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Abstract and Applied Analysis
Volume 2017, Article ID 7236103, 13 pages
https://doi.org/10.1155/2017/7236103
Research Article

A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem

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

Correspondence should be addressed to Teffera M. Asfaw; moc.oohay@mareffet

Received 19 April 2017; Accepted 20 July 2017; Published 12 September 2017

Academic Editor: Stanislaw Migorski

Copyright © 2017 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 locally uniformly convex reflexive Banach space with locally uniformly convex dual space . Let be maximal monotone, be bounded and of type and be compact with such that lies in (i.e., there exist and such that for all ). A new topological degree theory is developed for operators of the type . The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type , where is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.

1. Introduction: Preliminaries

In what follows, the norm of the spaces and will be denoted by For and , the pairing denotes the value Let and be real Banach spaces. For an operator , we define the domain of by , and the range of by . We also use the symbol for the graph of : An operator is “demicontinuous" if it is continuous from the strong topology of to the weak topology of . It is “compact" if it is strongly continuous and maps bounded subsets of to relatively compact subsets of An operator is “bounded" if it maps each bounded subset of into a bounded subset of . It is “finitely continuous" if it is upper semicontinuous from each finite dimensional subspace of to the weak topology of . Let be a continuous and strictly increasing function such that as . The mapping defined byis called the “duality mapping" associated with . As a consequence of the Hahn-Banach theorem, it is well-known that for all . Since and are locally uniformly convex, is single valued, bounded, monotone, and bicontinuous. The following definitions are needed throughout the paper.

Definition 1. An operator is said to be (i)“monotone" if for every , , , and , we have ;(ii)“maximal monotone" if is monotone and for every ; that is, is maximal monotone if and only if is monotone and for every implies and ;(iii)“coercive" if either is bounded or there exists a function such that as and for all and ;(iv)“weakly coercive” if either is bounded or as , where for each ,

It is important to note here that the class of weakly coercive operators includes the classes of coercive operators. For a maximal monotone operator , we know that for all and is single valued and demicontinuous. In addition, the operator , , defined by , is the “Yosida approximant" of . It is bounded, continuous, and maximal monotone with domain such that as , for every , where Furthermore, the operator , defined by , is called the “Yosida resolvent" of . It is continuous, for every , and for all , where is the convex hull of the set Furthermore, for each , for all Browder and Hess [1] introduced the following definitions. The original definition of single valued pseudomonotone operator is due to Brèzis [2].

Definition 2. An operator is said to be (a)“pseudomonotone" if the following conditions are satisfied:(i)For every , is nonempty, closed, convex, and bounded subset of ;(ii) is finitely continuous; that is, for every and every weak neighborhood of in , there exists a neighborhood of in such that ;(iii)for each sequence with such that and we have that, for every , there exists such that in particular, letting in place of in the above inequality, the pseudomonotonicity of implies (b)“of type " if (i) and (ii) of (a) hold and for each sequence in such that in as and every with we have and there exists a subsequence of , denoted again by , such that as ;(c)“of type ” if (i) and (ii) of (a) hold and for any sequence , such that as , it follows that there exists a subsequence of , denoted again by , such that as .

It is not difficult to see that the class of operators of type includes the classes of operators of type . Furthermore, it holds that is of type provided that is of type and is compact. The main goals of this paper are(i)to develop suitable degree theory for operators of the type , where is maximal monotone, is bounded of type , and is compact with and sublinear; that is, there exist and such that for all . The existing degree theories for operators of the type cannot be used to treat inclusions involving operators of the type because the compact operator is not everywhere defined. For recent degree theories for multivalued bounded or bounded pseudomonotone perturbations of arbitrary maximal monotone operators, the reader is referred to the papers by Asfaw and Kartsatos [3], Asfaw [4], Adhikari and Kartsatos [5], and the references therein. In these theories, the maximal monotone operator is arbitrary and and/or pseudomonotone operator is everywhere defined. The original degree mapping due to Browder [6] is for operators of the type , where is single valued bounded operator of type defined from the closure of a nonempty, bounded, and open subset of . Hu and Papageorgiou [7] generalized Browder’s theory for multivalued compact perturbation of , where the compact operator is defined on . All these theories do not include the case where is not defined on , in particular, when contains . In view of these, our work in developing a degree theory for operators of the type , where is a compact operator with , is essential. It is worth mentioning that the theory associated with (i) is a generalization of the previous degree theories for bounded perturbations of maximal monotone operators due to Browder [6], Kobayashi and Otani [8], Hu and Papageorgiou [7], Asfaw and Kartsatos [3], and the references therein. The most general degree theory currently available which is due to Asfaw [9] is for pseudomonotone perturbations of the sum of two maximal monotone operators with one of the maximal monotone operators which is of type ;(ii)to derive existence theorem(s) in order to establish solvability of operator inclusion problems involving operators of the type . Consequently, the theory developed in (i) is applied to prove existence of solution for the inclusion problem provided that there exists such that for all , , and ; that is, provided that is coercive. The result is a generalization of the existence result due to Asfaw and Kartsatos [3, Theorem ] for the operator . This result yields the surjectivity of provided that is coercive and either is bounded of type or is operator of type .

