Let E be a smooth Banach space with a norm ·. Let V(x,y)=x2+y2-2 x,Jy for any x,y∈E, where ·,· stands for the duality pair and J is the normalized duality mapping. We define a V-strongly nonexpansive mapping by V(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a V-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

1. Introduction

Let E be a smooth Banach space with a norm · and let E* be the dual space of E. We denote by ·,· a duality pair on E×E* and let J be the normalized duality mapping on E. It is well known that J is a continuous single-valued mapping in a smooth Banach space and a one-to-one mapping in a strictly convex Banach space (cf. [1]). We define a mapping V:E×E→R by V(x,y)=x2+y2-2x,Jy for all x,y∈E, where R is a set of real numbers. It is obvious that V(x,y)≥(x-y)2≥0. Let any y∈E be fixed, and then V(·,y) is a convex function because of convexity of ·2. Many nonlinear mappings which are defined by using V(·,·) are studied (see [2–4]). We also defined a nonlinear mapping which is called a V-strongly nonexpansive mapping in [5] as follows.

Definition 1.

Let C be a nonempty subset of a smooth Banach space E. A mapping T:C→E is called V-strongly nonexpansive if there exists a constant λ>0 such that for all x,y∈C(1)V(Tx,Ty)≤V(x,y)-λVI-Tx,I-Ty,where I is the identity mapping on E.

From this definition, it is obvious that the identity mapping I is also a V-strongly nonexpansive mapping. In a Hilbert space, it is trivial that this mapping is nonexpansive since V(x,y)=x-y2 and that any firmly nonexpansive mapping is a V-strongly nonexpansive mapping with λ=1 (see [5]). Moreover, we showed that if there exists a fixed point of a V-strongly nonexpansive mapping T, then T is strongly nonexpansive with a Bregman distance in [5]. However, in Banach spaces, as we give an example in the later section, we find that there exists a V-strongly nonexpansive mapping with fixed points which is not nonexpansive. We should point out that a guarantee of continuity of the V-strongly nonexpansive mappings has not been given in a generalized Banach space yet.

In this paper, we prove a weak convergence theorem and strong convergence theorems for finding fixed points of a V-strongly nonexpansive mapping in Banach spaces and show the existence theorem of fixed point with a dissipative property.

2. Preliminaries

In this section, at first we show the relationship between a V-strongly nonexpansive mapping and other nonlinear mappings, in a Hilbert space. Secondly, we state some properties of V-strongly nonexpansive mappings in a Banach space and give an example of a V-strongly nonexpansive mapping which is not a quasinonexpansive mapping in a Banach space although T has fixed points. We finally show some lemmas which are necessary in order to prove our theorems.

Let C be a subset of a Banach space E and let T:C→E be a mapping. Then a point p in the closure of C is said to be an asymptotically fixed point of T if C contains a sequence {xn} which converges weakly to p and the sequence {xn-Txn} converges strongly to 0. F^(T) denotes the set of asymptotically fixed points of T. In [6], Reich introduced a strongly nonexpansive mapping which is defined by using the Bregman distance D(·,·).

Definition 2.

Let E be a Banach space. The Bregman distance corresponding to a function f:E→R is defined by (2)Dx,y=fx-fy-f′yx-y,where f is Gâteaux differentiable and f′(x) stands for the derivative of f at the point x. Let C be a nonempty subset of E. We say that the mapping T:C→E is strongly nonexpansive if F^(T)≠∅ and (3)Dp,Tx≤Dp,x∀p∈F^(T)x∈C,and if it holds that limn→∞D(Txn,xn)=0 for a bounded sequence {xn} such that limn→∞(D(p,xn)-D(p,Txn))=0 for any p∈F^(T).

Taking the function ·2 as the convex, continuous, and Gâteaux differentiable function f, we obtain the fact that the Bregman distance D(·,·) coincides with V(·,·). In particular, in a Hilbert space, it is trivial that D(x,y)=V(x,y)=x-y2.

Proposition 3 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

In a Hilbert space, a V-strongly nonexpansive mapping with F^(T)≠∅ is strongly nonexpansive.

Next we recall two mappings of other nonlinear mappings (cf. [6–9]). A firmly nonexpansive mapping and an α-inverse strongly monotone mapping are defined as follows.

Definition 4.

