# Duality Fixed Point and Zero Point Theorems and Applications

Authors: Qingqing Cheng; Yongfu Su; Jingling Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The following main results have been given. (1) LetE 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.


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## Body

## 1. Introduction and Preliminaries

LetE 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
(1.1)〈Tx*,x*-x〉≥0,∀‖x‖≤‖x*‖(p-1)2.
Taking p=2, the problem (1.1) becomes the following variational inequality problem of finding an element x*∈E such that
(1.2)〈Tx*,x*-x〉≥0,∀‖x‖≤‖x*‖.
Secondly, for p≥2, we consider the optimal problem of finding an element x*∈E such that
(1.3)(‖x*‖(p-1)-‖Tx*‖)2=min⁡x∈E(‖x‖(p-1)-‖Tx‖)2.
Taking p=2, the problem (1.3) becomes the following optimal problem of finding an element x*∈E such that
(1.4)(‖x*‖-‖Tx*‖)2=min⁡x∈E(‖x‖-‖Tx‖)2.
Thirdly, for p≥2, we consider the operator equation problem of finding an element x*∈E such that
(1.5)〈Tx*,x*〉=‖Tx*‖p=‖x*‖p(p-1).
Taking p=2, the problem (1.5) becomes the following operator equation problem of finding an element x*∈E such that
(1.6)〈Tx*,x*〉=‖Tx*‖2=‖x*‖2.
Finally, we consider the operator equation problem of finding an element x*∈E such that
(1.7)Tx*=0.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
(1.8)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 surjective;(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].In this paper, we firstly present the definition of duality fixed point for a mappingT from E into its dual E* as follows.LetE 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.
LetE 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.
LetE 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.
Ifx* is a generalized duality fixed point of T, then x* must be a solution of variational inequality problem (1.1).Proof.
Supposex* is a generalized duality fixed point of T, then
(1.9)〈Tx*,x*〉=〈Jpx*,x*〉=‖Jpx*‖p=‖Tx*‖p=‖x*‖p(p-1).
Obverse that
(1.10)〈Tx*,x*-x〉=〈Tx*,x*〉-〈Tx*,x〉≥‖Tx*‖p-‖Tx*‖‖x‖=‖Tx*‖(‖Tx*‖p-1-‖x‖)=‖Tx*‖(‖x*‖(p-1)2-‖x‖)≥0
for all ∥x∥≤∥x*∥(p-1)2.Takingp=2, we have the following result.Conclusion 2.
Ifx* is a duality fixed point of T, then x* must be a solution of variational inequality problem (1.2).Conclusion 3.
Ifx* 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.
Ifx* is a generalized duality fixed point of T, then Tx*=Jpx*, so that
(1.11)〈Tx*,x*〉=〈Jpx*,x*〉=‖Jpx*‖p=‖Tx*‖p=‖x*‖p(p-1).
All conclusions are obvious.Takep=2, we have the following result.Conclusion 4.
Ifx* 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).
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 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:
(1.12)δ(ε)={1-‖x+y2‖:‖x‖=‖y‖=1,‖x-y‖≥ε}.
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)≥cεp 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 spaceE is said to be smooth if the limit
(1.13)lim⁡t→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 [4] 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 [3].
LetρE:[0,∞)→[0,∞) be the modulus of smoothness of E defined by
(1.14)ρE(t)=sup⁡{12(‖x+y‖+‖x-y‖)-1:x∈U,‖y‖≤t}.
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)≤ctq. 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 Lp, where p>1. More precisely, Lp is min{p,2}-uniformly smooth for every p>1.Lemma 1.4 (see [5, 6]).
LetE 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),
(1.15)〈x-y,j(x)-j(y)〉≥cpcp-2p‖x-y‖p,
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.
LetE 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,
(1.16)‖x-y‖≤(pc2)1/(p-1)‖Jp(x)-Jp(y)‖1/(p-1),
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.
LetE 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
(1.17)〈x-y,Jp(x)-Jp(y)〉≥cpcp-2p‖x-y‖p,
which implies that
(1.18)‖x-y‖‖Jp(x)-Jp(y)‖≥cpcp-2p‖x-y‖p.
That is
(1.19)‖Jp(x)-Jp(y)‖≥cpcp-2p‖x-y‖p-1.
Hence
(1.20)‖x-y‖≤(pc2)1/(p-1)‖Jp(x)-Jp(y)‖1/(p-1).
Then (1.6) has been proved. Therefore, from (1.6) we can see, for any x,y∈E, that  Jp(x)=Jp(y) implies that x=y.


