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Solvability for nonlocal boundary value problems on a half line with
Boundary Value Problems volume 2014, Article number: 167 (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 .
In this paper, we consider the existence of solutions to the following second-order nonlinear differential equation with nonlocal boundary conditions that contain integral and multi-point boundary conditions:
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. ;
(H3) let be the function such that . Then
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.
Letbe a Fredholm operator with index zero andbe L-compact on. Assume that the following conditions are satisfied:
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.
Let Z be the space of all bounded continuous vector-valued functions onand. Then S is relatively compact in Z if the following conditions hold:
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 X be the set of the functions such that and are uniformly bounded in . Here, w is the function in the assumption (H2). Then X is a Banach space equipped with a norm , where
Let Y denote the Banach space equipped with the usual norm, .
For any non-negative continuous function , we can choose a function which satisfies . For example, put
If , then , and the norm is equivalent to the norm . Here,
Define by , where
Clearly, . Now we define the linear operators under the hypothesis (H3) as follows:
Assume that (H1) holds. Then
Let . Then there exists such that . For ,
which imply that
and thus by the fact that . In a similar manner, . On the other hand, let satisfying . Take
Then , and . Thus the proof is complete. □
By Lemma 3.2, . Since , L is a Fredholm operator with index 0. Let be linear operators which are defined as follows:
Then, by simple calculations, , , , and . Define a bounded linear operator by
Then , i.e., is a linear projector. Since , , and .
Define a continuous projector by
Clearly, , and consequently . Define an operator by
Then is the inverse operator , and it satisfies
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.
Since Ω is a bounded open subset of X, there exists a constant such that for any . For any and for almost all , by (H2),
Then and for all . Thus is bounded in Y.
For any ,
Then and for all . Thus, is bounded in Y.
Next we will prove that is a relatively compact set in X. For , by (3),
Thus is bounded in X.
Let be given. For any and let with ,
which imply that
are equicontinuous on .
For any , by L’Hôspital’s rule,
Since for all ,
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;
(H5) there exists a positive constant C such that ifor, then, and only one of the following inequalities is satisfied:
If α and β satisfy
then problem (1) has at least one solution in X.
We divide the proof into four steps.
Step 1. Let
Then is bounded. In fact, means and . Thus, by Lemma 3.2, . By (H4), there exist and such that , . Then
On the other hand,
Since , for all , it follows from (3) and (H2) that
and, by (5),
which implies that is bounded.
Step 2. Set
Then is bounded. In fact, implies and . By (H5), we obtain and . Thus is bounded.
Step 3. Define an isomorphism by
Assume first that (i) in (H5) holds. Let
Then is bounded. Indeed, means that there exist constants and such that and . If , then . It follows from that . By (H5), we obtain and . If , clearly . For , by the facts that and , it follows that and . If or , by (i) in (H5),
which is a contradiction. Thus is bounded in X. On the other hand, if (ii) in (H5) holds, taking
one can show that is bounded in a similar manner.
Step 4. Take an open bounded set . By Step 1 and Step 2, in view of Theorem 2.1, we only need to prove that in order to show that has at least one solution in . Let
By Step 3, for all . Thus, by the homotopy invariance property of degree,
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.
Let, and assume that (H1)-(H5) hold. If α and β satisfy
then problem (1) has at least one solution in X.
Consider the following second-order nonlinear differential equation:
where for , satisfies ,
We also assume that the constants K and ρ satisfy
and . Then , and (H1)-(H2) hold for , , , and . For ,
Taking , then
Since, for any ,
and . Thus (H3) holds for any .
Take and let be arbitrary. Then (H4) holds. In fact, if for , then
If for , then for , and
i.e., for , and
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final draft.
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Jeong, J., Kim, CG. & Lee, E.K. Solvability for nonlocal boundary value problems on a half line with . Bound Value Probl 2014, 167 (2014). https://doi.org/10.1186/s13661-014-0167-6
- nonlocal boundary condition
- a half line