Journal of Function Spaces
Volume 2015 (2015), Article ID 478437, 7 pages
http://dx.doi.org/10.1155/2015/478437

Research Article

Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

Messaoud Bounkhel

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 6 April 2015; Accepted 30 May 2015

Academic Editor: Calogero Vetro

Copyright © 2015 Messaoud Bounkhel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The present paper is devoted to the study of the generalized projection , where is a uniformly convex and uniformly smooth Banach space and is a nonempty closed (not necessarily convex) set in . Our main result is the density of the points having unique generalized projection over nonempty close sets in . Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

1. Introduction

In 1994, Alber [1] (see also [2]) introduced and studied an appropriate extension of the projection operator over closed convex sets from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. It is called generalized projection operator. He proved various properties and extended many existing results from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces.

In 2005, Li [3] extended and studied this concept from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces. This concept has been used successfully in many applications such as variational inequalities, minimization principles, and differential inclusions (see [1, 2, 4–7] and the references therein). The main result in [1–3] is the existence property of the operator for closed convex sets in reflexive Banach spaces (resp., in uniformly convex and uniformly smooth Banach spaces) in [3] (resp., in [1]). Our main aim is to study the existence of for nonempty closed sets not necessarily convex. An application of our main result to variational problems with nonconvex sets is presented at the end of the paper.

2. Preliminaries

Let be a Banach space with topological dual space . We denote by and the closed unit ball in and , respectively. We recall some definitions and results on uniformly convex and uniformly smooth Banach spaces (see, e.g., [8, 9]). The moduli of convexity and smoothness of are defined, respectively, by

The space is said to be uniformly convex whenever for all and is said to be uniformly smooth whenever . Let , be real numbers. The space is said to be -uniformly convex (resp., -uniformly smooth) if there is a constant such that

Obviously from the definition of -uniform convexity and -uniform smoothness the constants and satisfy and . It is known (see, e.g., [8, 9]) that uniformly convex Banach spaces are reflexive strictly convex and that uniformly smooth Banach spaces are reflexive. If is a -uniformly convex Banach space, then is a -uniformly smooth Banach space, where is the conjugate number of . If is a -uniformly smooth Banach space, then is a -uniformly convex Banach space, where .

The normalized duality mapping is defined by

Many properties of the normalized duality mapping have been studied. For the details, one may see Takahashi’s book [10] or Vainberg’s book [11]. We list some properties of :

()For any , is nonempty.()For any and any real number , .()If is reflexive, then is a mapping of onto .()If is strictly convex (i.e., the unit sphere in is strictly convex; i.e., the inequality holds for all , such that , ), then is a single valued mapping.() is a continuous operator in smooth Banach spaces.()If is strictly convex, then is one-to-one.()If is a reflexive strictly convex space with strictly convex dual space and if is a normalized duality mapping in , then .() is the identity operator in Hilbert spaces.

It is known (see [8, 9]) that a reflexive Banach space is smooth if and only if is strictly convex. Hence by () and (), if is a reflexive smooth Banach space, then is a single valued mapping from onto . And, by (), if is reflexive smooth strictly convex Banach space, then . Let be defined by

First, we mention that, in Hilbert spaces (), the functional has the form , .

We list now some important properties of needed in our proofs, when is a reflexive smooth Banach space:

(i).(ii).(iii).(iv) is continuous and is convex with respect to when is fixed and convex with respect to when is fixed.(v) is differentiable with respect to when is fixed.(vi). This property is true whenever the space is smooth which is the case for uniformly convex spaces.(vii) if and only if .

Let be a function and where is finite. We recall from [4] that the -proximal subdifferential (called in [4] the analytical proximal subdifferential) is the set of all for which there exists such that

for all around . Recall also [4] that the -proximal normal cone (called in [4] the proximal normal cone) of a nonempty closed subset in at is defined by , where is the indicator function of . It has been proved in [4] that coincides with the normal cone in the sense of convex analysis given by .

Based on the functional , a set of generalized projections of onto is defined as follows (see [1]).

Definition 1. Let be a nonempty subset of and . If there exists a point satisfying

then is called a generalized projection of onto . The set of all such points is denoted by .

The following lemma is needed in our proofs and for its proof we refer to [1].

Lemma 2. If is a uniformly convex Banach space, then the inequality

holds for all and in , where .

We end this section with the following important result proved in [4]. It proves the density of the set in , that is, the set of points in at which is dense in .

