# Strong Convergence Theorems for Asymptotically WeakG-Pseudo-Ψ-Contractive Non-Self-Mappings with the Generalized Projection in Banach Spaces

Authors: Yuanheng Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A new concept of the asymptotically weakG-pseudo-Ψ-contractive non-self-mapping T:G↦B is introduced and some strong convergence theorems for the mapping are proved by using the generalized projection method combined with the modified successive approximation method or with the modified Mann iterative sequence method in a uniformly and smooth Banach space. The proof methods are also different from some past common methods.


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## Body

## 1. Introduction

LetB be a real Banach space with the norm ∥·∥, B* its dual space with the norm ∥·∥*. As usually, we introduce a dual product in B*×B by 〈x*,x〉, where x*∈B* and x∈B. Let J:B↦B* be the normalized duality mapping J in B defined as
(1.1)Jx={f∈B*:〈f,x〉=∥f∥*∥x∥=∥x∥2},∀x∈B.
It is clear that the operator J is well defined in a Banach space by the famous Hahn-Banach theorem.The concept of asymptotically nonexpansive mappings was first introduced by Goebel and Kirk [1] in 1972 and then Schu [2] introduced the asymptotically pseudocontractive mappings in 1991.Definition 1.1.
LetG be a nonempty subset of a real Banach space B and T:G↦G be a mapping. (1)
The mappingT is said to be asymptotically nonexpansive, if there exists a number sequence {kn} in [1,∞) with limn→∞kn=1 such that
(1.2)∥Tnx-Tny∥≤kn∥x-y∥,
for all x,y∈G and n≥1.(2)
The mappingT is said to be asymptotically pseudocontractive, if for all x,y∈G, there exists a number sequence {kn} in [1,∞) with limn→∞kn=1 and j(x-y)∈J(x-y) such that
(1.3)〈Tnx-Tny,j(x-y)〉≤kn∥x-y∥2.(3)
The mappingT is said to be asymptotically demi-pseudocontractive, if for all x∈G, p∈F(T), there exists a number sequence kn in [1,∞) with limn→∞kn=1 and j(x-y)∈J(x-y) such that
(1.4)〈Tnx-p,j(x-p)〉≤kn∥x-p∥2,
where F(T)≠∅ is the set of all fixed points of the mapping T.The iterative approximation problems for asymptotically nonexpansive and pseudocontractive mappingT were studied by many authors and we always assume that the fixed point set F(T) of the operator T is nonempty, such as see [1–6]. In 2011, Qin et al. [7] introduced a new concept of the asymptotically strict quasi-Φ-pseudocontractive mapping T:G→G. They combined the generalized projection ΠG to give a new iterative sequence for the T and proved that the sequence converges strongly to a point x′=ΠF(T)x0.But, all these arguments are not enough if the operatorT acts from G to B, which we called non-self-mappings, and the iterative methods we used to be, such as Mann iterative method and its some modifications, can not be used. Under this condition, it is natural for us to try to consider the metric projection operator PG:B↦G and the generalized projection operator ΠG:B↦G, and some authors have given relevant results and applications of the operator PG and πG (see [8–11]).Very recently, in 2012, Yao et al.[12] and Liou et al. [13] considered the non-self-mapping T:G⊆H→H in a Hilbert space H. They also proved their new iterative sequence for the T combined with the metric projection PG converges strongly to a point x′=PVI(G,T)(0) and the unique solution of a variational inequality, respectively.Motivated and inspired by the said above, we first introduce a new concept of the asymptotically weakG-pseudo-Ψ-contractive non-self-mapping T:G↦B. Then, in a uniformly convexand smooth Banach space, we prove some strong convergence theorems for the mapping by using the generalized projection method and the modified successive approximation method
(1.5)xn+1=(ΠGT)nxn,n=1,2,…,x1∈G,
or the modified Mann iterative sequence method
(1.6)xn+1=QG((1-αn)xn+αnT(ΠGT)n-1xn),n=1,2,…,x1∈G,
where QG:B↦G is a sunny nonexpansive retraction. So, in some ways, our results extend and improve some results of other authors (such as, see [1–5, 7, 9–13]), from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces.


