Duality Fixed Point and Zero Point Theorems and Applications

Abstract

The following main results have been given. (1) Let 𝐸 be a 𝑝-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with condition 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)<1. Then 𝑇 has a unique generalized duality fixed point 𝑥^∗∈𝐸 and (2) let 𝐸 be a 𝑝-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a 𝑞-𝛼-inverse strongly monotone mapping with conditions 1/𝑝+1/𝑞=1, 0<(𝑞/(𝑞−1)𝑐²)^𝑞−1<𝛼. Then 𝑇 has a unique generalized duality fixed point 𝑥^∗∈𝐸. (3) Let 𝐸 be a 2-uniformly smooth and uniformly convex Banach space with uniformly convex constant 𝑐 and uniformly smooth constant 𝑏 and let 𝑇∶𝐸→𝐸^∗ be a 𝐿-lipschitz mapping with condition 0<2𝑏/𝑐²<1. Then 𝑇 has a unique zero point 𝑥^∗. These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.

1. Introduction and Preliminaries

Let 𝐸 be a real Banach space with the dual 𝐸^∗ and let 𝑇 be an operator from 𝐸 into 𝐸^∗. Firstly, for 𝑝≥2, we consider the variational inequality problem of finding an element 𝑥^∗∈𝐸 such that ⟨𝑇𝑥^∗,𝑥^∗−𝑥⟩≥0,∀‖𝑥‖≤‖𝑥^∗‖^(𝑝−1)2.(1.1) Taking 𝑝=2, the problem (1.1) becomes the following variational inequality problem of finding an element 𝑥^∗∈𝐸 such that ⟨𝑇𝑥^∗,𝑥^∗−𝑥⟩≥0,∀‖𝑥‖≤‖𝑥^∗‖.(1.2) Secondly, for 𝑝≥2, we consider the optimal problem of finding an element 𝑥^∗∈𝐸 such that ‖𝑥^∗‖^(𝑝−1)−‖𝑇𝑥^∗‖²=min_𝑥∈𝐸‖𝑥‖^(𝑝−1)−‖𝑇𝑥‖².(1.3) Taking 𝑝=2, the problem (1.3) becomes the following optimal problem of finding an element 𝑥^∗∈𝐸 such that ‖𝑥^∗‖−‖𝑇𝑥^∗‖²=min_𝑥∈𝐸(‖𝑥‖−‖𝑇𝑥‖)².(1.4) Thirdly, for 𝑝≥2, we consider the operator equation problem of finding an element 𝑥^∗∈𝐸 such that ⟨𝑇𝑥^∗,𝑥^∗⟩=‖𝑇𝑥^∗‖^𝑝=‖𝑥^∗‖^{𝑝(𝑝−1)}.(1.5) Taking 𝑝=2, the problem (1.5) becomes the following operator equation problem of finding an element 𝑥^∗∈𝐸 such that ⟨𝑇𝑥^∗,𝑥^∗⟩=‖𝑇𝑥^∗‖²=‖𝑥^∗‖².(1.6) Finally, we consider the operator equation problem of finding an element 𝑥^∗∈𝐸 such that 𝑇𝑥^∗=0.(1.7)

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,(1.8) 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 surjective;(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].

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 𝐽_𝑝∶𝐸→𝐸^∗. Let 𝑇∶𝐸→𝐸^∗. An element 𝑥^∗∈𝐸 is said to be a generalized duality fixed point of 𝑇 if 𝑇𝑥^∗=𝐽_𝑝𝑥^∗. 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 𝐽_𝜆=(𝐽+𝜆𝐴)⁻¹𝐽∶𝐸→𝐸 (if 𝐴 is maximal monotone, then 𝐽_𝜆 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 𝐽_𝑝.