Theorem 3. Let , let , let be a -uniformly convex and -uniformly smooth Banach space, and let be a lower semicontinuous function. Let , and let be given. Then there exists a point satisfying and . Consequently, is dense in .

3. Minimization Principles in Banach Spaces

Given a lower semicontinuous function that is bounded below and , we define two functions and by

These functions and coincide, in Hilbert spaces, with the inf-convolution of the function and the function , which is due to the relation in Hilbert spaces. In [4], the authors studied the function and they derived some minimization principles in -uniformly convex and -uniformly smooth Banach spaces. In this section, we establish similar results for the function that will be used to prove our main theorem in this paper. We start with the following theorem proved in [4].

Theorem 4. Let , let , let be a -uniformly convex and -uniformly smooth Banach space, and let be a proper lower semicontinuous function. Suppose that is bounded below by some constant . Then is bounded below by and is Lipschitz on each bounded subset of . Furthermore, suppose that is such that is nonempty. Then there exists a point satisfying the following:

(i)If is a minimizing sequence in for the infimum in (8), then .(ii)The infimum in (8) is attained uniquely at .

Let be a -uniformly convex and -uniformly smooth Banach space with and . Then is reflexive; that is, , and is one-to-one from to with . Thus, observe that the function can be rewritten as follows:

where is defined by and is defined by

Using this observation together with Theorem 4 with and playing the role of we can prove the following theorem.

Theorem 5. Let , let , let be a -uniformly convex and -uniformly smooth Banach space, and let be a proper lower semicontinuous function which is bounded below by some constant . Then is bounded below by and is Lipschitz on each bounded subset of . Furthermore, for any with there exists a point satisfying the following:

If is a minimizing sequence in for the infimum in (9), then .The infimum in (9) is attained uniquely at ; that is,

Proof. Let and . Clearly is a proper l.s.c. function on and is bounded below by the same constant . Then by Theorem 4 the function is bounded below by and is Lipschitz on each bounded subset in . Furthermore, for any with , we have and so by Theorem 4 there exists a point such that

(i)if is a minimizing sequence in for the infimum in

 then ;(ii)the infimum in (13) is attained uniquely at ; that is,

The proof will be complete by taking and by using the fact that is continuous in smooth Banach spaces.

By taking different forms of the function , we can obtain various types of minimization principles. We state here the two following types. The first one is Stegall’s minimization principle and the second one is a variant of the smooth Borwein-Preiss variational principle in Banach spaces (see [4] for different variants in Banach spaces and see [12] for those principles in Hilbert spaces).

Theorem 6. Let and , let be a -uniformly convex and -uniformly smooth Banach space, and let be a lower semicontinuous function. Suppose that is bounded below on the bounded closed set , with . Then, for any , there exists with such that the function attains a unique minimum over .

Proof. Define on the function

which is of the form with and . Furthermore, expression (15) for can be rewritten as

Let and let . Then for any we have

that is, . Now, by the density theorem of the -proximal subdifferential in Theorem 3, there exists ; that is, with and

Using now Theorem 4(ii), we deduce that the infimum in (15) and (16) is attained at a unique point ; that is,

Therefore, by taking , we obtain and the function attains a unique minimum over at and so the proof is complete.

The following theorem is a different variant of the smooth Borwein-Preiss variational principle in which the perturbation is given in terms of the functional .

Theorem 7. Let , let , let be a -uniformly convex and -uniformly smooth Banach space, and let be a lower semicontinuous function bounded below, and . Suppose that is a point satisfying . Then for any there exist points and such that

(i), , ,(ii) has a unique minimum at .

Proof. Let be as in the statement of Theorem 7 and let . Put and consider the function

Since is l.s.c. on and by the density result in Theorem 3, there exists satisfying with . By Theorem 5 there is a unique point satisfying

and so

and so

Thus,

and so the function has a unique minimum at .

4. Generalized Projections on Closed Nonconvex Sets

Let us start with the following example showing that may be empty for nonconvex closed sets in uniformly convex and uniformly smooth Banach spaces.

Example 8. Let   (), let , and let with . Then is a closed nonconvex subset in with .

Proof. Clearly is closed and not convex. Let be any element in ; that is, for some , . Then for any we have and so , ; that is,

and so

This ensures that .

From the previous example, we see that even in uniformly convex and uniformly smooth Banach spaces the generalised projection may be empty for closed nonconvex sets and so there is no hope of getting the conclusion of Theorem  2.1 in [3] saying that , , whenever the set is closed convex in reflexive Banach spaces. However, we are going to prove that, for closed nonconvex sets, the set of points for which is dense in . We are going to prove our main result in the following theorem. It is an analogue result to Lau’s theorem for metric projections in reflexive Banach spaces [13].