## 2. Preliminaries

In the sequel, we will assume thatB is a real uniformly convex and uniformly smooth (hence reflexive) Banach space, then B* will be the same. If we denote by δB(ε) the modulus of convexity of the Banach space B and by ρB(τ) its modulus of smoothness, then
(2.1)δB(ε),ρB(τ),gB(ε)=ε-1δB(ε),hB(τ)=τ-1ρB(τ)
are all continuous and increasing on their domains, respectively, and δB(0)=ρB(0)=gB(0)=hB(0)=0 (see [9]). Also, under the conditions the normalized duality operator
(2.2)J:Jx=12grad{∥x∥2}
is single-valued, strictly monotone, continuous, coercive, bounded, and homogeneous, but not addible. In a Hilbert space, J is the Identity operator I:Ix=x.Definition 2.1 (see [10, 11]).
The operatorPG:B↦G⊆B is called metric projection operator if it assigns to each x∈B its nearest point x¯∈G, that is, the solution x¯ for the minimization problem
(2.3)PGx=x¯;x¯:∥x-x¯∥=infξ∈G∥x-ξ∥.The operatorΠG:B↦G⊆B is called the generalized projection operator if it assigns to each x∈B a minimum point x^∈G of the Lapunov function V(x,ξ):B×B↦[0,∞):
(2.4)V(x,ξ)=∥x∥2-2〈Jx,ξ〉+∥ξ∥2,
that is, a solution of the following minimization problem:
(2.5)ΠGx=x^;x^:V(x,x^)=infξ∈GV(x,ξ).Lemma 2.2 (see [10, 11]).
The pointx¯=PGx is the metric projection of x∈B on G⊆B if and only if the following inequality is satisfied:
(2.6)〈J(x-x¯),x¯-ξ〉≥0,∀ξ∈G,
and the operator PG is nonexpansive in Hilbert spaces.
The pointx^=ΠGx is the generalized projection of x∈B on G⊆B if and only if the following inequality is satisfied:
(2.7)〈Jx-Jx^,x^-ξ〉≥0,∀ξ∈G.
Furthermore, the inequality below also holds:
(2.8)V(ΠGx,ξ)≤V(x,ξ)-V(x,ΠGx),∀ξ∈G.
And thus, we have
(2.9)V(ΠGx,ξ)≤V(x,ξ),∀ξ∈G.Lemma 2.3 (see [8]).
For allx,y∈B, if ∥x∥≤R and ∥y∥≤R, then the following inequality is satisfied:
(2.10)(2L)-1R2δB(∥x-y∥4R)≤V(x,y)≤4LR2ρB(4∥x-y∥R),
where L:1<L<1.7 is a constant.In general, the operatorPG and ΠG are not nonexpansive in Banach spaces. It is easy to see PG=ΠG in Hilbert spaces because of J=I. In a uniformly convex and uniformly smooth Banach space, PG is well defined on a closed convex set G and ΠG is also well defined on a closed convex set G from the properties of the functional V(x,ξ) and strict monotonicity of the mapping J. More properties of the mappings J, V, PG, and ΠG and some of their applications can be found in [8–11].Definition 2.4 (see [14]).
LetB be a real Banach space, G⊆B be a subset. The operator QG:B↦G is called sunny nonexpansive retract if QG is nonexpansive, QG2=QG, and for any x∈G, t>0, tx+(1-t)QGx∈G holds QG(tx+(1-t)QGx)=QGx.IfB is a uniformly smooth Banach space and G⊂B, is a closed convex set, then the unique sunny nonexpansive retract QG exists.Definition 2.5.
LetB be a real Banach space, G be a nonempty subset of B, and T:G↦B be a non-self-mapping. If there exists a sequence {kn} in [1,∞) with limn→∞kn=1 and a continuous increasing function Ψ(t) for all t>0 with Ψ(0)=0, limt→∞Ψ(t)=∞, it is shown as follows, respectively:(1)
The mappingT is said to be asymptotically weak G-Ψ-contractive mapping, if
(2.11)V(T(ΠGT)n-1x,T(ΠGT)n-1y)≤knV(x,y)-Ψ(V(x,y)).(2)
The mappingT is said to be asymptotically weak G-quasi-Ψ-contractive mapping, if
(2.12)V(T(ΠGT)n-1x,x*)≤knV(x,x*)-Ψ(V(x,x*)),
for all x,y∈G,x*∈F(T), and n≥1, where F(T)≠∅.(3)
The mappingT is said to be asymptotically weak G-Ψ-pseudocontractive mapping, if
(2.13)〈T(ΠGT)n-1x-T(ΠGT)n-1y,j(x-y)〉≤kn∥x-y∥2-Ψ(∥x-y∥2),
for all x,y∈G,j(x-y)∈J(x-y).(4)
The mappingT is said to be asymptotically weak G-quasi-Ψ-pseudocontractive mapping, if
(2.14)〈T(ΠGT)n-1x-x*,j(x-x*)〉≤kn∥x-x*∥2-Ψ(∥x-x*∥2),
for all x,y∈G,x*∈F(T), j(x-x*)∈J(x-x*), and n≥1, where F(T)≠∅.Remark 2.6.
It is clear that one can omit each operatorΠG in (2.11)–(2.14) if the mapping T acts from G to G, that is, T:G↦G⊆B is a self-mapping. So, the class of asymptotically weak G-Ψ-contractive mappings contains that of asymptotically nonexpansive mappings and the class of asymptotically weak G-Ψ-pseudocontractive mappings contains that of asymptotically pseudocontractive mappings. Therefore, all the results and applications of asymptotically nonexpansive mappings can be as a part of the asymptotically weak G-Ψ-contractive mappings.In order to prove our main results, we also need the following lemmas.Lemma 2.7 (see [15]).
Let{λn}, {αn}, {βn}, and {γn} be sequences of nonnegative numbers satisfying the following conditions:
(2.15)αn>0,∑n=1∞‍αn=∞,∑n=1∞‍βn<∞,limn→∞γnαn=0.
Suppose the following recurse inequality holds:
(2.16)λn+1≤(1+βn)λn-αnψ(λn)+γn,n=1,2,…,
where ψ(t) is a continuous strictly increasing function for all t>0 with ψ(0)=0, limt→∞ψ(t)=∞. Then λn→0 as n→∞.Lemma 2.8 (see [16]).
LetB be a real Banach space and J be the normalized duality mapping. Then
(2.17)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,
for all x,y∈B and j(x+y)∈J(x+y).


