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

Asfaw, Teffera M.

doi:https://doi.org/10.1155/2016/3970621

Journal of Function Spaces

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Research Article | Open Access

Volume 2016 | Article ID 3970621 | https://doi.org/10.1155/2016/3970621

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

Teffera M. Asfaw¹

Academic Editor: Adrian Petrusel

Received30 Jun 2016

Accepted18 Oct 2016

Published05 Dec 2016

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 By using condition on and boundedness of , we can follow the arguments used in the proof of Theorem 5 to conclude that . Furthermore, is surjective provided that is coercive. The proof is completed.

Next we give the following important theorem on maximality of and without requiring (i) or to be quasibounded and and (ii) . The maximality condition (i) and (ii) are attributed to Browder and Hess [16] and Rockafellar [17], respectively.

Theorem 9. Let be densely defined maximal monotone of type and and be linear monotone, surjective, and closed such that is compact. Then is maximal monotone.

Proof. Let . By the boundedness and continuity of and monotonicity of and with , it follows that there exists such that for all and ; that is, is coercive. By Theorem 8, we conclude that ; that is, is maximal monotone. The maximality of follows by setting .

4. Degree Theory in a Real Hilbert Space with

The content of this section outlines the construction of the degree mapping for operators of the type in the setting of a real Hilbert space, where and are as in Section 3 and is linear densely defined, self-adjoint, closed, and range closed. The closedness of is achieved if we assume , where denotes the null space of . Under this condition, one can easily see that the restriction of to is one to one and onto . Let be the orthogonal projection onto . In addition, it is well-known that . For each , it follows that is linear and surjective. For further properties of operators of type , the reader is referred to the paper by Brézis and Nirenberg [18]. In the following lemma, we shall show that is compact for suitable .

Lemma 10. Let be linear, densely defined, and self-adjoint and be compact. Then there exists such that, for each , the operator is surjective and is compact.

Proof. By the property of the orthogonal projection onto , it is well-known that is nonexpansive; that is, for all and . Since is compact (i.e., it is continuous and linear), there exists such that For each , we see thatfor all and , which implies that is expansive; that is, is continuous. Next we show that is compact. Let be bounded in and for all ; that is, for all ; that is, for all . The boundedness of follows because of the expansiveness of . Since is compact, we assume by passing to a subsequence that as ; that is, the compactness of is proved.

As a consequence of Lemma 10, it follows that is a compact operator provided that is a bounded operator. Theorem 11 gives analogous result like that of Theorem 5.

Theorem 11. Let be a nonempty, bounded, and open subset of . Let , , where is bounded demicontinuous operator. Let be linear, densely defined, self-adjoint, closed, and range closed such that is compact. Suppose is a pseudomonotone homotopy of maximal monotone operators uniformly of type . Assume, further, that for all . Then degree is well-defined and independent of sufficiently small and .

Proof. Since is compact and and are bounded demicontinuous operators, it follows that is a compact operator. Suppose there exist and such that for all ; that is, for all . Since is surjective, it follows that , , and for all ; that is, we get Since , , and are bounded and is uniformly of type , we can follow the arguments in the proof of Theorem 5 to conclude that and are bounded. As a result, the compactness of implies the existence of a subsequence, denoted again by , such that Since and are bounded and the family is uniformly of type , the proof can be completed by following exactly similar arguments as in the proof of Theorem 5. The details are omitted here.

Based on Theorem 11, the definition of the degree mapping is given below.

Definition 12. Let be a nonempty, bounded, and open subset of . Let be maximal monotone of type , be bounded demicontinuous operator, and be linear, densely defined, self-adjoint, closed, and range closed such that is compact. Assume, further, that . Then the degree of at with respect to is defined by where , is the orthogonal projection onto , denotes the Leray-Schauder degree mapping for compact perturbations of the identity, and is the Yosida approximant of .

The basic properties and homotopy invariance results like that of Theorem 5 and existence theorems analogous to Theorems 8 and 9 can be proved in Hilbert space setting by using the surjectivity of instead of the surjectivity of . The degree satisfies the following properties.

Theorem 13. Let be a nonempty, bounded, and open subset of . Let be maximal monotone of type , be linear, densely defined, self-adjoint, closed, and range 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. The proofs can be easily completed as in the arguments used in the proofs of Theorems 5 and 7.

