# Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

Authors: Lu-Chuan Ceng; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Assume thatF is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset C of a real Hilbert space H. Assume also that Ω is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on C. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation F, which generates two sequences from an arbitrary initial point x0∈H. These two sequences are shown to converge in norm to the same point PΩx0 under very mild assumptions.


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## Body

## 1. Introduction and Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ||·|| and C a nonempty closed convex subset of H. Let T:C→C be a self-mapping of C. Recall that T is said to be a pseudocontractive mapping if (1.1)‖Tx-Ty‖2≤‖x-y‖2+‖(I-T)x-(I-T)y‖2,∀x,y∈C,
and T is said to be a strictly pseudo-contractive mapping if there exists a constant k∈[0,1) such that (1.2)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈C.
For such cases, we also say that T is a k-strict pseudo-contractive mapping. We use F(T) to denote the set of fixed points of T.It is well known that the class of strictly pseudo-contractive mappings strictly includes the class of nonexpansive mappings which are the mappingsT on C such that (1.3)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.Iterative methods for nonexpansive mappings have been extensively investigated; see [1–16] and the references therein.However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive mappings though Browder and Petryshyn initiated their work in 1967; the reason is probably that the second term appearing on the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strictly pseudo-contractive mapping T. However, on the other hand, strictly pseudo-contractive mappings have more powerful applications than nonexpansive mappings do in solving inverse problems; see Scherzer [17]. Therefore, it is interesting to develop iterative methods for strictly pseudo-contractive mappings. As a matter of fact, Browder and Petryshyn [18] showed that if a k-strict pseudo-contractive mapping T has a fixed point in C, then starting with an initial x0∈C, the sequence {xn} generated by the recursive formula: (1.4)xn+1=αxn+(1-α)Txn,∀n≥0,
where α is a constant such that k<α<1 converges weakly to a fixed point of T.Recently, Marino and Xu [19] have extended Browder and Petryshyn's result by proving that the sequence {xn} generated by the following Mann's algorithm: (1.5)xn+1=αnxn+(1-αn)Txn,∀n≥0
converges weakly to a fixed point of T, provided that the control sequence {αn} satisfies the condition that k<αn<1 for all n and ∑n=0∞‍(αn-k)(1-αn)=∞. However, this convergence is in general not strong. It is well known that if C is a bounded and closed convex subset of H, and T:C→C is a demicontinuous pseudocontraction, then T has a fixed point in C (Theorem  2.3 in [20]). However, all efforts to approximate such a fixed point by virtue of the normal Mann's iteration algorithm failed.In 1974, Ishikawa [21] introduced a new iteration algorithm and proved the following convergence theorem.Theorem I (see [21]).
IfC is a compact convex subset of a Hilbert space H,T:C→C is a Lipschitzian pseudocontraction and x0∈C is chosen arbitrarily, then the sequence {xn}n≥0 converges strongly to a fixed point of T, where {xn} is defined iteratively for each positive integer n≥0 by
(1.6)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,
where {αn} and {βn} are sequences of real numbers satisfying the conditions (i) 0≤αn≤βn<1; (ii) βn→0 as n→∞; (iii) ∑n=0∞‍αnβn=∞.Since its publication in 1974, it remains an open question whether or not Mann's iteration algorithm converges under the setting of Theorem I to a fixed point ofT if the mapping T is Lipschitzian pseudo-contractive. In [22], Chidume and Mutangadura gave an example of a Lipschitzian pseudocontraction with a unique fixed point for which Mann's iteration algorithm fails to converge.In an infinite-dimensional Hilbert space, Mann and Ishikawa's iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. So, in order to get strong convergence for strictly pseudo-contractive mappings, several attempts have been made based on the CQ method (see, e.g., [19, 23, 24]). The last scheme, in such a direction, seems for us to be the following due to Zhou [25]:
(1.7)x0∈Cchosenarbitrarily,yn=(1-αn)xn+αnTxn,zn=(1-βn)xn+βnTyn,Cn={z∈C:‖zn-z‖2≤‖xn-z‖2-αnβn(1-2αn-L2αn2)‖xn-Txn‖2},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0.