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^∗⟩=𝐽_𝑝𝑥^∗,𝑥^∗=‖‖𝐽_𝑝𝑥^∗‖‖^𝑝=‖𝑇𝑥^∗‖^𝑝=‖𝑥^∗‖^{𝑝(𝑝−1)}.(1.9) Obverse that ⟨𝑇𝑥^∗,𝑥^∗−𝑥⟩≥=‖⟨𝑇𝑇𝑥𝑥^∗_∗,𝑥^∗⟩−⟨𝑇𝑥^∗,𝑥⟩=‖𝑇𝑥_∗‖^𝑝−‖𝑇𝑥^∗‖‖𝑥‖=‖𝑇𝑥_∗‖‖𝑇𝑥^∗‖^𝑝−1−‖𝑥‖‖‖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^∗=𝐽_𝑝𝑥^∗, so that ⟨𝑇𝑥^∗,𝑥^∗⟩=𝐽_𝑝𝑥^∗,𝑥^∗=‖‖𝐽_𝑝𝑥^∗‖‖^𝑝=‖𝑇𝑥^∗‖^𝑝=‖𝑥^∗‖^{𝑝(𝑝−1)}.(1.11) All conclusions are obvious.

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+𝑦2‖‖‖∶‖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^𝑝 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^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 [3].
Let 𝜌_𝐸∶[0,∞)→[0,∞) be the modulus of smoothness of 𝐸 defined by 𝜌_𝐸(𝑡)=sup12(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶𝑥∈𝑈,‖𝑦‖≤𝑡.(1.14) A Banach space 𝐸 is said to be uniformly smooth if 𝜌_𝐸(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 𝜌_𝐸(𝑡)≤𝑐𝑡^𝑞. 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^𝑝, where 𝑝>1. More precisely, 𝐿^𝑝 is min{𝑝,2}-uniformly smooth for every 𝑝>1.

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)∈𝐽_𝑝(𝑥) and 𝑗(𝑦)∈𝐽_𝑝(𝑦), ⟨𝑥−𝑦,𝑗(𝑥)−𝑗(𝑦)⟩≥𝑐^𝑝𝑐^𝑝−2𝑝‖𝑥−𝑦‖^𝑝,(1.15) where 𝐽_𝑝 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 𝐽_𝑝 is one-to-one from 𝐸 onto 𝐽_𝑝(𝐸)⊂𝐸^∗ and for all 𝑥,𝑦∈𝐸, ‖𝑥−𝑦‖≤𝑝𝑐²^1/(𝑝−1)‖‖𝐽_𝑝(𝑥)−𝐽_𝑝(𝑦)‖‖^1/(𝑝−1),(1.16) where 𝐽_𝑝 is the generalized duality mapping from 𝐸 into 𝐸^∗ with range 𝐽_𝑝(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 𝐽_𝑝(𝑥)=‖𝑥‖^𝑝−2𝐽(𝑥), then 𝐽_𝑝(𝑥) is single valued. From (1.5) we have 𝑥−𝑦,𝐽_𝑝(𝑥)−𝐽_𝑝(𝑦)≥𝑐^𝑝𝑐^𝑝−2𝑝‖𝑥−𝑦‖^𝑝,(1.17) which implies that ‖𝑥−𝑦‖‖‖𝐽_𝑝(𝑥)−𝐽_𝑝(𝑦)‖‖≥𝑐^𝑝𝑐^𝑝−2𝑝‖𝑥−𝑦‖^𝑝.(1.18) That is ‖‖𝐽_𝑝(𝑥)−𝐽_𝑝(𝑦)‖‖≥𝑐^𝑝𝑐^𝑝−2𝑝‖𝑥−𝑦‖^𝑝−1.(1.19) Hence ‖𝑥−𝑦‖≤𝑝𝑐²^1/(𝑝−1)‖‖𝐽_𝑝(𝑥)−𝐽_𝑝(𝑦)‖‖^1/(𝑝−1).(1.20) Then (1.6) has been proved. Therefore, from (1.6) we can see, for any 𝑥,𝑦∈𝐸, that 𝐽_𝑝(𝑥)=𝐽_𝑝(𝑦) implies that 𝑥=𝑦.

2. Duality Contraction Mapping Principle and Applications

Let 𝐸 be a Banach space with the dual 𝐸^∗. An operator 𝑇∶𝐸→𝐸^∗ is said to be 𝑝-𝐿-Lipschitz, if ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖^𝑝,∀𝑥,𝑦∈𝐸,(2.1) where 𝐿∈(0,+∞),𝑝∈[1,+∞) are two constants. If 𝑝=1, the operator 𝑇 is said to be 𝐿-Lipschitz.

