- Open Access
Study on integro-differential equation with generalized p-Laplacian operator
© Wei et al.; licensee Springer 2012
- Received: 13 June 2012
- Accepted: 24 October 2012
- Published: 13 November 2012
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.
- maximal monotone operator
- pseudo-monotone operator
- generalized p-Laplacian operator
- integro-differential equation
- mixed boundary conditions
where , ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.
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 .
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 .
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 .
Definition 2.1 
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
for each finite-dimensional subspace F of X, the operator A is continuous from to in the weak topology.
Lemma 2.1 
Let Ω be a bounded conical domain in . If , then ; if and , then ; if and , then for , .
Lemma 2.2 
If is an everywhere defined, monotone, and hemi-continuous operator, then B is maximal monotone. If is a maximal monotone operator such that , then B is pseudo-monotone.
Lemma 2.3 
If X is a Banach space and is a proper convex and lower-semicontinuous function, then ∂ Φ is maximal monotone from X to .
Lemma 2.4 
If and are two maximal monotone operators in X such that , then is maximal monotone.
Lemma 2.5 
Let X and its dual be strictly convex Banach spaces. Suppose is a closed linear operator and is the conjugate operator of S. If and , then S is a maximal monotone operator possessing a dense domain.
Lemma 2.6 
Any hemi-continuous mapping is demi-continuous on .
Theorem 2.1 
the mapping is a maximal monotone operator;
the mapping is pseudo-monotone, bounded, and demi-continuous;
if the subset C is unbounded, then the operator B is A-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.
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 1 Green’s formula is available.
Assumption 2 For 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.
Then Φ is a proper, convex, and lower-semicontinuous mapping on V. 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.
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.
So, whenever in V. This completes the proof. □
Then S is a linear maximal monotone operator possessing a dense domain in V.
Proof It is obvious that S is closed and linear.
Then , where .
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.
Lemma 3.4 
Then is maximal monotone.
Lemma 3.5 
Here denotes the measure of Ω.
Lemma 3.6 The mapping is everywhere defined, bounded, monotone, and hemi-continuous. Therefore, Lemma 2.2 implies that it is also pseudo-monotone.
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.
which implies that A is monotone.
Hence, A is hemi-continuous.
This completes the proof. □
as in V.
In view of (3.2) and (3.3), we have shown that for , is equivalent to .
Therefore, it follows from (3.4), (3.5), and (3.6) that A satisfies (3.1) when .
Hence, the right side of (3.7) tends to +∞ as , which implies that A satisfies (3.1).
This completes the proof. □
Lemma 3.8 If , then a.e. on .
which implies that the result is true. □
We are now ready to prove the main result.
Theorem 3.1 The integro-differential equation (1.11) has a unique solution in V for .
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.
since S is monotone. But is monotone too, so , which implies that .
The proof is complete. □
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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