He proved, under suitable choice of the parameters αn and βn, that the sequence {xn} generated by (1.7) strongly converges to PF(T)x0.Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important. This is due to the relation between the class of pseudocontractions and the class of monotone mappings (we recall that a mappingA is monotone if 〈Ax-Ay,x-y〉≥0forallx,y∈H). A mapping A is monotone if and only if (I-A) is pseudo-contractive. It is well known (see, e.g., [26]) that if S is monotone, then the solutions of the equation Sx=0 correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 30 years or so, have been devoted to iterative methods for approximating fixed points of a pseudo-contractive mapping T (see e.g.,  [27–32] and the references therein).Very recently, motivated by the work in [19, 25, 33] and the related work in the literature, Yao et al. [34] suggested and analyzed a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, they proved the strong convergence of the proposed iterative algorithm for Lipschitzian pseudo-contractive mappings.Theorem YLM (see [34]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a L-Lipschitzian pseudo-contractive mapping such that F(T)≠∅. Assume that the sequence αn∈[a,b] for some a,b∈(0,1/(L+1)). Let x0∈H. For C1=C and x1=PC1x0, let {xn} be the sequence in C generated iteratively by
(1.8)yn=(1-αn)xn+αnTxn,Cn+1={z∈Cn:‖αn(I-T)yn‖2≤2αn〈xn-z,(I-T)yn〉},xn+1=PCn+1x0,n≥1.
Then {xn} converges strongly to PF(T)x0.Inspired by the above research work of Yao et al. [34], in this paper we will continue this direction of research. Let C be a nonempty closed convex subset of a real Hilbert space H. We will propose a new hybrid iterative scheme with perturbed mapping for approximating fixed points of a Lipschitzian pseudo-contractive self-mapping on C. We will establish a strong convergence theorem for this hybrid iterative scheme. To be more specific, let T:C→C be a L-Lipschitzian pseudo-contractive mapping and F:C→H a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Let {αn}⊂(0,1),{λn}⊂[0,1) and take a fixed number μ∈(0,2η/κ2). We introduce the following hybrid iterative process with perturbed mapping F. Let x0∈H. For C1=C and x1=PC1x0, two sequences {xn},{yn} are generated as follows:
(1.9)yn=(1-αn)xn+αnPC[Txn-λnμF(Txn)],Cn+1={‖αn(I-PC(I-λnμF)T)yn‖2z∈Cn:‖αn(I-PC(I-λnμF)T)yn‖2≤2αn[〈xn-z,(I-PC(I-λnμF)T)yn〉-〈Tyn-PC(I-λnμF)Tyn,yn-z〉]‖αn(I-PC(I-λnμF)T)yn‖2}xn+1=PCn+1x0,n≥1.
It is clear that if λn=0,foralln≥1, then the hybrid iterative scheme (1.9) reduces to the hybrid iterative process (1.8). Under very mild assumptions, we obtain a strong convergence theorem for the sequences {xn} and {yn} generated by the introduced method. Our proposed hybrid method with perturbation is quite general and flexible and includes the hybrid method considered in [34] and several other iterative methods as special cases. Our results represent the modification, supplement, extension, and improvement of [34, Algorithm  3.1 and Theorem  3.1]. Further, we consider the more general case, where {Ti}i=1N are NL-Lipschitzian pseudo-contractive self-mappings on C with N≥1 an integer. In this case, we propose another hybrid iterative process with perturbed mapping F for approximating a common fixed point of {Ti}i=1N. Let x0∈H. For C1=C and x1=PC1x0, two sequences {xn}and{yn} are generated as follows:
(1.10)yn=(1-αn)xn+αnPC[Tnxn-λnμF(Tnxn)],Cn+1={z∈Cn:‖αn(I-PC(I-λnμF)Tn)yn‖2≤2αn[〈xn-z,(I-PC(I-λnμF)Tn)yn〉-〈Tnyn-PC(I-λnμF)Tnyn,yn-z〉]‖αn(I-PC(I-λnμF)T)yn‖2}xn+1=PCn+1x0,n≥1,
where Tn:=TnmodN, for integer n≥1, with the mod function taking values in the set {1,2,…,N} (i.e., if n=jN+q for some integers j≥0 and 0≤q<N, then Tn=TN if q=0 and Tn=Tq if 1<q<N). It is clear that if N=1, then the hybrid iterative scheme (1.10) reduces to the hybrid iterative process (1.9). Under quite appropriate conditions, we derive a strong convergence theorem for the sequences {xn} and {yn} generated by the proposed method.We now give some preliminaries and results which will be used in the rest of this paper. A Banach spaceX is said to satisfy Opial's condition if whenever {xn} is a sequence in X which converges weakly to x, then (1.11)liminf⁡n→∞‖xn-x‖<liminf⁡n→∞‖xn-y‖,∀y∈X,y≠x.
It is well known that every Hilbert space H satisfies Opial's condition (see, e.g., [35]). Throughout this paper, we shall use the notations: “⇀” and “→” standing for the weak convergence and strong convergence, respectively. Moreover, we shall use the following notation: for a given sequence {xn}⊂X,ωw(xn) denotes the weak ω-limit set of {xn}, that is, (1.12)ωw(xn):={x∈X:xnj⇀xforsomesubsequence{nj}of{n}}.
