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Generalized setvalued variationallike inclusions involving H(\cdot ,\cdot )ηcocoercive operator in Banach spaces
Journal of Inequalities and Applications volume 2012, Article number: 149 (2012)
Abstract
The aim of this paper is to introduce a new H(\cdot ,\cdot )ηcocoercive operator and its resolvent operator. We study some of the properties of H(\cdot ,\cdot )ηcocoercive operator and prove the Lipschitz continuity of resolvent operator associated with H(\cdot ,\cdot )ηcocoercive operator. Finally, we apply the techniques of resolvent operator to solve a generalized setvalued variationallike inclusion problem in Banach spaces. Our results are new and generalize many known results existing in the literature. Some examples are given in support of definition of H(\cdot ,\cdot )ηcocoercive operator.
MSC:47H19, 49J40.
1 Introduction
Variational inclusion problems are interesting and intensively studied classes of mathematical problems and have wide applications in the field of optimization and control, economics and transportation equilibrium, and engineering sciences, etc., see for example [1–7]. Several authors used the resolvent operator technique to propose and analyze the iterative algorithms for computing the approximate solutions of different kinds of variational inclusions. Fang and Huang [8] studied variational inclusions by introducing a class of generalized monotone operators, called Hmonotone operators and defined the associated resolvent operator. Fang and Huang [9] further extended the notion of Hmonotone operators to Banach spaces, called Haccretive operators. Recently, Zou and Huang [10] introduced and studied H(\cdot ,\cdot )accretive operators and apply them to solve a variational inclusion problem in Banach spaces. After that Xu and Wang [11] introduced and studied (H(\cdot ,\cdot )\text{}\eta )monotone operators. Very recently, Ahmad et al. [12] introduced H(\cdot ,\cdot )cocoercive operators and apply them to solve a setvalued variational inclusion problem in Hilbert spaces.
By taking into account the fact that ηcocoercivity is an intermediate concept that lies between ηstrong monotonicity and ηmonotonicity, in this paper, we introduce H(\cdot ,\cdot )ηcocoercive operator and its resolvent operator. We then apply these new concepts to solve a generalized setvalued variationallike inclusion problem in Banach spaces.
2 Preliminaries
Throughout the paper, we assume that X is a real Banach space, {X}^{\ast} is the topological dual space of X, CB(X) is the family of all nonempty closed and bounded subsets of X, \mathcal{D}(\cdot ,\cdot ) is the Hausdörff metric on CB(X) defined by
\u3008\cdot ,\cdot \u3009 is the dual pair between X and {X}^{\ast}.
Definition 2.1 A continuous and strictly increasing function \phi :{R}^{+}\to {R}^{+} such that \phi (0)=0 and {lim}_{t\to \mathrm{\infty}}\phi (t)=\mathrm{\infty} is called a gauge function.
Definition 2.2 Given a gauge function φ, the mapping {J}_{\phi}:X\to {2}^{{X}^{\ast}} defined by
is called the duality mapping with gauge function φ, where X is any normed space.
In particular if \phi (t)=t, the duality mapping J={J}_{\phi} is called the normalized duality mapping.
Lemma 2.1 [13]
Let X be a real Banach space and J:X\to {2}^{{X}^{\ast}} be the normalized duality mapping. Then, for any x,y\in X,
for all j(x+y)\in J(x+y).
Definition 2.3 Let A:X\to X and \eta :X\times X\to X be two mappings and let J:X\to {2}^{{X}^{\ast}} be the normalized duality mapping. Then A is called

(i)
ηcocoercive, if there exists a constant {\mu}_{1}>0 such that
\u3008AxAy,j(\eta (x,y))\u3009\ge {\mu}_{1}{\parallel AxAy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X,j(\eta (x,y))\in J(\eta (x,y)); 
(ii)
ηaccretive, if
\u3008AxAy,j(\eta (x,y))\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X,j(\eta (x,y))\in J(\eta (x,y)); 
(iii)
ηstrongly accretive, if there exists a constant {\beta}_{1}>0 such that
\u3008AxAy,j(\eta (x,y))\u3009\ge {\beta}_{1}{\parallel xy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X,j(\eta (x,y))\in J(\eta (x,y)); 
(iv)
ηrelaxed cocoercive, if there exists a constant {\gamma}_{1}>0 such that
\u3008AxAy,j(\eta (x,y))\u3009\ge ({\gamma}_{1}){\parallel AxAy\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X,j(\eta (x,y))\in J(\eta (x,y)); 
(v)
Lipschitz continuous, if there exists a constant {\lambda}_{A}>0 such that
\parallel AxAy\parallel \le {\lambda}_{A}\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X; 
(vi)
αexpansive, if there exists a constant \alpha >0 such that
\parallel AxAy\parallel \ge \alpha \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X; 
(vii)
η is said to be Lipschitz continuous, if there exists a constant \tau >0 such that
\parallel \eta (x,y)\parallel \le \tau \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X.
If X=H, a Hilbert space, then definitions (i) to (iv) reduce to the definitions of ηcocoercive, ηmonotone, ηstrongly monotone and ηrelaxed cocoercive, respectively, introduced by Ansari and Yao [3].
If in addition, \eta (x,y)=xy, for all x,y\in X, then definitions (i) to (iv) reduce to the definitions of cocoercivity [14], monotonicity, strong monotonicity [15] and relaxed cocoercive, respectively.
Definition 2.4 Let A,B:X\to X, H:X\times X\to X, \eta :X\times X\to X be three singlevalued mappings and J:X\to {2}^{{X}^{\ast}} be a normalized duality mapping. Then

