Regularization iterative algorithms for monotone and strictly ... - EMIS
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 3909–3919 Research Article
Regularization iterative algorithms for monotone and strictly pseudocontractive mappings Sun Young Choa , Abdul Latifb , Xiaolong Qinc,∗ a
Department of Mathematics, Gyeongsang National University, Jinju, South Korea.
b
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
c
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan, P. R. China. Communicated by Y. J. Cho
Abstract In this article, the sum of a monotone mapping, an inverse strongly monotone mapping, and a strictly pseudocontractive mapping are investigated based on two regularization iterative algorithms. Strong conc vergence analysis of the two iterative algorithms is obtained in the framework of real Hilbert spaces. 2016 All rights reserved. Keywords: Monotone mapping, iterative algorithm, zero point, variational inclusion, resolvent technique. 2010 MSC: 47H05, 90C10.
1. Introduction and Preliminaries In this paper, p we always assume that H is a real Hilbert space with inner product hx, yi and induced norm kxk = hx, xi for x, y ∈ H and C is a nonempty convex and closed subset of H. Let S : C → C be a mapping. In this paper, we use F ix(S) to stand for the set of fixed points of mapping S. Recall that S is said to be contractive iff kSx − Syk ≤ αkx − yk,
∀x, y ∈ C.
We also say that S is an α-contractive mapping. S is said to be an Meir-Keeler contraction iff for every > 0, there exists η > 0 such that kx − yk ≤ + η implies kSx − Syk ≤ , ∀x, y ∈ C. In 1969, Meir and ∗
Corresponding author Email addresses: [email protected] (Sun Young Cho), [email protected] (Abdul Latif), [email protected] (Xiaolong Qin) Received 2016-01-10
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Keeler [16] proved that every Meir-Keeler contraction has a unique fixed point in complete metric spaces; see [16] and the references therein. S is said to be nonexpansive iff kSx − Syk ≤ kx − yk,
∀x, y ∈ C.
If C is bounded convex and closed, then the set of fixed points of S is not empty; see [5, 13] and the references therein. Fixed point theory of the class of nonexpansive mappings, which is powerful and efficient, has been applied to variational inclusion problems of maximal monotone operators in the framework of infinite dimensional Hilbert spaces. One of the most popular techniques for solving inclusion problems of nonlinear mapping B goes back to the work of Browder [6]. The basic ideas is to reduce the inclusion problems to a fixed point problem of mapping (I + hB)−1 , which is called the classical resolvent of B. If B has some monotonicity conditions, the classical resolvent of B is with full domain and nonexpansive, see [21, 22] and the references therein. S is said to be strictly pseudocontractive iff there is a constant λ ∈ [0, 1) such that kSx − Syk2 ≤ kx − yk2 + λkx − y − Sx + Syk2 ,
∀x, y ∈ C.
We also say S is λ-strictly pseudocontractive. The class of λ-strictly pseudocontractive mappings was introduced by Browder and Petryshyn [7] in 1967. It is clear that the class of λ-strictly pseudocontractive mappings strictly include the class of nonexpansive mappings as a special cases. It is also known that every λ-strict pseudocontraction is Lipschitz continuous; see [7] and the references therein. S is said to be pseudocontractive iff kSx − Syk2 ≤ kx − yk2 + kx − y − Sx + Syk2 ,
∀x, y ∈ C.
Let A : C → H be a mapping. Recall that A is said to be monotone iff hx − y, Ax − Ayi ≥ 0,
∀x, y ∈ C.
A is said to be strongly monotone iff there exists a positive constant κ > 0 such that hx − y, Ax − Ayi ≥ κkx − yk2 ,
∀x, y ∈ C.
We also say that A is κ-strongly monotone. A is said to be inverse strongly monotone iff there exists a positive constant κ > 0 such that hAx − Ay, x − yi ≥ κkAx − Ayk2 ,
∀x, y ∈ C.
We also say that A is inverse κ-strongly monotone. From the above, we also see that A is inverse strongly monotone iff A−1 is strongly monotone. Every inverse strongly monotone mapping is monotone and Lipschitz continuous. Recall that the classical variational inequality is to find a point x ¯ in C such that hA¯ x, y − x ¯i ≥ 0,
∀y ∈ C.
(1.1)
The solution set of variational inequality (1.1) is denoted by V I(C, A) in this paper. Projection methods have been recently investigated for solving variational inequality (1.1). It is known that x ¯ is a solution to (1.1) iff x ¯ is a fixed point of mapping P rojC (I − hA), where P rojC is the metric projection from H onto C, h is some positive real number and I denotes the identity on H. If A is strongly monotone, then the existence of solutions of variational inequality (1.1) is guaranteed by the contractivity of the mapping P rojC (I − hA). If A is inverse strongly monotone, then P rojC (I − hA) is nonexpansive. Mann iterative process is an efficient and powerful process to study fixed points of nonexpansive mappings. However, Mann iterative process is only weak convergence in infinite dimensional spaces; see [2] and the references therein. Halpern iterative process (HIP) generates a sequence {xn } in the following manner: xn+1 = αn x + (1 − αn )T xn ,
∀n ≥ 0,
where x is a fixed element, {αn } is a control sequence in (0, 1). HIP was initially introduced in [11]. Halpern showed that the following conditions
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(C1) limn→∞ αn = 0; P∞ (C2) n=0 αn = ∞, are necessary in the sense if Halpern iterative process is strongly convergent for all nonexpansive mappings, then {xn } must satisfy conditions (C1), and (C2). Recently, Halpbern iterative process has been extensively investigated by many author for solving solutions of variational inequality (1.1) and fixed points of nonexpansive mappings; see [8, 12, 15, 20, 25–27] and the references therein. Recall that an operator B : H ⇒ H is said to be monotone iff, for all x, y ∈ H, x0 ∈ Bx and y 0 ∈ By imply hx − y, x0 − y 0 i ≥ 0. In this paper, we use B −1 (0) to stand for the zero point of operator B. A monotone mapping B : H ⇒ H is maximal iff the graph Graph(B) of B is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping B is maximal if and only if, for any (x, x0 ) ∈ H × H, hx − y, x0 − y 0 i ≥ 0, for all (y, y 0 ) ∈ Graph(B) implies y 0 ∈ Bx. If B is maximal monotone, then (I + hB)−1 : H → Domain(B), where Domain(B) denote the domain of B, is single-valued and firmly nonexpansive. Moreover, B −1 (0) = F ix((I + hB)−1 ). One known example of maximal monotone mapping is N + M , where N is the normal cone mapping N x := {x∗ ∈ H : hx∗ , y − xi ≤ 0, ∀y ∈ C}, for x ∈ C and is empty otherwise, and M is a single valued maximal monotone mapping that is continuous on C. Then, 0 ∈ N x + M x iff x ∈ C satisfies variational inequalities of hM x, y − xi ≥ 0 for all y ∈ C. Another example of maximal monotone mapping is ∂B, the subdifferential of a proper closed convex function B : H → (−∞, ∞] which is defined by ∂Bx := {x∗ ∈ H : Bx + hy − x, x∗ i ≤ By, ∀y ∈ H},
∀x ∈ H.
