 1,
  • (iii)If{xn}is a sequence inHweakly converging to z, then
  •                  sup∥xn−y∥2=sup ∥xn−z∥2+ ∥z−y∥2, ∀y∈H.
Lemma 1.2 ( [3] ). Assume that ϕ : C × C → ℝ satisfies (A1)-(A4). For r > 0 and x H , define a mapping Sr : H C as follows;
  • Sr(x) = {z∈C:ϕ(z,y)+⟨y−z,z−x⟩ ≥ 0, ∀y∈C},
for all z H. Then the following hold;
  • (i)Sris single-valued;
  • (ii)Sris firmly nonexpansive, i.e., for any x,y∈H,
  •                ∥Srx−Sry∥2≤ ⟨Srx−Sry,x−y⟩;(iii)F(Sr) =EP(ϕ);
  • (iv)EP(ϕ)is closed and convex.
2. Main Results
Definition 2.1. A mapping T : C C is said to be asymptotically k-strictly pseudo-contractive if there exist a sequence { kn } ⊂ [1, ∞) with
PPT Slide
Lager Image
kn = 1 and k ∈ [0, 1) such that
Tnx Tny 2
PPT Slide
Lager Image
x y 2 + k ∥( I Tn ) x − ( I Tn ) y 2 , ∀ x , y C and n ∈ ℕ.
The following proposition by Osilike and Igbokwe [8] was considered by using infinite terms of the given sequences based on Lemma 1 in [9] , but our proof is considered by using only finite terms of the given sequences based on the basic concepts of limit superior and limit inferior.
Proposition 2.1. Let { an }, { cn } and { δn } be nonnegative real sequences satisfying the following condition:
  • an+1≤ (1 +δn)an+cn, ∀n∈ ℕ.
if
PPT Slide
Lager Image
and
PPT Slide
Lager Image
then
PPT Slide
Lager Image
an exists .
Proof . Consider
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
and
PPT Slide
Lager Image
for any ε > 0, take N ∈ ℕ such that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
for n ≥ N . Thus, lim sup m→∞ am eεan + εeε . Letting ε → 0, we have the wanted result. Hence lim sup m→∞ am ≤ lim inf n→∞ an , which shows the existence of
PPT Slide
Lager Image
an                                  ⧠
Putting δn = 0( ∀ n ∈ ℕ), we have the following known lemma as a corollary;
Lemma 2.1 ( [9] ). Let { an } and { bn } be nonnegative real sequences satisfying the following condition:
  • an+1≤an+bn, ∀n∈ ℕ
If
PPT Slide
Lager Image
then
PPT Slide
Lager Image
an exists .
Now, we prove our main result.
Theorem 2.1. Assume the conditions (1)-(5) in Theorem 1.1. Let ϕ : C × C → ℝ be a bifunction satisfying (A1)-(A4). Assume that F := (
PPT Slide
Lager Image
F ( Ti ))∩ EP ( ϕ )≠
PPT Slide
Lager Image
. For any x 0 C , let { xn } and { vn } be sequences generated by
PPT Slide
Lager Image
where { an }, { bn } and { cn } are sequences in [0; 1) such that an + bn + cn = 1, an k + ε , bn ε for some ε ∈ (0, 1),
PPT Slide
Lager Image
{ un } is a bounded sequence in C , { rn } is a sequence in (0, ∞) such that
PPT Slide
Lager Image
inf rn ≥ 0 and n = ( h −1) N + i ( n ≥ 1), where i = i ( n ) ∈ {1, 2,⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n → ∞. Then { xn } and { vn } converges weakly to an element of F .
Proof . Let p F . From (2.1) and Lemma 1.2, we have v n−1 = S rn−1 x n−1 and
  • ∥vn−1−p∥ = ∥Srn−1xn−1−Srn−1p∥ ≤ ∥xn−1−p∥, ∀n≥ 0.
From (2.1) and Lemma 1.1(ii),
PPT Slide
Lager Image
PPT Slide
Lager Image
From Proposition 2.1,
PPT Slide
Lager Image
xn p ∥ exists. Observe (2.2) again
PPT Slide
Lager Image
Since an k + ε , bn ε for all n ≥ 0 and some ε ∈ (0, 1),
PPT Slide
Lager Image
Taking the limits as n → ∞, we have
PPT Slide
Lager Image
Observe that
PPT Slide
Lager Image
It follows that
PPT Slide
Lager Image
Since S rn−1 is firmly nonexpansive, we have
PPT Slide
Lager Image
and hence
PPT Slide
Lager Image
Using (2.2) and (2.7), we have
  • ∥xn−p∥2≤∥vn−1−p∥2+cn-1∥un−1−p∥2
  •                          ≤∥xn−1−p∥2− ∥xn−1−vn−1∥2+cn−1∥un−1−p∥2,
hence
  • ∥xn−1−vn−1∥2≤∥xn−1−p∥2− ∥xn-1−p∥2+cn−1∥un−1−p∥2.