(i)
H(A,\cdot ) is said to be ηcocoercive with respect to A, if there exists a constant \mu >0 such that
\u3008H(Ax,u)H(Ay,u),j(\eta (x,y))\u3009\ge \mu {\parallel AxAy\parallel}^{2},
\mathrm{\forall}x,y\in X, j(\eta (x,y))\in J(\eta (x,y));

(ii)
H(\cdot ,B) is said to be ηrelaxed cocoercive with respect to B, if there exists a constant \gamma >0 such that
\u3008H(u,Bx)H(u,By),j(\eta (x,y))\u3009\ge (\gamma ){\parallel BxBy\parallel}^{2},
\mathrm{\forall}x,y\in X, j(\eta (x,y))\in J(\eta (x,y));

(iii)
H(A,\cdot ) is said to be {r}_{1}Lipschitz continuous with respect to A, if there exists a constant {r}_{1}>0 such that
\parallel H(Ax,u)H(Ay,u)\parallel \le {r}_{1}\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X; 
(iv)
H(\cdot ,B) is said to be {r}_{2}Lipschitz continuous with respect to B, if there exists a constant {r}_{2}>0 such that
\parallel H(u,Bx)H(u,By)\parallel \le {r}_{2}\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X.
Definition 2.5 A setvalued mapping M:X\to {2}^{X} is said to be ηcocoercive, if there exists a constant {\mu}_{2}>0 such that
Definition 2.6 A mapping T:X\to CB(X) is said to be \mathcal{D}Lipschitz continuous, if there exists a constant {\lambda}_{T}>0 such that
Definition 2.7 Let T,Q:X\to CB(X) be the mappings. A mapping N:X\times X\to X is said to be

(i)
Lipschitz continuous in the first argument with respect to T, if there exists a constant {t}_{1}>0 such that
\parallel N({w}_{1},\cdot )N({w}_{2},\cdot )\parallel \le {t}_{1}\parallel {w}_{1}{w}_{2}\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}{u}_{1},{u}_{2}\in X,{w}_{1}\in T({u}_{1}),{w}_{2}\in T({u}_{2}); 
(ii)
Lipschitz continuous in the second argument with respect to Q, if there exists a constant {t}_{2}>0 such that
\parallel N(\cdot ,{v}_{1})N(\cdot ,{v}_{2})\parallel \le {t}_{2}\parallel {v}_{1}{v}_{2}\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}{u}_{1},{u}_{2}\in X,{v}_{1}\in Q({u}_{1}),{v}_{2}\in Q({u}_{2}); 
(iii)
ηrelaxed Lipschitz in the first argument with respect to T, if there exists a constant {\tau}_{1}>0 such that
\u3008N({w}_{1},\cdot )N({w}_{2},\cdot ),j(\eta ({u}_{1},{u}_{2}))\u3009\le {\tau}_{1}{\parallel {u}_{1}{u}_{2}\parallel}^{2},
\mathrm{\forall}{u}_{1},{u}_{2}\in X, {w}_{1}\in T({u}_{1}), {w}_{2}\in T({u}_{2}), j(\eta ({u}_{1},{u}_{2}))\in J(\eta ({u}_{1},{u}_{2}));