In addition, for each point x∈H, there exists a unique nearest point in C, denoted by PCx, such that (1.13)‖x-PCx‖≤‖x-y‖,∀y∈C,
where PC is called the metric projection of H onto C. It is known that PC is a nonexpansive mapping.Now we collect some lemmas which will be used in the proof of the main result in the next section. We note that Lemmas1.1 and 1.2 are well known.Lemma 1.1.
LetH be a real Hilbert space. There holds the following identity:(1.14)
∥
x
-
y
∥
2
=
∥
x
∥
2
-
∥
y
∥
2
-
2
〈
x
-
y
,
y
〉
,
∀
x
,
y
∈
H
.Lemma 1.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Given x∈H and z∈C. Then z=PCx if and only if there holds the relation:(1.15)
〈
x
-
z
,
y
-
z
〉
≤
0
,
∀
y
∈
C
.Lemma 1.3 (see [23]).
LetC be a nonempty closed convex subset of H. Let {xn} be a sequence in H and u∈H. Let q=PCu. If {xn} is such that ωw(xn)⊂C and satisfies the condition:
(1.16)‖xn-u‖≤‖u-q‖,∀n≥0.
Then xn→q.Lemma 1.4 (see [27]).
LetX be a real reflexive Banach space which satisfies Opial's condition. Let C be a nonempty closed convex subset of X, and T:C→C be a continuous pseudo-contractive mapping. Then, I-T is demiclosed at zero.LetT:C→C be a nonexpansive mapping and F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strongly monotone, that is, F satisfies the following conditions: (1.17)‖Fx-Fy‖≤κ‖x-y‖,∀x,y∈C,〈Fx-Fy,x-y〉≥η‖x-y‖2,∀x,y∈C,
respectively. For any given numbers λ∈[0,1) and μ∈(0,2η/κ2), we define the mapping Tλ:C→H: (1.18)Tλx:=Tx-λμF(Tx),∀x∈C.Lemma 1.5 (see [36]).
If0≤λ<1 and 0<μ<2η/κ2, then there holds for Tλ:C→H:
(1.19)‖Tλx-Tλy‖≤(1-λτ)‖x-y‖,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1).In particular, wheneverT=I the identity operator of H, we have
(1.20)‖(I-λμF)x-(I-λμF)y‖≤(1-λτ)‖x-y‖,∀x,y∈C.


## 2.  Main Result

In this section, we introduce a hybrid iterative algorithm with perturbed mapping for pseudo-contractive mappings in a real Hilbert spaceH.Algorithm 2.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a pseudo-contractive mapping and F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Let {αn}⊂(0,1),{λn}⊂[0,1) and take a fixed number μ∈(0,2η/κ2). Let x0∈H. For C1=C and x1=PC1x0, define two sequences: {xn}and{yn} of C as follows:
(2.1)yn=(1-αn)xn+αnPC[Txn-λnμF(Txn)],Cn+1={z∈Cn:‖αn(I-PC(I-λnμF)T)yn‖2≤2αn[〈xn-z,(I-PC(I-λnμF)T)yn〉-〈Tyn-PC(I-λnμF)Tyn,yn-z〉]‖αn(I-PC(I-λnμF)T)yn‖2}xn+1=PCn+1x0,n≥1.Now we prove the strong convergence of the above iterative algorithm for Lipschitzian pseudo-contractive mappings.Theorem 2.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a L-Lipschitzian pseudo-contractive mapping such that F(T)≠∅, and let F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Assume that {αn}⊂[a,b] for some a,b∈(0,1/(L+1)) and {λn}⊂[0,1) such that limn→∞λn=0. Take a fixed number μ∈(0,2η/κ2). Then the sequences {xn}and{yn} generated by (2.1) converge strongly to the same point PF(T)x0.Proof.
Firstly, we observe thatPF(T) and {xn} are well defined. From [19, 27], we note that F(T) is closed and convex. Indeed, by [27], we can define a mapping g:C→C by g(x)=(2I-T)-1 for every x∈C. It is clear that g is a nonexpansive self-mapping such that F(T)=F(g). Hence, by [23, Proposition  2.1 (iii)], we conclude that F(g)=F(T) is a closed convex set. This implies that the projection PF(T) is well defined. It is obvious that {Cn} is closed and convex. Thus, {xn} is also well defined.
Now, we show thatF(T)⊂Cn for all n≥0. Indeed, taking p∈F(T), we note that (I-T)p=0, and (1.1) is equivalent to
(2.2)〈(I-T)x-(I-T)y,x-y〉≥0,∀x,y∈C.