Rockafellar [22] proved that ∂B is a maximal monotone operator. It is easy to verify that 0 ∈ ∂Bv if and only if Bv = minx∈H Bx. Recently, zero points of the sum of two monotone operators have been extensively investigated based on iterative techniques since the problem is applicable in image recovery, signal processing, and machine learning, which are mathematically modeled as a monotone mapping equation and this mapping is decomposed as the sum of two monotone mappings; see [3, 4, 9, 10, 17–19] and the references therein. In this article, we study the sum of a monotone mapping, an inverse strongly monotone mapping, and a strictly pseudocontractive mapping based on two viscosity regularization iterative algorithms. Strong convergence analysis of the iterative algorithms is obtained in the framework of real Hilbert spaces. Before giving the main results, we provide the following lemmas which play an important role in this article. Lemma 1.1 ([14]). Let {µn } be a sequence of nonnegative numbers satisfying P the condition µn+1 ≤ (1 − sn )µn +sn an +bn , ∀n ≥ 0, where {sn } is a number sequence in (0, 1) such that ∞ n→∞ sn = 0, n=0 sn = ∞, lim P∞ {an } is a sequence such that lim supn→∞ an ≤ 0, and {bn } is a positive sequence such that n=0 bn < ∞. Then limn→∞ µn = 0. Lemma 1.2 ([3]). Let C be a nonempty convex and closed subset of a real Hilbert space H. Let A : C → H be a monotone mapping, and B : H ⇒ H a maximal monotone operator. Then F ix((I + hB)−1 (I − hA)) = (B + A)−1 (0), where h is some positive real number. Lemma 1.3 ([1]). Let H be a Hilbert space, and B an mapping 0maximal monotone on H. For x ∈ E, h h0 0 −1 0 −1 −1 h > 0 and h > 0, we have (I + hB) x = (I + h B) h x + 1 − h (I + hB) x . Lemma 1.4 ([7]). Let C be a nonempty convex and closed subset of a real Hilbert space H. Let T : C → C be a λ-strict pseudo-contraction. Then I − T is demiclosed at zero, that is, {xn } converges weakly to some point x ¯ and {xn − T xn } converges strongly to 0, then x ¯ ∈ F ix(T ).
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Lemma 1.5 ([23]). Let H be a real Hilbert space and let {xn } and {yn } be bounded sequences in H. Let {λn } be a sequence in (0, 1) with 1 > lim supn→∞ λn ≥ lim inf n→∞ λn > 0. Suppose xn+1 = λn yn + (1 − λn )xn for all integers n ≥ 0 and lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞
Then limn→∞ kyn − xn k = 0. Lemma 1.6 ([24]). Let S be an Meir-Keeler contraction on a convex subset C of a Banach space E. Then for each > 0, there exists Q ∈ (0, 1) such that kx − yk ≥ implies kSx − Syk ≤ Qkx − yk, ∀x, y ∈ C.
(1.2)
2. Main results First, we give a strong convergence theorem with the aid of contractions. Theorem 2.1. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a fixed α-contractive mapping on C. Let B be a maximal monotone operator on H and A : C → H an inverse κ-strongly monotone mapping. Assume (A+B)−1 (0)∩F ix(T ) 6= ∅. Let {αn }, {βn } and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn − (I + hn B)−1 (zn − hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn } and {h } satisfy the following restrictions: lim n→∞ |hn − P∞ n hn−1 | = limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h0 < 2κ, 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h0 are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ (A + B)−1 (0) ∩ F ix(T ), where x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S x ¯. Proof. First, we show that {xn }, {yn }, and {zn } are bounded sequences. Since A is inverse κ-strongly monotone, we have k(I − hn A)x − (I − hn A)yk2 = kx − yk2 − 2hn hx − y, Ax − Ayi + hn 2 kAx − Ayk2 ≤ kx − yk2 − hn (2κ − hn )kAx − Ayk2 . By using the restriction imposed on {hn }, we have k(I − hn A)x − (I − hn A)yk ≤ kx − yk. That is, I − hn A is a nonexpansive mapping for every n. Fixing p ∈ (A + B)−1 (0) ∩ F ix(T ), we find that kβn xn + (1 − βn )T xn − pk2 = βn kxn − pk2 + (1 − βn )kT xn − pk2 − βn (1 − βn )kT xn − xn k2 ≤ βn kxn − pk2 + (1 − βn )kxn − pk2 − (1 − βn )(βn − λ)kT xn − xn k2 ≤ kxn − pk2 . Hence, we have kzn − pk ≤ αn kSxn − pk + (1 − αn )kβn xn + (1 − βn )T xn − pk ≤ αn kSxn − pk + (1 − αn )kxn − pk ≤ 1 − αn (1 − α) kxn − pk + αn kSp − pk. It follows from Lemma 1.2 that kxn+1 − pk ≤ γn kyn − pk + (1 − γn )kxn − pk
(2.1)
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≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k + γn k(I + hn B)−1 (zn − hn Azn ) − pk + (1 − γn )kxn − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk + (1 − γn )kxn − pk + γn µn ≤ γn kzn − pk + (1 − γn )kxn − pk + γn µn ≤ 1 − αn γn (1 − α) kxn − pk + αn γn kSp − pk + γn µn kSp − pk ≤ max kxn − pk, + µn 1−α .. . ∞
X kSp − pk ≤ max{kx0 − pk, }+ µi < ∞. 1−α i=0
This proves that sequence {xn } is bounded, so are {yn } and {zn }. Since P roj(A+B)−1 (0)∩F ix(T ) S is α-contractive, we find that it has a unique fixed point. Next, we denote the unique fixed point by x ¯. We are in a position to show that lim suph¯ x − Sx ¯, x ¯ − zn i ≤ 0. n→∞
To show this inequality, we choose a subsequence {zni } of {zn } such that lim suph¯ x − Sx ¯, x ¯ − zn i = lim h¯ x − Sx ¯, x ¯ − zni i ≤ 0. i→∞
n→∞
