Abstract and Applied Analysis
Volume 2017 (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

Teffera M. Asfaw

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

Correspondence should be addressed to Teffera M. Asfaw; tefferam@yahoo.com

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 by

is 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 that

where . 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 that

Since 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 that

Since , , 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 , denoted again by and , respectively, such that and as . Moreover, by the continuity of , it follows that as . From the continuity of and , we obtain that

as and as . Thus, we arrive at

as . Since for all , by the maximality of , we conclude that , , and as . Therefore, we get

as . The proofs of the cases and can be completed in an analogous manner. The details are omitted here. Thus the family is a homotopy of class ; that is, is independent of . This implies

Consequently, by the definition of , there exists such that

This proves that is independent of ; that is, the proof (iv) is complete.

(vi) Suppose the hypotheses in (vi) hold. Since, for each , is maximal monotone, let ,  , be Yosida approximant of and be the Yosida resolvent of . Since is bounded and of type and is compact with for all , we shall show that there exists such that is independent of and . Assume that this does not hold; that is, there exist , , , , and such that

for all , where and . By using the condition on , we arrive at

for all , where is an upper bound for . This gives the boundedness of and . Since as , and is bounded, it follows that as . Assume without loss of generality that as . Since as , the quasimonotonicity of and implies

Therefore, we get

Since , the result of Kobayashi and Otani [8] says that the family is a pseudomonotone homotopy of maximal monotone operators. By (i) of Definition 6, we conclude that , , and as . Applying analogous arguments to those of the proof of (iv) along with the condition on and , one can easily verify that , , , and so that ; that is, , which is impossible by the hypotheses. In conclusion, we have proved that is well-defined for all and sufficiently small .

Finally, we shall show that is independent of and . To this end, let , , , and , . To complete the proof, we consider the homotopy operator

It is sufficient to show that is a homotopy of class . For each , it is easy to see that is bounded and of type . Let be such that , as , , and so that , where . Let

We show that and are bounded. Since and for all , we conclude from Lemma 4 that is bounded. Since for all and , and are bounded, we get the boundedness of . Since is compact, we assume without loss of generality that as By the boundedness of , , and , we assume, by passing into subsequences if necessary, that , , and as . On the other hand, the pseudomonotonicity of and gives

Consequently, we get

Since is positively homogeneous of order , it is not difficult to see that is maximal monotone and positively homogeneous of order . It also holds that for all . In addition, we see that

In fact, it is true that, for all , and . For each , letting , we get

Since is continuous, it follows that as , where as . By the monotonicity of for all , we have

that is,

which implies as . Consequently, we arrive at

Since both and are bounded and of type , it follows that as . As a result of this, we get

as ; that is, we have Since is a pseudomonotone homotopy of maximal monotone operators, it follows that and . In conclusion, we obtain that , , , , and

Therefore, for any , the family is a homotopy of class . Thus, is independent of and ; that is,

On the other hand, by the definition of , we have that

is independent of and . In particular, for , we have for all . But, for , we see that for all . To complete the proof, it is sufficient to show that

For each , we consider the homotopy

Suppose that there exist , , , , and such that

We assume without loss of generality that , , , and as . By the condition on , we get

where is an upper bound for the sequence This shows the boundedness of and . By the maximality of along with Lemma 5, the compactness of , and the condition on and and analogous arguments to those in the proof of Theorem 9, we conclude that , , , and so that ; that is, . However, this is impossible. In addition, the boundary condition on in (v) implies that is an admissible homotopy; that is, for all . Since and satisfies the condition for all and and , it follows that for all and and is independent of . In particular, for all . Therefore, we conclude that is independent of all . This completes the proof.

2.3. Degree Theory for with Pseudomonotone

In this section we present a generalization of the theory developed in the previous section for operators of type , where is bounded pseudomonotone and satisfy the conditions of Section 2.1. For each , it is well-known that is bounded and of type . As a result of this, we may apply the arguments used in the proof of Lemma 7 to show that is well-defined and constant for all sufficiently small provided that , where is given in Definition 6. We thus give the following definition.

Definition 10. Let be a nonempty, bounded, and open subset of , be maximal monotone, be bounded pseudomonotone, 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 denotes the degree mapping constructed in Section 2.1.

The following theorem gives some basic properties and homotopy invariance results analogous to those of Theorem 9.

Theorem 11. Let be a nonempty, bounded, and open subset of . Let be maximal monotone, be bounded pseudomonotone, and be compact with and belonging to the class Then the following properties hold:

(i)(Normalization) if and if (ii)(Existence) if and , then . If is of type , then .(iii)(Decomposition) let and be nonempty and disjoint open subsets of such that Then

(iv)(Translation invariance) let Then we have

(v)Let , , and be bounded pseudomonotone such that for all . Then is independent of .(vi)Let , , and be densely defined maximal monotone and positively homogeneous of order , be bounded pseudomonotone, be bounded and of type , and for all . Assume, further, that for all . Then is independent of .

