# An Iterative Approximation Method for a Common Fixed Point of Two Pseudocontractive Mappings

Authors: Habtu Zegeye
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We introduce an iterative process for finding an element in the common fixed point sets of two continuous pseudocontractive
mappings. As a consequence, we provide an approximation method for a common fixed point of a finite family of pseudocontractive mappings. Furthermore,
our convergence theorem is applied to a convex minimization problem. Our
theorems extend and unify most of the results that have been proved for this
class of nonlinear mappings.


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## Body

## 1. Introduction

LetH be a real Hilbert space. A mapping T with domain D(T)⊂H and range R(T) in H is called pseudocontractive if for each x,y∈D(T) we have(1.1)〈Tx-Ty,x-y〉≤‖x-y‖2.T is called strongly pseudocontractive if there exists k∈(0,1) such that (1.2)〈x-y,Tx-Ty〉≤k‖x-y‖2,∀x,y∈D(T),
and T is said to be k-strict pseudocontractive if there exists a constant 0≤k<1 such that(1.3)〈x-y,Tx-Ty〉≤‖x-y‖2-k‖(I-T)x-(I-T)y‖2,∀x,y∈D(T).
The operator T is called Lipschitzian if there exists L≥0 such that ∥Tx-Ty∥≤L∥x-y∥ for all x,y∈D(T). If L=1, then T is called nonexpansive, and if L∈[0,1), then T is called a contraction. As a result of Kato [1], it follows from inequality (1.1) that T is pseudocontractive if and only if the inequality(1.4)‖x-y‖≤‖(1+t)(x-y)-t(Tx-Ty)‖
holds for each x,y∈D(T) and for all t>0.Apart from being an important generalization of nonexpansive, strongly pseudocontractive andk-strict pseudocontractive mappings, interest in pseudocontractive mappings stems mainly from their firm connection with the important class of nonlinear accretive operators, where a mapping A with domain D(A) and range R(A) in H is called accretive if the inequality(1.5)‖x-y‖≤‖x-y+s(Ax-Ay)‖
holds for every x,y∈D(A) and for all s>0. We observe that A is accretive if and only if T:=I-A is pseudocontractive, and thus a zero of A, N(A):={x∈D(A):Ax=0}, is a fixed point of T, F(T):={x∈D(T):Tx=x}. It is now well known that if A is accretive then the solutions of the equation Ax=0 correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts have been devoted to iterative methods for approximating fixed points of T when T is pseudocontractive (see, e.g., [2–4] and the references contained therein).Construction of fixed points of nonexpansive mappings via Mann's algorithm [5] has extensively been investigated recently in the literature (see, e.g., [6, 7] and references therein). Related works can also be found in [7–18]. Mann's algorithm is defined by x0∈K and(1.6)xn+1=αnxn+(1-αn)Txn,n≥0,
where {αn} is a real control sequence in the interval (0,1). If T is a nonexpansive mapping with a fixed point and if the control sequence {αn} is chosen so that ∑n=0∞αn(1-αn)=∞, then the sequence {xn} generated by Mann's algorithm (1.6) converges weakly to a fixed point of T (this is indeed true in a uniformly convex Banach space with a Fréchet differentiable norm [7]). However, this convergence is in general not strong (see the counterexample in [19]; see also [20]).For a sequence{αn} of real numbers in (0,1) and an arbitrary u∈C, let the sequence {xn} in K be iteratively defined by x0∈K and(1.7)xn+1:=αn+1u+(1-αn+1)T(xn),n≥0,
where T is a nonexpansive mapping of C into itself. Halpern [11] was the first to study the convergence of Algorithm (1.7) in the framework of Hilbert spaces. Lions [14] and Wittmann [21] improved the result of Halpern by proving strong convergence of {xn} to a fixed point of T if the real sequence {αn} satisfies certain conditions. Reich [22], Shioji and Takahashi [16], and Zegeye and Shahzad [23] extend the result of Wittmann [21] to the case of Banach space.In 2000, Moudafi [24] introduced viscosity approximation method and proved that if H is a real Hilbert space, for given x0∈C, the sequence {xn} generated by the algorithm(1.8)xn+1:=αnf(xn)+(1-αn)T(xn),n≥0,
where f:C→C is a contraction mapping and {αn}⊂(0,1) satisfies certain conditions, converges strongly to a common fixed point of T. Moudafi [24] generalizes Halpern’s theorems in the direction of viscosity approximations. In [25], Zegeye et al. extended Moudafi's result to the class of Lipschitz pseudocontractive mappings in Banach spaces more general than Hilbert spaces. Viscosity approximations are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations.Our concern now is the following. Is it possible to construct a viscosity approximation sequence that converges strongly to a fixed point of pseudocontractive mappings more general than nonexpansive mappings?In this paper, motivated and inspired by the work of Halpern [11], Moudafi [24], and the methods of Takahashi and Zembayashi [26], we introduce a viscosity approximation method for finding a common fixed point of two continuous pseudocontractive mappings. As a consequence, we provide an approximation method for a common fixed point of finite family of pseudocontractive mappings. This provides affirmative answer to the above concern. Furthermore, we apply our convergence theorem to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators.


