Fixed Point of Strong Duality Pseudocontractive Mappings and Applications

Abstract

Let E 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.

1. Introduction and Preliminaries

Let 𝐸 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 𝑇𝑥^∗,𝑥^∗=‖𝑇𝑥^∗‖²=‖𝑥^∗‖². (1.1) We also consider the second variational inequality problem of finding an element 𝑥^∗∈𝐸 such that 𝑇𝑥^∗,𝑥^∗−𝑥≥0,∀‖𝑥‖≤𝑥^∗. (1.2)

Let 𝐸 be a real Banach space with the dual 𝐸^∗. Let 𝑝 be a given real number with 𝑝>1. The generalized duality mapping 𝐽_𝑝 from 𝐸 into 2^𝐸∗ is defined by 𝐽_𝑝(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=𝑓^𝑝,𝑓=‖𝑥‖^𝑝−1,∀𝑥∈𝐸, (1.3) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. In particular, 𝐽=𝐽₂ is called the normalized duality mapping and 𝐽_𝑝(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)if 𝐸 is smooth, then 𝐽 is single valued;(ii)if 𝐸 is strictly convex, then 𝐽 is one to one;(iii)if 𝐸 is reflexive, then 𝐽 is a mapping of 𝐸 onto ^∗𝐸;(iv)if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸;(v)if 𝐸^∗ 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].

Let 𝐸 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 element 𝑥^∗∈𝐸 such that 𝑇𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1.4) 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 element 𝑥^∗∈𝐸 such that 𝐽𝑥^∗−𝑇𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1.5) 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. If 𝑥^∗ is a duality fixed point of 𝑇, then 𝑥^∗ must be a solution of problem (1.1).

Proof. If 𝑥^∗ is a normalized fixed point of 𝑇, then 𝑇𝑥^∗=𝐽𝑥^∗, so that 𝑇𝑥^∗,𝑥^∗=𝐽𝑥^∗,𝑥^∗=‖𝐽𝑥^∗‖²=‖𝑇𝑥^∗‖²=‖𝑥^∗‖². (1.6) This completes the proof.

Conclusion. If 𝑥^∗ is a duality fixed point of 𝑇, then 𝑥^∗ must be a solution of variational inequality problem (1.2).

Proof. Suppose 𝑥^∗ is a duality fixed point of 𝑇; then 𝑇𝑥^∗,𝑥^∗=‖𝑇𝑥^∗‖²=‖𝑥^∗‖². (1.7) Obverse that 𝑇𝑥^∗,𝑥^∗−𝑥=𝑇𝑥^∗,𝑥^∗−𝑇𝑥^∗,𝑥≥‖𝑇𝑥^∗‖²−𝑇𝑥^∗𝑥‖=‖𝑇𝑥^∗𝑇𝑥^∗−‖𝑥‖=𝑇𝑥^∗𝑥^∗−‖𝑥‖≥0, (1.8) for all ‖𝑥‖≤‖𝑥^∗‖. This completes the proof.

Let 𝑈={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: 𝛿(𝜀)=1−𝑥+𝑦2∶‖𝑥‖=𝑦=1,𝑥−𝑦≥𝜀. (1.9) 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)≥𝑐𝜀^𝑝 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 lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1.10) 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 𝐿^𝑞(1<𝑞≤2) spaces are 2-uniformly convex and uniformly smooth. Let 𝑋 be a Banach space, and let 𝐿^𝑞(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 𝐿^𝑞(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 generalized 𝛼-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.

Let 𝐸 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 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐸. (1.11)

(2) 𝛼-strong duality pseudocontractive mapping, if there exists a nonnegative real number 𝛼 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≤𝑥−𝑦,𝐽𝑥−𝐽𝑦−𝛼‖𝐽𝑥−𝐽𝑦−𝐴𝑥−𝐴𝑦‖² (1.12) for all 𝑥,𝑦∈𝐸.

It is easy to show that 𝐴 is 𝛼-strong duality pseudocontractive if and only if (𝐽−𝐴) is 𝛼-inverse-strongly monotone.

Let 𝐸 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]). Let 𝐸 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≤𝜆_𝑛≤𝛾𝑐²/2, where 1/𝑐 is the uniformly convexity constant. Define a sequence {𝑥_𝑛} in 𝐸 by the following algorithm: 𝑥₀𝑧∈𝐸chosenarbitrarily,_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑧_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐸∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑧₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐸,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, (2.1) where 𝐽 is the duality mapping on 𝐸. Then {𝑥_𝑛} converges strongly to Π_𝐴⁻¹_{0∩𝐹(𝑇)}𝑥₀, where Π_𝐴⁻¹_{0∩𝐹(𝑇)} is the generalized projection from 𝐸 onto 𝐴⁻¹0∩𝐹(𝑇).