(iv)
ηrelaxed Lipschitz in the second argument with respect to Q, if there exists a constant {\tau}_{2}>0 such that
\u3008N(\cdot ,{v}_{1})N(\cdot ,{v}_{2}),j(\eta ({u}_{1},{u}_{2}))\u3009\le {\tau}_{2}{\parallel {u}_{1}{u}_{2}\parallel}^{2},
\mathrm{\forall}{u}_{1},{u}_{2}\in X, {v}_{1}\in Q({u}_{1}), {v}_{2}\in Q({u}_{2}), j(\eta ({u}_{1},{u}_{2}))\in J(\eta ({u}_{1},{u}_{2})).
3 H(\cdot ,\cdot )ηcocoercive operator
In this section, we define a new H(\cdot ,\cdot )ηcocoercive operator and show some of its properties.
Definition 3.1 Let A,B:X\to X, H:X\times X\to X, \eta :X\times X\to X be four singlevalued mappings. Let M:X\to {2}^{X} be a setvalued mapping. M is said to be H(\cdot ,\cdot )ηcocoercive operator with respect to A and B, if M is ηcocoercive and (H(A,B)+\lambda M)(X)=X, for every \lambda >0.
Example 3.1 Let X=\mathbb{R} and A,B:\mathbb{R}\to \mathbb{R} be defined by
Let H(A,B),\eta :\mathbb{R}\times \mathbb{R}\to \mathbb{R} be defined by H(Ax,Bx)=A(x)B(x), \mathrm{\forall}x\in \mathbb{R} and
Let M:\mathbb{R}\to {2}^{\mathbb{R}} be a setvalued mapping defined by
Then

(i)
\u3008uv,\eta (x,y)\u3009\ge 2n{\parallel uv\parallel}^{2}, \mathrm{\forall}x,y\in \mathbb{R}, u\in M(x), v\in M(y).

