`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 210325, 14 pageshttp://dx.doi.org/10.1155/2012/210325`
Research Article

Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

1School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, China
2Institute of Applied Mathematics and Mechanics, Ordnance Engineering College, Shijiazhuang 050003, China

Received 4 September 2012; Accepted 15 October 2012

Copyright © 2012 Li Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a kind of nonlinear elliptic boundary value problems with generalized -Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

1. Introduction

The study on nonlinear boundary value problems with -Laplacian operator, , is a hot topic since it has a close relationship with practical problems. Some significant work has been done by us, see [18], and so forth.

Specifically, in 2004, we studied the following nonlinear elliptic boundary value problem involving the generalized -Laplacian operator:

In Wei and Hou [4], we proved that under some conditions (1.1) has solutions in , where . In [5, 6], we extended our work to the following problem: In Wei and Zhou [5], we established that (1.2) has solutions in , where , and in Wei [6] we proved that (1.2) has solutions in , where . As the summary and extension of [5, 6], we studied the following nonlinear boundary value problem:

It was shown by Wei and Agarwal [7] that (1.3) had a solution in , where , if and if , for .

Clearly, if , then (1.1), (1.2), and (1.3) reduce the cases of involving -Laplacian operators.

It is worth to mention that all of the work done in [47] is based on a perturbation result of the ranges of -accretive mappings by Calvert and Gupta [9].

In this paper, we will study the following nonlinear elliptic boundary value problem:

Necessary details of (1.4) will be provided in Section 3.

We may notice that the principal part of the concerned equation is almost the same as those in (1.1), (1.2), and (1.3) while the nonlinear item is replaced by the item , which is rather general. It seems that the difference is minor; however, the previous method cannot be employed. In this paper, we will use some perturbation results of the ranges for maximal monotone operators by Pascali and Sburlan [10] to prove that (1.4) has a unique solution in and later we will prove that the unique solution is the zero point of a suitably defined maximal monotone operator. Finally, we will employ an iterative scheme to approximate strongly to the unique solution. Some new ideas of combining the knowledge of the ranges of nonlinear operators, zero point of nonlinear operators, iterative schemes, and the solution of nonlinear boundary value problems are demonstrated in this paper.

2. Preliminaries

Now, we list some of the knowledge we need in the sequel.

Let be a real Banach space with a strictly convex dual space . We use to denote the generalized duality pairing between and . We use “” to denote strong convergence. Let “” denote the space embedded continuously in space . For any subset of , we denote by its interior.

Function is called a proper convex function on [11] if is defined from to , not identically such that , whenever and .

Function is said to be lower semicontinuous on [11] if , for any .

Given a proper convex function on and a point , we denote by the set of all such that , for every . Such elements are called subgradients of at , and is called the subdifferential of at [11].

A single-valued mapping is said to be hemicontinuous [11] if , for any .

A multivalued mapping is said to be monotone [10] if its graph is a monotone subset of in the sense that for any ,  . The mapping is said to be strictly monotone if the equality in (2.1) implies that . The monotone operator is said to be maximal monotone if is maximal among all monotone subsets of in the sense of inclusion. The mapping is said to be coercive [10] if for all such that . A point is said to be a zero point of if , and we denote by the set of zero points of .

Lemma 2.1 (Adams [12]). Let be a bounded conical domain in . If , then ; if and , then ; if and , then for .

Lemma 2.2 (Pascali and Sburlan [10]). If is an everywhere defined, monotone, and hemicontinuous operator, then is maximal monotone.

Lemma 2.3 (Pascali and Sburlan [10]). If is a proper convex and lower semicontinuous function, then is maximal monotone from to .

Lemma 2.4 (Pascali and Sburlan [10]). If and are two maximal monotone operators in such that , then is maximal monotone.

Lemma 2.5 (Pascali and Sburlan [10]). If is maximal monotone and coercive, then .