Throughout the paper, we shall use the following definition of a homotopy of class .

Definition 3. Let and The family is said to be a “homotopy of type ” if the following are true: (i)For each , , is a nonempty, closed, convex, and bounded subset of (ii)For each is finitely continuous.(iii)Let be such that and Let be such that Then and there exists a subsequence of , denoted again by , such that as

The following lemma is due to Ibrahimou and Kartsatos [10].

Lemma 4. Let be maximal monotone and be bounded. Let Let . Then there exists a constant , independent of and , such that for all , , and

For basic definitions and further properties of mappings of monotone type, the reader is referred to Barbu [11], Pascali and Sburlan [12], Browder and Hess [1], and Zeidler [13].

The content of the following important lemma is due to Brezis et al. [14].

Lemma 5. Let be a maximal monotone set in . If such that in , in , and then and as

Browder [6] introduced the concept of a pseudomonotone homotopy as given below.

Definition 6. Let be a family of maximal monotone operators from to such that ,   Then is called a “pseudomonotone homotopy" if it satisfies the following equivalent conditions: (i)Suppose that and are such that in , in and Then and (ii)The mapping defined by is continuous.(iii)For each , the mapping defined by is continuous.(iv)For any and any sequence , there exists a sequence such that and as .

For a maximal monotone operator , Kobayashi and Otani [8] proved that the family is a pseudomonotone homotopy of maximal monotone operators if and only if is densely defined. It is worth mentioning that the proof of this fact does not require the hypothesis . It is essential herein to mention that the original degree theory for single-value perturbations of maximal monotone operators is due to Browder [6]. For a generalization of Browder’s degree for multivalued compact perturbations of , where is maximal monotone and is bounded demicontinuous of type , the reader is referred to the paper due to Hu and Papageorgiou [7]. For existence results for compact perturbation of maximal monotone operators, the reader is referred to the paper due to Kartsatos [15]. For a relevant degree mapping for single multivalued operator of type , we cite the paper of Zhang and Chen [16]. Recent developments on degree theories for perturbations of the sum of two maximal monotone operators can be found in the papers due to Adhikari and Kartsatos [5] and Asfaw [4].

In Section 2 we construct a degree mapping for operators of the type , where is maximal monotone, is bounded and of type or bounded pseudomonotone, and is compact with and satisfies a sublinearity condition. The existence of solutions for operator inclusion problems of the type is included in Section 3. In Section 4, the theory is applied to establish existence of weak solution(s) for a nonlinear parabolic problem in appropriate Sobolve spaces.

2. Degree Theory for with

2.1. Degree Theory for with Bounded and of Type

The goal of this section is to develop a degree theory for operators of the type , where is maximal monotone, is bounded and of type , and is compact with . Throughout the paper, we assume that belongs to (i.e., there exist and such that for all ). To this end, we start by proving the following useful lemma.

Lemma 7. Let be a nonempty, bounded, and open subset of . Let be maximal monotone, be bounded and of type , and be compact with such that belongs to class . Assume, further, that . Then there exists such that is well-defined and independent of .