Proof. The proofs for (i) through (iv) follow as in the analogous items in the proof of Theorem 9. We shall give sketches of the proofs of (v) and (vi). To prove (v), for each , we consider the homotopy inclusion

Following the arguments used in the proof of (v) of Theorem 9, it can be shown that there exists such that for all and . Otherwise, we would get for some , which is impossible. On the other hand, for each , we see that

Since and are bounded operators of type , the proof of (v) of Theorem 9 implies that is independent of ; that is,

for all and . As a result of this, we get

for all . This proves that is independent of provided that for all . The proof of (vi) can be completed in analogous manner. The details are omitted here.

3. An Existence Theorem

As a consequence of the degree theory developed in Section 2, the following theorem gives a new existence result on the solvability of operator inclusions of the type in provided that is of type or is bounded of type .

Theorem 12. Let be maximal monotone with , be bounded pseudomonotone, and be compact with and belonging to the class Let . Assume, further, that there exists such that

for all , , and . Then Furthermore, provided that is coercive.

Proof. Let . We shall show that for all , where

Since , by using the boundary condition on , we see that

for all , , , and ; that is, for each , it follows that for all . Since and are continuous, bounded, and of type , (v) of Theorem 9 implies that is an admissible homotopy. Therefore, for each , we obtain

that is, . By (ii) of Theorem 9, we conclude that ; that is, for each , there exist , , and such that

Since is bounded, we have as , which implies that . If is coercive, then for each there exists such that the boundary condition holds. This implies that . Since is arbitrary, we conclude that . The proof is complete.

The arguments used in the proof of Theorem 12 gives the following existence result on the surjectivity of provided that either is bounded and of type or is of type .

Corollary 13. Let be maximal monotone with , , and be compact with and belonging to the class Let . Assume, further, that is coercive. Then is surjective provided that is bounded of type or is of type .

Proof. Let . Suppose is coercive; that is, there exists and as such that

Then, there exists such that

for all , , and Assume that is of type . By Theorem 12, we conclude that ; that is, there exists , , and such that as . Since is compact, we assume without loss of generality that as ; that is, as Since is of type , it follows that as . By the maximality of along with Lemma 5, the continuity of and generalized pseudomonotonicity of , and the arguments used in the proof of Lemma 7, we conclude that and ; that is, . Since is arbitrary, we conclude that is surjective. The case when is bounded and of type can be reached by following analogous arguments. The details are omitted here.

Theorem 12 is a new result and Corollary 13 gives a surjectivity result for operators of the type . For further existence results involving operators of the type , the reader is referred to Kenmochi [17], Le [18], and Asfaw [19]. For various examples on pseudomonotone and quasimonotone operators, we cite the paper due to Mustonen [20].

4. An Example

Let and . It is well-known that and are real Hilbert spaces with duality pairing between and denoted by which is given by

where denotes the duality pairing between and ,  ; that is, the norm of is given by , where denotes the norm of in . We shall apply the existence theorem(s) derived with the aid of the degree theory developed in this paper to establish existence of weak solution(s) in for nonlinear problem given by

where and and the functions () and satisfy the following measurability and sublinearity conditions:

() ( is Carathèodory function; that is, is measurable for almost all and is continuous for almost all Assume, further, that there exist and such that

 for all and .(C ₂)There exists such that for all , .(C ₃) is Carathèodory function and there exist and such that

 for all , and .

A weak solution is understood as follows.

Definition 14. A function is a weak solution of (82) if and such that the following are satisfied:

where is understood in the sense of distributions; that is,

Next we give the following theorem.

Theorem 15. Let . Assume that conditions through are satisfied. Then (82) admits at least one-weak solution.

Proof. Let be given by

By using and , it is well-known that is bounded continuous of type . For the proof of these facts and other relevant properties of pseudomonotone and type differential operators, the reader is referred to the papers by Browder [21], Berkovits and Mustonen [22], Hu and Papageorgiou [7], Landes and Mustonen [23], and the references therein. Let be defined by

where and such that , where , that is, It is well-known that is a densely defined maximal monotone operator. The proof of this result is due to Brèzis which can be found in the book by Zeidler [13, Theorem . L, pp. 897–899]. Since is compactly embedded in , it is known that is a completely continuous operator; that is, is a compact operator. Further reference on operators of the type and existence results for parabolic problems, the reader is referred to the recent book due to Carl et al. [24]. Next we shall use Theorem 12 using the compact operator , the maximal monotone operator , and the operator . It remains to show that lies in and for each , there exists such that for all . To this end, by applying condition (), Hòlder’s inequality and observing that for all and , we see that

that is, we get that

for all and . Consequently, taking supremum overall with , we conclude that for all , where and ; that is, belongs to . Next we show the boundary condition in Theorem 12. To this end, by using conditions through and monotonicity of ( for all ), we get

for all . Since the right side of the above inequality approaches as , for each there exists such that

for all . By applying Theorem 12, we conclude that the equation is solvable in ; that is, (82) admits at least one-weak solution.

In conclusion, we like to notice that the function depends on both and , sublinear, and possibly nonmonotone with respect to . Consequently, Theorem 15 improves those analogous results under monotonicity condition on with respect to . Existence results in elliptic as well as parabolic problems under monotone nonlinearities independent of ; the reader is referred to [7, 9, 17, 20, 21, 25–28] and the references therein.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of the paper.

Acknowledgments

The author is thankful to Virginia Tech for funding the article processing charge.

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