Theorem 2.1 (generalized duality contraction mapping principle). Let 𝐸 be a 𝑝-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with condition 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)<1. Then 𝑇 has a unique generalized duality fixed point 𝑥^∗∈𝐸 and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽^𝑝−1𝑇𝑥_𝑛 converges strongly to this generalized duality fixed point 𝑥^∗.

Proof. Let 𝐴=𝐽^𝑝−1𝑇, then 𝐴 is a mapping from 𝐸 into itself. By using Lemma 1.5, we have ‖𝐴𝑥−𝐴𝑦‖≤=‖‖𝐽𝑝^𝑝−1𝑇𝑥−𝐽^𝑝−1𝑇𝑦‖‖𝑐²^1/(𝑝−1)‖𝑇𝑥−𝑇𝑦‖^1/(𝑝−1)≤𝑝𝑐²^1/(𝑝−1)𝐿‖𝑥−𝑦‖^𝑝−1^1/(𝑝−1)≤𝑝𝐿𝑐²^1/(𝑝−1)‖𝑥−𝑦‖(2.2) for all 𝑥,𝑦∈𝐸, where 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)<1. By using Banach's contraction mapping principle, there exists a unique element 𝑥^∗∈𝐸 such that 𝐴𝑥^∗=𝑥^∗. That is, 𝑇𝑥^∗=𝐽_𝑝𝑥^∗, so 𝑥^∗ is a generalized unique duality fixed point of 𝑇. Further, the Picard iterative sequence 𝑥_𝑛+1=𝐴𝑥_𝑛=𝐽^𝑝−1𝑇𝑥_𝑛 (𝑛=0,1,2,…) converges strongly to this generalized duality fixed point 𝑥^∗.

Taking 𝑝=2, we have the following result.

Theorem 2.2 (duality contraction mapping principle). Let 𝐸 be a 2-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a 𝐿-Lipschitz mapping with condition 0<(2𝐿/𝑐²)<1. Then 𝑇 has a unique duality fixed point 𝑥^∗∈𝐸 and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽⁻¹𝑇𝑥_𝑛 converges strongly to this duality fixed point 𝑥^∗.

From Conclusions 1–4 and Theorem 2.1, we have the following result for solving the variational inequality problems (1.1) and (1.2), the optimal problems (1.3) and (1.4), and the operator equation problems (1.5) and (1.6).

Theorem 2.3. Let 𝐸 be a 𝑝-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with condition 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)<1. Then the variational inequality problem (1.1) (the optimal problem (1.2) and operator equation problem (1.3)) has solutions and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽^𝑝−1𝑇𝑥_𝑛 converges strongly to a solution of the variational inequality problem (1.1) (the optimal problem (1.3) and the operator equation problem (1.5)).

Taking 𝑝=2, we have the following result.

Theorem 2.4. Let 𝐸 be a 2-uniformly convex Banach space, let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with condition 0<(2𝐿/𝑐²)<1. Then the variational inequality problem (1.2) (the optimal problem (1.4) and operator equation problem (1.6)) has solutions and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽⁻¹𝑇𝑥_𝑛 converges strongly to a solution of the variational inequality problem (1.2) (the optimal problem (1.4) and the operator equation problem (1.6)).

Theorem 2.5 (generalized duality Mann weak convergence theorem). Let 𝐸 be a p-uniformly convex Banach space which satisfying Opial's condition, let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with nonempty generalized duality fixed point set. Assume 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)≤1, and the real sequence {𝛼_𝑛}⊂[0,1] satisfies the condition ∑^∞𝑛=0𝛼_𝑛(1−𝛼_𝑛)=+∞. Then for any given guess 𝑥₀∈𝐸, the generalized Mann iterative sequence 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐽^𝑝−1𝑇𝑥_𝑛(2.3) converges weakly to a generalized duality fixed point of 𝑇.