## 2. Preliminaries

LetC be closed and convex subset of a real Hilbert space H. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that (2.1)‖x-PCx‖≤‖x-y‖,∀y∈C.PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping of H onto C. In connection with metric projection, we have the following lemma.Lemma 2.1.
LetC be a nonempty convex subset of a Hilbert space H. Let x∈H and x0∈C. Then, x0=PCx if and only if
(2.2)〈z-x0,x0-x〉≥0,∀z∈C.Lemma 2.2 (see [27]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.3)an+1≤(1-γn)an+σn,n≥0,
where (i)  {γn}⊂[0,1], ∑γn=∞ and (ii)  limsupn→∞σn/γn≤0 or ∑|σn|<∞. Then, an→0 as n→∞.By a similar argument in [28], we have the following lemma.Lemma 2.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let A:C→H be a continuous accretive mapping. Then, for r>0 and x∈H, there exists z∈C such that
(2.4)〈y-z,Az〉+1r〈y-z,z-x〉≥0,∀y∈C.Moreover, by a similar argument of the proof of Lemmas 2.8 and 2.9 of [26], we get the following lemma.Lemma 2.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let A:C→H be a continuous accretive mapping. For r>0 and x∈H, define a mapping Tr:H→C as follows:
(2.5)Trx:={z∈C:〈y-z,Az〉+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then, the following hold:(1)
Tr is single valued;(2)
Tr is firmly nonexpansive type mapping, that is, for all x,y∈H,
(2.6)‖Trx-Try‖2≤〈Trx-Try,x-y〉;(3)
F(Tr)=VI(C,A);(4)
VI(C,A) is closed and convex.