Proof. In the first step, we claim that there exists such that is well-defined for all . Suppose that this is false; that is, there exist , , and such thatwhere . By the definitions of and , we have Since and are bounded, it follows that is bounded. Since belongs to , we get that for all , where is an upper bound for This yields the estimate for all . Since and is bounded, it follows that and are bounded. The compactness of implies the boundedness of . Now, assume without loss of generality that , , and as . Since is compact, we may assume, by passing into a subsequence if necessary, that as . The maximality of along with Lemma 5 gives that is, we obtain from (12) that Since is of type , we conclude that as and . Consequently, using (12) we arrive at The maximality of along with Lemma 5 yields and and as . Since is compact and as , it follows that as . Letting in (12), we get . However, this is impossible. Thus, there exists such that is well-defined for all .
Next, we shall prove that is independent of . Let    be such that , , . We consider the homotopy operator We will show that the family is a homotopy of class such that for all . To this end, let , ,  , , , and as be such that Since and are bounded, it follows that is also bounded. Since for all , we apply Lemma 4 to conclude that and are bounded. On the other hand, we see that By the compactness of , we may assume without loss of generality that as . Since as , we use the continuity of ([3], Lemma ) to conclude that as . Combining these along with the monotonicity of , we obtain Since is of type , we see that as and there exists a subsequence of , denoted again by , such that as . Since is continuous, we conclude that that is, is a homotopy of class such that for all . Therefore, is independent of ; that is, . Since and are arbitrary in , we conclude that is well-defined and independent of . This completes the proof.

Based on Lemma 7, the associated degree mapping is defined as follows.

Definition 8. Let be a nonempty, bounded, and open subset of , be maximal monotone, be bounded and of type , and be compact with and belonging to the class . Assume, further, that . Then the degree mapping for at with respect to is defined by where is the degree mapping for multivalued bounded operators of type from [16].

2.2. Basic Properties of the Degree

Theorem 9. Let be a nonempty, bounded, and open subset of . Let be maximal monotone, be bounded and of type , and be compact with such that belongs to . Then the following properties hold: (i)(Normalization) if and if (ii)(Existence) if and , then .(iii)(Decomposition) let and be nonempty, disjoint, and open subsets of such that Then (iv)(Translation invariance) let Then we have (v)Let , , where is bounded and of type and for all . Then is independent of .(vi)Let , , , where is densely defined maximal monotone and positively homogeneous of order , is bounded and of type such that for all , and for all . Assume, further, that for all . Then is independent of .

Proof. The proof of (i) follows by setting and . To prove (ii), assume that and . By the definition of , there exists such that for all ; that is, for each there exist and such thatSince is bounded, it follows that is bounded. By using condition on along with the arguments used in the proofs of Lemma 7, it is easy to see that and are bounded. Assume without loss of generality that , , , and as . By the maximality of , the condition on , and the arguments used in the proof of Lemma 7, we conclude that , , and such that . This shows that .
Next we prove (iii). Suppose the hypotheses in (iii) hold. By the definition of , we see that for all sufficiently small . Since is bounded and of type , the decomposition property of the degree mapping for multivalued operators implies that is, (iii) holds.

(iv) Suppose that ; that is, . This implies that is well-defined. Since , by the translation property of the degree mapping for multivalued bounded operators of type , we see that . Thus,

(v) Suppose that for all , where , For every , we consider

We shall show that there exists such that is well-defined and independent of all . To do this, we assume to the contrary that there exist , , , , and such thatSince , , and are bounded, it follows that and are bounded. By the condition on , the boundedness of and , and the arguments used in the proof of Lemma 7, we conclude that and are bounded. Assume without loss of generality that , , , , , , and as . Suppose that . We have as . Since is of type , it follows that , , and ; that is, , that is, . However, this is impossible. A similar proof covers the case . Assume . Suppose there exists a subsequence of , denoted again by , such that for all . Since is maximal monotone, Lemma 5 implies . As a result, (31) implies Since is of type , it follows that as and . Moreover, one can show that , , and so that ; that is, . However, this is a contradiction.

To show that is constant for all and , with as in the proof of (ii), we let and consider the homotopy operator Since for each is monotone, and are bounded and of type and is compact, it follows that is bounded demicontinuous and of type . It is not hard to verify that for all . As in the arguments used in the proof of Lemma 7, we shall show that is a homotopy of class . To this end, let and be such that and as , and be such that

Since is monotone with domain , it follows that for all . Since as and is continuous, we get as . As a result of this, we get

that is, Since for all , an application of Lemma 4 says that there exists independent of such that for all . In addition, by the definition of , we see that for all . Since is bounded, the boundedness of follows. By the compactness of , we may assume without loss of generality that as . As a result of this, we get

Let . The boundedness of and imply

Since and are bounded and of type , it follows that and there exist subsequences of and