Since {zni } is bounded, there exists a subsequence {znij } of {zni } which converges weakly to x ˆ. Without loss of generality, we assume that zni * x ˆ. Notice that kzn − zn−1 k ≤ 1 − αn (1 − α) kxn − xn−1 k + |αn − αn−1 |(kSxn−1 k + kβn−1 xn−1 + (1 − βn−1 )T xn−1 k) + (1 − αn )|βn − βn−1 |kxn−1 − T xn−1 k ≤ 1 − αn (1 − α) kxn − xn−1 k + (|αn − αn−1 | + |βn − βn−1 |)M, where M is an appropriate constant such that M ≥ supn≥1 {kSxn−1 k + kxn−1 k + kT xn−1 k}. Setting wn = zn − hn Azn , one further has kwn − wn−1 k ≤ khn − hn−1 kkAzn−1 k + kzn − zn−1 k ≤ (1 − αn (1 − α)) kxn − xn−1 k + |hn − hn−1 |kAzn−1 k
(2.2)
+ M (|αn − αn−1 | + |βn − βn−1 |). On the other hand, one has from Lemma 1.3 kyn−1 − yn k ≤ k(I + hn B)−1 wn − (I + hn−1 B)−1 wn−1 k + µn−1 + µn hn−1 −1 −1 −1 rn−1 wn + (1 − )(I + hn B) wn k = k(I + hn−1 B) wn−1 − (I + hn−1 B) hn hn hn−1 hn−1 ≤ k(1 − ) (I + hn B)−1 wn − wn−1 + (wn−1 − wn ) k hn hn |hn−1 − hn | ≤ k(I + hn B)−1 wn − wn k + kwn − wn−1 k. hn Combining (2.2) with (2.3), one finds that kyn−1 − yn k − kxn − xn−1 k ≤ |hn−1 − hn |kAzn−1 k + (|αn − αn−1 | + |βn − βn−1 |)M +
|hn−1 − hn |k(I + hn B)−1 wn − wn k . hn
(2.3)
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Using the restrictions imposed on the control sequences, one finds lim sup(kyn−1 − yn k − kxn − xn−1 k) ≤ 0. n→∞
Using Lemma (1.5), one has limn→∞ kyn − xn k = 0. Since kzn − pk2 ≤ αn kSxn − pk2 + (1 − αn )kβn xn + (1 − βn )T xn − pk2 ≤ αn kSxn − pk2 + (1 − αn )kxn − pk2 , one has from (2.1) kxn+1 − pk2 ≤ γn kyn − pk2 + (1 − γn )kxn − pk2 ≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k2 + γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn γn kSxn − pk2 + kxn − pk2 − hn (2κ − hn )γn kAzn − Apk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . It follows that hn (2κ − hn )γn kAzn − Apk2 ≤ αn γn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . ≤ (kxn − pk + kxn+1 − pk)kxn − xn+1 k + αn kSxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . Therefore, we have lim kAp − Azn k = 0.
n→∞
(2.4)
In view of the firm nonexpansivity of (I + hn B)−1 , one has k(I + hn B)−1 (zn − hn Azn ) − (I + hn B)−1 (p − hn Ap)k2 ≤ h(zn − hn Azn ) − (p − hn Ap), (I + hn B)−1 (zn − hn Azn ) − pi 1 ≤ k(I + hn B)−1 (zn − hn Azn ) − pk2 + kzn − pk2 2 − kzn − (I + hn B)−1 (zn − hn Azn ) − hn (Azn − Ap)k2 1 ≤ kzn − pk2 + k(I + hn B)−1 (zn − hn Azn ) − pk2 2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 − h2n kAzn − Apk2 + 2hn kzn − (I + hn B)−1 (zn − hn Azn )kkAzn − Apk 1 ≤ k(I + hn B)−1 (zn − hn Azn ) − pk2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 2 + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + kzn − pk2 . It follows that k(I + hn B)−1 (zn − hn Azn )n − pk2 ≤ kzn − pk2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk.
(2.5)
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Thanks to (2.5), one sees kxn+1 − pk2 ≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k2 + γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ γn kzn − pk2 − γn kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn γn kSxn − pk2 + kxn − pk2 − γn kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . This yields that γn kzn − (I + hn B)−1 (zn − hn Azn )k2 ≤ αn γn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn kSxn − pk2 + (kxn − pk + kxn+1 − pk)kxn − xn+1 k + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn k(I + hn B)−1 (zn − hn Azn ) − pk + µ2n . Since limn→∞ kyn − xn k = 0, we find that lim kxn+1 − xn k = 0.
(2.6)
lim kzn − (I + hn B)−1 (zn − hn Azn )k = 0.
(2.7)
n→∞
Using (2.4) and (2.6), we have n→∞
On the other hand, one has kzn − xn k ≤ kzn − (I + hn B)−1 (zn − hn Azn )k + kyn − xn k + µn . From (2.7), one sees lim kzn − xn k = 0.
n→∞
(2.8)
It follows that xni * x ˆ. Note that kT xn − xn k ≤
αn 1 kzn − xn k + kSxn − xn k. (1 − αn )(1 − βn ) (1 − αn )(1 − βn )
This yields from (2.8) and the restrictions imposed on control sequences {αn } and {βn } that lim kT xn − xn k = 0.
n→∞
From Lemma 1.4, we conclude x ˆ ∈ F ix(T ). We are in a position to conclude x ˆ ∈ (A + B)−1 (0). Putting 0 −1 0 0 yn = (I + hn B) (zn − hn Azn ), one has zn − hn Azn − yn ∈ hn Byn . Let w1 be an element in Bw2 . Since B is maximal monotone, we find that zn − yn0 0 w1 − + Azn , w2 − yn ≥ 0. hn
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Hence, one has 0 ≤ hw1 + Aˆ x, w2 − x ˆi. This implies that B x ˆ 3 −Aˆ x, that is, x ˆ ∈ (A + B)−1 (0). This proves x ¯ ∈ (A + B)−1 (0). Hence, we have lim suph¯ x − Sx ¯, x ¯ − zn i ≤ 0, (2.9) n→∞
Notice that kzn − x ¯k2 ≤ αn hSxn − S x ¯, zn − x ¯i + αn hS x ¯−x ¯, zn − x ¯i + (1 − αn )kxn − pkkzn − x ¯k ≤ (1 − αn (1 − α)) kxn − x ¯kkzn − x ¯k + αn h¯ x − Sx ¯, x ¯ − zn i, from which it follows that kzn − x ¯k2 ≤ (1 − αn (1 − α))kxn − x ¯k2 + 2αn h¯ x − Sx ¯, x ¯ − zn i. This yields that kxn+1 − x ¯k2 ≤ γn kyn − x ¯k2 + (1 − γn )kxn − x ¯k2 ≤ γn kyn − yn0 k2 + γn kyn0 − x ¯k2 + (1 − γn )kxn − x ¯k2 + 2γn µn kyn0 − x ¯k ≤ γn kzn0 − x ¯k2 + (1 − γn )kxn − x ¯k2 + 2γn µn kyn0 − x ¯k + γn µ2n ≤ (1 − αn γn (1 − α))kxn − x ¯k2 + 2αn γn h¯ x − Sx ¯, x ¯ − zn i + 2µn kyn0 − x ¯k + µ2n . Since
P∞
n=0 µn
< ∞, we conclude from Lemma 1.1 that {xn } converges in norm to x ¯, where x ¯ = P rojF ix(T )∩(A+B)−1 (0) S x ¯,
that is, x ¯ is the unique solution to the following variational inequality: hS x ¯−x ¯, x ¯ − x0 i ≥ 0, ∀x0 ∈ F ix(T ) ∩ (A + B)−1 (0).
