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Monotone type operators in nonreflexive Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 119 (2014)
Abstract
Let E be a real Banach space, be the dual space of E, be the dual space of . Let be a monotone type mapping. In this paper, first, we introduce the special case when T is the weak* sub-differential of a convex function ϕ and obtain a surjective result for the mapping , where . Second, we show the existence of solutions of the variational inequality problems for strictly quasi-monotone operators and semi-monotone operators. Finally, we construct a degree theory for mappings of the class and then construct a generalized degree for the weak* sub-differential of a convex function.
1 Introduction
Monotone operators in reflexive Banach spaces has many applications in nonlinear partial differential equations, nonlinear semi-group theory, variational inequality and so on (see [1–4]). The theory for monotone operators in reflexive Banach spaces has been well developed. In recent years, many authors have generalized the monotone operator theory to nonreflexive Banach spaces. For example, maximal monotone operators in nonreflexive Banach spaces has been studied in [5–8] and variational inequality problems related to monotone type mappings in nonreflexive Banach spaces have been studied in [9–14]. For more references on variational inequality problems, see [15–24] and [25]. Also, degree theory for monotone type mappings in nonreflexive separable Banach spaces has been studied in [26, 27]. Also, see [3, 28–36] for more references on degree theory of monotone type operators.
In this paper, we study variational inequality problems and degree theory for monotone type mappings in nonreflexive spaces. This paper is organized as follows:
Let E be a real Banach space, be the dual space of E and be the dual space of . In Section 2, we introduce the weak* sub-differential of a convex function , which is a subset of the classical sub-differential, and we obtain for the sum of a lower semi-continuous convex function in the weak* topology and , where . In Section 3, we show the existence of solutions of variational inequality problems related to strictly quasi-monotone operators and semi-monotone operators. In Section 4, we construct a degree theory for mappings of class and then construct a generalized degree for the weak* sub-differential of a convex function and obtain some degree results.
Through this paper, we use ⇀∗ to represent the convergence in the weak* topology, ⇀ to represent the convergence in the weak topology and → represent the convergence in norm topology.
2 The weak* sub-differential of convex functions
In this section, let E be a real Banach space, be the dual space of E and be the dual space of .
Now, we introduce the weak* sub-differential of a convex function and study the solvability problems related this mapping.
First, we recall that the classical sub-differential of a convex function at y is defined by
It is well known (Rockfellar [8]) that ∂ϕ is a maximal monotone mapping.
Definition 2.1 Let be a convex function. Then
is called the weak* sub-differential of ϕ at y.
It is obvious that , but when E is reflexive.
The following result is obvious.
Proposition 2.2 Let be a convex function. Then we have the following:
-
(1)
is a weak closed convex subset of ;
-
(2)
if and only if ;
-
(3)
is monotone.
Definition 2.3 (see [37])
Let X be a topological space. A function is said to be sequentially lower semi-continuous from above at if, for any sequence with , implies that .
Similarly, f is said to be sequentially upper semi-continuous from below at if, for any sequence with , implies that .
Remark 1 It is well known that a lower semi-continuous function is a lower semi-continuous from above function, but the converse is not true and a lower semi-continuous from above and convex function with the coercive condition in a reflexive Banach space attains its minimum (see [37]). Also, it is well known that, for a convex function in a reflexive Banach space, lower semi-continuity in the strong topology is equivalent to lower semi-continuity in the weak topology, but this is not true for lower semi-continuity from above (see [38]). For more on lower semi-continuous from above functions with its generalizations and applications in nonconvex equilibrium problems, variational problems and fixed point problems, see [38–50] and [51].
Proposition 2.4 Let be a convex function which is sequentially lower semi-continuous from above in the weak* topology and , then there exists such that .
Proof We take a sequence in such that
Since and is a bounded sequence in , it follows that has a subsequence of with in . By the assumption, since ϕ is sequentially lower semi-continuous from above, we have and so it follows that
This completes the proof. □
Proposition 2.5 The function defined by is sequentially lower semi-continuous in the weak* topology.
Proof Suppose . Then for all and so
for all . Thus we have
and so . This completes the proof. □
Theorem 2.6 Let be a convex function which is sequentially lower semi-continuous in the weak* topology. Then we have
for all .
Proof For any , we set for all . It is obvious that ψ is sequentially lower semi-continuous in the weak* topology. Thus ψ is sequentially lower semi-continuous from above in the weak* topology and
By Proposition 2.4, there exists such that . By (2) of Proposition 2.2, , which is equivalent to . This completes the proof. □
3 Existence of variational inequality problems
In this section, we study variational inequality problems related to monotone type operators in nonreflexive Banach spaces.
First, we recall the following.
Definition 3.1 ([11])
A mapping is said to be semi-monotone if it satisfies the following conditions:
-
(1)
for each , is monotone, i.e., for all ;
-
(2)
for each fixed , is completely continuous, i.e., if in weak* topology of , then has a subsequence with in norm topology of .