Since
PPT Slide
Lager Image
xn p ∥ exists and
PPT Slide
Lager Image
k h(n) = 1,
PPT Slide
Lager Image
From (2.6) and (2.8), we have
PPT Slide
Lager Image
It follows that
PPT Slide
Lager Image
Applying (2.8) and (2.9), we obtain
  • ∥xn−xn−1∥ = ∥xn−vn+vn−vn−1+vn−1−xn−1∥
  •                     ≤ ∥xn−vn∥+∥vn−vn−1∥+∥vn−1−xn−1∥ →0 asn→∞,
which implies that
PPT Slide
Lager Image
xn x n+j ∥ = 0, ∀ j ∈ {1, ⋯, N }. Since, for any positive integer n > N , it can be written as n = ( k ( n )−1) N + i ( n ), where i ( n ) ∈ {1, 2,⋯, N }, observe that
PPT Slide
Lager Image
Since, for each n > N , n n N (mod N ) and n = ( k ( n ) − 1) N + i ( n ), we have n N = ( k ( n )−1) N + i ( n ) = ( k ( n N )−1) N + i ( n N ), that is, i ( n N ) = i ( n ), k ( n N ) = k ( n ) − 1. Observe that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
It follows from (2.11)-(2.13) that
PPT Slide
Lager Image
Applying (2.5) and (2.10) to (2.14), we obtain
PPT Slide
Lager Image
From (2.9) and (2.15),
  • ∥vn−Tnvn∥ ≤ ∥vn−vn−1∥ + ∥vn−1−Tnvn−1∥ + ∥Tnvn−1−Tnvn∥
  •                           ≤ (1 +L) · ∥vn−vn−1∥ + ∥vn−1−Tnvn−1∥ → 0 asn→ ∞
Also, we have
  • ∥vn−Tn+jvn∥ ≤ ∥vn−vn+j∥ + ∥vn+j−Tn+jvn+j∥ + ∥Tn+jvn+j−Tn+jvn∥
  •                      ≤ (1 +L) · ∥vn−vn+j∥ + ∥vn+j−Tn+jvn+j∥ → 0 asn→ ∞
for any j = 1, ⋯, N , which gives that
  • ∥vn−Tlvn∥ = 0, ∀l∈ {1, ⋯,N}.
Moreover, for each l ∈ {1, 2,⋯, N }, we have
PPT Slide
Lager Image
Put W ( xn ) = { x H : xni x for some subsequence { xni } of { xn }}.
Firstly, W ( xn ) ≠ø. Indeed, since { xn } is bounded and H is reflexive, W ( xn ) ≠ø.
Secondly, we claim that
PPT Slide
Lager Image
Let w W ( xn ) be an arbitrary element. Then there exists a subsequence { xni } of { xn } converging weakly to w . Applying (2.8), we can obtain that vni w as i → ∞. It follows from
PPT Slide
Lager Image
vn Tlvn ∥ = 0 that Tlvni w , ∀ l ∈ {1, ⋯, N }. Let us show that w EP ( ϕ ). Since vn = Trn vn , we have
  • ϕ(vn,y) +⟨y−vn,vn−xn⟩ ≥ 0, ∀y∈C.
From (A2), we have
  • ⟨y−vn,vn−xn⟩ ≥ϕ(y,vn)
and hence
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
→ 0 and
PPT Slide
Lager Image
as i → ∞, from (A4), we have
  • ϕ(y,w) ≤ 0, ∀y∈C.
For t ∈ (0; 1] and y C , let yt = ty +(1− t ) w . Since y C and w C , yt C , and hence ϕ ( yt , w ) ≤ 0. So, from (A1) and (A4),
  • 0 =ϕ(yt,yt) ≤tϕ(yt,y) + ( 1−t)ϕ(yt,w) ≤tϕ(yt,y)
and hence 0 ≤ ϕ ( yt , y ). From (A3), 0 ≤ ϕ ( w , y ), ∀ y C and hence w EP ( ϕ )
Next, we prove that w ∈ (
PPT Slide
Lager Image
F ( Ti )). Suppose that w ∉ (
PPT Slide
Lager Image
F ( Ti )). Then there exists l ∈ {1, ⋯, N } such that w F ( Tl ). From (2.16) and the Opial’s condition,
PPT Slide
Lager Image
which derives a contradiction. Hence w ∈ (
PPT Slide
Lager Image
F ( Ti )).
Finally, we show that { xn } and { vn } converge weakly to an element of F . Indeed, it is sufficient to show that W ( xn ) is a single point set. We take w 1 , w 2 W ( xn ) arbitrarily and let { xni } and { xnj } be subsequences of { xn } such that xni w 1 and xnj w 2 . Since
PPT Slide
Lager Image
xn p ∥ exists for each p F and w 1 , w 2 F , by Lemma 1.1 (iii), we obtain
PPT Slide
Lager Image
Hence w 1 = w 2 , which shows that W ( xn ) is single point set.                                 ⧠
Remark 2.1. (1) If cn = 0(∀ n N ) in Theorem 2.1, then we obtain Theorem 1.2. (2) If ϕ ( x , y ) = 0(∀ x,y C ) and vn = xn , ∀ n ∈ ℕ in (2.1), then we obtain Theorem 1.1.
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