Abstract
We tackle the existence and uniqueness of the solution for a kind of integrodifferential equations involving the generalized pLaplacian operator with mixed boundary conditions. This is achieved by using some results on the ranges for maximal monotone operators and pseudomonotone operators. The method used in this paper extends and complements some previous work.
MSC: 47H05, 47H09.
Keywords:
maximal monotone operator; pseudomonotone operator; generalized pLaplacian operator; integrodifferential equation; mixed boundary conditions1 Introduction
Nonlinear boundary value problems (BVPs) involving the pLaplacian operator arise from a variety of physical phenomena such as nonNewtonian fluids, reactiondiffusion problems, petroleum extraction, flow through porous media, etc. Thus, the study of such problems and their generalizations have attracted numerous attention in recent years. Some of the BVPs studied in the literature include the following:
whose existence results in (for various ranges of p) can be found in [14]; a related BVP
was tackled in [57] and later generalized to one that contains a perturbation term [8,9]
Motivated by Tolksdorf’s work [10] where the following Dirichlet BVP has been discussed:
several generalizations have been investigated. These include [1114]
and
where , ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.
Inspired by all this research, recently we have studied the following nonlinear parabolic equation with mixed boundary conditions [15]:
We tackle the existence of solutions for (1.8) via the study of existence of solutions for two BVPs: (i) the elliptic equation with Dirichlet boundary conditions
and (ii) the elliptic equation with Neumann boundary conditions
By setting up the relations between the auxiliary equations (1.9) and (1.10) and by employing some results on ranges for maximal monotone operators, we showed that (1.8) has a unique solution in , where , if , and if .
In this paper, we shall employ the technique used in (1.8), viz. using the results on ranges for nonlinear operators, to study the existence and uniqueness of the solution to a nonlinear integrodifferential equation with the generalized pLaplacian operator. We note that most of the existing methods in the literature used to investigate such problems are based on the finite element method, hence our technique is new in tackling integrodifferential equations. We shall consider the following nonlinear integrodifferential problem with mixed boundary conditions:
Our discussion is based on some results on the ranges for maximal monotone operators and pseudomonotone operators in [1618]. Some new methods of constructing appropriate mappings to achieve our goal are employed. Moreover, we weaken the restrictions on p and q. The paper is outlined as follows. In Section 2 we shall state the definitions and results needed, and in Section 3 we shall establish the existence and uniqueness of the solution to (1.11).
2 Preliminaries
Let X be a real Banach space with a strictly convex dual space . We use to denote the generalized duality pairing between X and . For a subset C of X, we use IntC to denote the interior of C. We also use ‘→’ and ‘wlim’ to denote strong and weak convergences, respectively.
Let X and Y be Banach spaces. We use to denote that X is embedded continuously in Y.
The function Φ is called a proper convex function on X[17] if Φ is defined from X to , Φ is not identically +∞ such that , whenever and .
The function is said to be lowersemicontinuous on X[17] if for any .
Given a proper convex function Φ on X and a point , we denote by the set of all such that for every . Such elements are called subgradients of Φ at x, and is called the subdifferential of Φ at x[17].
A mapping is said to be demicontinuous on X if for any sequence strongly convergent to x in X. A mapping is said to be hemicontinuous on X if for any [17].
With each multivalued mapping , we associate the subset as follows [17]:
where . If is strictly convex, then and is singlevalued, which in this case is called the minimal section of A.
A multivalued mapping is said to be monotone[18] if its graph is a monotone subset of in the sense that for any , . The monotone operator B is said to be maximal monotone if is not properly contained in any other monotone subsets of .
Definition 2.1[18]
Let C be a closed convex subset of X, and let be a multivalued mapping. Then A is said to be a pseudomonotone operator provided that
(i) for each , the image Ax is a nonempty closed and convex subset of ;
(ii) if is a sequence in C converging weakly to and if is such that , then to each element , there corresponds an with the property that
(iii) for each finitedimensional subspace F of X, the operator A is continuous from to in the weak topology.
Lemma 2.1[19]
Let Ω be a bounded conical domain in. If, then; ifand, then; ifand, then for, .
Lemma 2.2[18]
Ifis an everywhere defined, monotone, and hemicontinuous operator, thenBis maximal monotone. Ifis a maximal monotone operator such that, thenBis pseudomonotone.
Lemma 2.3[18]
IfXis a Banach space andis a proper convex and lowersemicontinuous function, then∂Φ is maximal monotone fromXto.
Lemma 2.4[18]
Ifandare two maximal monotone operators inXsuch that, thenis maximal monotone.
Lemma 2.5[20]
LetXand its dualbe strictly convex Banach spaces. Supposeis a closed linear operator andis the conjugate operator ofS. Ifand, thenSis a maximal monotone operator possessing a dense domain.
Lemma 2.6[18]
Any hemicontinuous mappingis demicontinuous on.
Theorem 2.1[16]
LetXbe a real reflexive Banach space withbeing its dual space. LetCbe a nonempty closed convex subset ofX. Assume that
(i) the mappingis a maximal monotone operator;
(ii) the mappingis pseudomonotone, bounded, and demicontinuous;
(iii) if the subsetCis unbounded, then the operatorBisAcoercive with respect to the fixed element, i.e., there exists an elementand a numbersuch thatfor allwith.
