Convergence results for a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function
© Shahzad et al.; licensee Springer. 2013
Received: 27 August 2013
Accepted: 14 November 2013
Published: 13 December 2013
In this paper, we introduce an iterative process which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings with Bregman distance function. Our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.
MSC:47H05, 47J05, 47J25.
The function f is said to be Gâteaux differentiable at x if exists for any . In this case, coincides with , the value of the gradient ∇f of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in . We say that f is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for and .
is called the Bregman distance with respect to f .
If , a Hilbert space, J is the identity mapping, and hence the Bregman projection reduces to the metric projection of H on to C, .
Clearly, the class of monotone mappings includes the class of γ-inverse strongly monotone mappings.
is called the variational inequality problem. The set of solutions of the variational inequality is denoted by .
Variational inequality problems are related with the convex minimization problem, the zero of monotone mappings and the complementarity problem. Consequently, many researchers (see, e.g., [3, 5, 10–15]) have made efforts to obtain iterative methods for approximating solutions of variational inequality problems.
where is the metric projection from H onto C and is a sequence of positive real numbers. They proved that the sequence generated by (1.6) converges weakly to some element of provided that A is a γ-inverse strongly monotone mapping.
where is the generalized projection from E onto C, J is the normalized duality mapping from E into and is a sequence of positive real numbers. They proved that the sequence generated by (1.7) converges weakly to some element of .
It is worth mentioning that the convergence obtained above is weak convergence. Our concern now is to look for an iteration scheme which converges strongly to a solution of the variational inequality problem for a monotone mapping A.
where is a positive real sequence satisfying certain mild conditions and is the generalized projection from E onto , J is the duality mapping from E into . Then they proved that the sequence converges strongly to an element of .
where for all , , and satisfy certain mild conditions. Then they proved that the sequence converges strongly to , where is the generalized projection from E onto .
In 1967, Bregman  discovered an elegant and effective technique for using the so-called Bregman distance function in the process of designing and analyzing feasibility and optimization algorithms. Using Bregman’s distance function and its properties, authors have opened a growing area of research not only for iterative algorithms of solving feasibility and optimization problems but also for algorithms of solving nonlinear, equilibrium, variational inequality, fixed point problems and others (see, e.g., [19–25] and the references therein).
where , are error sequences in E with and is the Bregman projection with respect to f from E onto a closed and convex subset C of E. Those authors showed that the sequence defined by (1.10) converges strongly to a common element in under some mild conditions. Similar results are also available in [26, 27].
Remark 1.2 But it is worth mentioning that the iteration processes (1.8)-(1.10) seem difficult in the sense that at each stage of iteration, the set(s) and (or) is (are) computed and the next iterate is taken as the Bregman projection of onto the intersection of and (or ). This seems difficult to do in applications.
It is our purpose in this paper to introduce an iterative scheme which converges strongly to a common solution of a finite family of variational inequality problems for monotone mappings in real reflexive Banach spaces. Our scheme does not involve computations of or for each . Furthermore, we apply our convergence theorem to a convex minimization problem. Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators.
Let . The subdifferential of f at x is the convex set defined by , where the Fenchel conjugate of f is the function defined by .
Essentially smooth if ∂f is both locally bounded and single-valued on its domain.
Essentially strictly convex if is locally bounded on its domain and f is strictly convex on every convex subset of domf.
Legendre if it is both essentially smooth and essentially strictly convex.
When the subdifferential of f is single-valued, then (see ).
A function f on E is coercive  if the sublevel set of f is bounded; equivalently, .
We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see , Theorem 2.10). The following lemmas will be useful in the proof of our main result.
Lemma 2.1 
Lemma 2.2 
if and only if , .
Lemma 2.3 
Lemma 2.4 
Let be Gâteaux differentiable on such that is bounded on bounded subsets of . Let and . If is bounded, so is the sequence .
and (see ).
Lemma 2.5 
where and satisfy the following conditions: , , and . Then .
Lemma 2.6 
In fact, is the largest number n in the set such that the condition holds.
Following the agreement in , we have the following lemma.
is closed and convex.
3 Main result
where satisfies and , and for some .
Theorem 3.1 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of and , for , be a finite family of continuous monotone mappings with . Let be a sequence defined by (3.1). Then converges strongly to .
Now, we consider two possible cases.
Since is bounded and E is reflexive, we choose a subsequence of such that and . Then, from (3.5) and (3.4), we get that for each .
This implies that for all .
Therefore, we obtain that . Thus, by Lemma 2.2, we immediately obtain that . It follows from Lemma 2.5 and (3.3) that as . Consequently, .
It follows from (3.7) that as . This together with (3.8) implies that . Therefore, since for all , we conclude that as . Hence, both cases imply that converges strongly to and the proof is complete. □
If in Theorem 3.1 , then we get the following corollary.
where for all ; satisfies and and for some . Then the sequence converges strongly to a point .
If , then and hence the following corollary holds.
Corollary 3.3 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let , for , be a finite family of continuous monotone mappings. Let . Let be a sequence defined by (3.1). Then converges strongly to .
If in Theorem 3.1 we assume , then the scheme converges strongly to the common minimum-norm zero of a finite family of continuous monotone mappings. In fact, we have the following corollary.
Corollary 3.4 Let be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let C be a nonempty, closed and convex subset of , and let , for , be a finite family of continuous monotone mappings with . Let be a sequence defined by (3.1) with . Then converges strongly to , which is the common minimum-norm (with respect to the Bregman distance) solution of the variational inequalities.
In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Banach spaces.
Let , for , be continuously Fréchet differentiable convex functionals such that the gradients of , are continuous and monotone. For , let for all and for each . Then the following theorem holds.
where satisfies and and for some . Then the sequence converges strongly to .
Proof We note that from the convexity and Fréchet differentiability of f, we have for each . Thus, by Theorem 3.1, converges strongly to . □
Remark 4.2 Our results are new even if the convex function f is chosen to be () in uniformly smooth and uniformly convex spaces.
Remark 4.3 Our theorems extend and unify most of the results that have been proved for this important class of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.3 of , Theorem 3.1 of , Theorem 3.1 of  and Theorem 3.3 of  and Theorem 4.2 of  either to a more general class of continuous monotone operators or to a more general Banach space E. Moreover, in all our theorems and corollaries, the computation of or for each is not required.
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and third authors acknowledge with thanks DSR for financial support. The second author undertook this work when he was visiting the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy, as a regular associate.
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