## 3. Main Results

In the sequel, we will make use of the following lemmas.Lemma 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→H be a continuous pseudocontractive mapping. Then, for r>0 and x∈H, there exists z∈C such that
(3.1)〈y-z,Tz〉-1r〈y-z,(1+r)z-x〉≤0,∀y∈C.Proof.
Letx∈H and r>0. Let A:=I-T, where I is the identity mapping on C. Then, clearly A is continuous accretive mapping. Thus, by Lemma 2.3, there exists z∈C such that 〈y-z,Az〉+(1/r)〈y-z,z-x〉≥0, for all y∈C. But this is equivalent to 〈y-z,Tz〉-(1/r)〈y-z,(1+r)z-x〉≤0, for all y∈C. Hence, the lemma holds.Lemma 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be continuous pseudocontractive mapping. For r>0 and x∈H, define a mapping Fr:H→C as follows:
(3.2)Frx:={z∈C:〈y-z,Tz〉-1r〈y-z,(1+r)z-x〉≤0,∀y∈C}
for all x∈H. Then, the following hold:(1)
Fr is single valued;(2)
Fr is firmly nonexpansive type mapping, that is, for all x,y∈H,
(3.3)‖Frx-Fry‖2≤〈Frx-Fry,x-y〉;(3)
F(Fr)=F(T);(4)
F(T) is closed and convex.Proof.
We note that〈y-z,Tz〉-(1/r)〈y-z,(1+r)z-x〉≤0, for all y∈C, is equivalent to 〈y-z,Az〉+(1/r)〈y-z,z-x〉≥0, for all y∈C, where A:=I-T is continuous accretive mapping and I the identity mapping on C. Moreover, as T is self-map, we have that VI(C,A)=F(T). Thus, by Lemma 2.4, the conclusions of (1)–(4) hold.LetC be a nonempty closed convex subset of a real Hilbert space H. Let Ti:C→C, for i=1,2, be continuous pseudocontractive mappings. Then, in what follows, Trn,Frn:H→C are defined as follows. For x∈H and {rn}⊂(0,∞), define (3.4)Trnx:={z∈C:〈y-z,T1z〉-1rn〈y-z,(1+rn)z-x〉≤0,∀y∈C},Frnx:={z∈C:〈y-z,T2z〉-1rn〈y-z,(1+rn)z-x〉≤0,∀y∈C}.
Now, we prove our main convergence theorem.Theorem 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Ti:C→C, for i=1,2, be continuous pseudocontractive mappings such that F:=⋂i=12F(Ti)≠∅. Let f be a contraction of C into itself, and let {xn} be a sequence generated by x1∈C and
(3.5)xn+1=αnf(xn)+(1-αn)TrnFrnxn,
where {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈F, where z=PFf(z).Proof.
LetQ=PF. Then, Qf is a contraction of C into C. In fact, we have that
(3.6)‖Qf(x)-Qf(y)‖≤‖f(x)-f(y)‖≤β‖x-y‖,
for all x,y∈C, where β is contraction constant of f. So Qf is a contraction of C into itself. Since C is closed subset of H, there exists a unique element z of C such that z=Qf(z).
Letv∈F, and let un:=Trnwn, where wn:=Frnxn. Then, we have from Lemma 3.2 that
(3.7)‖un-v‖=‖Trnwn-Trnv‖≤‖wn-v‖=∥Frnxn-Frnv∥≤‖xn-v‖.
Moreover, from (3.5) and (3.7), we get that
(3.8)‖xn+1-v‖=‖αn(f(xn)-v)+(1-αn)(TrnFrnxn-v)‖≤αn‖f(xn)-v‖+(1-αn)‖un-v‖≤αn‖f(xn)-v‖+(1-αn)‖xn-v‖≤αn‖f(xn)-f(v)‖+αn‖f(v)-v‖+(1-αn)‖xn-v‖≤αnβ‖xn-v‖+αn‖f(v)-v‖+(1-αn)‖xn-v‖=(1-(1-β)αn)‖xn-v‖+(1-β)αn(11-β‖f(v)-v‖)≤max⁡{‖xn-v‖,11-β‖f(v)-v‖}.
By induction, we get that
(3.9)‖xn-v‖≤max⁡{‖x1-v‖,11-β‖f(v)-v‖},n≥1.
Therefore, {xn} is bounded. Consequently, we get that {wn}, {Trnwn}, {Frnxn}, and {f(xn)} are bounded. Next, we show that ∥xn+1-xn∥→0. But from (3.5) we have that
(3.10)‖xn+1-xn‖=‖αnf(xn)+(1-αn)un-αn-1f(xn-1)-(1-αn-1)un-1‖≤‖αnf(xn)-αnf(xn-1)+αnf(xn-1)-αn-1f(xn-1)+(1-αn)un-(1-αn)un-1+(1-αn)un-1-(1-αn-1)un-1‖≤αnβ‖xn-xn-1‖+|αn-αn-1|K+(1-αn)⋅‖un-un-1‖≤αnβ‖xn-xn-1‖+|αn-αn-1|K+(1-αn)⋅‖wn-wn-1‖,
where K=2sup{∥f(xn)∥+∥un∥:n∈N}. Moreover, since wn=Frnxn and wn+1=Frn+1xn+1, we get that
(3.11)〈y-wn,T2wn〉-1rn〈y-wn,(1+rn)wn-xn〉≤0,∀y∈C,(3.12)〈y-wn+1,T2wn+1〉-1rn+1〈y-wn+1,(1+rn+1)wn+1-xn+1〉≤0,∀y∈C.