Definition 3.2 ([15])
Let E be a real Banach space and be a mapping. T is said to be strictly quasi-monotone if for all and for some implies that for all .
Remark 2 For quasi-monotone mappings, see [21].
Lemma 3.3 Let E be a real Banach space and C be a nonempty bounded closed convex subset of . If is a finite dimensional weak* upper semi-continuous (i.e. for each finite dimensional subspace F of with , is upper semi-continuous in the weak topology) and strictly quasi-monotone mapping with bounded closed convex values, then for all and for some if and only if for all and .
Proof The proof is similar to Lemma 2.3 in [15], we omit the details. □
Remark 3 For the results of Lemma 3.3 in monotone case, we refer to [10].
Theorem 3.4 Let E be a real Banach space and C be a nonempty weak* closed convex bounded subset of . If is a finite dimensional weakly upper semi-continuous and strictly quasi-monotone mapping with bounded closed convex values, then there exists such that
for all and for some .
Proof For any finite dimensional subspace F of E with , let be the natural inclusion and be the conjugate mapping of . Consider the following variational inequality problem:
Find such that
for all and for some .
Since T is finite dimensional weakly upper semi-continuous and is upper semi-continuous on , there exists such that
for all and for some , i.e., for all and for some . By Lemma 3.3, we get
for all and . Now, we put
It is obvious that is weak* closed convex. One can easily check that
for . Hence , where
Take . We claim that satisfies the conclusion of Theorem 3.4. In fact, for all and . By Lemma 3.3, it follows that
for all and for some . This completes the proof.
From Theorem 3.4, we have the following. □
Corollary 3.5 Let E be a real Banach space and C be a nonempty weak* closed convex unbounded subset of . If is a finite dimensional weakly upper semi-continuous and strictly quasi-monotone mapping with bounded closed convex values and there exist and such that
for all and with , then there exists such that
for all and for some .
Proof If , then, by Theorem 3.4, there exists such that
for all and for some . By Lemma 3.3, we know that
for all and for some . By the assumption, we know that for each and thus we may assume that as . Otherwise, we take a subsequence. Consequently, it follows that
for all and . Again, if we use Lemma 3.3, we get the conclusion. This completes the proof. □
Corollary 3.6 Let E be a real Banach space, is the open ball centered at 0 with radius R. If is a finite dimensional weakly continuous and strictly quasi-monotone mapping and
for all , then there exists such that .
Proof It is obvious that is weak* closed and convex. By Theorem 3.4, there exists such that
for all . Now, we claim that . First, we prove that . In fact, if , then, by the assumption, and thus there exists such that . But we have
which is a contradiction. Therefore, we have . Since there exists such that for all with , we have
for all and so . This completes the proof. □
Theorem 3.7 Let be a bounded weak* closed convex subset. Suppose that is a lower semi-continuous convex function in the weak* topology , is semi-monotone, and is finite dimensional continuous for each . Then there exists such that
for all .
Proof For each finite dimensional subspace F of with , set and for . By Theorem 2.5 in [11], there exists such that
for all . Let
and
By (3.1) and the monotonicity of , is a nonempty bounded subset. Denote by the weak* closure of . For any for each , it is easy to see that for each . So, we have
Let . Now, we prove that
for all . For each , take such that and . There exists such that and
for each . By letting , the complete continuity of and weak* lower semi-continuity of ϕ imply that
Set for all and , by using the convexity of ϕ and letting , we get
This completes the proof. □
4 Degree theory for monotone type mapping
In this section, assume that E is always a real Banach space, is the dual space of E and is the dual space of .
Definition 4.1 A set-valued operator is said to be strong to weak upper semi-continuous at if, for each weak open neighborhood V of 0 in (i.e., open in the weak topology of ), there exists an open neighborhood W of 0 in such that for all .
Definition 4.2 A set-valued operator is said to be a mapping of class if the following conditions are satisfied:
-
(1)
for each , Tx is a bounded closed convex subset;
-
(2)
T is strong to weak upper semi-continuous;
-
(3)
if , for each and such that
then and has a subsequence with .
Definition 4.3 A family of set-valued operators for all is said to be a homotopy of mappings of class if the following conditions are satisfied:
-
(1)
for each , , is a bounded closed convex subset;
-
(2)
is strong to weak upper semi-continuous;
-
(3)
if , , for each , and such that
then and has a subsequence with .
Definition 4.4 Let be a mapping satisfying the conditions (1) and (2) in Definition 4.1. Let with and with . If implies that
then T is called a generalized pseudo-monotone mapping.
Proposition 4.5 Let be a mapping of class and be a mapping with closed convex values. Then the following conclusions hold:
-
(1)
if S is an upper semi-continuous and compact mapping, then is a mapping of class ;
-
(2)
if S is a generalized pseudo-monotone mapping and weak compact, i.e., S maps bounded subsets in to weak compact subsets in , then is a mapping of class .