Let C be a nonempty, closed, and convex subset of a Banach space E. A mapping T:C→E is said to be firmly nonexpansive if (4)Tx-Ty2≤x-y,jfor all x,y∈C and some j∈J(Tx-Ty).

It is trivial that a firmly nonexpansive mapping is nonexpansive.

Definition 5.

Let H be a Hilbert space. A mapping T:C→H is said to be α-inverse strongly monotone if (5)αTx-Ty2≤x-y,Tx-Tyfor all x,y∈C.

The relation among firmly nonexpansive mappings, α-inverse strongly monotone mappings and V-strongly nonexpansive mappings is shown in the following proposition.

Proposition 6 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

In a Hilbert space, the following hold.

A firmly nonexpansive mapping is V-strongly nonexpansive with λ=1.

Let A be an α-inverse strongly monotone mapping for α>1/2; then S=(I-A) is V-strongly nonexpansive with (2α-1).

The above (b) is obvious by showing that, for all x,y∈H,(6)Sx-Sy,x-y≤x-y2-α(I-S)x-I-Sy2.We will introduce some properties of V-strongly nonexpansive mappings in [5].

Proposition 7 (see [<xref ref-type="bibr" rid="B10">5</xref>]).

In a smooth Banach space E, the following hold.

For c∈(-1,1], T=cI is V-strongly nonexpansive. For c=1, T=I is V-strongly nonexpansive for any λ>0. For c∈(-1,1), T=cI is V-strongly nonexpansive for any λ∈(0,(1+c)/(1-c)].

If T is V-strongly nonexpansive with λ, then, for any α∈[-1,1] with α≠0, αT is also V-strongly nonexpansive with α2λ.

If T is V-strongly nonexpansive with λ≥1, then A=I-T is V-strongly nonexpansive with λ-1.

Suppose that T is V-strongly nonexpansive with λ and that α∈[-1,1] satisfies α2λ≥1. Then (I-αT) is V-strongly nonexpansive with (α2λ)-1. Moreover, if Tα=I-αT, then (7)V(Tαx,Tαy)≤V(x,y)-λ-1V(Tx,Ty).

Now we give an example of a V-strongly nonexpansive mapping in a Banach space.

Example 8 (see [<xref ref-type="bibr" rid="B11">10</xref>]).

Let 1<p,q<∞ such that 1/p+1/q=1. Let E=R×R be a real Banach space with a norm ·p defined by (8)xp=x1p+x2p1/p∀x=(x1,x2)∈E.Then E is smooth, and the normalized duality mapping J is single-valued. J is given by (9)Jx=xp2-px1x1p-2,x2x2p-2∈lqR×R∀x=x1,x2∈E.Hence, we have for x,y∈E that (10)Vx,y=xp2+yp2-2x,Jy=xp2+yp2-2yp2-p·x1y1y1p-2+x2y2y2p-2.We define a mapping T:E→E as follows: (11)Tx=xifxp≤1,1xpxifxp>1.In a case of p=1, we have shown that the mapping T defined by (11) is a V-strongly nonexpansive mapping (see [5]). We will show that T is V-strongly nonexpansive with any λ≤1, for p>1.

Proposition 9.

Suppose that T is defined by the formula (11) under the above situation. Then, T is a V-strongly nonexpansive mapping with any λ≤1.

Proof.

Case (a): suppose that x,y∈E with xp≤1 and yp>1.

Since Ty=(Ty)1,(Ty)2=y1yp-1,y2yp-1, we have that (12)VTx,Ty=Vx,Ty=xp2+Typ2-2Typ2-p·x1Ty1Ty1p-2+x2Ty2Ty2p-2=xp2+1-2yp1-p·x1y1y1p-2+x2y2y2p-2.Since (13)y-Ty=yp-1ypy1,yp-1ypy2,we have that (14)Vx-Tx,y-Ty=V(0,y-Ty)=y-Typ2=(yp-1)ypyp2=yp-12.Hence, we obtain that (15)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=xp2+yp2-2yp2-px1y1y1p-2+x2y2y2p-2-xp2-1+2yp1-px1y1y1p-2+x2y2y2p-2-λyp-12=yp2-1-2yp1-p(yp-1)·x1y1y1p-2+x2y2y2p-2-λyp-12≥(yp-1)yp+1-2yp1-pmmmmmmmmm·(x1y1p-1+x2y2p-1)mmmmmmmmmyp1-p-λ(yp-1).Hölder’s inequality implies that (16)x1y1p-1+x2y2p-1≤xpy1p-1q+y2p-1q1/q=xpy1p+y2p1/q=xpypp-1.Therefore, we obtain that (17)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)≥(yp-1)·yp+1-2yp1-pxpypp-1-λyp+λ=(yp-1)yp+1-2xp-λyp+λ≥(yp-1)(1-λ)yp+1-2+λ=(yp-1)(1-λ)(yp-1)=(1-λ)yp-12≥0,foranyλ∈[0,1].