Next, we give another strong convergence theorem with the aid of Meir-Keeler contractions. Theorem 2.2. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S¯ a Meir-Keeler contraction on C. Let B be a maximal monotone operator on H and A : C → H an inverse κ-strongly monotone mapping. Assume (A+B)−1 (0)∩F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( ¯ n + (1 − αn )(1 − βn )T xn , zn = βn (1 − αn )xn + αn Sx xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn −(I +hn B)−1 (zn −hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and {h } satisfy the following restrictions: limn→∞ |hn − n P 0 hn−1 | = limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, ∞ α = ∞, 0 < γ ≤ γ , 0 < h ≤ h n n ≤ h < 2κ, n=0 n 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h, and h0 are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ (A + B)−1 (0) ∩ F ix(T ), where x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S¯x ¯. Proof. Set ( z¯n = βn (1 − αn )¯ xn + αn S x ¯ + (1 − αn )(1 − βn )T x ¯n , x ¯n+1 = γn y¯n + (1 − γn )¯ xn , ∀n ≥ 0, where {¯ yn } is a sequence in C such that k¯ yn − (I + hn B)−1 (¯ zn − hn A¯ zn )k ≤ µn and x ¯ is a fixed element in C. From Theorem 2.1, one sees that {¯ xn } converges in norm to x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S x ¯.
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Finally, we prove that xn → x ¯ as n → ∞. To end this, we need to show that xn − x ¯n → 0 as n → ∞. Arguing by contradiction, we assume lim supn→∞ kxn − x ¯n k = d > 0. Then we choose with 0 < < d. xn −¯ xk Using Lemma 1.6, for such , there exists Q ∈ (0, 1) satisfying (1.2). We also choose m such that Qk¯ 0, i.e., (I + h∂iC )−1 . Letting x = (I + h∂iC )−1 y, we find that y ∈ x + r∂iC x ⇐⇒ y ∈ hNC x + x ⇐⇒ P rojC y = x, where P rojC is the metric projection from H onto C and NC x := {e ∈ H : he, v − xi, ∀v ∈ C}. Corollary 2.5. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a fixed α-contractive mapping on C. Let A : C → H be an inverse κ-strongly monotone mapping. Assume V I(C, A) ∩ F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn −P rojC (zn −hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and P {hn } satisfy the following restrictions: limn→∞ |hn − hn−1 | = ∞ 0 limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h < 2κ, and 0 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ V I(C, A) ∩ F ix(T ), where x ¯ = P rojV I(C,A)∩F ix(T ) S x ¯. Corollary 2.6. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a Meir-Keeler contraction on C. Let A : C → H be an inverse κ-strongly monotone mapping. Assume V I(C, A) ∩ F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn − P rojC (zn − hn Azn )k ≤ µn where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and P {hn } satisfy the following restrictions: limn→∞ |hn − hn−1 | = ∞ 0 limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h < 2κ, and 0 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ V I(C, A) ∩ F ix(T ), where x ¯ = P rojV I(C,A)∩F ix(T ) S x ¯.
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Acknowledgements This article was supported by the National Natural Science Foundation of China under Grant No.11401152. References [1] V. Barbu, Nonlinear semigroups and differential equations in Banach space, Translated from the Romanian. Editura Academiei Republicii Socialiste Romˆ ania, Bucharest, Noordhoff International Publishing, Leiden, (1976). 1.3 [2] H. H. Bauschke, E. Matouskova, S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal., 56 (2004), 715–738. 1 [3] B. A. Bin Dehaish, A. Latif, H. O. Bakodah, X. Qin, A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces, J. Inequal. Appl., 2015 (2015), 14 pages. 1, 1.2 [4] B. A. Bin Dehaish, X. Qin, A. Latif, H. O. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321–1336. 1 [5] F. E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1272–1276. 1 [6] F. E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Nat. Acad. Sci. U.S.A., 56 (1966), 1080–1086. 1 [7] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228. 1, 1.4 [8] S. Y. Cho, X. Qin, On the strong convergence of an iterative process for asymptotically strict pseudo-contractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430–438. 1 [9] S. Y. Cho, X. Qin, L. Wang, Strong convergence of a splitting algorithm for treating monotone operators, Fixed Point Theory Appl., 2014 (2014), 15 pages. 1 [10] X. P. Ding, J. C. Yao, Existence and algorithm of solutions for a new system of generalized mixed quasi-variational inclusions involving (h, η)-monotone mappings, Nonlineaer Funct. Anal. Appl., 12 (2007), 645–657. 1 [11] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957–961. 1 [12] J. K. Kim, S. Y. Cho, X. Qin, Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci. Ser. B Engl. Ed., 31 (2015), 2041–2057. 1 [13] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006. 1 [14] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194 (1995), 114–125. 1.1 [15] G. Marino, L. Muglia, Y. Yao, Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points, Nonlinear Anal., 75 (2012), 1787–1798. 1 [16] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326–329. 1 [17] M. A. Noor, Two-step approximation schemes for multivalued quasi variational inclusions, Nonlinear Funct. Anal. Appl., 7 (2002), 1–14. 1 [18] X. Qin, S. Y. Cho, L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014), 10 pages. [19] X. Qin, S. Y. Cho, L. Wang, Convergence of splitting algorithms for the sum of two accretive operators with applications, Fixed Point Theory Appl., 2014 (2014), 12 pages. 1 [20] X. Qin, S. Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl., 2013 (2013), 17 pages. 1 [21] R. T. Rockfellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116. 1 [22] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization, 14 (1976), 877–898. 1, 1 [23] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239. 1.5 [24] T. Suzuki, Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl., 325 (2005), 342–352. 1.6 [25] Z. M. Wang, X. Zhang, Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems, J. Nonlinear Funct. Anal., 214 (2014), 15 pages. 1 [26] L. Yang, F. Zhao, Strong convergence theorems by Halpern-type iterations for relatively nonexpansive multi-valued mappings in Banach spaces, Nonlinear Funct. Anal. Appl., 17 (2012), 433–441. [27] J. Zhao, S. Wang, Viscosity approximation method for the split common fixed point problem of quasi-strict pseudocontractions without prior knowledge of operator norms, Noninear Funct. Anal. Appl., 20 (2015), 199–213. 1
Regularization iterative algorithms for monotone and strictly pseudocontractive mappings Sun Young Choa , Abdul Latifb , Xiaolong Qinc,∗ a
Department of Mathematics, Gyeongsang National University, Jinju, South Korea.