For any subspace F of , let be the natural inclusion and be the conjugate mapping of . Note that, under the canonical injection mapping , i.e., for all and , can be injected as a subspace of and so, in the following, we always regard as a subspace of .
First, we need the following result from [36] (also, see [3]).
Lemma 4.6 Let F be a finite dimensional subspace, be an open bounded subset and let . Let be an upper semi-continuous mapping with compact convex values, be a proper subspace of F, and be the Galerkin approximation of T, where is the adjoint mapping of the natural inclusion . If , then there exist and such that and for all , where is the topological degree for upper semi-continuous mappings with compact convex values in finite dimensional spaces (see Ma [52]).
Remark See [53, 54] for more references on degree theory of multivalued mappings.
Lemma 4.7 Let be a bounded mapping of and let . Then there exists a finite dimensional subspace of such that
for all finite dimensional subspace F of with , where .
Under the condition of Lemma 4.7, we know that is well defined for the whole finite dimensional subspace F of with , where is the same as in Lemma 4.7.
Lemma 4.8 Under the condition of Lemma 4.7, there exists a finite dimensional subspace of such that does not depend on F.
Now, let be a nonempty open bounded subset and be a mapping of class . Suppose that . In view of Lemmas 4.6 and 4.8, we may define the topological degree as follows:
where F is a finite dimensional subspace of such that and is the same as in Lemma 4.8.
Theorem 4.9 If , then has a solution in Ω.
Proof The proof can be seen from the following proof of Theorem 4.10. □
Theorem 4.10 Let be a homotopy of mappings of class . If for all , then does not depends on .
Proof First, we claim that there exist finite dimensional subspaces of such that for all finite dimensional subspaces F with . Suppose that this is not true. For any finite dimensional subspaces F, we define a set as follows:
Then is nonempty. Let be the closure of in with endowed with weak* topology. Consider the following family of sets:
It is easy to show that . Let . If, for each , we take a finite dimensional subspace F such that and , then there exist and such that
for each . Hence we have
But, since is a homotopy of mappings of class , it follows that and has a subsequence that converges weakly to . Therefore, we have . By Mazur’s separation theorem (see [55]), we get , which is a contradiction. The claim is completed. So, it follows that is well defined for the whole finite dimensional subspace F with .
Next, we prove that there exist a finite dimensional subspace and such that does not depend on for all finite dimensional subspace F of with .
Suppose that this is not true. For any finite dimensional subspace F with , we define
Then is nonempty by Lemma 4.6. Let be the closure of in with endowed with the weak∗∗ topology. Consider again the following family of sets:
It is easy to show that . Let . Then, for each , we take a finite dimensional subspace F such that , and . Then there exist and such that
for . Hence we have
But, since is a homotopy of mappings of class , we have and has a subsequence which converges weakly to . Therefore, we have . Again, by Mazur’s separation theorem, , which is a contradiction. This completes the proof. □
Theorem 4.11 Let be a mapping of class , where is an open bounded subset. If and for all and , then
Proof Assume that F is a finite dimensional subspaces of . It is straightforward to check that
for all and . Therefore, we have and so, by (4.1),
□
Theorem 4.12 Let be a bounded mapping of class . If
then .
Proof For each , we set for all . Then it is easy to see that is a mapping of class . One can easily see that for all , and sufficiently large R. Thus, by Theorem 4.11, and so, by Theorem 4.9, has a solution in , i.e., has a solution in . This completes the proof. □
In the following, we assume that is separable and so we take any sequence of finite dimensional subspaces of such that
Lemma 4.13 Let be a lower semi-continuous convex function in the weak* topology, be open bounded and let . Suppose that for all . Then there exists a positive integer N such that
where is a mapping defined by for all and for all .
Proof Suppose that the conclusion is not true. There exists such that and so we have for all , which contradicts for all .
Under the assumption of Lemma 4.13, we know that there exists a positive integer N such that
for all and so, by [32], is well defined. Now, we define a generalized degree as follows:
□
Remark For generalized degree theory, see [56].
Theorem 4.14 Let be a lower semi-continuous convex function in the weak* topology. If , then
for sufficiently large r.
Proof By the assumption , it follows from Proposition 2.4 that there exists such that if we take a large enough r such that for all .
For any () satisfying (4.2), we put . We may easily see that
and so we have
for all . Thus we have
and, consequently, we have
This completes the proof. □
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The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2013053358).
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Chen, Y., Cho, Y.J. Monotone type operators in nonreflexive Banach spaces. Fixed Point Theory Appl 2014, 119 (2014). https://doi.org/10.1186/1687-1812-2014-119
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DOI: https://doi.org/10.1186/1687-1812-2014-119