3 Existence and uniqueness of the solution to (1.11)
We begin by stating some notations and assumptions used in this paper. Throughout, we shall assume that
Let and be the dual space of V. The duality pairing between V and will be denoted by . The norm in V will be denoted by , which is defined by
Let and be the dual space of W. The norm in W will be denoted by , which is defined by
In the integrodifferential equation (1.11), Ω is a bounded conical domain of a Euclidean space where , Γ is the boundary of Ω with [5], ϑ denotes the exterior normal derivative to Γ. Here, and denote the Euclidean norm and the innerproduct in , respectively. Also, , is a given function, T and a are positive constants, and ε is a nonnegative constant. Moreover, is the subdifferential of , where for , and is a given function.
To tackle (1.11), we need the following assumptions which can be found in [5,14].
Assumption 1Green’s formula is available.
Assumption 2For each, is a proper, convex, and lowersemicontinuous function and.
Assumption 3and for each, the functionis measurable for.
We shall present a series of lemmas before we prove the main result.
Lemma 3.1Define the functionby
Then Φ is a proper, convex, and lowersemicontinuous mapping onV. Therefore, , the subdifferential of Φ, is maximal monotone.
Proof The proof of this lemma is analogous to that of Lemma 3.1 in [1]. We give the outline of the proof as follows.
Note that for each , the function is measurable, where denotes the minimal section of . Since for all we have
it implies that for , the function is measurable on Γ. Then from the property of , we know that Φ is proper and convex on V.
To see that Φ is lowersemicontinuous on V, let in V. We may assume that there exists a subsequence of , for simplicity, we still denote it by , such that for and a.e. This yields
for all and each a.e. since is lowersemicontinuous for each . It then follows from Fatou’s lemma that for each ,
So, whenever in V. This completes the proof. □
ThenSis a linear maximal monotone operator possessing a dense domain inV.
Proof It is obvious that S is closed and linear.
For , integrating by parts gives
which implies that
which implies that
Thus,
In the same manner, we have for . Therefore, noting Lemma 2.5 the result follows. □
In view of Lemmas 2.3 and 2.4, we have the following result.
Lemma 3.4[14]
Lemma 3.5[14]
Letdenote the closed subspace of all constant functions in. LetXbe the quotient space. For, define the mappingby
Then, there is a constantsuch that for every,
Definition 3.1 Define as follows:
Lemma 3.6The mappingis everywhere defined, bounded, monotone, and hemicontinuous. Therefore, Lemma 2.2 implies that it is also pseudomonotone.
Proof From Lemma 2.1, we know that when , and when . If , then since . Thus, for all , , where is a constant. Therefore, for , we have
and
Moreover, since , then , which implies that and for .
which implies that A is everywhere defined and bounded.
which also implies that A is everywhere defined and bounded.
Since is monotone, we can easily see that for ,
which implies that A is monotone.
To show that A is hemicontinuous, it suffices to show that for any and , , as . Noting the fact that is hemicontinuous and using the Lebesgue’s dominated convergence theorem, we have
Hence, A is hemicontinuous.
This completes the proof. □
Lemma 3.7The mappingsatisfies that for,
Proof First, we shall show that for ,
is equivalent to
In fact, from Lemma 3.5, we know that for ,
where C is a positive constant. Thus,
which implies that
On the other hand, we have
which implies that
Hence,
In view of (3.2) and (3.3), we have shown that for , is equivalent to .
Next, we shall show that A satisfies (3.1). In fact, we have
From (3.2) and (3.3), we know that
Also,
It follows from (3.5) that
Moreover, we have
Therefore, it follows from (3.4), (3.5), and (3.6) that A satisfies (3.1) when .
where M is a positive constant. We can easily see that
Hence, the right side of (3.7) tends to +∞ as , which implies that A satisfies (3.1).
This completes the proof. □
Proof If , then from the definition of subdifferential, we have
which implies that the result is true. □
We are now ready to prove the main result.
Theorem 3.1The integrodifferential equation (1.11) has a unique solution inVfor.
Proof First, we shall show the existence of a solution. Noting Lemmas 2.6, 3.6, 3.7 and 3.3, and by using Theorem 2.1, we know that there exists such that
The definition of subdifferential implies that
From the definition of S, we have
Moreover,
From the properties of a generalized function, we get
Noting (3.10) again, by using Green’s formula, we have
Then using (3.10), we obtain
In view of Lemma 3.8, we have a.e. on . Combining it with (3.8) and (3.11), we know that (1.11) has a solution in V.
Next, we shall prove the uniqueness of the solution. Let and be two solutions of (1.11). By (3.8), we have
since S is monotone. But is monotone too, so , which implies that .
The proof is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors approve the final manuscript.
Acknowledgements
Li Wei is supported by the National Natural Science Foundation of China (No. 11071053), the Natural Science Foundation of Hebei Province (No. A2010001482) and the Project of Science and Research of Hebei Education Department (the second round in 2010).
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