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Journal of Function Spaces
Volume 2016 (2016), Article ID 3970621, 15 pages
http://dx.doi.org/10.1155/2016/3970621
Research Article

A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Received 30 June 2016; Accepted 18 October 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Teffera M. Asfaw. 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

Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space . Let be maximal monotone of type (i.e., there exist and a nondecreasing function with such that for all , , and be linear, surjective, and closed such that is compact, and be a bounded demicontinuous operator. A new degree theory is developed for operators of the type . The surjectivity of can be omitted provided that is closed, is densely defined and self-adjoint, and , a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for , where is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when is monotone, a maximality result is included for and . The theory is applied to prove existence of weak solutions in of the nonlinear equation given by ,  ;  ,  ; and ,  , where , , , , , is a nonempty, bounded, and open subset of with smooth boundary, and satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.

1. Introduction and Preliminaries

Throughout the paper, denotes a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space For and , the duality pairing denotes the value Let be the normalized duality mapping given by It is well-known that for all because of the Hahn-Banach Theorem. Since and are locally uniformly convex reflexive Banach spaces, it is well-known that is single valued and homeomorphism. For a multivalued operator from into , the domain of denoted by is given as The range of , denoted by , is given by and graph of , denoted by , is given by The following definition is used in the squeal.

Definition 1. A multivalued operator is called (i)“monotone” if, for all , , , and , we have ,(ii)“maximal monotone” if is monotone and for every implies and This is equivalent to saying that is “maximal monotone” if and only if for every ,(iii)“coercive” if either is bounded or there exists a function such that as and

It is the goal of the paper to develop a topological degree theory for classes of operators of the type , where , , and satisfy one of the following conditions:(i) is linear, surjective, and closed such that is compact, is maximal monotone of type , and is bounded demicontinuous operator.(ii) , a real Hilbert space, is linear, densely defined, self-adjoint, closed, and range closed such that is compact, is maximal monotone of type , and is bounded demicontinuous operator. The main reason for the need of such a theory is the existence of nonlinear problems (i.e., nonlinear equations and variational inequality problems) which cannot be addressed by the existing theories under minimal assumptions on , , and . In addition, considering the classes of operators of the type , it is an essential contribution to have a theory useful to drive existence theorems to treat larger class of problems. Therefore, Section 2 gives a preliminary lemma, which will be useful to extend the definition of pseudomonotone homotopy of maximal monotone operators initially introduced by Browder [1, 2]. Section 3 deals with the construction of the degree mapping along with basic properties and homotopy invariance results. The main contribution of this work is providing a new degree theory for treating nonlinear problems involving operators of type , where , , and satisfy condition (i) or (ii). In this theory, the operator might not be pseudomonotone type and is just bounded demicontinuous operator. The well-known degree for monotone type operators, which is attributed to Browder [1, 2], is for operators of type , where is maximal monotone and is bounded demicontinuous operator of type . In view of this, the degree mapping constructed herein allows to be bounded demicontinuous operator not necessarily compact, bounded of type , or pseudomonotone. To the best of the author’s knowledge, this degree mapping is new and has the potential to address new classes of problems such as wave equations with nonmonotone nonlinearities. As a consequence of the theory, new existence results are given for the solvability of operator inclusions of the type , . In the last section, examples are provided proving existence of weak solutions for nonlinear parabolic as well as hyperbolic problems in appropriate Sobolev spaces. For degree theories for bounded demicontinuous perturbations of maximal monotone operators, the reader is referred to the papers of Browder [1, 2], Kobayashi and Ôtani [3], Hu and Papageorgiou [4], Berkovits and Mustonen [5, 6], Berkovits [7], Kartsatos and Skrypnik [8], and Kien et al. [9] and the references therein. For recent topological degree theories for bounded pseudomonotone perturbations of maximal monotone operators, the reader is referred to the recent papers of Asfaw and Kartsatos [10] and Asfaw [11]. Basic definitions, properties, and existence theorems concerning operators of monotone type can be found in the books of Barbu [12, 13], Pascali and Sburlan [14], and Zeidler [15].

2. A Preliminary Lemma

The following lemma is useful towards the extension of the definition of a pseudomonotone homotopy of maximal monotone operators introduced by Browder [1, 2].

