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Existence and monotone iteration of symmetric positive solutions for integral boundary-value problems with ϕ-Laplacian operator
Journal of Inequalities and Applications volume 2014, Article number: 463 (2014)
The purpose of this paper is to investigate the existence and iteration of symmetric positive solutions for integral boundary-value problems. An existence result of positive, concave and symmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. An example is worked out to demonstrate the main result.
MSC:34B10, 34B18, 39A10.
The existence and multiplicity of positive solutions for linear and nonlinear multi-point boundary-value problems have been widely studied by many authors using a fixed point theorem in cones. To identify a few, we refer a reader to [1–4] and references therein.
At the same time, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two-, three-, multi-point and nonlocal boundary-value problems as special cases. Hence, increasing attention is paid to boundary-value problems with integral boundary conditions. For an overview of the literature on integral boundary-value problems, see [5–9].
Recently an increasing interest has been observed in investigating the existence of positive solutions of boundary-value problems for differential equations by using the monotone iterative method. In [10–14], the authors used the monotone iterative method to get a positive solution of multi-point boundary-value problems for differential equations. In particular, we would like to mention some excellent results.
In , Pang and Tong concentrated on the following problem:
They constructed a specific form of the symmetric upper and lower solutions, and by applying monotone iterative techniques, they constructed successive iterative schemes for approximating solutions.
In , Sun et al. considered the positive solutions to the p-Laplacian boundary-value problem
They investigated the iteration and existence of positive solutions for the multi-point boundary-value problem with p-Laplacian. By applying monotone iterative techniques, they constructed some successive iterative schemes to approximate the solutions.
In , Ding considered the integral boundary-value problem
By applying classical monotone iterative techniques, they not only obtained the existence of positive solutions, but also gave iterative schemes for approximating the solutions.
So, motivated by all the works above, in this study, we consider the following second-order multi-point integral boundary-value problem (BVP):
where is an increasing homeomorphism and homomorphism with . A projection is called an increasing homeomorphism and homomorphism if the following conditions are satisfied:
If , then for all ;
ϕ is a continuous bijection and its inverse mapping is also continuous;
for all .
We will assume that the following assumptions are satisfied:
(C1) , for all and for all ;
(C2) , for and with for ;
(C3) and g is symmetric about on . In addition .
We note that the m-point boundary condition is related to intervals of the area under the curve of the solution from to for . We investigate here the iteration and existence of symmetric positive solutions for the multi-point integral boundary-value problems with ϕ-Laplacian (1.1). We do not require the existence of lower and upper solutions. By applying monotone iterative techniques, we construct successive iterative schemes for approximating solutions. To the best of our knowledge, no contribution exists concerning the existence of symmetric positive solutions for multi-point boundary-value problems with integral boundary conditions by applying monotone iterative techniques.
In this section, we give some lemmas which help to simplify the presentation of our main result.
Lemma 2.1 If is nonnegative on and on any subinterval of , then the BVP
has a unique solution given by
Proof Suppose that u is a solution of BVP (1.1). By integrating both sides of
According to boundary conditions (2.1), we have
where satisfies (2.2). □
Lemma 2.2 If is nonnegative on and on any subinterval of , then there exists a unique satisfying (2.2). Moreover, there is a unique such that .
Proof For any h in Lemma 2.1, define
From the expression of it is easy to see that is continuous and strictly increasing, and
Therefore there exists a unique satisfying (2.2). Furthermore, let
then is continuous and strictly increasing on , and , . Thus
implies that there exists a unique such that
Remark 2.1 By Lemmas 2.1 and 2.2 the unique solution of BVP (2.1) is given by
Lemma 2.3 Suppose that (C1)-(C3) hold. If is symmetric, nonnegative on and on any subinterval of , then the unique solution of BVP (2.1) is concave and symmetric with on .
Proof Let be a solution of BVP (2.1). Then
Therefore, is strictly decreasing. It follows that is also strictly decreasing. Thus, is strictly concave on . For symmetry of , it is easy to show that , i.e., . So from (2.4) and for , then , by the transformation , we have
Again let and by (C2), for . Then
We note that for . So,
If , then . From (2.5), it follows that , . This together with (2.5) implies that , . Without loss of generality, we assume that . By the concavity of u, we know that , . So we obtain
By , it is obvious that . Hence, , . □
Let be the Banach space with the norm
Define the cone by
We can define an operator by
It is easy to prove that each fixed point of T is a solution for (1.1).
Lemma 2.4 Let (C1)-(C3) hold. Then is completely continuous.
Proof From the definition of T, it is easy to check that Tu is nonnegative on and satisfies boundary conditions (1.1) for all . Furthermore, since , we obtain the concavity of on . From the definition of the symmetry of f, holds for . In summary, , and . Next, by standard methods and the Arzela-Ascoli theorem, one can easily prove that the operator T is completely continuous. □
Lemma 2.5 Assume that (C1)-(C3) hold. Suppose also that there exists such that for , , ,
Then, for with , , , we have
Proof First we prove that, for all ,
From our assumptions, we have
Since is increasing on ℝ, then for all , we obtain
Thus, for ,
Therefore, , hold for . In fact, if , then and hence from the fact that and are symmetric about on , it follows that, for ,
3 Main results
In this section, we establish our existence result of positive, concave, and symmetric solutions and its monotone iterative scheme for BVP (1.1). For convenience, we denote
Theorem 3.1 Assume that (C1)-(C3) hold. If there exists such that
for any , , ;
Then (1.1) has at least two positive, concave, and symmetric solutions and satisfying
where , .
Proof Let . Define two functionals on as follows:
If , then . Hence,
From (i) and (ii), we have that
So we have . Let
Then and . Let , then . Now we denote a sequence by the iterative scheme
Since and , we have , . From Lemma 2.4, T is compact, we assert that has a convergent subsequence and there exists such that . On the other hand, since
We note that
By Lemma 2.5, we know that T is increasing, it follows that
Moreover, we have
Therefore, . Let in (3.3) to obtain since T is continuous. It is well known that the fixed point of the operator T is a solution of BVP (1.1). Therefore, is a concave, symmetric, positive solution of BVP (1.1).
Let , , then . Let , then . We denote
Similar to , we assert that has a convergent subsequence and there exists such that , that is,
Since , then
By induction, it is easy to see that for ,
Thus and .
Since every fixed point of T in P is a solution of BVP (1.1), then and are two positive, concave, and symmetric solutions of BVP (1.1). □
Corollary 3.1 Assume that (C1)-(C3) and Theorem 3.1(i) hold. If there exists such that
Then BVP (1.1) has at least 2n positive, concave solutions and satisfying
where , .
Consider the following second-order boundary-value problem with integral boundary conditions:
and , , , , , , , , . We note that , for and for . Then , , , . Choosing , by calculations we obtain . It is easy to verify that satisfies
for any , , ;
Hence, by Theorem 3.1, (4.1) has two positive solutions and . For , the two iterative schemes are:
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The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.