## 3. Main Results

Theorem 3.1.
LetB be a uniformly convex and uniformly smooth Banach space, G be a closed convex subset of B, T:G↦B be an asymptotically weak G-quasi-Ψ-contractive mapping with a sequence {kn}⊆[1,∞), ∑n=1∞‍(kn-1)<∞, and x*∈G is its fixed point. Then the iterative sequence {xn} generated by the modified successive approximation method (1.5) is bounded for all n≥1 and converges strongly to x*.Proof.
Ifx*∈F(T) is the fixed point of T in G, that is, Tx*=x*, then we get by (1.5) and (2.8) in Lemma 2.2 for x*∈G,
(3.1)V(xn+1,x*)=V((ΠGT)nxn,x*)=V(ΠGT(ΠGT)n-1xn,x*)≤V(T(ΠGT)n-1xn,x*).
We use the condition (2.12) of asymptotically weak G-quasi-Ψ-contractive of the operator T and get
(3.2)V(xn+1,x*)≤knV(xn,x*)-ψ(V(xn,x*))≤knV(xn,x*)≤knkn-1V(xn-1,x*)≤⋯≤knkn-1⋯k1V1(x1,x*).
Because∑n=1∞‍(kn-1)<∞, we know limn→∞knkn-1⋯k1= constant and {knkn-1⋯k1} is bounded, say by K:1≤knkn-1⋯k1≤K for all n≥1.
It is obviously thatV(x,ξ)=∥x∥2-2〈Jx,ξ〉+∥ξ∥2 satisfies the inequality
(3.3)(∥x∥-∥ξ∥)2≤V(x,ξ)≤(∥x∥+∥ξ∥)2.
Therefore by (3.2) and (3.3), we have
(3.4)∥xn∥-∥x*∥≤knkn-1⋯k1V1(x1,x*)≤K(∥x1∥+∥x*∥),∥xn∥≤K∥x1∥+(1+K)∥x*∥,
for all n≥1, that is, the sequence {xn} is bounded.
The sequence of positive number{λn} defined by λn=V(xn,x*) are bounded and from (3.2) it satisfies the following inequality:
(3.5)λn+1≤(1+βn)λn-αnΨ(λn)+γn,n=1,2,…,
where βn=kn-1, ∑n=1∞‍βn<∞, αn=1, ∑n=1∞‍αn=∞, γn=0. So, using Lemma 2.7 we get
(3.6)limn→∞V(xn,x*)=0.
BecauseR=K∥x1∥+(1+K)∥x*∥ is a constant and ∥xn∥≤R,∥yn∥≤R, we obtain from the left part of the estimate of (2.10) in Lemma 2.3 the following:(3.7)0≤limn→∞2L-1R2δB(∥xn-x*∥4R)≤limn→∞V(xn,x*)=0.
By the properties of δB(ε), this implies
(3.8)limn→∞∥xn-x*∥=0,
that is, the sequence {xn} converges strongly to fixed point x*.Corollary 3.2.
LetG be a closed convex set in B, T:G↦B be an asymptotically weak G-Ψ-contractive mapping with a sequence {kn}⊆[1,∞), ∑n=1∞‍(kn-1)<∞, and x*∈G its fixed point. Then the iterative sequence {xn} defined by modified successive approximation method (1.5) converges strongly to x*.Proof.
If we takey=x*∈G as the fixed point of T, then we have
(3.9)T(πGT)n-1x*=T(πGT)n-2(πGTx*)=T(πGT)n-2(πGx*)=T(πGT)n-2x*=⋯=Tx*=x*.So the asymptotically weakG-Ψ-contractive mapping T:G↦B is also an asymptotically weak G-quasi-Ψ-contractive mapping and the results of Theorem 3.1 still hold.Theorem 3.3.