Definition 2.6 (Kamimura and Takahashi [13]). Let be a real smooth Banach space. Then the Lyapunov functional is defined as follows: where is the duality mapping defined by , for .

Lemma 2.7 (Kamimura and Takahashi [13]). Let be a real reflexive, strictly convex, and smooth Banach space, let be a nonempty closed and convex subset of , and let . Then there exists a unique element such that

Define a mapping from onto by for all . Then is called the generalized projection mapping from onto . It is easy to see that coincides with the metric projection in a Hilbert space.

3. Main Results

3.1. Notations and Assumptions of (1.4)

In the following of this paper, unless otherwise stated, we will assume that ,   if , and if , for . Let . We use , and to denote the norm of spaces , and , respectively.

In nonlinear boundary value problem (1.4), is a bounded conical domain of an Euclidean space with its boundary (see Wei and He [1]). We will assume that Green’s formula is available. is a given function. ,   is a nonnegative constant and denotes the exterior normal derivative of .

Let be a given function such that, for each , is a proper, convex, and lower semicontinuous function with . Let be the subdifferential of , that is, . Suppose that and for each , the function is measurable for .

Suppose that is a given function satisfying the following conditions, which can be found in Zeidler [14].(a)Carathéodory's conditions: (b)Growth condition: where ,   and is a fixed positive constant.(c)Monotone condition: is monotone with respect to , that is, for all and .

3.2. Existence and Uniqueness of the Solution of (1.4)

Lemma 3.1. Define the mapping by for any . Then, is everywhere defined, strictly monotone, hemicontinuous, and coercive.

Moreover, Lemma 2.2 implies that is maximal monotone.

Proof. From Lemma  3.2 by Wei and Agarwal [7], we know that is everywhere defined, monotone, hemicontinuous, and coercive. Then we only need to show that is strictly monotone.
In fact, for any ,
Define , for all , where is a nonnegative constant.
Case 1. If , then , for all . That is, is strictly increasing. Thus we can see from (3.5) that if , then and , a.e. in , which implies that in . Therefore, is strictly monotone.
Case 2. If , then for all , which implies that is also strictly increasing. In the same way as Case 1, we know that is strictly monotone.
This completes the proof.

Lemma 3.2 (see Wei and Agarwal [7]). The mapping defined by , for any , is proper, convex, and lower semicontinuous on .

Therefore, is maximal monotone in view of Lemma 2.3.

Lemma 3.3. Define the mapping by for all ; then is everywhere defined, monotone, and hemicontinuous on .

Moreover, Lemma 2.2 implies that is maximal monotone.

Proof. We split our proof into four steps.
Step 1. For ,   is measurable on .
From the facts that ,  , we know that is measurable on . Combining with the fact that satisfies Carathéodory’s conditions, we know that is measurable on .
Step 2. is everywhere defined.
From Lemma 2.1, we know that , when , and , when . Thus, for all , where is a constant.
When , we know from Lemma 2.1 that . Since , then and , which implies that (3.8) is still true.
Now, for , we have from (3.8) that which implies that is everywhere defined.
Step 3. is monotone.
Since is monotone with respect to , then is monotone.
Step 4. is hemicontinuous.
In fact, it suffices to show that, for any and ,  , as .
Noticing the facts that is measurable on and satisfies Carathéodory’s conditions, by using Lebesque’s dominated convergence theorem, we have and hence is hemicontinuous.
This completes the proof.

In view of Lemma 2.4, we can easily obtain the following result.

Lemma 3.4. is maximal monotone.

Lemma 3.5. Define the mapping by , for any . Define the mapping by for any . Then , where is the same as that in Lemma 3.2.

Proof. We will prove the result under the additional condition , where and . Refer to the result of Brezis [15] for the general case.
It is obvious that is continuous. For all , since is monotone, then , which implies that is monotone. Thus, is maximal monotone in view of Lemma 2.2.
Define by , for all ; then it is easy to see that is a proper, convex, and lower semicontinuous function on , which implies that is maximal monotone. Since , for all , then . So .
Now clearly, is maximal monotone since both are continuous. Finally, for any , we have Hence we get and so .
This completes the proof.