Using Lemma 1.1 and (2.2), we obtain
(2.3)‖xn-p-αn(I-PC(I-λnμF)T)yn‖2=‖xn-p‖2-‖αn(I-PC(I-λnμF)T)yn‖2-2αn〈(I-PC(I-λnμF)T)yn,xn-p-αn(I-PC(I-λnμF)T)yn〉=‖xn-p‖2-‖αn(I-PC(I-λnμF)T)yn‖2-2αn〈(I-T)yn-(I-T)p,yn-p〉-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉-2αn〈(I-PC(I-λnμF)T)yn,xn-yn-αn(I-PC(I-λnμF)T)yn〉≤‖xn-p‖2-‖αn(I-PC(I-λnμF)T)yn‖2-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉-2αn〈(I-PC(I-λnμF)T)yn,xn-yn-αn(I-PC(I-λnμF)T)yn〉=‖xn-p‖2-‖xn-yn+yn-xn+αn(I-PC(I-λnμF)T)yn‖2-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉-2αn〈(I-PC(I-λnμF)T)yn,xn-yn-αn(I-PC(I-λnμF)T)yn〉=‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2-2〈xn-yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉+2αn〈(I-PC(I-λnμF)T)yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉=‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2-2〈xn-yn-αn(I-PC(I-λnμF)T)yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉≤‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2+2|〈xn-yn-αn(I-PC(I-λnμF)T)yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉|-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉.
Since T is L-Lipschitzian, utilizing Lemma 1.5 we derive
(2.4)‖(I-PC(I-λnμF)T)xn-(I-PC(I-λnμF)T)yn‖≤‖xn-yn‖+‖PC(I-λnμF)Txn-PC(I-λnμF)Tyn‖≤‖xn-yn‖+‖(I-λnμF)Txn-(I-λnμF)Tyn‖≤‖xn-yn‖+(1-λnτ)‖Txn-Tyn‖≤‖xn-yn‖+‖Txn-Tyn‖≤(L+1)‖xn-yn‖.
From (2.1), we observe that xn-yn=αn(I-PC(I-λnμF)T)xn. Hence, utilizing Lemma 1.5 and (2.4) we obtain
(2.5)|〈xn-yn-αn(I-PC(I-λnμF)T)yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉|=αn|〈(I-PC(I-λnμF)T)xn-(I-PC(I-λnμF)T)yn,yn-xn+αn(I-PC(I-λnμF)T)yn〉|≤αn‖(I-PC(I-λnμF)T)xn-(I-PC(I-λnμF)T)yn‖×‖yn-xn+αn(I-PC(I-λnμF)T)yn‖≤αn(L+1)‖xn-yn‖‖yn-xn+αn(I-PC(I-λnμF)T)yn‖≤αn(L+1)2(‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2).
Combining (2.3) and (2.5), we get
(2.6)‖xn-p-αn(I-PC(I-λnμF)T)yn‖2≤‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2+αn(L+1)(‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2)-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉=‖xn-p‖2+[αn(L+1)-1](‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)T)yn‖2)-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉≤‖xn-p‖2-2αn〈Tyn-PC(I-λnμF)Tyn,yn-p〉.
At the same time, we observe that
(2.7)‖xn-p-αn(I-PC(I-λnμF)T)yn‖2=‖xn-p‖2-2αn〈xn-p,(I-PC(I-λnμF)T)yn〉+‖αn(I-PC(I-λnμF)T)yn‖2.
Therefore, from (2.6) and (2.7) we have
(2.8)‖αn(I-PC(I-λnμF)T)yn‖2≤2αn[〈xn-p,(I-PC(I-λnμF)T)yn〉-〈Tyn-PC(I-λnμF)Tyn,yn-p〉],
which implies that
(2.9)p∈Cn,
that is,
(2.10)F(T)⊂Cn,∀n≥0.
Fromxn=PCnx0, we have
(2.11)〈x0-xn,xn-y〉≥0,∀y∈Cn.
Utilizing F(T)⊂Cn, we also have
(2.12)〈x0-xn,xn-u〉≥0,∀u∈F(T).
So, for all u∈F(T) we have
(2.13)0≤〈x0-xn,xn-u〉=〈x0-xn,xn-x0+x0-u〉=-‖x0-xn‖2+〈x0-xn,x0-u〉≤-‖x0-xn‖2+‖x0-xn‖‖x0-u‖,
which hence implies that
(2.14)‖x0-xn‖≤‖x0-u‖,∀u∈F(T).
Thus, {xn} is bounded and so are {yn} and {Tyn}.