LetB be a real uniformly convex and uniformly smooth Banach space, G be a nonempty closed convex subset of B, T:G↦B be an asymptotically weak G-quasi-Ψ-pseudocontractive mapping with a sequence {kn}⊆[1,∞), ∑n=1∞‍(kn-1)<∞, and x*∈G its fixed point. Consider the iterative sequence {xn} defined by the modified Mann iterative sequence method (1.6). Suppose the sequence {xn} and {T(ΠGT)n-1xn} are bounded, {αn} is a number sequence in (0,1] satisfing the conditions below:
(3.10)∑n=1∞‍αn=∞,∑n=1∞‍αn2<∞,
where QG:B↦G is a sunny nonexpansive retraction. Then the iterative sequence {xn} converges strongly to x*.Proof.
By the virtue of (2.17) in Lemma 2.8, it follows that
(3.11)∥xn+1-x*∥2=∥QG((1-αn)xn+αnT(ΠGT)n-1xn)-QGx*∥2≤∥(1-αn)xn+αnT(ΠGT)n-1xn-x*∥2=∥(1-αn)(xn-x*)+αn(T(ΠGT)n-1xn-x*)∥2≤(1-αn)2∥(xn-x*)∥2+2αn〈T(ΠGT)n-1xn-x*,j(xn+1-x*)〉=(1-αn)2∥(xn-x*)∥2+2αn〈T(ΠGT)n-1xn-x*,j(xn-x*)〉+2αn〈T(ΠGT)n-1xn-x*,j(xn+1-x*)-j(xn-x*)〉.
Since {xn-T(PGT)n-1xn} is bounded, say by K, we have
(3.12)∥(xn+1-x*)-(xn-x*)∥=∥QG((1-αn)xn+αnT(PGT)n-1xn)-QGxn∥≤∥(1-αn)xn+αnT(PGT)n-1xn-xn∥=αn∥xn-T(PGT)n-1xn∥≤αnK.From (3.10) we know limn→∞αn=0 and then one gets
(3.13)limn→∞∥xn+1-xn∥=limn→∞∥(xn+1-x*)-(xn-x*)∥=0.
By using the uniform continuity of j=J in the uniformly convex and uniform smooth Banach space B and the bound of the sequence {T(ΠGT)n-1xn-x*}, we have
(3.14)γn:=〈T(ΠGT)n-1xn-x*,j(xn+1-x*)-j(xn-x*)〉→0,n→∞.
Substituting (3.14) into (3.11) and using (2.14), we get
(3.15)∥xn+1-x*∥2≤(1-αn)2∥xn-x*∥2+2αn〈T(ΠGT)n-1xn-x*,j(xn-x*)〉+2αnγn≤(1-αn)2∥xn-x*∥2+2αn(kn∥xn-x*∥2-Ψ(∥xn-x*∥2))+2αnγn=(1+2αn(kn-1)+αn2)∥xn-x*∥2-2αnΨ(∥xn-x*∥2)+2αnγn≤(1+2(kn-1)+αn2)∥xn-x*∥2-2αnΨ(∥xn-x*∥2)+2αnγn.Thus, the sequence of positive number{λn}∞n=1 defined by λn=∥xn-x*∥2 satisfies the recursive inequality
(3.16)λn+1≤(1+βn)λn-2αnψ(∥xn-x*∥2)+2αnγn,
where βn=2(kn-1)+αn2, ∑n=1∞‍βn=2∑n=1∞‍(kn-1)+∑n=1∞‍αn2<∞, ∑n=1∞‍(2αn)=∞, γn→0 as n→∞. Therefore by the virtue of Lemma 2.7, it is clear that the assertion λn→0 as n→∞ holds, that is,
(3.17)limn→∞∥xn-x*∥2=0,limn→∞∥xn-x*∥=0.Corollary 3.4.