(ii)
(H(A,B)+\lambda M)(\mathbb{R})=\mathbb{R}.
Thus M is H(\cdot ,\cdot )ηcocoercive with respect to A and B.
Now, we show that the mapping M need not be H(\cdot ,\cdot )ηcocoercive with respect to A and B.
Let X=C[0,1] be space of all real valued continuous functions defined over closed interval [0,1] with the norm
Let A,B:X\to X be defined by
Let H(A,B):X\times X\to X be defined as
Suppose that M=I, where I is the identity mapping. Then for \lambda =1, we have
which means that 0\notin (H(A,B)+M)(X) and thus M is not H(\cdot ,\cdot )ηcocoercive with respect to A and B.
Proposition 3.1 Let H(A,B) be ηcocoercive with respect to A with constant \mu >0 and ηrelaxed cocoercive with respect to B with constant \gamma >0, A is αexpansive and B is βLipschitz continuous and \mu >\gamma, \alpha >\beta. Let M:X\to {2}^{X} be H(\cdot ,\cdot )ηcocoercive operator. If the following inequality
then x\in Mu, where Graph(M)=\{(u,x)\in X\times X:x\in Mu\}.
Proof Suppose that there exists some ({u}_{0},{x}_{0}) such that
Since M is H(\cdot ,\cdot )ηcocoercive with respect to A and B, we know that (H(A,B)+\lambda M)(X)=X holds for every \lambda >0 and so there exists ({u}_{1},{x}_{1})\in Graph(M) such that
It follows from (3.1) and (3.2) that
which gives {u}_{1}={u}_{0}, since \mu >\gamma and \alpha >\beta. By (3.2), we have {x}_{1}={x}_{0}. Hence ({u}_{0},{x}_{0})=({u}_{1},{x}_{1})\in Graph(M) and so {x}_{0}\in M{u}_{0}. □
Theorem 3.1 Let H(A,B) be ηcocoercive with respect to A with constant \mu >0 and ηrelaxed cocoercive with respect to B with constant \gamma >0, A is αexpansive and B is βLipschitz continuous, \mu >\gamma and \alpha >\beta. Let M be an H(\cdot ,\cdot )ηcocoercive operator with respect to A and B. Then the operator {(H(A,B)+\lambda M)}^{1} is singlevalued.
Proof For any given u\in X, let x,y\in {(H(A,B)+\lambda M)}^{1}(u). It follows that
As M is ηcocoercive (thus ηaccretive), we have
Since H is ηcocoercive with respect to A with constant μ and ηrelaxed cocoercive with respect to B with constant γ, A is αexpansive and B is βLipschitz continuous, thus (3.3) becomes
since \mu >\gamma and \alpha >\beta. Thus, we have x=y and so {(H(A,B)+\lambda M)}^{1} is singlevalued. □
Definition 3.2 Let H(A,B) be ηcocoercive with respect to A with constant \mu >0 and ηrelaxed cocoercive with respect to B with constant \gamma >0, A is αexpansive and B is βLipschitz continuous, \mu >\gamma and \alpha >\beta. Let M be H(\cdot ,\cdot )ηcocoercive operator with respect to A and B. Then the resolvent operator {R}_{\lambda ,M}^{H(\cdot ,\cdot )\eta}:X\to X is defined by
Now, we show the Lipschitz continuity of the resolvent operator defined by (3.5) and calculate its Lipschitz constant.
Theorem 3.2 Let H(A,B) be ηcocoercive with respect to A with constant \mu >0 and ηrelaxed cocoercive with respect to B with constant \gamma >0, A is αexpansive, B is βLipschitz continuous and η is τLipschitz continuous and \mu >\gamma, \alpha >\beta. Let M be an H(\cdot ,\cdot )ηcocoercive operator with respect to A and B. Then the resolvent operator {R}_{\lambda ,M}^{H(\cdot ,\cdot )\eta}:X\to X is \frac{\tau}{\mu {\alpha}^{2}\gamma {\beta}^{2}}Lipschitz continuous, that is
Proof Let u and v be any given points in X. It follows from (3.5) that
This implies that
For the sake of convenience, we take
Since M is ηcocoercive (thus ηaccretive), we have
which implies that
It follows that
and so
i.e.
i.e.
This completes the proof. □
4 Existence result for generalized setvalued variationallike inclusion problem
In this section, we apply H(\cdot ,\cdot )ηcocoercive operators to find a solution of generalized setvalued variationallike inclusion problem.
Let N:X\times X\to X, \eta :X\times X\to X, H:X\times X\to X, A,B:X\to X be the singlevalued mappings and T,Q:X\to CB(X), M:X\to {2}^{X} be the setvalued mappings such that M is H(\cdot ,\cdot )ηcocoercive with respect to A and B. Then we consider the following problem. Find u\in X, w\in T(u), v\in Q(u) such that
We call problem (4.1), a generalized setvalued variationallike inclusion problem.
Let X be a Hilbert space. If Q\equiv 0 and \eta (u,v)=uv, \mathrm{\forall}u,v\in X and N(\cdot ,\cdot )=T(\cdot ), where T:X\to CB(X) be a setvalued mapping. Then problem (4.1) reduces to the problem of finding u\in X, w\in T(u) such that
A problem similar to (4.2) is studied by Ahmad et al. [12] by applying H(\cdot ,\cdot )cocoercive operators.
If H(\cdot ,\cdot )=H(\cdot ) and M is Haccretive mapping, then problem (4.1) is introduced and studied by Chang et al. [5]. It is clear that for suitable choices of operators involved in the formulation of (4.1), one can obtain many variational inclusions studied in literature.
Lemma 4.1 The (u,w,v), where u\in X, w\in T(u), v\in Q(u) is a solution of problem (4.1) if and only if (u,w,v) is a solution of the following equation
where \lambda >0 is a constant.
Proof Proof is straightforward by the use of definition of resolvent operator. □
Based on Lemma 4.1, we define the following algorithm for approximating a solution of generalized setvalued variationallike inclusion problem (4.1).
Algorithm 4.1 For any {u}_{0}\in X, {w}_{0}\in T({u}_{0}), {v}_{0}\in Q({u}_{0}), compute the sequences \{{u}_{n}\}, \{{w}_{n}\}, and \{{v}_{n}\} by the following iterative scheme:
for all n=0,1,2,\dots and \lambda >0 is a constant.
Theorem 4.1 Let X be a real Banach space. Let A,B:X\to X, H:X\times X\to X, N:X\times X\to X, \eta :X\times X\to X be the singlevalued mappings. Suppose that T,Q:X\to CB(X) and M:X\to {2}^{X} are setvalued mappings such that M is H(\cdot ,\cdot )ηcocoercive operator with respect to A and B. Let

(i)
T is \mathcal{D}Lipschitz continuous with constant {\lambda}_{T} and Q is \mathcal{D}Lipschitz continuous with constant {\lambda}_{Q};

(ii)
H(A,B) is ηcocoercive with respect to A with constant \mu >0 and ηrelaxed cocoercive with respect to B with constant \gamma >0;

(iii)
A is αexpansive and B is βLipschitz continuous;

(iv)
H(A,B) is {r}_{1}Lipschitz continuous with respect to A and {r}_{2}Lipschitz continuous with respect to B;

(v)
N is {t}_{1}Lipschitz continuous with respect to T in the first argument and {t}_{2}Lipschitz continuous with respect to Q in the second argument;

(vi)
η is τLipschitz continuous;