Proof. Letting 𝐴=𝐽⁻¹𝑇, by using Lemma 1.4, we have ‖𝐴𝑥−𝐴𝑦‖=‖‖𝐽⁻¹𝑇𝑥−𝐽⁻¹𝑇𝑦‖‖≤2𝑐²‖𝑇𝑥−𝑇𝑦‖≤2𝐿𝑐²‖𝑥−𝑦‖≤‖𝑥−𝑦‖,(2.4) for all 𝑥,𝑦∈𝐸. Hence 𝐴 is a nonexpansive mapping from 𝐸 into itself. In addition, we have 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐽⁻¹𝑇𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐴𝑥_𝑛.(2.5) By using the well-known result, we know that the sequence {𝑥_𝑛} converges weakly to a fixed point 𝑥^∗ of 𝐴(𝐴𝑥^∗=𝑥^∗). This point 𝑥^∗ is also a duality fixed point of 𝑇(𝑇𝑥^∗=𝐽𝑥^∗).

Take 𝑝=2, we have the following result.

Theorem 2.6 (duality Mann weak convergence theorem). Let 𝐸 be a 2-uniformly convex Banach space which satisfy Opial's condition and let 𝑇∶𝐸→𝐸^∗ be a 𝐿-Lipschitz mapping with nonempty duality fixed point set. Assume 0<2𝐿/𝑐²≤1, and the real sequence {𝛼_𝑛}⊂[0,1] satisfies the condition ∑^∞𝑛=0𝛼_𝑛(1−𝛼_𝑛)=+∞. Then for any given guess 𝑥₀∈𝐸, the generalized Mann iterative sequence 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐽⁻¹𝑇𝑥_𝑛(2.6) converges weakly to a duality fixed point of 𝑇.

Theorem 2.7 (duality Halpern strong convergence theorem). Let 𝐸 be a p-uniformly convex Banach space which satisfying Opial's condition, let 𝑇∶𝐸→𝐸^∗ be a (𝑝−1)-𝐿-Lipschitz mapping with nonempty generalized duality fixed point set. Assume 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)≤1, and the real sequence {𝛼_𝑛}⊂[0,1] satisfies the condition: (C₁): lim_𝑛→∞𝛼_𝑛=0; (C₂): ∑^∞𝑛=0𝛼_𝑛=∞; (C₃): lim_𝑛→∞(𝛼_𝑛+1−𝛼_𝑛)/𝛼_𝑛+1=0 or lim_𝑛→∞(𝛼_𝑛/𝛼_𝑛+1)=1.Let 𝑢,𝑥₀ be given, then iterative sequence 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝐽^𝑝−1𝑇𝑥_𝑛,(2.7) converges strongly to a generalized duality fixed point of 𝑇.

Proof. Let 𝐴=𝐽^𝑝−1𝑇, then 𝐴 is a mapping from 𝐸 into itself. By using Lemma 1.5, we have ‖𝐴𝑥−𝐴𝑦‖≤=‖‖𝐽𝑝^𝑝−1𝑇𝑥−𝐽^𝑝−1𝑇𝑦‖‖𝑐²^1/(𝑝−1)‖𝑇𝑥−𝑇𝑦‖^1/(𝑝−1)≤𝑝𝑐²^1/(𝑝−1)𝐿‖𝑥−𝑦‖^𝑝−1^1/(𝑝−1)≤𝑝𝐿𝑐²^1/(𝑝−1)‖𝑥−𝑦‖(2.8) for all 𝑥,𝑦∈𝐸, where 0<(𝑝𝐿/𝑐²)^1/(𝑝−1)≤1. Hence 𝐴 is a nonexpansive mapping from 𝐸 into itself. In addition, we have 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝐽^𝑝−1𝑇𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝐴𝑥_𝑛.(2.9) By using the well-known result of Xu [7, Theorem  2.3], we know that the iterative sequence {𝑥_𝑛} converges strongly to a fixed point of nonexpansive mapping 𝐴. Hence the sequence {𝑥_𝑛} converges strongly to a generalized duality fixed point of 𝑇.

Theorem 2.8. Letting 𝐻 be a Hilbert space, then one has its uniformly convexity constant 1/𝑐≥√2/2, that is 𝑐≤√2.

Proof. If 𝑐>√2. For any 𝑥≠𝑦, by using Lemma 1.4, we have ‖𝑥−𝑦‖=‖‖𝐽⁻¹𝑥−𝐽⁻¹𝑦‖‖≤2𝑐²‖𝑥−𝑦‖<‖𝑥−𝑦‖.(2.10) This is a contradiction.

