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.
Keywordsmaximal 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.
3 Existence and uniqueness of the solution to (1.11)
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).
- Calvert BD, Gupta CP: Nonlinear elliptic boundary value problems in -spaces and sums of ranges of accretive operators. Nonlinear Anal. 1978, 2: 1-26.MathSciNetView Article
- Gupta CP, Hess P: Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems. J. Differ. Equ. 1976, 22: 305-313. 10.1016/0022-0396(76)90030-9MathSciNetView Article
- Wei L, He Z: The applications of sums of ranges of accretive operators to nonlinear equations involving the p -Laplacian operator. Nonlinear Anal. 1995, 24: 185-193. 10.1016/0362-546X(94)E0051-HMathSciNetView Article
- Wei L: The existence of solution of nonlinear elliptic boundary value problem. Math. Pract. Theory 2001, 31: 360-364. in Chinese
- Wei L, He Z: The applications of theories of accretive operators to nonlinear elliptic boundary value problems in -spaces. Nonlinear Anal. 2001, 46: 199-211. 10.1016/S0362-546X(99)00457-5MathSciNetView Article
- Wei L: The existence of a solution of nonlinear elliptic boundary value problems involving the p -Laplacian operator. Acta Anal. Funct. Appl. 2002, 4: 46-54. in ChineseMathSciNet
- Wei L: Study of the existence of the solution of nonlinear elliptic boundary value problems. Math. Pract. Theory 2004, 34: 123-130. in Chinese
- Wei L, Zhou H: The existence of solutions of nonlinear boundary value problem involving the p -Laplacian operator in -spaces. J. Syst. Sci. Complex. 2005, 18: 511-521.MathSciNet
- Wei L, Zhou H: Research on the existence of solution of equation involving the p -Laplacian operator. Appl. Math. J. Chin. Univ. Ser. B 2006, 21(2):191-202. 10.1007/BF02791356MathSciNetView Article
- Tolksdorf P: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equ. 1983, 8(7):773-817. 10.1080/03605308308820285MathSciNetView Article
- Wei L, Hou W: Study of the existence of the solution of nonlinear elliptic boundary value problems. J. Hebei Norm. Univ. 2004, 28(6):541-544. in ChineseMathSciNet
- Wei L, Zhou H: Study of the existence of the solution of nonlinear elliptic boundary value problems. J. Math. Res. Expo. 2006, 26(2):334-340. in ChineseMathSciNet
- Wei L: The existence of solutions of nonlinear boundary value problems involving the generalized p -Laplacian operator in a family of spaces. Acta Anal. Funct. Appl. 2005, 7(4):354-359. in ChineseMathSciNet
- Wei L, Agarwal RP: Existence of solutions to nonlinear Neumann boundary value problems with generalized p-Laplacian operator. Comput. Math. Appl. 2008, 56(2):530-541. 10.1016/j.camwa.2008.01.013MathSciNetView Article
- Wei L, Agarwal RP, Wong PJY: Existence of solutions to nonlinear parabolic boundary value problems with generalized p -Laplacian operator. Adv. Math. Sci. Appl. 2010, 20(2):423-445.MathSciNet
- Zeilder E: Nonlinear Functional Analysis and Its Applications. Springer, New York; 1990.
- Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden; 1976.View Article
- Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Sijthoff and Noordhoff, The Netherlands; 1978.View Article
- Adams RA: The Sobolev Space. People’s Education Press, China; 1981. Version of Chinese Translation
- Lions JL: Quelques Methods de Resolution des Problems aux Limites Nonlineaires. Dunod Gauthier-Villars, Paris; 1969.
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