Fromxn=PCnx0 and xn+1=PCn+1x0∈Cn+1⊂Cn, we have
(2.15)〈x0-xn,xn-xn+1〉≥0.
Hence,
(2.16)0≤〈x0-xn,xn-xn+1〉=〈x0-xn,xn-x0+x0-xn+1〉=-‖x0-xn‖2+〈x0-xn,x0-xn+1〉≤-‖x0-xn‖2+‖x0-xn‖‖x0-xn+1‖,
and therefore
(2.17)‖x0-xn‖≤‖x0-xn+1‖.
This implies that limn→∞||xn-x0|| exists.
From Lemma1.1 and (2.15), we obtain
(2.18)‖xn+1-xn‖2=‖(xn+1-x0)-(xn-x0)‖2=‖xn+1-x0‖2-‖xn-x0‖2-2〈xn+1-xn,xn-x0〉≤‖xn+1-x0‖2-‖xn-x0‖2⟶0.
Since xn+1∈Cn+1⊂Cn, from ||xn-xn+1||→0 and λn→0 it follows that
(2.19)‖αn(I-PC(I-λnμF)T)yn‖2≤2αn[〈xn-xn+1,(I-PC(I-λnμF)T)yn〉-〈Tyn-PC(I-λnμF)Tyn,yn-xn+1〉]≤2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tyn‖+‖Tyn-PC(I-λnμF)Tyn‖‖yn-xn+1‖]≤2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tyn‖+‖Tyn-(I-λnμF)Tyn‖‖yn-xn+1‖]=2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tyn‖+λnμ‖F(Tyn)‖‖yn-xn+1‖]⟶0.
Noticing that αn∈[a,b] for some a,b∈(0,1/(L+1)), thus, we obtain
(2.20)‖yn-PC(I-λnμF)Tyn‖⟶0.
Also, we note that ||Tyn-PC(I-λnμF)Tyn||≤λnμ||F(Tyn)||→0. Therefore, we get
(2.21)‖yn-Tyn‖≤‖yn-PC(I-λnμF)Tyn‖+‖Tyn-PC(I-λnμF)Tyn‖⟶0.
On the other hand, utilizing Lemma1.5 we deduce that
(2.22)‖xn-PC(I-λnμF)Txn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tyn‖+‖PC(I-λnμF)Tyn-PC(I-λnμF)Txn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tyn‖+‖(I-λnμF)Tyn-(I-λnμF)Txn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tyn‖+(1-λnτ)‖Tyn-Txn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tyn‖+L‖yn-xn‖=(L+1)‖xn-yn‖+‖yn-PC(I-λnμF)Tyn‖=αn(L+1)‖xn-PC(I-λnμF)Txn‖+‖yn-PC(I-λnμF)Tyn‖,
that is,
(2.23)‖xn-PC(I-λnμF)Txn‖≤11-αn(L+1)‖yn-PC(I-λnμF)Tyn‖⟶0.
Meantime, it is clear that
(2.24)‖Txn-PC(I-λnμF)Txn‖≤λnμ‖F(Txn)‖⟶0.
Consequently,
(2.25)‖xn-Txn‖≤‖xn-PC(I-λnμF)Txn‖+‖Txn-PC(I-λnμF)Txn‖⟶0.
Now (2.25) and Lemma 1.4 guarantee that every weak limit point of {xn} is a fixed point of T, that is, ωw(xn)⊂F(T). In fact, the inequality (2.14) and Lemma 1.3 ensure the strong convergence of {xn} to PF(T)x0. Since ||xn-yn||=||αn(I-PC(I-λnμF)T)xn||→0, it is immediately known that {yn} converges strongly to PF(T)x0. This completes the proof.Corollary 2.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a nonexpansive mapping such that F(T)≠∅, and let F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Assume that {αn}⊂[a,b] for some a,b∈(0,1/2) and {λn}⊂[0,1) such that limn→∞λn=0. Take a fixed number μ∈(0,2η/κ2). Then the sequences {xn}and{yn} generated by (2.1) converge strongly to the same point PF(T)x0.Corollary 2.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a L-Lipschitzian pseudo-contractive mapping such that F(T)≠∅. Assume that {αn}⊂[a,b] for some a,b∈(0,1/(L+1)) and {λn}⊂[0,1) such that limn→∞λn=0. Then the sequences {xn}and{yn} generated by the scheme
(2.26)yn=(1-αn)xn+αnPC((1-λn)Txn),Cn+1={z∈Cn:‖αn(yn-PC((1-λn)Tyn))‖2≤2αn[〈xn-z,yn-PC((1-λn)Tyn)〉-〈Tyn-PC((1-λn)Tyn),yn-z〉]‖αn(yn-PC((1-λn)Tyn))‖2}xn+1=PCn+1x0
converge strongly to the same point PF(T)x0.Proof.