That is, the inequality (1) holds.

Case (b): suppose that x,y∈E with xp≥1 and yp≤1.

Then we have that (18)VTx,Ty=VTx,y=1+yp2-2xp-1yp2-p·x1y1y1p-2+x2y2y2p-2,(19)V(x-Tx,y-Ty)=V(xp-1)xpx,0=xp-12.Hence, we have that (20)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=xp2+yp2-2yp2-p·x1y1y1p-2+x2y2y2p-2-1-yp2+2yp2-pxp-1x1y1y1p-2+x2y2y2p-2-λxp-12≥xp2-1-2yp2-p·x1y1p-1+x2y2p-11-xp-1-λ1-xp2.As (a), we obtain from Hölder’s inequality that (21)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)≥xp2-1-2xpyp2-pypp-1·1-xp-1-λxp-12=xp-1xp+1-2ypxp-1-λxp-12=xp-1xp+1-2yp-λxp+λ≥xp-1(1-λ)xp-1=(1-λ)xp-12≥0,foranyλ∈[0,1].That is, the inequality (1) holds.

Case (c): suppose that x,y∈E with xp,yp≥1.

Then we have that (22)VTx,Ty=1+1-2xp-1x1,x2,mmmmimmmmyp1-py1y1p-2,y2y2p-2=2-2xp-1yp1-px1y1y1p-2+x2y2y2p-2,V(x-Tx,y-Ty)=Vxp-1xpx,yp-1ypy=xp-12+yp-12-2(xp-1)(yp-1)xp-1yp-1yp2-p·(x1,x2),y1p-2y1,y2p-2y2=xp-12+yp-12-2xp-1yp-1xp-1yp1-p·x1y1y1p-2+x2y2y2p-2.Hence, we have that (23)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=xp2+yp2-2yp2-px1y1y1p-2+x2y2y2p-2-2+2xp-1yp1-px1y1y1p-2+x2y2y2p-2-λxp-12-λyp-12+2λxp-1yp-1xp-1yp1-p·x1y1y1p-2+x2y2y2p-2=xp2+yp2-2-λxp-12-λyp-12-2xp-1yp1-px1y1y1p-2+x2y2y2p-2·xpyp-1-λxp-1yp-1.It is obvious that (24)xpyp-1-λxp-1yp-1≥0for any λ∈[0,1] and xp,yp≥1. Thus, we have from Hölder’s inequality that (25)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)≥xp2+yp2-2-λxp-12-λyp-12-2xp-1yp1-pxpypp-1·xpyp-1-λxp-1yp-1=xp2+yp2-2-λxp-12-λyp-12-2xpyp-1-λxp-1yp-1=xp2+yp2-2-λ·xp2-2xp+1+yp2-2yp+1-2xpyp+2+2λxpyp-xp-yp+1=xp-yp2-λxp-yp2=(1-λ)xp-yp2≥0,foranyλ∈[0,1].That is, the inequality (1) holds.

It is clear that if xp,yp≤1 then inequality (1) holds. Therefore, from Cases (a), (b), and (c), we obtain the conclusion that T is V-strongly nonexpansive for any λ∈(0,1].

Remark 10.

When p=1, we have given the result in [5]. When p=2, we already know that E is a Hilbert space and a V-strongly nonexpansive mapping T is nonexpansive.

Theorem 11.

There exists a V-strongly nonexpansive mapping T with a nonempty subset of fixed points such that T is not nonexpansive for some Banach space.

Proof.

It is enough to show that the V-strongly nonexpansive mapping which is given in the previous proposition is not nonexpansive.

