# Fixed Point of Strong Duality Pseudocontractive Mappings and Applications

Authors: Baowei Liu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetE be a smooth Banach space with the dual E∗, an operator T:E→E∗ is said to be α-strong duality pseudocontractive if x-y,Tx-Ty≤x-y,Jx-Jy-α∥Jx-Jy-Tx-Ty∥2, for all x,y∈E, where α is a nonnegative constant. An element x∈E is called a duality fixed point of T if Tx=Jx. The purpose of this paper is to introduce the definition of α-strong duality pseudocontractive mappings and to study its fixed point problem and applications for operator equation and variational inequality problems.


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## Body

## 1. Introduction and Preliminaries

LetE be a real Banach space with the dual E*: let T be an operator from E into E*. We consider the first operator equation problem of finding an element x*∈E such that
(1.1)〈Tx*,x*〉=∥Tx*∥2=∥x*∥2.
We also consider the second variational inequality problem of finding an element x*∈E such that
(1.2)〈Tx*,x*-x〉≥0,∀∥x∥≤∥x*∥.LetE be a real Banach space with the dual E*. Let p be a given real number with p>1. The generalized duality mapping Jp from E into 2E* is defined by
(1.3)Jp(x)={f∈E*:〈x,f〉=∥f∥p,∥f∥=∥x∥p-1},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing. In particular, J=J2 is called the normalized duality mapping and Jp(x)=∥x∥p-2J(x) for all x≠0. If E is a Hilbert space, then J=I, where I is the identity mapping. The duality mapping J has the following properties: (i)
ifE is smooth, then J is single valued;(ii)
ifE is strictly convex, then J is one to one;(iii)
ifE is reflexive, then J is a mapping of E onto *E;(iv)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;(v)
ifE* is uniformly convex, then J is uniformly continuous on each bounded subsets of E and J is single valued and also one to one.For more details, see [1, 2].LetE be a smooth Banach space with the dual E*. Let T:E→E* be an operator; an element x*∈E is called a duality fixed point of T, if Tx*=Jx*.We also consider the third variational inequality problem of finding an elementx*∈E such that
(1.4)〈Tx*,x-x*〉≥0,∀x∈C,
where C is a closed convex subset of E. The set of solutions of the variational inequality problem (1.4) is denoted by VI(C,T).We also consider the fourth variational inequality problem of finding an elementx*∈E such that
(1.5)〈Jx*-Tx*,x-x*〉≥0,∀x∈C,
where C is a closed convex subset of E. The set of solutions of the variational inequality problem (1.5) is denoted by VI(C,J,T).Conclusion.
Ifx* is a duality fixed point of T, then x* must be a solution of problem (1.1).Proof.
Ifx* is a normalized fixed point of T, then Tx*=Jx*, so that
(1.6)〈Tx*,x*〉=〈Jx*,x*〉=∥Jx*∥2=∥Tx*∥2=∥x*∥2.
This completes the proof.Conclusion.
Ifx* is a duality fixed point of T, then x* must be a solution of variational inequality problem (1.2).Proof.
Supposex* is a duality fixed point of T; then
(1.7)〈Tx*,x*〉=∥Tx*∥2=∥x*∥2.
Obverse that
(1.8)〈Tx*,x*-x〉=〈Tx*,x*〉-〈Tx*,x〉≥∥Tx*∥2-∥Tx*∥∥x∥=∥Tx*∥(∥Tx*∥-∥x∥)=∥Tx*∥(∥x*∥-∥x∥)≥0,
for all ∥x∥≤∥x*∥. This completes the proof.LetU={x∈E:∥x∥=1}. A Banach space E is said to be strictly convex if for any x,y∈U, x≠y implies ∥(x+y)/2∥<1. It is also said to be uniformly convex if for each ε∈(0,2], there exists δ>0 such that for any x,y∈U, ∥x-y∥≥ε implies ∥(x+y)/2∥<1-δ. It is known that a uniformly convex Banach space is reflexive and strictly convex. And we define a function δ:[0,2]→[0,1] called the modulus of convexity of E as follows:
(1.9)δ(ε)={1-∥x+y2∥:∥x∥=∥y∥=1,∥x-y∥≥ε}.
It is known that E is uniformly convex if and only if δ(ε)>0 for all ε∈(0,2]. Let p be a fixed real number with p≥2. Then E is said to be p-uniformly convex if there exists a constant c>0 such that δ(ε)≥cεp for all ε∈[0,2]. For example, see [3, 4] for more details. The constant 1/c is said to be uniformly convexity constant of E.A Banach space E is said to be smooth if the limit(1.10)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U. It is also said to be uniformly smooth if the above limit is attained uniformly for x,y∈U. One should note that no Banach space is p-uniformly convex for 1<p<2; see [5] for more details. It is well known that the Hilbert and the Lebesgue Lq(1<q≤2) spaces are 2-uniformly convex and uniformly smooth. Let X be a Banach space, and let Lq(X)={Ω,Σ,μ;X},1<q≤∞ be the Lebesgue-Bochner space on an arbitrary measure space (Ω,Σ,μ). Let 2≤p<∞, and let 1<q≤p. Then Lq(X) is p-uniformly convex if and only if X is p-uniformly convex; see [4].In this paper, we first propose the definition of generalizedα-strongly pseudocontractive mappings from a smooth Banach E into its dual E* as follows. We also discuss the problem of fixed point for generalized α-strongly pseudocontractive mappings and its applications.LetE be a smooth Banach space and E* denote the dual of E. An operator A:E→E* is said to be(1)α-inverse-strongly monotone if there exists nonnegative real number α such that
(1.11)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2,∀x,y∈E.(2)α-strong duality pseudocontractive mapping, if there exists a nonnegative real number α such that
(1.12)〈x-y,Ax-Ay〉≤〈x-y,Jx-Jy〉-α∥Jx-Jy-(Ax-Ay)∥2
for all x,y∈E.It is easy to show thatA is α-strong duality pseudocontractive if and only if (J-A) is α-inverse-strongly monotone.LetE be a smooth Banach space and E* denote the dual of E. Let A:E→E* be an operator. The set of zero points of A is defined by A-10={x∈E:Ax=0}. The set of duality fixed points of A is defined by F(A)={x∈E:Ax=Jx}. It is also easy to show that, an element u∈E is a zero point of an α-inverse-strongly monotone operator A if and only if u is a duality fixed point of the α-strong duality pseudocontractive mapping (J-A).