Putμ=2 and F=(1/2)I in Theorem 2.2. Then, in this case we have κ=η=1/2, and hence
(2.27)(0,2ηκ2)=(0,4).
This implies that μ=2∈(0,2η/κ2)=(0,4). Meantime, it is easy to see that the scheme (2.1) reduces to (2.26). Therefore, by Theorem 2.2, we obtain the desired result.Corollary 2.5 ([34, Corollary  3.2]).
LetA:H→H be a L-Lipschitzian monotone mapping for which A-1(0)≠∅. Assume that the sequence {αn}⊂[a,b] for some a,b∈(0,1/(L+2)). Then the sequence {xn} generated by the scheme
(2.28)yn=xn-αnAxn,Cn+1={z∈Cn:‖αnAyn‖2≤2αn〈xn-z,Ayn〉},xn+1=PCn+1x0
strongly converges to PA-1(0)x0.Proof.
Putλn=0 and T=I-A in Corollary 2.4. Then, it is easy to see that the scheme (2.26) reduces to (2.28). Therefore, by Corollary 2.4, we derive the desired result.Next, consider the more general case whereΩ is expressed as the intersection of the fixed-point sets of N pseudo-contractive mappings Ti:C→C with N≥1 an integer, that is,
(2.29)Ω=⋂i=1N‍F(Ti).
In this section, we propose another hybrid iterative algorithm with perturbed mapping for a finite family of pseudo-contractive mappings in a real Hilbert space H.Algorithm 2.6.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1N be N pseudo-contractive self-mappings on C with N≥1 an integer, and let F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Let {αn}⊂(0,1),{λn}⊂[0,1), and take a fixed number μ∈(0,2η/κ2). Let x0∈H. For C1=C and x1=PC1x0, define two sequences {xn},{yn} of C as follows:
(2.30)yn=(1-αn)xn+αnPC[Tnxn-λnμF(Tnxn)],Cn+1={z∈Cn:‖αn(I-PC(I-λnμF)Tn)yn‖2≤2αn[〈xn-z,(I-PC(I-λnμF)Tn)yn〉-〈Tnyn-PC(I-λnμF)Tnyn,yn-z〉]‖αn(I-PC(I-λnμF)Tn)yn‖2},xn+1=PCn+1x0,n≥1,
where
(2.31)Tn:=Tnmod⁡N,
for integer n≥1, with the mod function taking values in the set {1,2,…,N} (i.e., if n=jN+q for some integers j≥0 and 0≤q<N, then Tn=TN if q=0 and Tn=Tq if 1<q<N).Theorem 2.7.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1N be NL-Lipschitzian pseudo-contractive self-mappings on C such that Ω=⋂i=1N‍F(Ti)≠∅, and let F:C→H be a mapping such that for some constants κ,η>0,F is κ-Lipschitzian and η-strong monotone. Assume that {αn}⊂[a,b] for some a,b∈(0,1/(L+1)) and {λn}⊂[0,1) such that limn→∞λn=0. Take a fixed number μ∈(0,2η/κ2). Then the sequences {xn},{yn} generated by (2.30) converge strongly to the same point PΩx0.Proof.
Firstly, as stated in the proof of Theorem2.2, we can readily see that each F(Ti) is closed and convex for i=1,2,…,N. Hence, Ω is closed and convex. This implies that the projection PΩ is well defined. It is clear that the sequence {Cn} is closed and convex. Thus, {xn} is also well defined.
Now let us show thatΩ⊂Cn for all n≥0. Indeed, taking p∈Ω, we note that (I-Tn)p=0 and
(2.32)〈(I-Tn)x-(I-Tn)y,x-y〉≥0,∀x,y∈C.
Using Lemma 1.1 and (2.32), we obtain
(2.33)‖xn-p-αn(I-PC(I-λnμF)Tn)yn‖2=‖xn-p‖2-‖αn(I-PC(I-λnμF)Tn)yn‖2-2αn〈(I-PC(I-λnμF)Tn)yn,xn-p-αn(I-PC(I-λnμF)Tn)yn〉=‖xn-p‖2-‖αn(I-PC(I-λnμF)Tn)yn‖2-2αn〈(I-Tn)yn-(I-Tn)p,yn-p〉-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉-2αn〈(I-PC(I-λnμF)Tn)yn,xn-yn-αn(I-PC(I-λnμF)Tn)yn〉≤‖xn-p‖2-‖αn(I-PC(I-λnμF)Tn)yn‖2-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉-2αn〈(I-PC(I-λnμF)Tn)yn,xn-yn-αn(I-PC(I-λnμF)Tn)yn〉=‖xn-p‖2-‖xn-yn+yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉-2αn〈(I-PC(I-λnμF)Tn)yn,xn-yn-αn(I-PC(I-λnμF)Tn)yn〉=‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2-2〈xn-yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉+2αn〈(I-PC(I-λnμF)Tn)yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉=‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2-2〈xn-yn-αn(I-PC(I-λnμF)Tn)yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉≤‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2+2|〈xn-yn-αn(I-PC(I-λnμF)Tn)yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉|-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉.