3. Fixed Point Theorem of Inverse Strongly Monotone Mappings

Definition 3.1. Letting 𝐸 be a Banach space, the mapping 𝑇∶𝐸→𝐸^∗ is called 𝑞-𝛼-inverse strongly monotone, if ⟨𝑇𝑥−𝑇𝑦,𝑥−𝑦⟩≥𝛼‖𝑇𝑥−𝑇𝑦‖^𝑞,∀𝑥,𝑦∈𝐸.(3.1)

Lemma 3.2. Let 𝐸 be a Banach space and let 𝑇∶𝐸→𝐸^∗ be a 𝑞-𝛼-inverse strongly monotone mapping. Then 𝑇 is 1/(𝑞−1)−(1/𝛼)^(1/(q−1))-Lipschitz.

Proof. Let 𝑇∶𝐸→𝐸^∗ be a 𝑞-𝛼-inverse strongly monotone mapping, that is, ⟨𝑇𝑥−𝑇𝑦,𝑥−𝑦⟩≥𝛼‖𝑇𝑥−𝑇𝑦‖^𝑞,∀𝑥,𝑦∈𝐸.(3.2) It follows from the above inequality that 𝛼‖𝑇𝑥−𝑇𝑦‖^𝑞≤⟨𝑇𝑥−𝑇𝑦,𝑥−𝑦⟩≤‖𝑇𝑥−𝑇𝑦‖‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸,(3.3) which leads to ‖𝑇𝑥−𝑇𝑦‖^𝑞≤1𝛼‖𝑇𝑥−𝑇𝑦‖‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(3.4) Further ‖𝑇𝑥−𝑇𝑦‖^𝑞−1≤1𝛼‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸,(3.5) and hence ‖𝑇𝑥−𝑇𝑦‖≤1𝛼^1/(𝑞−1)‖𝑥−𝑦‖^1/(𝑞−1),∀𝑥,𝑦∈𝐸.(3.6) Then 𝑇 is 1/(𝑞−1)−(1/𝛼)^1/(𝑞−1)-Lipschitz.

Theorem 3.3 (fixed point theorem of inverse strongly monotone mappings). Let 𝐸 be a 𝑝-uniformly convex Banach space and let 𝑇∶𝐸→𝐸^∗ be a 𝑞-𝛼-inverse strongly monotone mapping with conditions 1/𝑝+(1/𝑞)=1, 0<(𝑞/(𝑞−1)𝑐²)^𝑞−1<𝛼. Then 𝑇 has a unique generalized duality fixed point 𝑥^∗∈𝐸 and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽^𝑝−1𝑇𝑥_𝑛 converges strongly to this generalized duality fixed point 𝑥^∗.

Proof. Letting 𝐴=𝐽^𝑝−1𝑇, then 𝐴 is a mapping from 𝐸 into itself. By using Lemma 1.5 and Lemma 3.2, we have ‖𝐴𝑥−𝐴𝑦‖≤=‖‖𝐽𝑝^𝑝−1𝑇𝑥−𝐽^𝑝−1𝑇𝑦‖‖𝑐²^1/(𝑝−1)‖𝑇𝑥−𝑇𝑦‖^1/(𝑝−1)≤𝑝𝑐²^1/(𝑝−1)𝛼1^1/(𝑞−1)‖𝑥−𝑦‖^1/(𝑞−1)^1/(𝑝−1)=𝑝𝑐²^1/(𝑝−1)1=𝑞𝛼‖𝑥−𝑦‖(𝑞−1)𝑐²^𝑞−11𝛼‖𝑥−𝑦‖(3.7) for all 𝑥,𝑦∈𝐸. It follows from the condition 0<(𝑞/(𝑞−1)𝑐²)^𝑞−1<𝛼 that 0<(𝑞/(𝑞−1)𝑐²)^𝑞−1(1/𝛼)<1. By using Banach's contraction mapping principle, there exists a unique element 𝑥^∗∈𝐸 such that 𝐴𝑥^∗=𝑥^∗. That is, 𝑇𝑥^∗=𝐽_𝑝𝑥^∗, so 𝑥^∗ is a generalized unique duality fixed point of 𝑇. Further, the Picard iterative sequence 𝑥_𝑛+1=𝐴𝑥_𝑛=𝐽^𝑝−1𝑇𝑥_𝑛 (𝑛=0,1,2,…) converges strongly to this generalized duality fixed point 𝑥^∗.