Theorem 9. Let and , let be a -uniformly convex and -uniformly smooth Banach space, and let be any closed nonempty set of . Then there is a dense set of points in admitting unique generalised projection on ; that is, for any , there exists with , .

Proof. Observe that

which means that has the form with and . Since is proper l.s.c. and is bounded below, we can apply Theorem 5 to get for any with the existence of some satisfying

that is, and , which means that . Using now the density result in Theorem 3, to get the density of the set in , that is, for any , there exists with . Therefore, by what precedes, there exists ; that is, , . This proves the conclusion of the theorem.

5. Applications to Nonconvex Variational Problems

Let and and let be a -uniformly convex and -uniformly smooth Banach space. Let be a set-valued mapping and let be a nonempty closed set not necessarily convex. Our aim is to use the main result in the previous section to study the following nonconvex variational problem:

First we show that in the convex case (29) coincides with the usual variational inequality

Proposition 10. Whenever is a closed convex set, one has (29) ⇔ (30).

Proof. The proof follows from the fact that coincides with the convex normal cone which can be characterised as .

We suggest the following algorithm to solve the proposed problem (29) under some natural and appropriate assumptions on and .

Algorithm 11. Let with being too small:

(i)Select , , and .(ii)For ,

(a)compute ;(b)choose with ;(c)compute and .

Since is not necessarily convex, the generalised projection does not exist necessarily for any . However, our previous algorithm is well defined as we will prove in the following proposition.

Proposition 12. Assume that is uniformly convex and uniformly smooth Banach space. The above algorithm is well defined.

Proof. Let and let with be given. The point is well defined since and are well defined and one-to-one because the space is assumed to be uniformly convex and uniformly smooth. Now, since the generalised projection of is not ensured we use our main result in Theorem 9 to choose some point too close to so that and . Then by the same theorem we have the uniqueness of the generalised projection so we can take and then we are done.

After proving the well definedness of the algorithm without any additional assumptions on and we add some natural assumptions on the data to prove the convergence of the sequence to a solution of (29).

In our analysis we need the following assumptions on and :

Assumptions

(1)The solution set of (29) is nonempty.(2)The set is ball compact; that is, any bounded set in is relatively compact.(3) is bounded on by some constant .(4) is -Lipschitz on ; that is,

(5) is --strongly monotone on ; that is,

(6)There exist some constants and such that

(7)The constants , , , , , and satisfy

Theorem 13. Assume that is 2-uniformly smooth. Let be a sequence generated by Algorithm 11. Assume that Assumptions hold and that the parameter satisfies the inequalities

where . Then there exists a subsequence of converging to a solution of (29).

Proof. Let be a solution of (29); that is, there exists such that . Hence by definition of the -proximal normal cone there exists such that . Without loss of generality we may assume that is too small so that . First we claim that ; that is, . Let . Then for any we have

We distinguish two cases.
Case  1. If , then obviously we have

Case  2. If , then since we have

and so we obtain

Therefore, in both cases, we have

Hence

On the other hand, simple decomposition yields

Consequently, we have

which means that . Set . Since is 2-uniformly smooth we have the 2-uniform convexity of the dual space and so for some constant depending only on the space . On the other hand, by Lemma 2, we have

where . Hence

Therefore, we obtain

Using now the -Lipschitz property and the --strong monotony of , we write

and so

Since and belong to (by construction) we have , and so by our assumptions on the constants , , and and the choice of we obtain

which ensures that and belong to . This yields with the Lipschitz assumption of the generalised projection on that

And consequently inequality (48) becomes

Set , , and . Then for any we have

By mathematical induction we get

Our Assumptions and the choice of ensure that and hence the sequence is bounded and so the sequences and are bounded and since the set is ball compact then the sequence is compact and hence there exists a subsequence converging to some limit . By Lipschitz property of we can check easily that the sequence is convergent to some limit belonging to and so . Set . To complete the proof we have to prove that is a solution of (29). By Algorithm 11 the subsequence satisfies and so by the Lipschitz property of the generalised projection on the set we can write

and so which ensures by definition of the -proximal normal cone that

Thus

that is, is a solution of (29). Thus the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank both referees for the complete reading of the first version of this work and for their suggestions allowing him to improve the presentation of the paper. The author extends his appreciations to the Deanship of Scientific Research at King Saud University for funding the work through the Research Group Project no. RGP-024.

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