Since each Ti is L-Lipschitzian for i=1,2,…,N, utilizing Lemma 1.5 we derive
(2.34)‖(I-PC(I-λnμF)Tn)xn-(I-PC(I-λnμF)Tn)yn‖≤‖xn-yn‖+‖PC(I-λnμF)Tnxn-PC(I-λnμF)Tnyn‖≤‖xn-yn‖+‖(I-λnμF)Tnxn-(I-λnμF)Tnyn‖≤‖xn-yn‖+(1-λnτ)‖Tnxn-Tnyn‖≤‖xn-yn‖+‖Tnxn-Tnyn‖≤(L+1)‖xn-yn‖.
From (2.30), we observe that xn-yn=αn(I-PC(I-λnμF)Tn)xn. Hence, utilizing Lemma 1.5 and (2.34) we obtain
(2.35)|〈xn-yn-αn(I-PC(I-λnμF)Tn)yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉|=αn|〈(I-PC(I-λnμF)Tn)xn-(I-PC(I-λnμF)Tn)yn,yn-xn+αn(I-PC(I-λnμF)Tn)yn〉|≤αn‖(I-PC(I-λnμF)Tn)xn-(I-PC(I-λnμF)Tn)yn‖×‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖≤αn(L+1)‖xn-yn‖‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖≤αn(L+1)2(‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2).
Combining (2.33) and (2.35), we get
(2.36)‖xn-p-αn(I-PC(I-λnμF)Tn)yn‖2≤‖xn-p‖2-‖xn-yn‖2-‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2+αn(L+1)(‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2)-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉=‖xn-p‖2+[αn(L+1)-1](‖xn-yn‖2+‖yn-xn+αn(I-PC(I-λnμF)Tn)yn‖2)-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉≤‖xn-p‖2-2αn〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉.
Meantime, we observe that
(2.37)‖xn-p-αn(I-PC(I-λnμF)Tn)yn‖2=‖xn-p‖2-2αn〈xn-p,(I-PC(I-λnμF)Tn)yn〉+‖αn(I-PC(I-λnμF)Tn)yn‖2.
Therefore, from (2.36) and (2.37) we have
(2.38)‖αn(I-PC(I-λnμF)Tn)yn‖2≤2αn[〈xn-p,(I-PC(I-λnμF)Tn)yn〉-〈Tnyn-PC(I-λnμF)Tnyn,yn-p〉],
which implies that
(2.39)p∈Cn,
that is,
(2.40)Ω⊂Cn,∀n≥0.
Fromxn=PCnx0, we have
(2.41)〈x0-xn,xn-y〉≥0,∀y∈Cn.
Utilizing Ω⊂Cn, we also have
(2.42)〈x0-xn,xn-u〉≥0,∀u∈Ω.
So, for all u∈Ω we have
(2.43)0≤〈x0-xn,xn-u〉=〈x0-xn,xn-x0+x0-u〉=-‖x0-xn‖2+〈x0-xn,x0-u〉≤-‖x0-xn‖2+‖x0-xn‖‖x0-u‖,
which hence implies that
(2.44)‖x0-xn‖≤‖x0-u‖,∀u∈Ω.
Thus {xn} is bounded and so are {yn} and {Tnyn}.
Fromxn=PCnx0 and xn+1=PCn+1x0∈Cn+1⊂Cn, we have
(2.45)〈x0-xn,xn-xn+1〉≥0.
Hence,
(2.46)0≤〈x0-xn,xn-xn+1〉=〈x0-xn,xn-x0+x0-xn+1〉=-‖x0-xn‖2+〈x0-xn,x0-xn+1〉≤-‖x0-xn‖2+‖x0-xn‖‖x0-xn+1‖,
and therefore
(2.47)‖x0-xn‖≤‖x0-xn+1‖.
This implies that limn→∞||xn-x0|| exists.
From Lemma1.1 and (2.45), we obtain
(2.48)‖xn+1-xn‖2=‖(xn+1-x0)-(xn-x0)‖2=‖xn+1-x0‖2-‖xn-x0‖2-2〈xn+1-xn,xn-x0〉≤‖xn+1-x0‖2-‖xn-x0‖2⟶0asn⟶∞.
