Solvability for nonlocal boundary value problems on a half line with
© Jeong et al.; licensee Springer 2014
Received: 3 March 2014
Accepted: 25 June 2014
Published: 26 September 2014
The existence of at least one solution to the second-order nonlocal boundary value problems on a half line is investigated by using Mawhin’s continuation theorem.
MSC: 34B10, 34B40, 34B15.
Boundary value problems on an infinite interval arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and in various applications such as an unsteady flow of gas through a semi-infinite porous media, theory of drain flows and plasma physics. For an extensive collection of results as regards boundary value problems on unbounded domains, we refer the reader to a monograph by Agarwal and O’Regan . For more recent results on unbounded domains, we refer the reader to – and the references therein.
A boundary value problem is called to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. Resonance problems can be expressed as an abstract equation , where L is a noninvertible operator. When L is linear, Mawhin’s continuation theorem  is an efficient tool in finding solutions for these problems. Recently, there have been many works concerning the existence of solutions for multi-point boundary value problems at resonance. For example, see , – in the case that , and see , – in the case that .
where , , , , and is a Carathéodory function, i.e., is Lebesgue measurable in t for all and continuous in for almost all . Throughout this paper, we assume that the following conditions hold:
(H1) , , for , and ;
(H2) let , and there exist non-negative measurable functions α, β, and γ such that , and , a.e. ;
where the linear operators will be defined later in Section 3.
If , then w is continuous in , and in (H3), there exists a function k satisfying (see, e.g., Remark 3.1(1)). The boundary conditions in problem (1) are crucial since the differential operator under the boundary conditions in (1) satisfies . The purpose of this paper is to establish the sufficient conditions for the existence of solutions to problem (1) on a half line at resonance with by using Mawhin’s continuation theorem .
The remainder of this paper is organized as follows: some preliminaries are provided in Section 2, the main result is presented in Section 3, and finally an example to illustrate the main result is given in Section 4.
In this section, we recall some notations and two theorems which will be used later. Let X and Y be two Banach spaces with the norms and , respectively. Let be a Fredholm operator with index zero, and let , be projectors such that and . Then and . It follows that is invertible. We denote the inverse of it by . If Ω is an open bounded subset of X with , then the map will be called L-compact on if is bounded in Y and is compact.
for every ;
for every ;
, where is a projector such that .
Then the equationhas at least one solution in.
Since the Arzelá-Ascoli theorem fails in the noncompact interval case, we use the following result in order to show that is compact.
S is bounded in Z;
S is equicontinuous on any compact interval of ;
S is equiconvergent at ∞, that is, given , there exists a such that for all and all .
3 Main results
Let Y denote the Banach space equipped with the usual norm, .
- (1)For any non-negative continuous function , we can choose a function which satisfies . For example, put
- (2)If , then , and the norm is equivalent to the norm . Here,
Then , and . Thus the proof is complete. □
Then , i.e., is a linear projector. Since , , and .
Let a nonlinear operator be defined by , . Then problem (1) is equivalent to , .
From now on, we consider the case . The case can be dealt in a similar manner.
Let, and assume that (H1)-(H3) hold. Assume that Ω is a bounded open subset of X such that. Then N is L-compact on.
Then and for all . Thus is bounded in Y.
Then and for all . Thus, is bounded in Y.
Thus is bounded in X.
are equicontinuous on .
uniformly on as . In view of Theorem 2.2, is a relatively compact set in X, and thus N is L-compact on . □
The following theorem is the main result in this paper.
Let, and assume that (H1)-(H3) hold. Assume also that the following hold:
(H4) there exist positive constants A and B such that iffor everyorfor every, then;
then problem (1) has at least one solution in X.
We divide the proof into four steps.
which implies that is bounded.
Then is bounded. In fact, implies and . By (H5), we obtain and . Thus is bounded.
one can show that is bounded in a similar manner.
By Theorem 2.1, has at least one solution in , i.e., problem (1) has at least one solution in X. □
In the case that , , and using the norm on X in Remark 3.1(2), we have a similar result to Theorem 3.4. We omit the proof.
then problem (1) has at least one solution in X.
and . Thus (H3) holds for any .
This study was supported by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2014, Project No. RIBS-PNU-2014-101.
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