Lemma 3.6. One has .

Proof. From Lemma 3.5, we know that for all , Since , then for all ,  .
Since is monotone, then . Moreover, in view of (3.8), we have Then by using Lemma 3.1, we have as , which implies that is coercive. Then Lemmas 3.4 and 2.5 ensure that the result is true.
This completes the proof.

Theorem 3.7. For , nonlinear boundary value problem (1.4) has a unique solution .

Proof. From Lemma 3.6, we know that for , there exists such that
Now, we will show that this is unique.
Otherwise, there exists satisfying (3.15). Then notice the facts that and are all monotone, we have Since is strictly monotone, then .
Next, we will show that this is the solution of (1.4).
Since for any and , where and , we have . Since , then for all , which implies that the result is true.
From (3.18), we know that . By using Green's formula, we have that for any ,
Then , where is the space of the traces of .
Combining with the results of Lemma 3.5, (3.18), and (3.19), we have
Then
From (3.19) and (3.21) we know that is the solution of (1.4).
This completes the proof.

3.3. Iterative Construction of the Solution of (1.4)

Lemma 3.8 (Wei and Zhou [16]). Suppose that is a real smooth and uniformly convex Banach space and is a maximal monotone operator with . Let be a sequence generated by the following iterative scheme:
If with ,   with , and as , then converges strongly to , where is the generalized projection operator from onto .

Definition 3.9. Define the mapping by for any , where is the same as that in (1.4).

Similarly to Lemma 3.4, we know that is also a maximal monotone operator. Moreover, we can easily get the following result.

Lemma 3.10. is the solution of (1.4) if and only if is the zero point of .

Proof. Let be the solution of (1.4), then for all , by using Green’s formula and Lemma 3.5, we have
Thus .
If , then for all ,
which implies the result , is true.
Copying the last part of Theorem 3.7, we can obtain (3.21), which implies that is the solution of (1.4).
This completes the proof.

Remark 3.11. From Lemma 3.10 we can see that . It is a good example to show that the assumption that in Lemma 3.8 is valid.

Lemma 3.12 (Takahashi [17]). Let be a Banach space and a duality mapping defined on . If is single valued, then is smooth.

Based on the facts of Lemmas 3.8, 3.10, and 3.12, we can construct an iterative sequence to approximate strongly to the solution of (1.4).

Theorem 3.13. Let be a sequence generated by the following iterative scheme:
If with ,   with , and as , then converges strongly to .

Remark 3.14. Theorem 3.13 not only tells us that the sequence generated by (3.26) converges strongly to the solution of (1.4), but also tells us that the unique solution of (1.4) is the generalized projection of the initial function onto .

Remark 3.15. Compared to the work done in [8], we may find that different techniques are employed to show the existence and uniqueness of the solution of the desired equation because the nonlinear item is involved.

Remark 3.16. We can get the following special cases in our paper.

Corollary 3.17. For , the following equation has a unique solution in :

Corollary 3.18. Let be a sequence generated by where is a special case of defined in Definition 3.9 if .
If with ,   with , and as , then converges strongly to , where denotes the metric projection from onto . And,   is the unique solution of (3.27).

Corollary 3.19. For , the following equation has a unique solution in :

Corollary 3.20. Let be a sequence generated by (3.26), where is defined by for .

Then under the conditions of Theorem 3.13, we know that converges strongly to both the unique solution of (3.29) and the zero point of .

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11071053), the Natural Science Foundation of Hebei Province (Grant no. A2010001482), the Key Project of Science and Research of Hebei Education Department (Grant no. ZH2012080), and the Youth Project of Science and Research of Hebei Education Department (Grant no. Q2012054).

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