- Open Access
Existence and monotone iteration of symmetric positive solutions for integral boundary-value problems with ϕ-Laplacian operator
© Cerdik and Hamal; licensee Springer. 2014
- Received: 10 July 2014
- Accepted: 31 October 2014
- Published: 26 November 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.
- integral boundary conditions
- monotone iterative technique
- positive solutions
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.
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.
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.
By applying classical monotone iterative techniques, they not only obtained the existence of positive solutions, but also gave iterative schemes for approximating the solutions.
If , then for all ;
ϕ is a continuous bijection and its inverse mapping is also continuous;
for all .
(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.
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 .
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 .
By , it is obvious that . Hence, , . □
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. □
for any , , ;
where , .
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).
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). □
where , .
for any , , ;
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