WEAK CONVERGENCE OF A HYBRID ITERATIVE SCHEME WITH ERRORS FOR EQUILIBRIUM PROBLEMS AND COMMON FIXED POINT PROBLEMS

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. 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

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 space H, (2) Ti : C → C is an asymptotically ki-strictly pseudo-contractive mapping,       where 1 ≤ i ≤ N for some natural number N and 0 ≤ 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 := (F(Ti)) ≠ . 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) ϕ is monotone, i.e., ϕ(x, y) + ϕ(y, x) ≤ 0 for any x, y ∈ C; (A3) ϕ is upper-hemicontinuous, i.e., for each x, y, z ∈C,                   ϕ(tz + (1 − t)x, y) ≤ ϕ(x, y);(A4) ϕ(x, · ) is convex and lower semicontinuous for each x ∈ C.

Assume that F := (F(Ti)) ∩ EP(ϕ) ≠ , where EP(ϕ) is a set of solutions of equilibrium problem (1.1). For any x0 ∈ C, let {xn} and {υn} be sequences generated by

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 some α, 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 in H weakly 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) Sr is single-valued; (ii) Sr is 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 kn = 1 and k ∈ [0, 1) such that

∥Tnx − Tny∥2 ≤ ∥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 and then an exists.

Proof. Consider

Thus Since and for any ε > 0, take N ∈ ℕ such that and for n ≥ N. Thus, lim supm→∞ am ≤ eεan + εeε. Letting ε → 0, we have the wanted result. Hence lim supm→∞ am ≤ lim infn→∞ an, which shows the existence of 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 then 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 := (F(Ti))∩EP(ϕ)≠. For any x0 ∈ C, let {xn} and {vn} be sequences generated by

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), {un} is a bounded sequence in C, {rn} is a sequence in (0, ∞) such that 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 vn−1 = Srn−1xn−1 and

∥vn−1 − p∥ = ∥Srn−1xn−1 − Srn−1p∥ ≤ ∥xn−1 − p∥, ∀ n ≥ 0.

From (2.1) and Lemma 1.1(ii),

From Proposition 2.1, ∥xn − p∥ exists. Observe (2.2) again

Since an ≥ k + ε, bn ≥ ε for all n ≥ 0 and some ε ∈ (0, 1),

Taking the limits as n → ∞, we have

Observe that

It follows that

Since Srn−1 is firmly nonexpansive, we have

and hence

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 ∥xn − p∥ exists and kh(n) = 1,

From (2.6) and (2.8), we have

It follows that

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 as n→∞,

which implies that ∥xn − xn+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

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

and

It follows from (2.11)-(2.13) that

Applying (2.5) and (2.10) to (2.14), we obtain

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 as n → ∞

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 as n → ∞

for any j = 1, ⋯, N, which gives that

∥vn − Tlvn∥ = 0, ∀l ∈ {1, ⋯, N}.

Moreover, for each l ∈ {1, 2,⋯, N}, we have

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

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 ∥vn − Tlvn∥ = 0 that Tlvni → w, ∀l ∈ {1, ⋯, N}. Let us show that w∈ EP(ϕ). Since vn = Trnvn, 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

Since → 0 and 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 ∈ (F(Ti)). Suppose that w ∉ (F(Ti)). Then there exists l ∈ {1, ⋯, N} such that w ∉ F(Tl). From (2.16) and the Opial’s condition,

which derives a contradiction. Hence w∈ (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 w1, w2 ∈ W(xn) arbitrarily and let {xni} and {xnj} be subsequences of {xn} such that xni ⇀ w1 and xnj ⇀ w2. Since ∥xn − p∥ exists for each p ∈ F and w1, w2 ∈ F, by Lemma 1.1 (iii), we obtain

Hence w1 = w2, 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.