This part of the theory improves the degree theory developed by Berkovits and Mustonen [19] for operators of the type , where is bounded demicontinuous pseudomonotone. In the present paper, we only assumed that is bounded demicontinuous operator. Berkovits and Mustonen [19, Theorem , p. 959], gave an existence result for solvability of operator equations of the type , where is bounded demicontinuous operator with bounded range, such that (where denotes the set of all eigenvalues of ), is pseudomonotone, and the recession function (cf. Brézis and Nirenberg [18]) corresponding to given by satisfies for all with However, for , these conditions on and exclude the possibility that . If is bounded and is any constant, we can easily see that is sublinear for all satisfying (i.e., for all , where and ). As a result of Theorem 14 below, the surjectivity of follows under mild assumption on the constant omitting both conditions such that is pseudomonotone and .

Theorem 14. Let be linear, densely defined, self-adjoint, closed, and range closed such that is compact and be bounded demicontinuous operator. Assume, further, that there exist nonnegative constants and such that for all with sufficiently large , where is the largest positive constant satisfying for all . Then is surjective. If a reflexive Banach space is used instead of a real Hilbert space , then the same conclusion holds provided that is surjective.

Proof. Let and (i.e., ). Consider the homotopy equation given by Consequently, we have for all and . Since (i.e., ) and the right hand side of this inequality is independent of , letting implies that there exists such that for all and . Consequently, by using (i) and (v) of Theorem 13, we conclude that that is, is solvable in . Since is arbitrary, we conclude that is surjective. The proof is completed.

In the case when is bounded, we can apply Theorem 14 to conclude that is surjective because of the sublinearity of with and is demicontinuous operator. As a result, it follows that can be zero and is surjective if without being pseudomonotone. In [19], Berkovits and Mustonen gave an existence theorem for the surjectivity of operators of the type , where satisfies conditions of Theorem 14 and is bounded demicontinuous pseudomonotone satisfying the following: (i) for all , where is the largest positive constant, (ii) there exist and such that for all , and (iii) there exist and such that for all . By combining (ii) and (iii), we can easily see that for all satisfying , which is a sublinearity condition used in Theorem 14. This shows that conditions (ii) and (iii) used by Berkovits and Mustonen [19, Theorem 8, p. 957] give stronger conditions on as compared with the sublinearity of . However, Theorem 14 does not need condition (ii) or (iii). The last but the main improvement of Theorem 14 over that of Berkovits and Mustonen [19] is dropping the requirement of to be pseudomonotone. It is worth mentioning here that the same conclusion holds in Theorem 14 if the sublinearity condition on holds for all with sufficiently large . As a result, we get the following corollary.

Corollary 15. Let be linear, densely defined, self-adjoint, closed, and range closed such that is compact and be bounded demicontinuous operator. Assume, further, that there exist nonnegative constants and such that and where and is the largest positive constant such that for all . If as , then is surjective. If a reflexive Banach space is used instead of a real Hilbert space , then the same conclusion holds provided that is surjective.

Proof. By the side condition on , we see thatConsequently, we get Since as , there exists such that for all ; that is, we have where Since by the hypotheses, we can apply Theorem 14 with and to conclude that is surjective. The proof is completed.

We can observe that the largest positive constant used in the hypotheses of Theorem 14 satisfies the condition In [18], Brézis and Nirenberg [18, Theorem III.2, p. 270] proved that provided that the following conditions hold: (i) and there are positive constants , , and such that where , , and .(ii) for all with , where , , and is the recession function of . In view of these, Corollary 15 does not require or (ii). In conclusion, Theorem 14 and its corollary gave new surjectivity results with weaker assumptions on and . In addition, we note here that Berkovits and Mustonen [19] proved surjectivity of under weak coercivity condition of the type for all , for some and , and condition of type (i) with the possibility of having infinite dimensional null space of .

5. Applications

In this section, we shall apply the abstract existence results to prove existence of weak solutions for nonlinear parabolic and hyperbolic problems such as wave and minimal surface equations. In these examples, the main contribution is that the Leray-Lion condition which guarantees pseudomonotonicity of the nonlinear term(s) is dropped. This will help to treat larger class of nonlinear equations and inequalities in appropriate Sobolev spaces. In the following examples, the norm of , where , where , is denoted by .

Example 16 (nonlinear parabolic equation). Let , where . We prove the existence of weak solution for the nonlinear parabolic problem given bywhere , is a nonempty, bounded, and open subset of with smooth boundary, , , , , and () for all . Suppose the following hypotheses are satisfied: (i), i = 1,2,…, N are Carathéodory functions; that is, , , and are measurable functions for almost all and , , and are continuous for almost all .(ii)There exist and such that for all , where and (iii)There exist and such that for all and (iv)There exist and such that for all (v)There exists such that for all , where . Let be given by , , where and that is, we have, , where is understood in the sense that for all and is the inner product in . The norm of in is given by where . It is well-known that is surjective maximal monotone and exists and is continuous. As a consequence of the compact embedding of into , it is not difficult to show that is compact. In addition, the maximal monotonicity of implies the closedness of graph of ; that is, is closed. Let be given by, and be given by, . By conditions (i) through (iv), it well-known that and are bounded continuous operators; that is, given by is bounded continuous operator. A weak solution in of (66) is understood in the sense of the following definition.