Putting y:=wn+1 in (3.11) and y:=wn in (3.12), we get that
(3.13)〈wn+1-wn,T2wn〉-1rn〈wn+1-wn,(1+rn)wn-xn〉≤0,(3.14)〈wn-wn+1,T2wn+1〉-1rn+1〈wn-wn+1,(1+rn+1)wn+1-xn+1〉≤0.
Adding (3.13) and (3.14), we have
(3.15)〈wn+1-wn,T2wn-T2wn+1〉-〈wn+1-wn,(1+rn)wn-xnrn-(1+rn+1)wn+1-xn+1rn+1〉≤0,
which implies that
(3.16)〈wn+1-wn,(wn+1-T2wn+1)-(wn-T2wn)〉-〈wn+1-wn,wn-xnrn-wn+1-xn+1rn+1〉≤0.
Now, using the fact that T2 is pseudocontractive, we get that
(3.17)〈wn+1-wn,wn-xnrn-wn+1-xn+1rn+1〉≥0,
and hence
(3.18)〈wn+1-wn,wn-wn+1+wn+1-xn-rnrn+1(wn+1-xn+1)〉≥0.
Without loss of generality, let us assume that there exists a real number b such that rn>b>0 for all n∈N. Then, we have
(3.19)‖wn+1-wn‖2≤〈wn+1-wn,xn+1-xn+(1-rnrn+1)(wn+1-xn+1)〉≤‖wn+1-wn‖{‖xn+1-xn‖+|(1-rnrn+1)|⋅‖wn+1-xn+1‖},
and hence from (3.19) we obtain that
(3.20)‖wn+1-wn‖≤‖xn+1-xn‖+1rn+1|rn+1-rn|⋅‖wn+1-xn+1‖≤‖xn+1-xn‖+1b|rn+1-rn|L,
where L=sup{∥wn-xn∥:n∈N}. Furthermore, from (3.10) and (3.20), we have that
(3.21)‖xn+1-xn‖≤αnβ‖xn-xn-1‖+|αn-αn-1|K+(1-αn)(‖xn-xn-1‖+1b|rn-rn-1|L)=(1-αn(1-β))‖xn-xn-1‖+K|αn-αn-1|+(1-αn)Lb|rn-rn-1|.
Now, using conditions of {αn}, {rn} and Lemma 2.2, we have that
(3.22)lim⁡n→∞‖xn+1-xn‖=0.
Consequently, from (3.20) and (3.22), we obtain that
(3.23)lim⁡n→∞‖wn+1-wn‖=0.
Similarly, taking un=Trnwn and un+1=Trn+1wn+1 and following the method used for wn, we get that limn→∞∥un+1-un∥=0. Furthermore, since xn=αn-1f(xn-1)+(1-αn-1)un-1, we have that
(3.24)‖xn-un‖≤‖xn-un-1‖+‖un-1-un‖≤αn-1‖f(xn-1)-un-1‖+‖un-1-un‖.
Thus, since αn→0, we obtain that
(3.25)‖xn-un‖⟶0.
Moreover, for v∈F, using Lemma 3.2, we get that
(3.26)‖wn-v‖2=‖Frnxn-Frnv‖2≤〈Frnxn-Frnv,xn-v〉=〈wn-v,xn-v〉=12(‖wn-v‖2+‖xn-v‖2-‖xn-wn‖2),
and hence
(3.27)‖wn-v‖2≤‖xn-v‖2-‖xn-wn‖2.
Therefore, from (3.5), the convexity of ∥·∥2, (3.7) and (3.27) we get that
(3.28)‖xn+1-v‖2=‖αnf(xn)+(1-αn)un-v‖2≤αn‖f(xn)-v‖2+(1-αn)‖un-v‖2≤αn‖f(xn)-v‖2+(1-αn)‖wn-v‖2≤αn‖f(xn)-v‖2+(1-αn)(‖xn-v‖2-‖xn-wn‖2)≤αn‖f(xn)-v‖2+‖xn-v‖2-(1-αn)‖xn-wn‖2,
and hence
(3.29)(1-αn)‖xn-wn‖2≤αn‖f(xn)-v‖2+‖xn-v‖2-‖xn+1-v‖2≤αn‖f(xn)-v‖2+‖xn-xn+1‖(‖xn-v‖+‖xn+1-v‖).
So we have ∥xn-wn∥→0 as n→∞. This implies with (3.25) that ∥un-wn∥≤∥un-xn∥+∥xn-wn∥→0 as n→∞.
Next, we show that(3.30)limsup⁡n→∞〈f(z)-z,xn-z〉≤0,
where z=PFf(z). To show this inequality, we choose a subsequence {xni} of {xn} such that
(3.31)limsup⁡n→∞〈f(z)-z,xn-z〉=lim⁡i→∞〈f(z)-z,xni-z〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} and w∈H such that xnij⇀w. Without loss of generality, we may assume that xni⇀w. Since {xni}⊂C and C is convex and closed, we get that w∈C. Moreover, since xn-wn→0 as n→∞, we have that wni⇀w. Now, we show that w∈F. Note that, from the definition of wni, we have
(3.32)〈y-wni,T2wni〉-1rni〈y-wni,(rni+1)wni-xni〉≤0,∀y∈C.
Put zt=tv+(1-t)w for all t∈(0,1] and v∈C. Consequently, we get that zt∈C. From (3.32) and pseudocontractivity of T2, it follows that
(3.33)〈wni-zt,T2zt〉≥〈wni-zt,T2zt〉+〈zt-wni,T2wni〉-1rni〈zt-wni,(1+rni)wni-xni〉=-〈zt-wni,T2zt-T2wni〉-1rni〈zt-wni,wni-xni〉-〈zt-wni,wni〉≥-‖zt-wni‖2-1rni〈zt-wni,wni-xni〉-〈zt-wni,wn〉=〈wni-zt,zt〉-〈zt-wni,wni-xnirni〉.
Then, since wn-xn→0, as n→∞, we obtain that (wni-xni)/rni→0 as i→∞. Thus, as i→∞, it follows that
(3.34)〈w-zt,T2zt〉≥〈w-zt,zt〉,
and hence
(3.35)-〈v-w,T2zt〉≥-〈v-w,zt〉∀v∈C.
Letting t→0 and using the fact that T2 is continuous, we obtain that