Let x=(0,1)∈E. Suppose that y=(y1,y2) satisfies that ypp=|y1|p+|y2|p>1 and 0<y1,y2<1. Then Ty=yp-1y. Let h=y2/y1 and t=yp-1y1-y1. We have that t<0 and yp-1y2-y2=ht<0. Then we obtain that Ty=(yp-1y1,yp-1hy1). Then, we have that (26)Tx-Typp=-yp-1y1,1-yp-1hy1p=-yp-1y1p+1-yp-1hy1p=yp-1y1p+1-yp-1hy1p=y1+tp+1-hy1+tp,and since x-ypp=y1p+(1-hy1)p, we have that (27)Tx-Typp-x-ypp=y1+tp-y1p+1-hy1+tp-1-hy1p.Therefore, we will show that (28)Tx-Typp-x-ypp>0⟺y1+tp-y1p+1-hy1+tp-1-hy1p>0⟺{y1+tp-y1p}t-1+1-hy1+tp-1-hy1pt-1<0,since t<0. Let h be fixed. As ypp=y1p+(hy1)p→1, t=yp-1y1-y1→0. Thus, we have for a sufficiently small |t| that (29)y1+tp-y1pt-1+{1-hy1+tp-1-hy1p}t-1<0⟺py1p-1-ph1-hy1p-1<0.It is trivial that (30)py1p-1-ph1-hy1p-1<0⟺y1p-1<h1-hy1p-1⟺y1p<y21-y2p-1.Let p=3/2. For y=(0.2,0.95), we have that (31)y1p=0.23/2<0.950.051/2=y21-y2p-1.We obtain that ypp=(0.2)3/2+(0.95)3/2>1 and that (32)Tx-Typp=yp-p{0.23/2+yp-0.953/2}>0.23/2+0.053/2=x-ypp.Therefore, we obtain the conclusion.

We remark that the symbols xn→u and xn⇀u mean that {xn} converges strongly and weakly to u, respectively. We will introduce the following important lemmas for proofs of our theorems.

Lemma 12.

(a) For all x,y,z∈E, (33)Vx,y≤Vx,y+Vy,z=V(x,z)-2x-y,Jy-Jz.(b) Let {xn} be a sequence in E such that there exists limn→∞V(xn,p)<∞ for some p∈E; then {xn} is bounded.

Lemma 13 (see [<xref ref-type="bibr" rid="B5">3</xref>]).

Let E be a smooth and uniformly convex Banach space and C a nonempty, convex, and closed subset of E. Suppose that T:C→E satisfies (34)VTx,Ty≤Vx,y∀x,y∈C.If a weakly convergent sequence {zn}n≥1⊂C satisfies that limn→∞V(Tzn,zn)=0, it holds that zn⇀z∈F(T).

Theorem 14 (see [<xref ref-type="bibr" rid="B16">1</xref>, <xref ref-type="bibr" rid="B15">11</xref>]).

Let Y be a compact subset of a topological vector space E and let X be a convex subset of Y. Let A:X→2Y be an operator such that, for each y∈Y, A-1y is convex. Suppose that B:X→2Y satisfies the following:

Bx⊂Ax for each x∈X,

B-1y≠∅ for each y∈Y,

Bx is open for each x∈X.

Then there exists a point x0∈X such that x0∈Ax0.

Lemma 15 (see [<xref ref-type="bibr" rid="B17">12</xref>]).

Let s>0 and let E be a Banach space. Then E is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞), g(0)=0, such that (35)x+y2≥x2+2y,j+g(y)for all x,y∈{z∈E:z≤s} and j∈Jx.

Lemma 16 (see [<xref ref-type="bibr" rid="B9">13</xref>]).

Let E be a smooth and uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞) such that g(0)=0 and, for each real number r>0, (36)0≤g(x-y)≤V(x,y)for all x,y∈Br={z∈E:z≤r}.

Lemma 17 (see [<xref ref-type="bibr" rid="B9">13</xref>]).

Let E be a smooth and uniformly convex Banach space and {yn} and {zn} in E. If limn→∞V(yn,zn)=0 and either {yn} or {zn} is bounded, then {yn-zn}→0.

3. Main Results

In this section, we prove a weak convergence theorem and strong convergence theorems for finding fixed points of a V-strongly nonexpansive mapping T in Banach spaces, and then we show the existence theorem for fixed points of T with a dissipative property (cf. [10]).