Taking 𝑝=2, we have the following results.

Lemma 3.4. Let 𝐸 be a Banach space and let 𝑇∶𝐸→𝐸^∗ be a 2-𝛼-inverse strongly monotone mapping. Then 𝑇 is (1/𝛼)-Lipschitz.

Theorem 3.5 (fixed point theorem of inverse strongly monotone mappings). Let 𝐸 be a 2-uniformly convex Banach space, let 𝑇∶𝐸→𝐸^∗ be a 2-𝛼-inverse strongly monotone mapping with condition 0<2/𝑐²<𝛼. Then 𝑇 has a unique duality fixed point 𝑥^∗∈𝐸 and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽⁻¹𝑇𝑥_𝑛 converges strongly to this generalized duality fixed point 𝑥^∗.

4. Application for Zero Point of Operators

Lemma 4.1 (see [8]). Let 𝐸 be a 𝑝-uniformly smooth Banach space with uniformly smooth constant 𝑏. Then ‖𝐽𝑥−𝐽𝑦‖≤𝑏‖𝑥−𝑦‖^𝑝−1,∀𝑥,𝑦∈𝐸.(4.1)

Theorem 4.2. Let 𝐸 be a 2-uniformly smooth and uniformly convex Banach space with uniformly convex constant 𝑐 and uniformly smooth constant 𝑏 and let 𝑇∶𝐸→𝐸^∗ be a 𝐿-lipschitz mapping with condition 0<2𝑏/𝑐²<1. Then 𝑇 has a unique zero point 𝑥^∗ and for any given guess 𝑥₀∈𝐸, the iterative sequence 𝑥_𝑛+1=𝐽⁻¹(𝐽+𝑟𝑇)𝑥_𝑛 converges strongly to this zero point 𝑥^∗.

Proof. Let 𝐴=𝐽⁻¹(𝐽+𝑟𝑇), then 𝐴 is a mapping from 𝐸 into itself. By using Lemma 1.5 and Lemma 4.1, we have ‖𝐴𝑥−𝐴𝑦‖≤=‖‖2𝐽⁻¹(𝐽+𝑟𝑇)𝑥−𝐽⁻¹(𝐽+𝑟𝑇)𝑦‖‖𝑐²=2‖(𝐽+𝑟𝑇)𝑥−(𝐽+𝑟𝑇)𝑦‖𝑐²≤2‖(𝐽𝑥−𝐽𝑦)−𝑟(𝑇𝑥−𝑇𝑦)‖𝑐²≤2(‖𝐽𝑥−𝐽𝑦‖+𝑟‖𝑇𝑥−𝑇𝑦‖)𝑐²(𝑏+𝑟𝐿)‖𝑥−𝑦‖(4.2) for all 𝑥,𝑦∈𝐸. Observing the condition 0<2𝑏/𝑐²<1, it follows that, there exists a positive number 𝑟>0 such that 0<2/𝑐²(𝑏+𝑟𝐿)<1. By using Banach's contraction mapping principle, there exists a unique element 𝑥^∗∈𝐸 such that 𝐴𝑥^∗=𝑥^∗. That is, (𝐽+𝑟𝑇)𝑥^∗=𝐽𝑥^∗ which implies 𝑟𝑇𝑥^∗=0, so 𝑥^∗ is a zero point of 𝑇. Further, the Picard iterative sequence 𝑥_𝑛+1=𝐴𝑥_𝑛=𝐽⁻¹(𝐽+𝑟𝑇)𝑥_𝑛 (𝑛=0,1,2,…) converges strongly to this zero point 𝑥^∗.

Remark 4.3. Under the conditions of Theorem 4.2, we know that the operator equation 𝑇𝑥=0 has a unique solution which can be computed by the iterative scheme 𝑥_𝑛+1=𝐴𝑥_𝑛=𝐽⁻¹(𝐽+𝑟𝑇)𝑥_𝑛 (𝑛=0,1,2,…) starting any given guess 𝑥₀∈𝐸.

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).