## 2. Duality Contraction Mapping Principle and Applications

LetE be a Banach space with the dual E*. An operator T:E→E* is said to be p-L-Lipschitz, if
(2.1)‖Tx-Ty‖≤L‖x-y‖p,∀x,y∈E,
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).
LetE 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 xn+1=Jp-1Txn converges strongly to this generalized duality fixed point x*.Proof.
LetA=Jp-1T, then A is a mapping from E into itself. By using Lemma 1.5, we have
(2.2)‖Ax-Ay‖=‖Jp-1Tx-Jp-1Ty‖≤(pc2)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‖
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*=Jpx*, so x* is a generalized unique duality fixed point of T. Further, the Picard iterative sequence xn+1=Axn=Jp-1Txn (n=0,1,2,…) converges strongly to this generalized duality fixed point x*.Takingp=2, we have the following result.Theorem 2.2 (duality contraction mapping principle).
LetE 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 xn+1=J-1Txn converges strongly to this duality fixed point x*.From Conclusions1–4 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.
LetE 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 xn+1=Jp-1Txn converges strongly to a solution of the variational inequality problem (1.1) (the optimal problem (1.3) and the operator equation problem (1.5)).Takingp=2, we have the following result.Theorem 2.4.
LetE 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 xn+1=J-1Txn 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).
LetE 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 {αn}⊂[0,1] satisfies the condition ∑n=0∞‍αn(1-αn)=+∞. Then for any given guess x0∈E, the generalized Mann iterative sequence
(2.3)xn+1=(1-αn)xn+αnJp-1Txn
converges weakly to a generalized duality fixed point of T.Proof.
LettingA=J-1T, by using Lemma 1.4, we have
(2.4)‖Ax-Ay‖=‖J-1Tx-J-1Ty‖≤2c2‖Tx-Ty‖≤2Lc2‖x-y‖≤‖x-y‖,
for all x,y∈E. Hence A is a nonexpansive mapping from E into itself. In addition, we have
(2.5)xn+1=(1-αn)xn+αnJ-1Txn=(1-αn)xn+αnAxn.
By using the well-known result, we know that the sequence {xn} converges weakly to a fixed point x* of A(Ax*=x*). This point x* is also a duality fixed point of T(Tx*=Jx*).Takep=2, we have the following result.Theorem 2.6 (duality Mann weak convergence theorem).
LetE 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 {αn}⊂[0,1] satisfies the condition ∑n=0∞αn(1-αn)=+∞. Then for any given guess x0∈E, the generalized Mann iterative sequence
(2.6)xn+1=(1-αn)xn+αnJ-1Txn
converges weakly to a duality fixed point of T.Theorem 2.7 (duality Halpern strong convergence theorem).
LetE 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 {αn}⊂[0,1] satisfies the condition:(C1): limn→∞αn=0;(C2): ∑n=0∞αn=∞;(C3): limn→∞(αn+1-αn)/αn+1=0 or limn→∞(αn/αn+1)=1.Let u,x0 be given, then iterative sequence
(2.7)xn+1=αnu+(1-αn)Jp-1Txn,
converges strongly to a generalized duality fixed point of T.Proof.
LetA=Jp-1T, then A is a mapping from E into itself. By using Lemma 1.5, we have
(2.8)‖Ax-Ay‖=‖Jp-1Tx-Jp-1Ty‖≤(pc2)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‖
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
(2.9)xn+1=αnu+(1-αn)Jp-1Txn=αnu+(1-αn)Axn.
By using the well-known result of Xu [7, Theorem  2.3], we know that the iterative sequence {xn} converges strongly to a fixed point of nonexpansive mapping A. Hence the sequence {xn} converges strongly to a generalized duality fixed point of T.Theorem 2.8.
LettingH be a Hilbert space, then one has its uniformly convexity constant 1/c≥2/2, that is c≤2.Proof.
Ifc>2. For any x≠y, by using Lemma 1.4, we have
(2.10)‖x-y‖=‖J-1x-J-1y‖≤2c2‖x-y‖<‖x-y‖.
This is a contradiction.