Theorem 18.

Let E be a smooth and uniformly convex Banach space and C a nonempty, closed, and convex subset of E. Suppose that a mapping T:C→C is V-strongly nonexpansive with λ and that F(T)≠∅. One defines a Mann iterative sequence {xn} as follows: for any x1∈C and n≥1, (37)xn+1=βnxn+(1-βn)Txn,where {βn}⊂(0,1) and limn→∞βn=0. Then xn⇀p0 for some p0∈F(T).

Proof.

Suppose that p∈F(T). Then we have from the convexity of V that (38)Vxn+1,p=V(βnxn+(1-βn)Txn,p)≤βnV(xn,p)+(1-βn)V(Txn,p)=βnV(xn,p)+(1-βn)V(Txn,Tp).Since T is V-strongly nonexpansive with λ, we have that (39)V(xn+1,p)≤βnV(xn,p)+(1-βn)·{V(xn,p)-λV((I-T)xn,(I-T)p)}=V(xn,p)-(1-βn)λV(xn-Txn,0)≤V(xn,p).Hence, we have limn→∞V(xn,p)=α<∞. From Lemma 12 (b), {xn} is bounded. Furthermore, we have that (40)(1-βn)λV(xn-Txn,0)≤V(xn,p)-V(xn+1,p).Since limn→∞βn=limn→∞{V(xn,p)-V(xn+1,p)}=0, we obtain that (41)limn→∞V(xn-Txn,0)=limn→∞xn-Txn2=0.This means that {xn-Txn} converges strongly to 0. Hence, {Txn} is also bounded, and there exists M>0 such that xn,Txn≤M-p for all n≥1.

On the other hand, we have from Lemma 12 (a) that (42)0≤V(xn,Txn)=V(xn,p)-V(Txn,p)-2xn-Txn,JTxn-Jp≤V(xn,p)-V(Txn,p)+2xn-Txn(Txn+p)≤V(xn,p)-V(Txn,p)+2Mxn-Txn=xn2-Txn2-2xn-Txn,Jp+2Mxn-Txn=(xn-Txn)(xn+Txn)-2xn-Txn,Jp+2Mxn-Txn≤xn-Txn(xn+Txn+2M)-2xn-Txn,Jp.Hence, we obtain that limn→∞V(xn,Txn)=limn→∞V(Txn,xn)=0. From Lemma 13, there exists a point p0∈F(T) such that xn⇀p0 and Txn⇀p0.

The duality mapping J of a Banach space E with Gâteaux differentiable norm is said to be weakly sequentially continuous if xn⇀x in E implies that {Jxn} converges weak star to Jx in E* (cf. [14]). This happens, for example, if E is a Hilbert space, or finite-dimensional and smooth, or lp if 1<p<∞ (cf. [15]). Next we prove a strong convergence theorem.

Theorem 19.

Let E be a reflexive, smooth, and strictly convex Banach space. Suppose that the duality mapping J of E is weakly sequentially continuous. Suppose that C is a nonempty, closed, and convex subset of E, T:C→C is V-strongly nonexpansive with λ, and F(T)≠∅. One defines a Mann iterative sequence {xn} as follows: for any x1∈C and n≥1,(43)xn+1=βnxn+(1-βn)Txn,where {βn}⊂(0,1) and limn→∞βn=0. If T satisfies that (44)x,JTx≤0∀x∈C,then xn→p0 and Txn→p0 for some p0∈F(T).

Proof.

As in the proof of Theorem 18, we obtain that limn→∞V(xn,Txn)=0 and xn⇀p0 and Txn⇀p0 for some p0∈F(T). Furthermore, from Lemma 12 (a), we have that (45)0≤V(xn,p0)+V(p0,Txn)=V(xn,Txn)-2xn-p0,Jp0-JTxn=V(xn,Txn)-2xn-p0,Jp0+2xn,JTxn-2p0,JTxn.Hence, the assumptions imply that (46)Vxn,p0⟶0,Vp0,Txn⟶0asn⟶∞.From Lemma 17, we have the conclusion that xn→p0 and Txn→p0.

Condition (44) is a definition of a linear dissipative mapping T (cf. [16]). Moreover, we give a definition of a J-dissipative mapping for nonlinear mappings in a Banach space.

Definition 20.