b
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
c
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Sichuan, P. R. China. Communicated by Y. J. Cho
Abstract In this article, the sum of a monotone mapping, an inverse strongly monotone mapping, and a strictly pseudocontractive mapping are investigated based on two regularization iterative algorithms. Strong conc vergence analysis of the two iterative algorithms is obtained in the framework of real Hilbert spaces. 2016 All rights reserved. Keywords: Monotone mapping, iterative algorithm, zero point, variational inclusion, resolvent technique. 2010 MSC: 47H05, 90C10.
1. Introduction and Preliminaries In this paper, p we always assume that H is a real Hilbert space with inner product hx, yi and induced norm kxk = hx, xi for x, y ∈ H and C is a nonempty convex and closed subset of H. Let S : C → C be a mapping. In this paper, we use F ix(S) to stand for the set of fixed points of mapping S. Recall that S is said to be contractive iff kSx − Syk ≤ αkx − yk,
∀x, y ∈ C.
We also say that S is an α-contractive mapping. S is said to be an Meir-Keeler contraction iff for every > 0, there exists η > 0 such that kx − yk ≤ + η implies kSx − Syk ≤ , ∀x, y ∈ C. In 1969, Meir and ∗
Corresponding author Email addresses: [email protected] (Sun Young Cho), [email protected] (Abdul Latif), [email protected] (Xiaolong Qin) Received 2016-01-10
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Keeler [16] proved that every Meir-Keeler contraction has a unique fixed point in complete metric spaces; see [16] and the references therein. S is said to be nonexpansive iff kSx − Syk ≤ kx − yk,
∀x, y ∈ C.
If C is bounded convex and closed, then the set of fixed points of S is not empty; see [5, 13] and the references therein. Fixed point theory of the class of nonexpansive mappings, which is powerful and efficient, has been applied to variational inclusion problems of maximal monotone operators in the framework of infinite dimensional Hilbert spaces. One of the most popular techniques for solving inclusion problems of nonlinear mapping B goes back to the work of Browder [6]. The basic ideas is to reduce the inclusion problems to a fixed point problem of mapping (I + hB)−1 , which is called the classical resolvent of B. If B has some monotonicity conditions, the classical resolvent of B is with full domain and nonexpansive, see [21, 22] and the references therein. S is said to be strictly pseudocontractive iff there is a constant λ ∈ [0, 1) such that kSx − Syk2 ≤ kx − yk2 + λkx − y − Sx + Syk2 ,
∀x, y ∈ C.
We also say S is λ-strictly pseudocontractive. The class of λ-strictly pseudocontractive mappings was introduced by Browder and Petryshyn [7] in 1967. It is clear that the class of λ-strictly pseudocontractive mappings strictly include the class of nonexpansive mappings as a special cases. It is also known that every λ-strict pseudocontraction is Lipschitz continuous; see [7] and the references therein. S is said to be pseudocontractive iff kSx − Syk2 ≤ kx − yk2 + kx − y − Sx + Syk2 ,
∀x, y ∈ C.
Let A : C → H be a mapping. Recall that A is said to be monotone iff hx − y, Ax − Ayi ≥ 0,
∀x, y ∈ C.
A is said to be strongly monotone iff there exists a positive constant κ > 0 such that hx − y, Ax − Ayi ≥ κkx − yk2 ,
∀x, y ∈ C.
We also say that A is κ-strongly monotone. A is said to be inverse strongly monotone iff there exists a positive constant κ > 0 such that hAx − Ay, x − yi ≥ κkAx − Ayk2 ,
∀x, y ∈ C.
We also say that A is inverse κ-strongly monotone. From the above, we also see that A is inverse strongly monotone iff A−1 is strongly monotone. Every inverse strongly monotone mapping is monotone and Lipschitz continuous. Recall that the classical variational inequality is to find a point x ¯ in C such that hA¯ x, y − x ¯i ≥ 0,
∀y ∈ C.
(1.1)
The solution set of variational inequality (1.1) is denoted by V I(C, A) in this paper. Projection methods have been recently investigated for solving variational inequality (1.1). It is known that x ¯ is a solution to (1.1) iff x ¯ is a fixed point of mapping P rojC (I − hA), where P rojC is the metric projection from H onto C, h is some positive real number and I denotes the identity on H. If A is strongly monotone, then the existence of solutions of variational inequality (1.1) is guaranteed by the contractivity of the mapping P rojC (I − hA). If A is inverse strongly monotone, then P rojC (I − hA) is nonexpansive. Mann iterative process is an efficient and powerful process to study fixed points of nonexpansive mappings. However, Mann iterative process is only weak convergence in infinite dimensional spaces; see [2] and the references therein. Halpern iterative process (HIP) generates a sequence {xn } in the following manner: xn+1 = αn x + (1 − αn )T xn ,
∀n ≥ 0,
where x is a fixed element, {αn } is a control sequence in (0, 1). HIP was initially introduced in [11]. Halpern showed that the following conditions
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(C1) limn→∞ αn = 0; P∞ (C2) n=0 αn = ∞, are necessary in the sense if Halpern iterative process is strongly convergent for all nonexpansive mappings, then {xn } must satisfy conditions (C1), and (C2). Recently, Halpbern iterative process has been extensively investigated by many author for solving solutions of variational inequality (1.1) and fixed points of nonexpansive mappings; see [8, 12, 15, 20, 25–27] and the references therein. Recall that an operator B : H ⇒ H is said to be monotone iff, for all x, y ∈ H, x0 ∈ Bx and y 0 ∈ By imply hx − y, x0 − y 0 i ≥ 0. In this paper, we use B −1 (0) to stand for the zero point of operator B. A monotone mapping B : H ⇒ H is maximal iff the graph Graph(B) of B is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping B is maximal if and only if, for any (x, x0 ) ∈ H × H, hx − y, x0 − y 0 i ≥ 0, for all (y, y 0 ) ∈ Graph(B) implies y 0 ∈ Bx. If B is maximal monotone, then (I + hB)−1 : H → Domain(B), where Domain(B) denote the domain of B, is single-valued and firmly nonexpansive. Moreover, B −1 (0) = F ix((I + hB)−1 ). One known example of maximal monotone mapping is N + M , where N is the normal cone mapping N x := {x∗ ∈ H : hx∗ , y − xi ≤ 0, ∀y ∈ C}, for x ∈ C and is empty otherwise, and M is a single valued maximal monotone mapping that is continuous on C. Then, 0 ∈ N x + M x iff x ∈ C satisfies variational inequalities of hM x, y − xi ≥ 0 for all y ∈ C. Another example of maximal monotone mapping is ∂B, the subdifferential of a proper closed convex function B : H → (−∞, ∞] which is defined by ∂Bx := {x∗ ∈ H : Bx + hy − x, x∗ i ≤ By, ∀y ∈ H},
∀x ∈ H.