(vii)
N is ηrelaxed Lipschitz continuous with respect to T in the first argument and ηrelaxed Lipschitz continuous with respect to Q in the second argument with constants {\tau}_{1} and {\tau}_{2}, respectively.
Suppose that the following condition is satisfied:
Then there exist u\in X, w\in T(u) and v\in Q(u) satisfying the generalized setvalued variationallike inclusion problem (4.1) and the iterative sequences \{{u}_{n}\}, \{{w}_{n}\} and \{{v}_{n}\} generated by Algorithm 4.1 converge strongly to u, w and v, respectively.
Proof Since T is \mathcal{D}Lipschitz continuous with constant {\lambda}_{T} and Q is \mathcal{D}Lipschitz continuous with constant {\lambda}_{Q}, it follows from Algorithm 4.1 that
By using Algorithm 4.1 and Lipschitz continuity of resolvent operator {R}_{\lambda ,M}^{H(\cdot ,\cdot )\eta}, we have
Using Lemma 2.1, we have
As H(\cdot ,\cdot ) is {r}_{1}Lipschitz continuous with respect to A, we have
Since N is {t}_{1}Lipschitz continuous with respect to T in the first argument and {t}_{2}Lipschitz continuous with respect to Q in the second argument and T is {\lambda}_{T}Lipschitz continuous and Q is {\lambda}_{Q}Lipschitz continuous, we have
As η is τLipschitz continuous, we have
Since N is ηrelaxed Lipschitz continuous with respect to T and ηrelaxed Lipschitz continuous with respect to Q in first and second arguments with constants {\tau}_{1} and {\tau}_{2}, respectively, we have
Using (4.12)(4.15), (4.11) becomes
Using {r}_{2}Lipschitz continuity of H(\cdot ,\cdot ) with respect to B and (4.16), (4.10) becomes
where
and
From (4.7), it follows that \theta <1, so \{{u}_{n}\} is a Cauchy sequence in X, thus there exists a u\in X such that {u}_{n}\to u as n\to \mathrm{\infty}. Also from (4.8) and (4.9), it follows that \{{w}_{n}\} and \{{v}_{n}\} are also Cauchy sequences in X, thus there exist w and v in X such that {w}_{n}\to w, {v}_{n}\to v as n\to \mathrm{\infty}. By the continuity of {R}_{\lambda ,M}^{H(\cdot ,\cdot )\eta}, H, A, B, η, N, T and Q and from (4.4) of Algorithm 4.1, it follows that
Now, we prove that w\in T(u). In fact, since {w}_{n}\in T({u}_{n}), we have
which means that d(w,T(u))=0. Since T(u)\in CB(X), it follows that w\in T(u). Similarly, we can show that v\in Q(u). By Lemma 4.1, we conclude that (u,w,v) is a solution of generalized setvalued variationallike inclusion problem (4.1). This completes the proof. □
From Theorem 4.1, we can obtain the following theorem which is similar to the Theorem 4.3 of Ahmad et al. [12].
Theorem 4.2 Let X be a Hilbert space. Let A, B, H and T be the same as in Theorem 4.1 and M:X\to {2}^{X} be the setvalued, H(\cdot ,\cdot )cocoercive mapping with respect to A and B. Assume that

(i)
T is \mathcal{D}Lipschitz continuous with constant {\lambda}_{T};

(ii)
H(A,B) is cocoercive with respect to A with constant \mu >0 and relaxed cocoercive with respect to B with constant \gamma >0;

(iii)
A is αexpansive;

(iv)
B is βLipschitz continuous;

(v)
H(A,B) is {r}_{1}Lipschitz continuous with respect to A and {r}_{2}Lipschitz continuous with respect to B;

(vi)
({r}_{1}+{r}_{2})<[(\mu {\alpha}^{2}\gamma {\beta}^{2})\lambda ]; \mu >\gamma, \alpha >\beta.
Then the problem (4.2) admits a solution (u,w) with u\in X and w\in T(u) and the iterative sequences \{{u}_{n}\} and \{{w}_{n}\} converge strongly to u and w, respectively.
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Acknowledgements
The first author is supported by the Department of Science and Technology, Government of India under grant no. SR/S4/MS: 577/09. The fourth author is supported by Grant no. NSC 992221E037007MY3.
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Ahmad, R., Dilshad, M., Wong, MM. et al. Generalized setvalued variationallike inclusions involving H(\cdot ,\cdot )ηcocoercive operator in Banach spaces. J Inequal Appl 2012, 149 (2012). https://doi.org/10.1186/1029242X2012149
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DOI: https://doi.org/10.1186/1029242X2012149
Keywords
 H(\cdot ,\cdot )ηcocoercive
 Lipschitz continuity
 algorithm
 variationallike inclusion