Let J be a single-valued duality mapping on E and let C be a nonempty subset of E. Then a mapping T:C→E is called J-dissipative if it holds that (47)x-y,JTx-JTy≤0for all x,y∈C.

In a Hilbert space, such a mapping T is called dissipative. In Banach spaces, we remark that the J-dissipative mapping is not equal to the dissipative mapping (cf. [17]). Next we give a characterization of J-dissipative mappings by using V(·,·).

Theorem 21.

Let E be a smooth Banach space, C a nonempty subset of E, and T:C→E a mapping. Then, the following are equivalent.

T is J-dissipative.

For all x,y∈C, (48)V(x,Ty)+V(y,Tx)≤V(x,Tx)+V(y,Ty).

Proof.

For any x,y∈C, (49)x-y,JTx-JTy≤0is equal to (50)-2x,JTy-2y,JTx≤-2x,JTx-2y,JTy,-2x,JTy-2y,JTx+x2+Ty2+y2+Tx2≤-2x,JTx-2y,JTy+x2+Tx2+y2+Ty2.From the definition of V, this inequality is equivalent to (51)V(x,Ty)+V(y,Tx)≤V(x,Tx)+V(y,Ty).

Furthermore, we have the following result by this theorem.

Lemma 22.

Suppose that E is a smooth and strictly convex Banach space and that C⊂E is a nonempty convex subset. Assume that a mapping T:C→E is J-dissipative. If there are fixed points of T, then F(T) is singleton.

Proof.

Assume that there exist p0 and q0 such that Tp0=p0 and Tq0=q0. Since T is J-dissipative, we have by Theorem 21 that (52)0≤Vp0,Tq0+Vq0,Tp0≤V(p0,Tp0)+V(q0,Tq0)=V(p0,p0)+V(q0,q0)=0.Thus, we have that V(p0,q0)=V(q0,p0)=0. This implies that (53)0≤p0-q02≤V(p0,q0)=0,p0=q0.

Furthermore, we have (54)Vp0,q0=p02+q02-2p0,Jq0=p02+p02-2p0,Jq0=0,and we have p02=p0,Jq0. Since E is strictly convex and J is one-to-one, we obtain that p0=q0.

We give a result before proving an existence theorem for fixed points.

Theorem 23 (see [<xref ref-type="bibr" rid="B11">10</xref>]).

Let E be a smooth and uniformly convex Banach space, and let T:E→E be a V-strongly nonexpansive mapping with λ. Then, one has that (55)limx-y→0Tx-Ty=0,for x,y,Tx,Ty≤r, where r>0.

Proof.

Since T is a V-strongly nonexpansive with λ, we have (56)0≤VTx,Ty+λVx-Tx,y-Ty≤V(x,y)=x2+y2-2x,Jy=x2-y2-2x-y,Jy≤x-y(x+y+2y)=x-y(x+3y),foranyx,y∈E.Thus, we obtain, for x,y with x, y≤r, (57)V(Tx,Ty)⟶0,V(x-Tx,y-Ty)⟶0asx-y⟶0.From Lemma 16, we have that (58)0≤gTx-Ty≤VTx,Ty.Therefore, we have from (57) that limx-y→0g(Tx-Ty)=0. From the definition of g, we obtain that (59)limx-y→0Tx-Ty=0.

Remark 24.

If x∈E satisfies that Tx<r0 for r0>0, the (57) implies that Ty<r0+1 for y in the neighborhood of x.

We will prove the following existence theorem by using Theorem 14.

Theorem 25.

Let E be a reflexive, strictly convex, and smooth Banach space and C a nonempty, bounded, closed, and convex subset of E. Suppose T:C→C is a V-strongly nonexpansive and J-dissipative mapping. Then, there exists a unique fixed point of T.

Proof.