Rockafellar [22] proved that ∂B is a maximal monotone operator. It is easy to verify that 0 ∈ ∂Bv if and only if Bv = minx∈H Bx. Recently, zero points of the sum of two monotone operators have been extensively investigated based on iterative techniques since the problem is applicable in image recovery, signal processing, and machine learning, which are mathematically modeled as a monotone mapping equation and this mapping is decomposed as the sum of two monotone mappings; see [3, 4, 9, 10, 17–19] and the references therein. In this article, we study the sum of a monotone mapping, an inverse strongly monotone mapping, and a strictly pseudocontractive mapping based on two viscosity regularization iterative algorithms. Strong convergence analysis of the iterative algorithms is obtained in the framework of real Hilbert spaces. Before giving the main results, we provide the following lemmas which play an important role in this article. Lemma 1.1 ([14]). Let {µn } be a sequence of nonnegative numbers satisfying P the condition µn+1 ≤ (1 − sn )µn +sn an +bn , ∀n ≥ 0, where {sn } is a number sequence in (0, 1) such that ∞ n→∞ sn = 0, n=0 sn = ∞, lim P∞ {an } is a sequence such that lim supn→∞ an ≤ 0, and {bn } is a positive sequence such that n=0 bn < ∞. Then limn→∞ µn = 0. Lemma 1.2 ([3]). Let C be a nonempty convex and closed subset of a real Hilbert space H. Let A : C → H be a monotone mapping, and B : H ⇒ H a maximal monotone operator. Then F ix((I + hB)−1 (I − hA)) = (B + A)−1 (0), where h is some positive real number. Lemma 1.3 ([1]). Let H be a Hilbert space, and B an mapping 0maximal monotone on H. For x ∈ E, h h0 0 −1 0 −1 −1 h > 0 and h > 0, we have (I + hB) x = (I + h B) h x + 1 − h (I + hB) x . Lemma 1.4 ([7]). Let C be a nonempty convex and closed subset of a real Hilbert space H. Let T : C → C be a λ-strict pseudo-contraction. Then I − T is demiclosed at zero, that is, {xn } converges weakly to some point x ¯ and {xn − T xn } converges strongly to 0, then x ¯ ∈ F ix(T ).
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Lemma 1.5 ([23]). Let H be a real Hilbert space and let {xn } and {yn } be bounded sequences in H. Let {λn } be a sequence in (0, 1) with 1 > lim supn→∞ λn ≥ lim inf n→∞ λn > 0. Suppose xn+1 = λn yn + (1 − λn )xn for all integers n ≥ 0 and lim sup(kyn+1 − yn k − kxn+1 − xn k) ≤ 0. n→∞
Then limn→∞ kyn − xn k = 0. Lemma 1.6 ([24]). Let S be an Meir-Keeler contraction on a convex subset C of a Banach space E. Then for each > 0, there exists Q ∈ (0, 1) such that kx − yk ≥ implies kSx − Syk ≤ Qkx − yk, ∀x, y ∈ C.
(1.2)
2. Main results First, we give a strong convergence theorem with the aid of contractions. Theorem 2.1. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a fixed α-contractive mapping on C. Let B be a maximal monotone operator on H and A : C → H an inverse κ-strongly monotone mapping. Assume (A+B)−1 (0)∩F ix(T ) 6= ∅. Let {αn }, {βn } and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn − (I + hn B)−1 (zn − hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn } and {h } satisfy the following restrictions: lim n→∞ |hn − P∞ n hn−1 | = limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h0 < 2κ, 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h0 are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ (A + B)−1 (0) ∩ F ix(T ), where x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S x ¯. Proof. First, we show that {xn }, {yn }, and {zn } are bounded sequences. Since A is inverse κ-strongly monotone, we have k(I − hn A)x − (I − hn A)yk2 = kx − yk2 − 2hn hx − y, Ax − Ayi + hn 2 kAx − Ayk2 ≤ kx − yk2 − hn (2κ − hn )kAx − Ayk2 . By using the restriction imposed on {hn }, we have k(I − hn A)x − (I − hn A)yk ≤ kx − yk. That is, I − hn A is a nonexpansive mapping for every n. Fixing p ∈ (A + B)−1 (0) ∩ F ix(T ), we find that kβn xn + (1 − βn )T xn − pk2 = βn kxn − pk2 + (1 − βn )kT xn − pk2 − βn (1 − βn )kT xn − xn k2 ≤ βn kxn − pk2 + (1 − βn )kxn − pk2 − (1 − βn )(βn − λ)kT xn − xn k2 ≤ kxn − pk2 . Hence, we have kzn − pk ≤ αn kSxn − pk + (1 − αn )kβn xn + (1 − βn )T xn − pk ≤ αn kSxn − pk + (1 − αn )kxn − pk ≤ 1 − αn (1 − α) kxn − pk + αn kSp − pk. It follows from Lemma 1.2 that kxn+1 − pk ≤ γn kyn − pk + (1 − γn )kxn − pk
(2.1)
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≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k + γn k(I + hn B)−1 (zn − hn Azn ) − pk + (1 − γn )kxn − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk + (1 − γn )kxn − pk + γn µn ≤ γn kzn − pk + (1 − γn )kxn − pk + γn µn ≤ 1 − αn γn (1 − α) kxn − pk + αn γn kSp − pk + γn µn kSp − pk ≤ max kxn − pk, + µn 1−α .. . ∞