Definition 2. A family is said to be uniformly of type (i) if there exist and a nondecreasing function such that and uniformly for all and , , and ,(ii) if there exists such that uniformly for all and and . If for all , then the operator is said to be of type or if it satisfies either (i) or (ii), respectively.

It is easy to see that a family of monotone operators is uniformly of type if for all . It is also true that the class includes the class . The lemma below is used in the construction of the degree.

Lemma 3. Let be a family of maximal monotone operators uniformly of type . Then the following four conditions are equivalent: (i)For any sequences in and such that in , and and in as with , it follows that , and as (ii)For each , the operator defined by is continuous from to (iii)For each fixed , the operator defined by is continuous from to (iv)For any given pair and any sequence as , there exist sequences and such that and and as

Proof. The proof for the implications follows from the result attributed to to Browder [1, 2] without requiring the condition for all . Next we give the proof of the implication . Fix . Let such that and as For each , let and . It follows that for some , for all , and for some We shall show that is bounded. By applying condition on the family , we see that that is, we getSince is bounded, (83) implies the boundedness of the sequence . Assume, without loss of generality, that as Since , it follows that as By the condition in (i) and monotonicity of and as and boundedness of , we obtain that Consequently, we arrive at Thus, by using conditions in (i), it follows that , , and as , which implies Since is of type and continuous, we have and as , which implies ; that is, , implying in turn that ; that is, as This shows that is continuous from into Therefore, the equivalency of the four statements is proved.

A larger class of pseudomonotone homotopies of maximal monotone operators is introduced below. The original definition of pseudomonotone homotopy of maximal monotone operators is attributed to Browder [2] which requires the family to satisfy for all .

Definition 4. A family of maximal monotone operators uniformly of type is called a “pseudomonotone homotopy of type ” if one of the equivalent conditions of Lemma 3 holds.

3. Degree Theory in Reflexive Banach Space with

The section deals with the main contribution of the paper. A new topological degree mapping is constructed for operators of type , where is maximal monotone of type , is bounded demicontinuous operator, and is linear, surjective, and closed such that is compact. The construction is based on the Leray-Schauder degree mapping for the operator , where is the Yosida approximant of . Since is surjective, is compact, and is bounded continuous operator, it follows that is a well-defined compact operator. Next we prove the following theorem.

Theorem 5. Let be a nonempty, bounded, and open subset of . Let be a pseudomonotone homotopy of maximal monotone operators uniformly of type and with is bounded demicontinuous operator and let be linear, surjective, and closed such that is compact. Assume, further, that for all . Then there exists such that is well-defined and independent of and , where denotes the Leray-Schauder degree mapping for compact displacement of the identity and is the Yosida approximant of .

Proof. Suppose the hypotheses hold. Assume to the contrary that there exist , , and such thatSince is surjective, it follows that and for all . The uniform boundedness of the family implies the boundedness of . Since is of type , let and be as in Definition 2. Let . It is well-known that , , and for all . For each , we see that where for all and is an upper bound for . Now, setting in place of , we obtain thatfor all . Since is nondecreasing and is bounded, we see that where is an upper bound for . By similar argument, setting in place of , we get for all ; that is, , where is an upper bound for For each , combining these two inequalities shows that there exists such that for all . By applying the well-known uniform boundedness principle, we conclude that is bounded. Consequently, we obtain the boundedness of . Since is bounded and is compact, we assume without loss of generality that there exists a subsequence, denoted again by , such that as ; that is, as . Assume without loss of generality that , , , and as . Since , it follows that as ; that is, Since is a pseudomonotone homotopy of type , (iv) of Lemma 3 implies that , , and as . Since is closed, we conclude that and , which implies that . However, this is impossible. Therefore, there exists such that is well-defined for all and . Next we prove that there exists such that is independent of and . Suppose this is false; that is, there exist , , and such thatfor all For each , we consider the homotopy Since is compact and , , and are bounded continuous operators, we observe that is Leray-Schauder type homotopy. We shall show that is an admissible homotopy for all large ; that is, for all large , we have for all . Suppose there exists a subsequence of , denoted again by , such that there exist , , and such that for all ; that is, we have for all . Assume without loss of generality that and as . For each , let , , and . It is well-known that , , , and for all . Let . By the definition of pseudomonotone homotopy of type and uniform boundedness condition on , we see that where , , and is upper bound for . By following the argument used in the first part of this proof, it follows that is bounded; that is, is bounded. By the compactness of , there exists a subsequence, denoted again by , such that as . Assume without loss of generality that and as . Since is closed, we have and . Since and are bounded and as , we get as . To complete the proof, we consider the following cases.
Case I. is bounded. Since is bounded, is also bounded. Since is of type (i.e., of type ), it follows that for all and for all . Let . Since is a pseudomonotone homotopy of type , by (iv) of Lemma 3, there exists a sequence such that and as On the other hand, the monotonicity of implies that is,In a similar manner, we getfor all Multiplying (22) and (23) by and , respectively, and adding the resulting inequalities, we getSince is bounded and , it follows that as . Consequently, using (24), we obtainfor all , which yields By the maximal monotonicity of , we conclude that and Therefore, we obtain that and such that . However, this is a contradiction.
Case II. Suppose is unbounded. Then there exists a subsequence, denoted again by , such that as Then and being unbounded. Assume without loss of generality that and as If either or is unbounded, (24) impliesAssuming or and taking limits in (26) imply thatwhich is impossible. Thus, and are bounded. Consequently, we get as Similarly, we have as In all cases, (24) and (25) yield a contradiction. Therefore, by using the compactness of and boundedness of and , we proved that the family is an admissible homotopy of Leray-Schauder type; that is, is independent of for all large ; that is, However, this is impossible. Therefore, there exists such that is well-defined and independent of and . The proof is completed.