At first, we will show that there exists y0∈C such that (60){x∈C:V(x,Tx)<V(y0,Tx)}=∅.Assume that, for all y∈C, (61){x∈C:V(x,Tx)<V(y,Tx)}≠∅.Let Ax={y∈C:V(x,Ty)<V(y,Ty)} and Bx={y∈C:V(x,Tx)<V(y,Tx)} for all x∈C. Then, from the assumption, B-1y is nonempty for all y∈C. Since T is J-dissipative, Theorem 21 implies that (62)V(x,Ty)-V(y,Ty)≤V(x,Tx)-V(y,Tx)for all y∈Bx. This means that Bx⊂Ax for any x∈C. For any y∈C, let vj∈A-1y with j∈{1,2,…,n}, and suppose that v=∑j=1nαjvj and ∑j=1nαj=1 with αj>0. From the convexity of V, we have (63)Vv,Ty=V∑j=1nαjvj,Ty≤∑j=1nαjVvj,Ty≤∑j=1nαjV(y,Ty)=V(y,Ty).Thus, we obtain that A-1y is convex for all y∈C. Since it is obvious that Bx is open for each x∈C, Theorem 14 implies that there exists a point x0∈C such that x0∈Ax0. This means that (64)V(x0,Tx0)<V(x0,Tx0).This is a contradiction. Thus, we have for some y0∈C that (65){x∈C:V(x,Tx)<V(y0,Tx)}=∅.This means that there exists y0∈C such that (66)V(y0,Tx)≤V(x,Tx)for all x∈C.

Furthermore, we will show V(y0,Ty0)≤V(x,Ty0) for all x∈C if y0 satisfies (66). Let yt=(1-t)y0+tx for any t∈(0,1) and x∈C. Since C is convex, then yt∈C. Thus, we obtain that (67)Vy0,Tyt≤V(yt,Tyt)=V((1-t)y0+tx,Tyt).From the convexity of V(·,y) for y∈C, (68)V(y0,Tyt)≤(1-t)V(y0,Tyt)+tV(x,Tyt)and we have V(y0,Tyt)≤V(x,Tyt). From the definition of V(·,·), we have that (69)Vx,Tyt-Vx,Ty0=Tyt2-Ty02-2x,JTyt-JTy0≤(Tyt+Ty0)Tyt-Ty0+2xJTyt-JTy0.Therefore, we have, by Theorem 23 and the continuity of J on a smooth Banach space, that limt→0+V(x,Tyt)=V(x,Ty0) and (70)Vy0,Ty0=limt→0+Vy0,Tyt≤limt→0+V(x,Tyt)=V(x,Ty0)for all x∈C. Letting x=Ty0, we have that (71)V(y0,Ty0)≤V(Ty0,Ty0)=0.Hence, V(y0,Ty0)=0. This implies that (72)y02+Ty02=2y0,JTy0≤2y0Ty0,and then we obtain that (73)y0-Ty02≤0.Thus, we have y0=Ty0 and we have by (72) that y02=y0,JTy0. Since J is one-to-one on a strictly convex Banach space, JTy0=Jy0 implies that Ty0=y0. Therefore, we have the conclusion.

Finally, we will prove a strong convergence theorem for finding fixed points of a V-strongly nonexpansive mapping T in a Banach space, without the assumption that F(T)≠∅.

Theorem 26.

Let E be a smooth and uniformly convex Banach space, and let C be a nonempty, compact, and convex subset of E. Suppose that T:C→C is J-dissipative and V-strongly nonexpansive with λ. One defines a Mann iterative sequence {xn} as follows: for any x1∈C and n≥1,(74)xn+1=βnxn+(1-βn)Txn,where {βn}⊂(0,1) and limn→∞βn=0. Then, there exists a unique fixed point p0∈C such that xn→p0 and Txn→p0.

Proof.

From Theorem 25, we have that F(T)≠∅. As in the proof of Theorem 18, we obtain that limn→∞V(xn,Txn)=0 and that there exists a point p0∈F(T) such that xn⇀p0 and Txn⇀p0. Since T is J-dissipative, Theorem 21 implies that (75)0≤V(xn,Tp0)+V(p0,Txn)≤V(xn,Txn)+V(p0,Tp0).From Tp0=p0, we have for n≥1 that (76)0≤Vxn,p0+Vp0,Txn≤V(xn,Txn)+V(p0,p0)=V(xn,Txn).Since limn→∞V(xn,Txn)=0, we have that (77)limn→∞V(xn,p0)=limn→∞V(p0,Txn)=0.By Lemma 17, we obtain that xn→p0 and Txn→p0. We have the conclusion.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the reviewers of this paper for their enduring reading and Professor N. Hirano and Professor A. T. Lau for their encouragement. Finally, the author deeply appreciates her mother Ayako’s hearty support.

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