X kSp − pk ≤ max{kx0 − pk, }+ µi < ∞. 1−α i=0
This proves that sequence {xn } is bounded, so are {yn } and {zn }. Since P roj(A+B)−1 (0)∩F ix(T ) S is α-contractive, we find that it has a unique fixed point. Next, we denote the unique fixed point by x ¯. We are in a position to show that lim suph¯ x − Sx ¯, x ¯ − zn i ≤ 0. n→∞
To show this inequality, we choose a subsequence {zni } of {zn } such that lim suph¯ x − Sx ¯, x ¯ − zn i = lim h¯ x − Sx ¯, x ¯ − zni i ≤ 0. i→∞
n→∞
Since {zni } is bounded, there exists a subsequence {znij } of {zni } which converges weakly to x ˆ. Without loss of generality, we assume that zni * x ˆ. Notice that kzn − zn−1 k ≤ 1 − αn (1 − α) kxn − xn−1 k + |αn − αn−1 |(kSxn−1 k + kβn−1 xn−1 + (1 − βn−1 )T xn−1 k) + (1 − αn )|βn − βn−1 |kxn−1 − T xn−1 k ≤ 1 − αn (1 − α) kxn − xn−1 k + (|αn − αn−1 | + |βn − βn−1 |)M, where M is an appropriate constant such that M ≥ supn≥1 {kSxn−1 k + kxn−1 k + kT xn−1 k}. Setting wn = zn − hn Azn , one further has kwn − wn−1 k ≤ khn − hn−1 kkAzn−1 k + kzn − zn−1 k ≤ (1 − αn (1 − α)) kxn − xn−1 k + |hn − hn−1 |kAzn−1 k
(2.2)
+ M (|αn − αn−1 | + |βn − βn−1 |). On the other hand, one has from Lemma 1.3 kyn−1 − yn k ≤ k(I + hn B)−1 wn − (I + hn−1 B)−1 wn−1 k + µn−1 + µn hn−1 −1 −1 −1 rn−1 wn + (1 − )(I + hn B) wn k = k(I + hn−1 B) wn−1 − (I + hn−1 B) hn hn hn−1 hn−1 ≤ k(1 − ) (I + hn B)−1 wn − wn−1 + (wn−1 − wn ) k hn hn |hn−1 − hn | ≤ k(I + hn B)−1 wn − wn k + kwn − wn−1 k. hn Combining (2.2) with (2.3), one finds that kyn−1 − yn k − kxn − xn−1 k ≤ |hn−1 − hn |kAzn−1 k + (|αn − αn−1 | + |βn − βn−1 |)M +
|hn−1 − hn |k(I + hn B)−1 wn − wn k . hn
(2.3)
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Using the restrictions imposed on the control sequences, one finds lim sup(kyn−1 − yn k − kxn − xn−1 k) ≤ 0. n→∞
Using Lemma (1.5), one has limn→∞ kyn − xn k = 0. Since kzn − pk2 ≤ αn kSxn − pk2 + (1 − αn )kβn xn + (1 − βn )T xn − pk2 ≤ αn kSxn − pk2 + (1 − αn )kxn − pk2 , one has from (2.1) kxn+1 − pk2 ≤ γn kyn − pk2 + (1 − γn )kxn − pk2 ≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k2 + γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn γn kSxn − pk2 + kxn − pk2 − hn (2κ − hn )γn kAzn − Apk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . It follows that hn (2κ − hn )γn kAzn − Apk2 ≤ αn γn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . ≤ (kxn − pk + kxn+1 − pk)kxn − xn+1 k + αn kSxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . Therefore, we have lim kAp − Azn k = 0.
n→∞
(2.4)
In view of the firm nonexpansivity of (I + hn B)−1 , one has k(I + hn B)−1 (zn − hn Azn ) − (I + hn B)−1 (p − hn Ap)k2 ≤ h(zn − hn Azn ) − (p − hn Ap), (I + hn B)−1 (zn − hn Azn ) − pi 1 ≤ k(I + hn B)−1 (zn − hn Azn ) − pk2 + kzn − pk2 2 − kzn − (I + hn B)−1 (zn − hn Azn ) − hn (Azn − Ap)k2 1 ≤ kzn − pk2 + k(I + hn B)−1 (zn − hn Azn ) − pk2 2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 − h2n kAzn − Apk2 + 2hn kzn − (I + hn B)−1 (zn − hn Azn )kkAzn − Apk 1 ≤ k(I + hn B)−1 (zn − hn Azn ) − pk2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 2 + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + kzn − pk2 . It follows that k(I + hn B)−1 (zn − hn Azn )n − pk2 ≤ kzn − pk2 − kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk.
(2.5)
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Thanks to (2.5), one sees kxn+1 − pk2 ≤ γn kyn − (I + hn B)−1 (zn − hn Azn )k2 + γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk ≤ γn k(I + hn B)−1 (zn − hn Azn ) − pk2 + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ γn kzn − pk2 − γn kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + (1 − γn )kxn − pk2 + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn γn kSxn − pk2 + kxn − pk2 − γn kzn − (I + hn B)−1 (zn − hn Azn )k2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n . This yields that γn kzn − (I + hn B)−1 (zn − hn Azn )k2 ≤ αn γn kSxn − pk2 + kxn − pk2 − kxn+1 − pk2 + 2hn γn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn γn k(I + hn B)−1 (zn − hn Azn ) − pk + γn µ2n ≤ αn kSxn − pk2 + (kxn − pk + kxn+1 − pk)kxn − xn+1 k + 2hn k(I + hn B)−1 (zn − hn Azn ) − zn kkAzn − Apk + 2µn k(I + hn B)−1 (zn − hn Azn ) − pk + µ2n . Since limn→∞ kyn − xn k = 0, we find that lim kxn+1 − xn k = 0.
(2.6)
lim kzn − (I + hn B)−1 (zn − hn Azn )k = 0.