Thus,
(2.49)lim⁡n→∞‖xn+1-xn‖=0.
Obviously, it is easy to see that limn→∞||xn-xn+i||=0 for each i=1,2,…,N. Since xn+1∈Cn+1⊂Cn, from ||xn-xn+1||→0 and λn→0 it follows that
(2.50)‖αn(I-PC(I-λnμF)Tn)yn‖2≤2αn[〈xn-xn+1,(I-PC(I-λnμF)Tn)yn〉-〈Tnyn-PC(I-λnμF)Tnyn,yn-xn+1〉]≤2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tnyn‖+‖Tnyn-PC(I-λnμF)Tnyn‖‖yn-xn+1‖]≤2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tnyn‖+‖Tnyn-(I-λnμF)Tnyn‖‖yn-xn+1‖]=2αn[‖xn-xn+1‖‖yn-PC(I-λnμF)Tnyn‖+λnμ‖F(Tnyn)‖‖yn-xn+1‖]⟶0.
Noticing that αn∈[a,b] for some a,b∈(0,1/(L+1)), thus, we obtain
(2.51)‖yn-PC(I-λnμF)Tnyn‖⟶0.
Also, we note that ||Tnyn-PC(I-λnμF)Tnyn||≤λnμ||F(Tnyn)||→0. Therefore, we get
(2.52)‖yn-Tnyn‖≤‖yn-PC(I-λnμF)Tnyn‖+‖Tnyn-PC(I-λnμF)Tnyn‖⟶0.
On the other hand, utilizing Lemma1.5 we deduce that
(2.53)‖xn-PC(I-λnμF)Tnxn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tnyn‖+‖PC(I-λnμF)Tnyn-PC(I-λnμF)Tnxn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tnyn‖+‖(I-λnμF)Tnyn-(I-λnμF)Tnxn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tnyn‖+(1-λnτ)‖Tnyn-Tnxn‖≤‖xn-yn‖+‖yn-PC(I-λnμF)Tnyn‖+L‖yn-xn‖=(L+1)‖xn-yn‖+‖yn-PC(I-λnμF)Tnyn‖=αn(L+1)‖xn-PC(I-λnμF)Tnxn‖+‖yn-PC(I-λnμF)Tnyn‖,
that is,
(2.54)‖xn-PC(I-λnμF)Tnxn‖≤11-αn(L+1)‖yn-PC(I-λnμF)Tnyn‖⟶0.
Furthermore, it is clear that
(2.55)‖Tnxn-PC(I-λnμF)Tnxn‖≤λnμ‖F(Tnxn)‖⟶0asn⟶∞.
Consequently,
(2.56)‖xn-Tnxn‖≤‖xn-PC(I-λnμF)Tnxn‖+‖Tnxn-PC(I-λnμF)Tnxn‖⟶0,
and hence for each i=1,2,…,N:
(2.57)‖xn-Tn+ixn‖≤‖xn-xn+i‖+‖xn+i-Tn+ixn+i‖+‖Tn+ixn+i-Tn+ixn‖≤(L+1)‖xn-xn+i‖+‖xn+i-Tn+ixn+i‖⟶0asn⟶∞.
So, we obtain limn→∞||xn-Tn+ixn||=0 for each i=1,2,…,N. This implies that
(2.58)lim⁡n→∞‖xn-Tlxn‖=0foreachl=1,2,…,N.
Now (2.58) and Lemma 1.4 guarantee that every weak limit point of {xn} is a fixed point of Tl. Since l is an arbitrary element in the finite set {1,2,…,N}, it is known that every weak limit point of {xn} lies in Ω, that is, ωw(xn)⊂Ω. This fact, the inequality (2.44) and Lemma 1.3 ensure the strong convergence of {xn} to PΩx0. Since ||xn-yn||=||αn(I-PC(I-λnμF)Tn)xn||→0, it follows immediately that {yn} converges strongly to PΩx0. This completes the proof.Remark 2.8.
Algorithm  3.1 in [34] for a Lipschitzian pseudocontraction is extended to develop our hybrid iterative algorithm with perturbation for N-Lipschitzian pseudocontractions; that is, Algorithm 2.6. Theorem 2.7 is more general and more flexible than Theorem  3.1 in [34]. Also, the proof of Theorem 2.7 is very different from that of Theorem  3.1 in [34] because our technique of argument depends on Lemma 1.5. Finally, we observe that several recent results for pseudocontractive and related mappings can be found in [37–42].



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*Source: 250538-2012-07-17.xml*