LetB be a real uniformly convex and uniformly smooth Banach space, G be a nonempty closed convex subset of B, T:G↦B be an asymptotically weak G-Ψ-pseudocontractive mapping with a sequence {kn}⊆[1,∞), ∑n=1∞‍(kn-1)<∞, and x*∈G its fixed point. Consider the iterative sequence {xn} defined by the modified Mann iterative sequence (1.6). Suppose the sequence {xn} and {T(ΠGT)n-1xn} are bounded, {αn} is a real number sequence in (0,1] satisfying the conditions (3.10). Then one has
(3.18)limn→∞∥xn+1-xn∥=0,
and the iterative sequence {xn} converges in norm to x*.Proof.
Following Theorem3.3, we can have the assertions of the corollary.Remark 3.5.
Because a Hilbert space must be a uniformly convex and uniformly smooth Banach space, the above results still hold in a Hilbert space. In fact, if we noticeΠG=PG in Hilbert spaces, we can abate some conditions in Corollary 3.4 and have the following theorem.Theorem 3.6.
LetG be a closed convex set of a Hilbert space H. T:G↦H is said to be an asymptotically weak G-quasi-Ψ-pseudocontractive mapping with a sequence {kn}⊆[1,∞), ∑n=1∞‍(kn-1)<∞, and x*∈G its fixed point, if
(3.19)∥T(PGT)n-1x-x*∥≤kn∥x-x*∥-ψ(∥x-x*∥),
where Ψ is a continuous increasing function for all t>0 with Ψ(0)=0, limt→∞Ψ(t)=∞. Consider the new modified Mann iterative sequence {xn} defined by the modified Mann iterative sequence (1.6). If the number sequence {αn} satisfies the conditions
(3.20)0<αn≤1,∑n=1∞‍αn=∞,
then the iterative sequence {xn} converges strongly to x*.Proof.
BecausePG=ΠG is nonexpansive in Hilbert spaces, {αn} satisfies (3.20) and the operator T satisfies (3.19), we get
(3.21)∥xn+1-x*∥=∥PG((1-αn)xn+αnT(PGT)n-1xn)-PGx*∥≤∥(1-αn)xn+αnT(PGT)n-1xn-x*∥=∥(1-αn)(xn-x*)+αn(T(PGT)n-1xn-x*)∥≤(1-αn)∥(xn-x*)∥+αn∥T(PGT)n-1xn-x*∥≤(1-αn)∥(xn-x*)∥+αn(kn∥(xn-x*)∥-ψ(∥(xn-x*)∥))=(1+αn(kn-1))∥(xn-x*)∥-αnψ(∥(xn-x*)∥)≤kn∥(xn-x*)∥-αnψ(∥(xn-x*)∥).
Denoteλn=∥xn-x*∥ and we have the following inequality:
(3.22)λn+1≤(1+βn)λn-αnψ(λn)+γn,
where βn=kn-1, ∑n=1∞‍βn<∞, αn∈(0,1], ∑n=1∞‍αn=∞, γn=0. Therefore we know λn→0 as n→∞ by using Lemma 2.7, that is,
(3.23)limn→∞∥xn-x*∥=0.Remark 3.7.
It is clear that the above results, in some ways, extend and improve some results of other authors (such as, see [1–5, 7, 9–13]), from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces. And in the proof process, our methods are different from some past common methods.



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