Next we give the definition of the required degree mapping.

Definition 6. Let be a nonempty, bounded, and open subset of . Let be maximal monotone of type , be bounded demicontinuous operator, and be linear, surjective, and closed such that is compact. Assume, further, that . Then the degree of at with respect to is given by where denotes the Leray-Schauder degree mapping for compact perturbations of the identity and is the Yosida approximant of .

The degree satisfies the following basic properties and homotopy invariance result.

Theorem 7. Let be a nonempty, bounded, and open subset of . Let be maximal monotone of type , be linear, surjective, and closed such that is compact, and be bounded demicontinuous operator. Then (i)(normalization) there exists such that if and if . If is monotone, then if and if ;(ii)(existence) if and , then ;(iii)(decomposition) let and be nonempty and disjoint open subsets of   such that Then (iv)(translation invariance) let Then we have (v)(homotopy invariance) let be a pseudomonotone homotopy of maximal monotone operators uniformly of type and is bounded demicontinuous operator. Let Then is independent of provided that for all .

Proof. (i) Suppose the hypotheses hold. Since is continuous, there exists such that for all . Let and , . If there exist and such that , then it follows that and . Since (i.e., ), this gives or . But these are impossible because and . Since is a family of compact operators from into such that ,  , is uniformly continuous in uniformly for all , it follows that is an admissible homotopy of Leray-Schauder type; that is, is independent of . Therefore, we obtain that if and if .
(ii) Suppose and . By the definition of the degree mapping , we see that for all sufficiently small ; that is, for each , there exists such that By the arguments used in the proof of Theorem 5, one can easily show that . The details are omitted here.
(iii) Suppose the hypotheses hold. The definition of the degree mapping and decomposition property of the Leray-Schauder degree imply for all sufficiently small . This completes the proof of (iii). The proof of (iv) follows from the translation invariance property of Leray-Schauder degree.
(v) Let ,. The proof of Theorem 5 confirms the existence of such that is well-defined and independent of and ; that is, by the definition of the degree, we get thatis well-defined and independent of . The details are omitted here.

Consequently, we prove the following new existence result.

Theorem 8. Let be densely defined maximal monotone with and of type , be linear monotone, surjective, and closed such that is compact, and be bounded demicontinuous operator. Let . Assume, further, that there exists such that for all and . Then . In addition, is surjective if is coercive.

Proof. By the continuity of , there exists such that for all . Let and Since and , we see that can be rewritten as . Since and are monotone with , we get for all , , and ; that is, we have for all , , and ; that is, for all . Since is a pseudomonotone homotopy of maximal monotone operators of type , (v) of Theorem 7 implies that Next we show that . We consider Following the above arguments, it is not difficult to show that for all . Since is densely defined, it is well-known that is a pseudomonotone homotopy of type ; that is, (v) of Theorem 7 gives Consequently, we get that is, for each , there exist and such that