- Research Article
- Open Access
General Comparison Principle for Variational-Hemivariational Inequalities
© S. Carl and P. Winkert. 2009
- Received: 13 March 2009
- Accepted: 18 June 2009
- Published: 2 August 2009
We study quasilinear elliptic variational-hemivariational inequalities involving general Leray-Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke's generalized gradient is required. Based on these results, in the second part the theory is extended to discontinuous variational-hemivariational inequalities.
- Trace Operator
- Maximal Monotone Operator
- Extremal Solution
- Monotonicity Condition
- Hemivariational Inequality
Furthermore, we denote the trace operator by which is known to be linear, bounded, and even compact.
In the second part we consider (1.1) with a discontinuous Nemytskij operator involved, which extends results in  and partly of . Let us consider next some special cases of problem (1.1), where we suppose .
where denotes the conormal derivative of . The method of sub- and supersolution for this kind of problems is a special case of .
which has been discussed in detail in .
whose method of sub- and supersolution has been developed in [7, Chapter 5].
Existence results for variational-hemivariational inequalities with or without the method of sub- and supersolutions have been obtained under different structure and regularity conditions on the nonlinear functions by various authors. For example, we refer to [9–16]. In case that is the whole space or , respectively, problem (1.1) reduces to a hemivariational inequality which has been treated in [17–25].
for all , for a.a. , and for all with can be found in . Inspired by results recently obtained in [8, 26], we prove the existence of (extremal) solutions for the variational-hemivariational inequality (1.1) within a sector of an ordered pair of sub- and supersolutions without assuming a one-sided growth condition on Clarke's gradient of the form (1.13).
For functions we use the notation , and and introduce the following definitions.
A function is said to be a subsolution of (1.1) if the following holds:
A function is said to be a supersolution of (1.1) if the following holds:
Let , and assume for the coefficients the following conditions.
for a.e. and for all , where denotes the Euclidian norm of the vector .
for a.e. , for all , and for all with .
for a.e. , for all and for all .
where stands for the duality pairing between and , and assumption (A3) is a coercivity type condition.
Let be an ordered pair of sub- and supersolutions of problem (1.1). We impose the following hypotheses on and the nonlinearity in problem (1.1).
(j1) and are measurable in and , respectively, for all .
(j2) and are locally Lipschitz continuous in for a.a. and for a.a. , respectively.
(i) is measurable in for all .
(ii) is continuous in for a.a. .
for a.e. , for all , and for all .
Note that the associated Nemytskij operator defined by is continuous and bounded from to (cf. ). We recall that the normed space is equipped with the natural partial ordering of functions defined by if and only if , where is the set of all nonnegative functions of .
The following lemma provides some properties of the functions and .
Let the assumptions in (j1)–(j3) be satisfied. Then the modified functions and have the following qualities.
() and are measurable in and , respectively, for all , and and are locally Lipschitz continuous in for a.a. and for a.a. , respectively.
and the inclusions and are valid for .
With a view to the assumptions (j1)–(j3) and the definition of in (3.8), one verifies the lemma in few steps.
An important tool in our considerations is the following surjectivity result for multivalued pseudomonotone mappings perturbed by maximal monotone operators in reflexive Banach spaces.
Let be a real reflexive Banach space with the dual space , a maximal monotone operator, and . Let be a pseudomonotone operator, and assume that either is quasibounded or is strongly quasibounded. Assume further that is -coercive, that is, there exists a real-valued function with as such that for all one has . Then is surjective, that is, .
The proof of the theorem can be found, for example, in [28, Theorem ]. The notation and stand for and , respectively. Note that any bounded operator is, in particular, also quasibounded and strongly quasibounded. For more details we refer to . The next proposition provides a sufficient condition to prove the pseudomonotonicity of multivalued operators and plays an important part in our argumentations. The proof is presented, for example, in [28, Chapter 2].
Let be a reflexive Banach space, and assume that satisfies the following conditions:
(i)for each one has that is a nonempty, closed, and convex subset of ;
(ii) is bounded;
(iii)if in and in with and if , then and .
Then the operator is pseudomonotone.
where are defined as mentioned above. The operators have the following properties (see, e.g., [5, Lemmas and ]).
The multivalued operators and are bounded and pseudomonotone.
is bounded and continuous. Since the embedding is compact, the composed operator is completely continuous.
We are going to prove the following properties for the operator .
The operator is bounded, pseudomonotone, and coercive for sufficiently large.
Setting for and provides the estimate in (3.24) satisfying (3.23). This proves the coercivity of and completes the proof of the lemma.
