We tackle the existence and uniqueness of the solution for a kind of integro-differential equations involving the generalized p-Laplacian operator with mixed boundary conditions. This is achieved by using some results on the ranges for maximal monotone operators and pseudo-monotone operators. The method used in this paper extends and complements some previous work.
MSC: 47H05, 47H09.
Keywords:maximal monotone operator; pseudo-monotone operator; generalized p-Laplacian operator; integro-differential equation; mixed boundary conditions
Nonlinear boundary value problems (BVPs) involving the p-Laplacian operator arise from a variety of physical phenomena such as non-Newtonian fluids, reaction-diffusion 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:
Motivated by Tolksdorf’s work  where the following Dirichlet BVP has been discussed:
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 :
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 integro-differential equation with the generalized p-Laplacian 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 integro-differential equations. We shall consider the following nonlinear integro-differential problem with mixed boundary conditions:
Our discussion is based on some results on the ranges for maximal monotone operators and pseudo-monotone operators in [16-18]. 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).
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 ‘w-lim’ 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 if Φ is defined from X to , Φ is not identically +∞ such that , whenever and .
The function is said to be lower-semicontinuous on X 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.
A mapping is said to be demi-continuous on X if for any sequence strongly convergent to x in X. A mapping is said to be hemi-continuous on X if for any .
With each multi-valued mapping , we associate the subset as follows :
where . If is strictly convex, then and is single-valued, which in this case is called the minimal section of A.
A multi-valued mapping is said to be monotone 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 .
Let C be a closed convex subset of X, and let be a multi-valued mapping. Then A is said to be a pseudo-monotone 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 finite-dimensional subspace F of X, the operator A is continuous from to in the weak topology.
Let Ω be a bounded conical domain in . If , then ; if and , then ; if and , then for , .
If is an everywhere defined, monotone, and hemi-continuous operator, thenBis maximal monotone. If is a maximal monotone operator such that , thenBis pseudo-monotone.
IfXis a Banach space and is a proper convex and lower-semicontinuous function, then∂Φ is maximal monotone fromXto .
If and are two maximal monotone operators inXsuch that , then is maximal monotone.
LetXand its dual be strictly convex Banach spaces. Suppose is a closed linear operator and is the conjugate operator ofS. If and , thenSis a maximal monotone operator possessing a dense domain.
Any hemi-continuous mapping is demi-continuous on .
LetXbe a real reflexive Banach space with being its dual space. LetCbe a nonempty closed convex subset ofX. Assume that
(i) the mapping is a maximal monotone operator;
(ii) the mapping is pseudo-monotone, bounded, and demi-continuous;
(iii) if the subsetCis unbounded, then the operatorBisA-coercive with respect to the fixed element , i.e., there exists an element and a number such that for all with .
Then the equation has a solution.
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 integro-differential equation (1.11), Ω is a bounded conical domain of a Euclidean space where , Γ is the boundary of Ω with , ϑ denotes the exterior normal derivative to Γ. Here, and denote the Euclidean norm and the inner-product 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.
Assumption 1Green’s formula is available.
Assumption 2For each , is a proper, convex, and lower-semicontinuous function and .
Assumption 3 and for each , the function is measurable for .
We shall present a series of lemmas before we prove the main result.
Lemma 3.1Define the function by
Then Φ is a proper, convex, and lower-semicontinuous 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 . 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 lower-semicontinuous 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 lower-semicontinuous for each . It then follows from Fatou’s lemma that for each ,
So, whenever in V. This completes the proof. □
Lemma 3.2Define by
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
Then , where .
For , we find
which implies that
Similarly, for ,
which implies that
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 3.3 is maximal monotone.
Define the mapping as follows:
Then is maximal monotone.
Let denote the closed subspace of all constant functions in . LetXbe the quotient space . For , define the mapping by
Then, there is a constant such that for every ,
Here denotes the measure of Ω.
Definition 3.1 Define as follows:
Lemma 3.6The mapping is everywhere defined, bounded, monotone, and hemi-continuous. Therefore, Lemma 2.2 implies that it is also pseudo-monotone.
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
Moreover, since , then , which implies that and for .
If , then for , we have
which implies that A is everywhere defined and bounded.
If , then for , we have
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 hemi-continuous, it suffices to show that for any and , , as . Noting the fact that is hemi-continuous and using the Lebesgue’s dominated convergence theorem, we have
Hence, A is hemi-continuous.
This completes the proof. □
Lemma 3.7The mapping satisfies 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
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
If , then
From (3.2) and (3.3), we know that
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 .
If , then
where M is a positive constant. We can easily see that
as . Moreover, if , then
as ; while if ,
Hence, the right side of (3.7) tends to +∞ as , which implies that A satisfies (3.1).
This completes the proof. □
Lemma 3.8If , then a.e. on .
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 integro-differential 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
Then we have for all ,
The definition of subdifferential implies that
From the definition of S, we have
Let , where . Then we have
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. □
The authors declare that they have no competing interests.
All authors approve the final manuscript.
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).
Gupta, CP, Hess, P: Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems. J. Differ. Equ.. 22, 305–313 (1976). Publisher Full Text
Wei, L, He, Z: The applications of sums of ranges of accretive operators to nonlinear equations involving the p-Laplacian operator. Nonlinear Anal.. 24, 185–193 (1995). Publisher Full Text
Wei, L, He, Z: The applications of theories of accretive operators to nonlinear elliptic boundary value problems in -spaces. Nonlinear Anal.. 46, 199–211 (2001). Publisher Full Text
Wei, L, Zhou, H: Research on the existence of solution of equation involving the p-Laplacian operator. Appl. Math. J. Chin. Univ. Ser. B. 21(2), 191–202 (2006). Publisher Full Text
Tolksdorf, P: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equ.. 8(7), 773–817 (1983). Publisher Full Text
Wei, L, Agarwal, RP: Existence of solutions to nonlinear Neumann boundary value problems with generalized p-Laplacian operator. Comput. Math. Appl.. 56(2), 530–541 (2008). Publisher Full Text