(3.36)-〈v-w,T2w〉≥-〈v-w,w〉∀v∈C.
Now, let v=T2w. Then, we obtain that w=T2w, and hence w∈F(T2). Furthermore, the fact that un-wn→0 and wni⇀w imply that uni⇀w, following the method used for wn, we obtain that w∈F(T1), and hence w∈⋂i=12F(Ti). Therefore, since z=PFf(z), by Lemma 2.1, we have
(3.37)limsup⁡n→∞〈f(z)-z,xn-z〉=lim⁡i→∞〈f(z)-z,xni-z〉=〈f(z)-z,w-z〉≤0.
Now, we show that xn→z as n→∞. From xn+1-z=αn(f(xn)-z)+(1-αn)(un-z), we have that
(3.38)‖xn+1-z‖2≤(1-αn)2‖un-z‖2+2αn〈f(xn)-z,xn+1-z〉=(1-αn)2‖un-z‖2+2αn〈f(xn)-f(z),xn+1-z〉+2αn〈f(z)-z,xn+1-z〉≤(1-αn)2‖xn-z‖2+2αnβ‖xn-z‖⋅‖xn+1-z‖+2αn〈f(z)-z,xn+1-z〉≤(1-αn)2‖xn-z‖2+αnβ{‖xn-z‖2+‖xn+1-z‖2}+2αn〈f(z)-z,xn+1-z〉.
This implies that,
(3.39)‖xn+1-z‖2≤(1-αn)2+αnβ1-αnβ‖xn-z‖2+2αn1-αnβ〈f(z)-z,xn+1-z〉=1-2αn+αnβ1-αnβ‖xn-z‖2+αn21-αnβ‖xn-z‖2+2αn1-αnβ〈f(z)-z,xn+1-z〉≤(1-γn)‖xn-z‖2+σn,
where γn:=2(1-β)αn/(1-αnβ), σn:=(2(1-β)αn/(1-αnβ)){αnM/2(1-β)+(1/(1-β))〈f(z)-z,xn+1-z〉}, for M=sup{∥xn-z∥2:n∈N}. But note that ∑n=1∞γn=∞, limn→∞γn=0, and limsupn→∞σn/γn≤0. Therefore, by Lemma 2.2, we conclude that {xn} converges to z∈F, where z=PFf(z). This completes the proof.If, in Theorem3.3, f=u∈C is a constant mapping, then we get z=PF(u). In fact, we have the following corollary.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Ti:C→C, for i=1,2, be continuous pseudocontractive mappings such that F:=⋂i=12F(Ti)≠∅. Let {xn} be a sequence generated by x1,u∈C and
(3.40)xn+1=αnu+(1-αn)TrnFrnxn,
where {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈F, where z=PF(u).If, in Theorem3.3, we have that T2≡I, identity mapping on C, then we obtain the following corollary.Corollary 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T1:C→C be continuous pseudocontractive mapping such that F(T1)≠∅. Let f be a contraction of C into itself, and let {xn} be a sequence generated by x1∈C and
(3.41)xn+1=αnf(xn)+(1-αn)Trnxn,
where {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈F, where z=PF(T1)f(z).LetH be a real Hilbert space. Let Ai:H→H, for i=1,2, be accretive mappings. Let Trn′x:={z∈H:〈y-z,(I-A1)z〉-(1/rn)〈y-z,(1+rn)z-x〉≤0, for all y∈H}, Frn′x:={z∈H:〈y-z,(I-A2)z〉-(1/rn)〈y-z,(1+rn)z-x〉≤0, for all y∈H}. Then we have the following convergence theorem for a zero of two accretive mappings.Theorem 3.6.
LetH be a real Hilbert space. Let Ai:H→H, for i=1,2, be continuous accretive mappings such that N:=⋂i=12N(Ai)≠∅. Let f be a contraction of H into itself, and let {xn} be a sequence generated by x1∈H and
(3.42)xn+1=αnf(xn)+(1-αn)Trn′Frn′xn,
where {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈N, where z=PN(f(z)).Proof.
LetTi:=(I-Ai), for i=1,2. Then, we get that Ti, for i=1,2, are continuous pseudocontractive mappings with ⋂i=12N(Ai)=⋂i=12F(Ti). Thus, the conclusion follows from Theorem 3.3.The proof of the following theorem can be easily obtained from the method of proof of Theorem3.3.Theorem 3.7.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Ti:C→C, for i=1,2,…,L, be continuous pseudocontractive mappings such that F:=⋂i=1LF(Ti)≠∅. Let f be a contraction of C into itself, and let {xn} be a sequence generated by x1∈C and
(3.43)xn+1=αnf(xn)+(1-αn)K1,rnK2,rn,…,KN,rnxn,
where Ki,rnx:={z∈C:〈y-z,Tiz〉-(1/rn)〈y-z,(1+rn)z-x〉≤0, for all y∈C}, for i=1,2,…,L, and {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈F, where z=PF(f(z)).