## 2. Main Results and Applications

Recently, Zegeye and Shahzad [6] proved the following result.Theorem 2.1 (see, [6]).
LetE be a uniformly smooth and 2-uniformly convex real Banach space with the dual E*: let A:E→E* be a γ-inverse-strongly monotone mapping and T:E→E a relatively weak nonexpansive mapping with A-10∩F(T)≠∅. Assume that 0<α≤λn≤γc2/2, where 1/c is the uniformly convexity constant. Define a sequence {xn} in E by the following algorithm:
(2.1)x0∈Echosenarbitrarily,zn=J-1(Jxn-λnAxn),yn=Tzn,Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,zn)≤ϕ(z,xn)},C0={z∈E:ϕ(z,y0)≤ϕ(z,z0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=E,xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E. Then {xn} converges strongly to ΠA-10∩F(T)x0, where ΠA-10∩F(T) is the generalized projection from E onto A-10∩F(T).IF takingT=I, then Theorem 2.1 reduces to the following result.Theorem 2.2.
LetE be a uniformly smooth and 2-uniformly convex real Banach space with the dual E*, let A:E→E* be a γ-inverse strongly monotone mapping with A-10≠∅. Assume that 0<α≤λn≤γc2/2, where 1/c is the uniformly convexity constant. Define a sequence {xn} in E by the following algorithm:
(2.2)x0∈Echosenarbitrarily,yn=J-1(Jxn-λnAxn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,xn)},C0={z∈E:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=E,xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E. Then {xn} converges strongly to ΠA-10x0, where ΠA-10 is the generalized projection from E onto A-10.Theorem 2.3.
LetE be a uniformly smooth and 2-uniformly convex real Banach space; let A:E→E* be an α-strong duality pseudocontractive mapping with nonempty set of duality fixed points F(A). Let T:E→E be a relatively weak nonexpansive mapping and F(A)∩F(T)=∅. Assume 0<a≤λn≤αc2/2L. Define a sequence {xn} in E by the following algorithm:
(2.3)x0∈Echosenarbitrarily,zn=J-1((1-λn)Jxn+λnAxn),yn=Tzn,Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,zn)≤ϕ(z,xn)},C0={z∈E:ϕ(z,y0)≤ϕ(z,z0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=E,xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E. Then {xn} converges strongly to a common element x*∈F(A)∩F(T). This element is also a common solution of operator equation (1.1) and variational inequality (1.2).Proof.
LetB=J-A, then B:E→E* is α/L-inverse-strongly monotone and α-strongly monotone, so that B-10=F(A) has only one element. On the other hand, we have
(2.4)zn=J-1((1-λn)Jxn+λnAxn)=J-1(Jxn-λnBxn).
By using Theorem 2.1 and Conclusions 1 and 2, we obtain the conclusion of Theorem 2.3.
TakingT=I in Theorem 2.3, we get the following result.Theorem 2.4.
LetE be a uniformly smooth and 2-uniformly convex real Banach space; let A:E→E* be a L-Lipschitz and α-strongly duality pseudocontractive mapping with nonempty set of duality fixed points F(A). Assume 0<a≤λn≤αc2/2L. Define a sequence {xn} in E by the following algorithm:
(2.5)x0∈Echosenarbitrarily,yn=J-1((1-λn)Jxn+λnAxn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,xn)},C0={z∈E:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=E,xn+1=ΠCn∩Qn(x0),
where J is the duality mapping on E. Then {xn} converges strongly to a duality fixed point x*∈F(A). This element x* is also a common solution of operator equation (1.1) and variational inequality (1.2).Iiduka and Takahashi [7] introduce an iterative scheme for finding a solution of the variational inequality problem for an operator A that satisfies the following conditions (i)–(iii) in a 2-uniformly convex and uniformly smooth Banach space E:(i)A is α-inverse-strongly monotone;(ii)VI(C,A)≠∅;(iii)∥Ay∥≤∥Ay-Au∥ for all y∈E and u∈VI(C,A).They proved the following convergence theorem.Theorem 2.5 (see, [7]).
LetE be a 2-uniformly convex and uniformly smooth Banach space, whose duality mapping J is weakly sequentially continuous, and C a nonempty, closed convex subset of E. Assume that A is an operator of C into E*, that satisfies the conditions (i)–(iii). Suppose that x1=x∈C and {xn} is given by
(2.6)xn+1=ΠCJ-1(Jxn-λnAxn),
for every n=1,2,…, where {λn} is a sequence of positive numbers. If {λn} is chosen so that λn∈[a,b] for some a,b with 0<a<b<c2α/2, then the sequence {xn} converges weakly to some element z∈VI(C,A), where 1/c is the 2-uniformly convexity constant of E. Further z=limn→∞ΠVI(C,A)xn.In this paper, we introduce an iterative scheme for finding a solution of the variational inequality problem for an operatorT that satisfies the following conditions (iv)–(vi) in a 2-uniformly convex and uniformly smooth Banach space E:(iv)T is α-strong duality pseudocontractive,(v)VI(C,J,T)≠∅,(vi)∥Jy-Ty∥≤∥(J-T)y-(J-T)u∥ for all y∈E and u∈VI(C,J,T).By using Theorem2.5, we prove the following convergence theorem.Theorem 2.6.
LetE be a 2-uniformly convex and uniformly smooth Banach space, whose duality mapping J is weakly sequentially continuous, and C a nonempty, closed convex subset of E. Assume that T is an operator of C into E*. that satisfies the conditions (iv)–(vi). Suppose that x1=x∈C and {xn} is given by
(2.7)xn+1=ΠCJ-1((1-λn)Jxn+λnTxn),
for every n=1,2,…, where {λn} is a sequence of positive numbers. If {λn} is chosen so that λn∈[a,b] for some a,b with 0<a<b<c2α/2, then the sequence {xn} converges weakly to some element z∈VI(C,J,T), where 1/c is the 2-uniformly convexity constant of E. Further z=limn→∞ΠVI(C,J,T)xn.Proof.
LetA=J-T, then B:E→E* is α-inverse-strongly monotone, so that B-10=F(A). On the other hand, we have
(2.8)xn+1=ΠCJ-1((1-λn)Jxn+λnTxn)=J-1(Jxn-λnAxn).
By using Theorem 2.5, we obtain the conclusion of Theorem 2.6.In fact, from condition (vi), we haveF(T)=VI(C,J,T), so that under the conditions of Theorem 2.6, the {xn} converges strongly to a duality fixed point z∈F(T). This element z is also a common solution of operator equation (1.1) and variational inequality (1.2). where {xn} is defined by Algorithm (2.7).



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