Duality Fixed Point and Zero Point Theorems and Applications

and Applied Analysis 3 In this paper, we firstly present the definition of duality fixed point for a mapping T from E into its dual E∗ as follows. Let E be a Banach space with a single-valued generalized duality mapping Jp : E → E∗. Let T : E → E∗. An element x∗ ∈ E is said to be a generalized duality fixed point of T if Tx∗ Jpx∗. An element x∗ ∈ E is said to be a duality fixed point of T if Tx∗ Jx∗. Example 1.1. Let E be a smooth Banach space with the dual E∗, and let A : E → E∗ be an operator, then an element x∗ ∈ E is a zero point of A if and only if x∗ is a duality fixed point of J λA for any λ > 0. Namely, the x∗ is a duality fixed point of J λA for any λ > 0 if and only if x∗ is a fixed point of Jλ J λA −1J : E → E if A is maximal monotone, then Jλ is, namely, the resolvent of A . Example 1.2. In Hilbert space, the fixed point of an operator is always duality fixed point. Example 1.3. Let E be a p-uniformly convex Banach space with the dual E∗, then any element of E must be the generalized duality fixed point of the generalized normalized duality mapping Jp. Conclusion 1. If x∗ is a generalized duality fixed point of T , then x∗ must be a solution of variational inequality problem 1.1 . Proof. Suppose x∗ is a generalized duality fixed point of T , then 〈Tx∗, x∗〉 Jpx∗, x∗ 〉 ∥Jpx∗ ∥∥p ‖Tx∗‖p ‖x∗‖p p−1 . 1.9 Obverse that 〈Tx∗, x∗ − x〉 〈Tx∗, x∗〉 − 〈Tx∗, x〉 ≥ ‖Tx∗‖p − ‖Tx∗‖‖x‖ ‖Tx∗‖ ( ‖Tx∗‖p−1 − ‖x‖ ) ‖Tx∗‖ ( ‖x∗‖ p−1 2 − ‖x‖ ) ≥ 0 1.10 for all ‖x‖ ≤ ‖x∗‖ p−1 2 . Taking p 2, we have the following result. Conclusion 2. If x∗ is a duality fixed point of T , then x∗ must be a solution of variational inequality problem 1.2 . Conclusion 3. If x∗ is a generalized duality fixed point of T , then x∗ must be a solution of the optimal problem 1.3 . Therefore, x∗ is also a solution of operator equation problem 1.5 . Proof. If x∗ is a generalized duality fixed point of T , then Tx∗ Jpx∗, so that 〈Tx∗, x∗〉 Jpx∗, x∗ 〉 ∥Jpx∗ ∥∥p ‖Tx∗‖p ‖x∗‖p p−1 . 1.11 All conclusions are obvious. 4 Abstract and Applied Analysis Take p 2, we have the following result. Conclusion 4. If x∗ is a duality fixed point of T , then x∗ must be a solution of the optimal problem 1.4 . Therefore, x∗ is also a solution of operator equation problem 1.6 . 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 well 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 y 2 ∥∥∥∥ : ‖x‖ ∥∥y∥∥ 1, ∥∥x − y∥∥ ≥ ε } . 1.12 It is well 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 for all ε ∈ 0, 2 . For example, see 2, 3 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→ 0 ∥∥x ty∥∥ − ‖x‖ t 1.13 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 4 for more details. It is well known that the Hilbert and the Lebesgue L 1 < q ≤ 2 spaces are 2-uniformly convex and uniformly smooth. Let X be a Banach space and let L 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 X is p-uniformly convex if and only if X is p-uniformly convex; see 3 . Let ρE : 0,∞ → 0,∞ be the modulus of smoothness of E defined by ρE t sup { 1 2 (∥∥x y∥∥ ∥∥x − y∥∥) − 1 : x ∈ U, ∥∥y∥∥ ≤ t } . 1.14 A Banach space E is said to be uniformly smooth if ρE t /t → 0 as t → 0. Let q > 1. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρE t ≤ ct. It is well known that E is uniformly smooth if and only if the norm of E is uniformly Fréchet differentiable. If E is q-uniformly smooth, then q ≤ 2 and E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable, in particular, the norm of E is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are L, where p > 1.More precisely, L is min{p, 2}-uniformly smooth for every p > 1. Abstract and Applied Analysis 5 Lemma 1.4 see 5, 6 . Let E be a p-uniformly convex Banach space with p ≥ 2. Then, for all x, y ∈ E, j x ∈ Jp x and j y ∈ Jp y , 〈 x − y, j x − j(y)〉 ≥ c p cp−2p ∥∥x − y∥∥p, 1.15and Applied Analysis 5 Lemma 1.4 see 5, 6 . Let E be a p-uniformly convex Banach space with p ≥ 2. Then, for all x, y ∈ E, j x ∈ Jp x and j y ∈ Jp y , 〈 x − y, j x − j(y)〉 ≥ c p cp−2p ∥∥x − y∥∥p, 1.15 where Jp is the generalized duality mapping from E into E∗ and 1/c is the p-uniformly convexity constant of E. Lemma 1.5. Let E be a p-uniformly convex Banach space with p ≥ 2. Then Jp is one-to-one from E onto Jp E ⊂ E∗ and for all x, y ∈ E, ∥∥x − y∥∥ ≤ ( p c2 )1/ p−1 ∥Jp x − Jp ( y )∥∥1/ p−1 , 1.16 where Jp is the generalized duality mapping from E into E∗ with range Jp E , and 1/c is the p-uniformly convexity constant of E. Proof. Let E be a p-uniformly convex Banach space with p ≥ 2, then J J2 is one-to-one from E onto E∗. Since Jp x ‖x‖p−2J x , then Jp x is single valued. From 1.5 we have 〈 x − y, Jp x − Jp ( y )〉 ≥ c p cp−2p ∥∥x − y∥∥p, 1.17