Let hypotheses (A1)–(A3), (j1)–(j3), and (F1) be satisfied, and assume the existence of sub- and supersolutions and , respectively, satisfying and (2.1). Then, there exists a solution of (1.1) in the order interval .
where and , which proves that is also a solution of our original problem (1.1). This completes the proof of the theorem.
We are going to show that possesses the smallest and the greatest element with respect to the given partial ordering.
Let the hypothesis of Theorem 4.1 be satisfied. Then the solution set is compact.
Hence, . This shows the compactness of the solution set .
In order to prove the existence of extremal elements of the solution set , we drop the -dependence of the operator . Then, our assumptions read as follows.
for a.e. and for all , where denotes the Euclidian norm of the vector .
for a.e. , and for all with .
for a.e. , and for all .
Let us recall the definition of a directed set.
Let be a partially ordered set. A subset of is said to be upward directed if for each pair there is a such that and . Similarly, is downward directed if for each pair there is a such that and . If is both upward and downward directed, it is called directed.
Let hypotheses ( )–( ) and (j1)–(j3) be fulfilled, and assume that (F1) and (4.24) are valid. Then the solution set of problem (1.1) is a directed set.
and hence, meaning that for . This proves . The proof for can be shown in a similar way. More precisely, we obtain a solution of (4.48) satisfying which implies and . The same arguments as at the end of the proof of Theorem 4.1 apply, which shows that is in fact a solution of problem (1.1) belonging to the interval . Thus, the solution set is upward directed. Analogously, one proves that is downward directed.
Theorems 4.2 and 4.4 allow us to formulate the next theorem about the existence of extremal solutions.
Let the hypotheses of Theorem 4.4 be satisfied. Then the solution set possesses extremal elements.
Therefore, as , we conclude that is the greatest element in . The existence of the smallest solution of (1.1) in can be proven in a similar way.
If depends on , we have to require additional assumptions. For example, if satisfies in a monotonicity condition, the existence of extremal solutions can be shown, too. In case , a Lipschitz condition with respect to is sufficient for proving extremal solutions. For more details we refer to .
where we will allow to depend discontinuously on its third argument. The aim of this section is to deal with discontinuous Nemytskij operators by combining the results of Section 4 with an abstract fixed point result for not necessarily continuous operators, cf. [30, Theorem ]. This will extend recent results obtained in . Let us recall the Definitions of sub- and supersolutions.
A function is called a subsolution of (5.1) if the following holds:
A function is called a supersolution of (5.1) if the following holds:
The conditions for Clarke's generalized gradient and the functions are the same as in (j1)–(j3). We only change the property (F1) to the following.
(i) is measurable for all , for all , and for all measurable functions .
(ii) is continuous in for all and for a.a. .
(iii) is decreasing for all and for a.a. .
for a.e. , for all , and for all .
By  the mapping is measurable for ; however, the associated Nemytskij operator is not necessarily continuous. An important tool in extending the previous result to discontinuous Nemytskij operators is the next fixed point result. The proof of this lemma can be found in [30, Theorem ].
Let be a subset of an ordered normed space, an increasing mapping, and .
(1)If has a lower bound in and the increasing sequences of converge weakly in , then has the least fixed point , and .
(2)If has an upper bound in and the decreasing sequences of converge weakly in , then has the greatest fixed point , and .
Our main result of this section is the following theorem.
Assume that hypotheses ( )–( ), (j1)–(j3), (F2), and (4.24) are valid, and let and be sub- and supersolutions of (5.1) satisfying and (2.1). Then there exist extremal solutions and of (5.1) with .
for all . Consequently, is a supersolution of (5.1). This shows .
for all . Hence, is a subsolution of (5.8). By Theorem 4.5, we know that there exists the greatest solution of (5.8) in . But is the greatest solution of (5.8) in and therefore, . This shows that is increasing.
which shows that is a supersolution of (5.1), that is, . As is an upper bound of , we can apply Lemma 5.3, which yields the existence of the greatest fixed point of in . This implies that must be the the greatest solution of (5.1) in . By analogous reasoning, one shows the existence of the smallest solution of (5.1). This completes the proof of the theorem.
Thus, hypothesis ( ) is satisfied with and . Hypothesis ( ) is a consequence of the inequalities from the vector-valued function (see [7, page 37]), and ( ) is satisfied with and . The construction is done by using solutions of simple auxiliary elliptic boundary value problems and the eigenfunction of the -Laplacian which belongs to its first eigenvalue.
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