Definition 17. Let . An element is called a “weak solution” of (66) if that is, satisfies the functional equation for all . The following existence result holds.

Theorem 18. Suppose conditions (i) through (v) are satisfied. Then, for each , (66) admits at least one weak solution in .

Proof. Let , , and be as defined in (73), (76), and (77), respectively. It is well-known that is surjective maximal monotone (i.e., graph closed) and is compact and is bounded continuous operator. By applying Hólder’s inequalities along with conditions (i) through (v), monotonicity of , and coercivity of we can see that for all . Since the right side of the above inequality tends to as , there exists such that for all ; that is, the boundary condition in Theorem 8 is satisfied with linear operator , maximal monotone operator , and bounded continuous operator . Therefore, we conclude that the problem is solvable in , where is a functional on generated by . Since is arbitrary, there exists such that that is, (66) admits at least one weak solution in . The proof is completed.

One of the main advantages of this theory concerning parabolic problems of type (66) is dropping the requirement for to be pseudomonotone type, that is, dropping the conditionfor all and in .

Example 19 (minimal surface equation). Let , , , and be as in Example 16. Let be given by for all . It follows that () satisfies (i) and , that is, condition (iv) of Example 16 with for all and . We notice here that Let , , , and be defined by and . By using the operators and and following analogous arguments used in the proof of Theorem 18, for each , we can show that is solvable in ; that is, satisfies for all provided that . Equivalently, we conclude that the minimal surface equation, given by admits at least one weak solution in provided that , and satisfies conditions (i) and (iv) of Example 16, and for all .

Example 20 (nonlinear wave equation). Let , . We shall show existence of weak solutions in of the wave equation given by where and (a) is Carathéodory function;(b)there exist and such that A weak solution of (88) is understood in the sense of the following definition.

Definition 21. Let . An element is called a “weak solution of (88)” if satisfieswhere that is, for all

Next we prove the following existence theorem.

Theorem 22. Suppose satisfies conditions (a) and (b), and let be given by , . Then, for each , (88) admits at least one weak solution in .

Proof. By the sublinearity of , we can easily see that is bounded continuous operator and for all . The abstract representation of the wave operator in is the linear operator given by where , , and If , then, by using successive integration by parts, we see that is a solution of if and only if satisfies (90). It is well-known that is linear densely defined, self-adjoint, and closed and (i.e., is closed), the restriction of to , denoted again by , is one to one onto , and is compact. Since is continuous, there exists such that for all . Let be the largest positive constant to satisfy this condition. If , then, by applying Theorem 14 with and , we conclude that there exists such that that is, (88) admits a weak solution in . The proof is completed.

It is important to notice that the monotonicity assumption on is not required to establish existence of weak solutions for (88). For existence of weak solutions under the requirement that is monotone, the reader is referred to the papers by Rabinowitz [20], Brézis and Nirenberg [18, 21], Brezis [22], and Barbu and Pavel [23] and the references therein. In an attempt to remove the monotonicity assumption on , Coron [24] used additional assumption on ; that is, he assumed the existence of a closed subspace of such that and is invariant under and . For further results on nonmonotone , the reader is referred to the papers by Coron [24] and Hofer [25]. Consequently, Theorem 22 provides a new result concerning existence of weak solution for the nonlinear wave equation with lower order nonlinear part satisfying only continuity and sublinearity conditions. In conclusion, we like to mention that various examples of pseudomonotone type operators under Leray-Lion type growth conditions along with (83) can be found in the papers of Landes and Mustonen [26], Mustonen [27], and Mustonen and Tienari [28] and the references therein. Existence results for perturbations of maximal monotone operator by bounded demicontinuous operator of type or bounded pseudomonotone can be found in the papers of Browder and Hess [16], Kenmochi [29], Kartsatos [30], Asfaw [31–33], and Le [34] and the references therein. For detailed study of ranges of sums of perturbed operators in Hilbert space and more examples and properties of elliptic, parabolic, and hyperbolic linear operators, we mention the papers by Brézis and Nirenberg [18] and Berkovits and Mustonen [19] and the references therein.

Competing Interests

The author declares that there are no competing interests regarding the publication of the paper.

Acknowledgments

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

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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.

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