(2.7)
n→∞
Using (2.4) and (2.6), we have n→∞
On the other hand, one has kzn − xn k ≤ kzn − (I + hn B)−1 (zn − hn Azn )k + kyn − xn k + µn . From (2.7), one sees lim kzn − xn k = 0.
n→∞
(2.8)
It follows that xni * x ˆ. Note that kT xn − xn k ≤
αn 1 kzn − xn k + kSxn − xn k. (1 − αn )(1 − βn ) (1 − αn )(1 − βn )
This yields from (2.8) and the restrictions imposed on control sequences {αn } and {βn } that lim kT xn − xn k = 0.
n→∞
From Lemma 1.4, we conclude x ˆ ∈ F ix(T ). We are in a position to conclude x ˆ ∈ (A + B)−1 (0). Putting 0 −1 0 0 yn = (I + hn B) (zn − hn Azn ), one has zn − hn Azn − yn ∈ hn Byn . Let w1 be an element in Bw2 . Since B is maximal monotone, we find that zn − yn0 0 w1 − + Azn , w2 − yn ≥ 0. hn
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Hence, one has 0 ≤ hw1 + Aˆ x, w2 − x ˆi. This implies that B x ˆ 3 −Aˆ x, that is, x ˆ ∈ (A + B)−1 (0). This proves x ¯ ∈ (A + B)−1 (0). Hence, we have lim suph¯ x − Sx ¯, x ¯ − zn i ≤ 0, (2.9) n→∞
Notice that kzn − x ¯k2 ≤ αn hSxn − S x ¯, zn − x ¯i + αn hS x ¯−x ¯, zn − x ¯i + (1 − αn )kxn − pkkzn − x ¯k ≤ (1 − αn (1 − α)) kxn − x ¯kkzn − x ¯k + αn h¯ x − Sx ¯, x ¯ − zn i, from which it follows that kzn − x ¯k2 ≤ (1 − αn (1 − α))kxn − x ¯k2 + 2αn h¯ x − Sx ¯, x ¯ − zn i. This yields that kxn+1 − x ¯k2 ≤ γn kyn − x ¯k2 + (1 − γn )kxn − x ¯k2 ≤ γn kyn − yn0 k2 + γn kyn0 − x ¯k2 + (1 − γn )kxn − x ¯k2 + 2γn µn kyn0 − x ¯k ≤ γn kzn0 − x ¯k2 + (1 − γn )kxn − x ¯k2 + 2γn µn kyn0 − x ¯k + γn µ2n ≤ (1 − αn γn (1 − α))kxn − x ¯k2 + 2αn γn h¯ x − Sx ¯, x ¯ − zn i + 2µn kyn0 − x ¯k + µ2n . Since
P∞
n=0 µn
< ∞, we conclude from Lemma 1.1 that {xn } converges in norm to x ¯, where x ¯ = P rojF ix(T )∩(A+B)−1 (0) S x ¯,
that is, x ¯ is the unique solution to the following variational inequality: hS x ¯−x ¯, x ¯ − x0 i ≥ 0, ∀x0 ∈ F ix(T ) ∩ (A + B)−1 (0).
Next, we give another strong convergence theorem with the aid of Meir-Keeler contractions. Theorem 2.2. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S¯ a Meir-Keeler contraction on C. Let B be a maximal monotone operator on H and A : C → H an inverse κ-strongly monotone mapping. Assume (A+B)−1 (0)∩F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( ¯ n + (1 − αn )(1 − βn )T xn , zn = βn (1 − αn )xn + αn Sx xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn −(I +hn B)−1 (zn −hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and {h } satisfy the following restrictions: limn→∞ |hn − n P 0 hn−1 | = limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, ∞ α = ∞, 0 < γ ≤ γ , 0 < h ≤ h n n ≤ h < 2κ, n=0 n 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h, and h0 are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ (A + B)−1 (0) ∩ F ix(T ), where x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S¯x ¯. Proof. Set ( z¯n = βn (1 − αn )¯ xn + αn S x ¯ + (1 − αn )(1 − βn )T x ¯n , x ¯n+1 = γn y¯n + (1 − γn )¯ xn , ∀n ≥ 0, where {¯ yn } is a sequence in C such that k¯ yn − (I + hn B)−1 (¯ zn − hn A¯ zn )k ≤ µn and x ¯ is a fixed element in C. From Theorem 2.1, one sees that {¯ xn } converges in norm to x ¯ = P roj(A+B)−1 (0)∩F ix(T ) S x ¯.
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Finally, we prove that xn → x ¯ as n → ∞. To end this, we need to show that xn − x ¯n → 0 as n → ∞. Arguing by contradiction, we assume lim supn→∞ kxn − x ¯n k = d > 0. Then we choose with 0 < < d. xn −¯ xk Using Lemma 1.6, for such , there exists Q ∈ (0, 1) satisfying (1.2). We also choose m such that Qk¯ 0, i.e., (I + h∂iC )−1 . Letting x = (I + h∂iC )−1 y, we find that y ∈ x + r∂iC x ⇐⇒ y ∈ hNC x + x ⇐⇒ P rojC y = x, where P rojC is the metric projection from H onto C and NC x := {e ∈ H : he, v − xi, ∀v ∈ C}. Corollary 2.5. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a fixed α-contractive mapping on C. Let A : C → H be an inverse κ-strongly monotone mapping. Assume V I(C, A) ∩ F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn −P rojC (zn −hn Azn )k ≤ µn , where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and P {hn } satisfy the following restrictions: limn→∞ |hn − hn−1 | = ∞ 0 limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h < 2κ, and 0 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ V I(C, A) ∩ F ix(T ), where x ¯ = P rojV I(C,A)∩F ix(T ) S x ¯. Corollary 2.6. Let C be a nonempty convex closed subset of a real Hilbert space H. Let T be a λ-strictly pseudocontractive mapping on C and S a Meir-Keeler contraction on C. Let A : C → H be an inverse κ-strongly monotone mapping. Assume V I(C, A) ∩ F ix(T ) 6= ∅. Let {αn }, {βn }, and {γn } be real number sequences in [0, 1] and let {hn } be a positive real number sequence in (0, 2κ). Let {xn } be a sequence in C in the following process: x0 ∈ C and ( zn = βn (1 − αn )xn + αn Sxn + (1 − αn )(1 − βn )T xn , xn+1 = γn yn + (1 − γn )xn , ∀n ≥ 0, P where {yn } is a sequence in C such that kyn − P rojC (zn − hn Azn )k ≤ µn where ∞ n=1 µn < ∞. Assume that the control sequences {αn }, {βn }, {γn }, and P {hn } satisfy the following restrictions: limn→∞ |hn − hn−1 | = ∞ 0 limn→∞ |βn − βn−1 | = 0, limn→∞ αn = 0, n=0 αn = ∞, 0 < γ ≤ γn , 0 < h ≤ hn ≤ h < 2κ, and 0 0 ≤ λ ≤ βn ≤ β < 1, where β, γ, h and h are four real numbers. Then {xn } converges in norm to a point x ¯ ∈ V I(C, A) ∩ F ix(T ), where x ¯ = P rojV I(C,A)∩F ix(T ) S x ¯.
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