IF taking 𝑇=𝐼, then Theorem 2.1 reduces to the following result.

Theorem 2.2. Let 𝐸 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≤𝜆_𝑛≤𝛾𝑐²/2, where 1/𝑐 is the uniformly convexity constant. Define a sequence {𝑥_𝑛} in 𝐸 by the following algorithm: 𝑥₀𝑦∈𝐸chosenarbitrarily,_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐸∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐸,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, (2.2) where 𝐽 is the duality mapping on 𝐸. Then {𝑥_𝑛} converges strongly to Π_𝐴⁻¹₀𝑥₀, where Π_𝐴⁻¹₀ is the generalized projection from 𝐸 onto 𝐴⁻¹0.

Theorem 2.3. Let 𝐸 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≤𝜆_𝑛≤𝛼𝑐²/2𝐿. Define a sequence {𝑥_𝑛} in 𝐸 by the following algorithm: 𝑥₀𝑧∈𝐸chosenarbitrarily,_𝑛=𝐽⁻¹1−𝜆_𝑛𝐽𝑥_𝑛+𝜆_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝑇𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑧_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐸∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑧₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄⟩≥0₀𝑥=𝐸,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, (2.3) where 𝐽 is the duality mapping on 𝐸. Then {𝑥_𝑛} 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. Let 𝐵=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 𝑧_𝑛=𝐽⁻¹1−𝜆_𝑛𝐽𝑥_𝑛+𝜆_𝑛𝐴𝑥_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛. (2.4) By using Theorem 2.1 and Conclusions 1 and 2, we obtain the conclusion of Theorem 2.3.
Taking 𝑇=𝐼 in Theorem 2.3, we get the following result.

Theorem 2.4. Let 𝐸 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≤𝜆_𝑛≤𝛼𝑐²/2𝐿. Define a sequence {𝑥_𝑛} in 𝐸 by the following algorithm: 𝑥₀𝑦∈𝐸chosenarbitrarily,_𝑛=𝐽⁻¹1−𝜆_𝑛𝐽𝑥_𝑛+𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝐶₀=𝑧∈𝐸∶𝜙𝑧,𝑦₀≤𝜙𝑧,𝑥₀,𝑄_𝑛=𝑧∈𝐶_𝑛−1𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐸,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, (2.5) where 𝐽 is the duality mapping on 𝐸. Then {𝑥_𝑛} 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]). Let 𝐸 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 {𝑥_𝑛} is given by 𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛, (2.6) for every 𝑛=1,2,…, where {𝜆_𝑛} is a sequence of positive numbers. If {𝜆_𝑛} is chosen so that 𝜆_𝑛∈[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, then the sequence {𝑥_𝑛} converges weakly to some element 𝑧∈𝑉𝐼(𝐶,𝐴), where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Further 𝑧=lim_𝑛→∞Π_{𝑉𝐼(𝐶,𝐴)}𝑥_𝑛.

In this paper, we introduce an iterative scheme for finding a solution of the variational inequality problem for an operator 𝑇 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 Theorem 2.5, we prove the following convergence theorem.

Theorem 2.6. Let 𝐸 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 {𝑥_𝑛} is given by 𝑥_𝑛+1=Π_𝐶𝐽⁻¹1−𝜆_𝑛𝐽𝑥_𝑛+𝜆_𝑛𝑇𝑥_𝑛, (2.7) for every 𝑛=1,2,…, where {𝜆_𝑛} is a sequence of positive numbers. If {𝜆_𝑛} is chosen so that 𝜆_𝑛∈[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, then the sequence {𝑥_𝑛} converges weakly to some element 𝑧∈𝑉𝐼(𝐶,𝐽,𝑇), where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Further 𝑧=lim_𝑛→∞Π_{𝑉𝐼(𝐶,𝐽,𝑇)}𝑥_𝑛.

Proof. Let 𝐴=𝐽−𝑇, then 𝐵∶𝐸→𝐸^∗ is 𝛼-inverse-strongly monotone, so that 𝐵⁻¹0=𝐹(𝐴). On the other hand, we have 𝑥_𝑛+1=Π_𝐶𝐽⁻¹1−𝜆_𝑛𝐽𝑥_𝑛+𝜆_𝑛𝑇𝑥_𝑛=𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛. (2.8) By using Theorem 2.5, we obtain the conclusion of Theorem 2.6.

In fact, from condition (vi), we have 𝐹(𝑇)=𝑉𝐼(𝐶,𝐽,𝑇), so that under the conditions of Theorem 2.6, the {𝑥_𝑛} 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 {𝑥_𝑛} is defined by Algorithm (2.7).