## 3. Fixed Point Theorem of Inverse Strongly Monotone Mappings

Definition 3.1.
LettingE be a Banach space, the mapping T:E→E* is called q-α-inverse strongly monotone, if
(3.1)〈Tx-Ty,x-y〉≥α‖Tx-Ty‖q,∀x,y∈E.Lemma 3.2.
LetE 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.
LetT:E→E* be a q-α-inverse strongly monotone mapping, that is,
(3.2)〈Tx-Ty,x-y〉≥α‖Tx-Ty‖q,∀x,y∈E.
It follows from the above inequality that
(3.3)α‖Tx-Ty‖q≤〈Tx-Ty,x-y〉≤‖Tx-Ty‖‖x-y‖,∀x,y∈E,
which leads to
(3.4)‖Tx-Ty‖q≤1α‖Tx-Ty‖‖x-y‖,∀x,y∈E.
Further
(3.5)‖Tx-Ty‖q-1≤1α‖x-y‖,∀x,y∈E,
and hence
(3.6)‖Tx-Ty‖≤(1α)1/(q-1)‖x-y‖1/(q-1),∀x,y∈E.
Then T is 1/(q-1)-(1/α)1/(q-1)-Lipschitz.Theorem 3.3 (fixed point theorem of inverse strongly monotone mappings).
LetE 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 xn+1=Jp-1Txn converges strongly to this generalized duality fixed point x*.Proof.
LettingA=Jp-1T, then A is a mapping from E into itself. By using Lemma 1.5 and Lemma 3.2, we have
(3.7)‖Ax-Ay‖=‖Jp-1Tx-Jp-1Ty‖≤(pc2)1/(p-1)‖Tx-Ty‖1/(p-1)≤(pc2)1/(p-1)((1α)1/(q-1)‖x-y‖1/(q-1))1/(p-1)=(pc2)1/(p-1)1α‖x-y‖=(q(q-1)c2)q-11α‖x-y‖
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*=Jpx*, so x* is a generalized unique duality fixed point of T. Further, the Picard iterative sequence xn+1=Axn=Jp-1Txn (n=0,1,2,…) converges strongly to this generalized duality fixed point x*.Takingp=2, we have the following results.Lemma 3.4.
LetE 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).
LetE 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 xn+1=J-1Txn converges strongly to this generalized duality fixed point x*.


## 4. Application for Zero Point of Operators

Lemma 4.1 (see [8]).
LetE be a p-uniformly smooth Banach space with uniformly smooth constant b. Then
(4.1)‖Jx-Jy‖≤b‖x-y‖p-1,∀x,y∈E.Theorem 4.2.
LetE 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 xn+1=J-1(J+rT)xn converges strongly to this zero point x*.Proof.
LetA=J-1(J+rT), then A is a mapping from E into itself. By using Lemma 1.5 and Lemma 4.1, we have
(4.2)‖Ax-Ay‖=‖J-1(J+rT)x-J-1(J+rT)y‖≤2c2‖(J+rT)x-(J+rT)y‖=2c2‖(Jx-Jy)-r(Tx-Ty)‖≤2c2(‖Jx-Jy‖+r‖Tx-Ty‖)≤2c2(b+rL)‖x-y‖
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 xn+1=Axn=J-1(J+rT)xn (n=0,1,2,…) converges strongly to this zero point x*.Remark 4.3.
Under the conditions of Theorem4.2, we know that the operator equation Tx=0 has a unique solution which can be computed by the iterative scheme xn+1=Axn=J-1(J+rT)xn (n=0,1,2,…) starting any given guess x0∈E.



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