## 4. Application

In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Hilbert spaces. Leth and g be continuously Fréchet differentiable convex functionals such that the gradient of h, (∇h) and the gradient of g, (∇g) are continuous and accretive. For γ>0 and x∈H, let Trn′′x:={z∈H:〈y-z,(I-(∇h))z〉-(1/rn)〈y-z,(1+rn)z-x〉≤0, for all y∈H} and Frn′′x:={z∈H:〈y-z,(I-(∇g))z〉-(1/rn)〈y-z,(1+rn)z-x〉≤0, for all y∈H} for all x∈H. Then, the following theorem holds.Theorem 4.1.
LetH be a real Hilbert space. Let h and g be continuously Fréchet differentiable convex functionals such that the gradient of h, (∇h) and the gradient of g, (∇g) are continuous and accretive such that N:=N(∇h)∩N(∇g)≠∅. Let f be a contraction of H into itself, and let {xn} be a sequence generated by x1∈H and
(4.1)xn+1=αnf(xn)+(1-αn)Trn′′Frn′′xn,
where {αn}⊂[0,1] and {rn}⊂(0,∞) such that limn→∞αn=0, ∑n=1∞αn=∞, ∑n=1∞|αn+1-αn|<∞, liminfn→∞rn>0, and ∑n=1∞|rn+1-rn|<∞. Then, the sequence {xn}n≥1 converges strongly to z∈F, where z=PN(f(z)).Proof.
The conclusion follows from Theorem3.6. We note that from the convexity and Fréchet differentiability of h and g we have N(∇h)=argminy∈Ch(y) and N(∇g)=argminy∈Cg(y).Remark 4.2.
Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators. In particular, Theorem3.3 extends Theorem 2.2 of Moudafi [24] and Theorem 4.1 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudocontractive mappings. Moreover, this provides affirmative answer to the concern raised.



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