Solvability for second-order nonlocal boundary value problems with
© Jeong et al.; licensee Springer. 2014
Received: 1 August 2014
Accepted: 28 November 2014
Published: 20 December 2014
The existence of at least one solution to the second-order nonlocal boundary value problems on the real line is investigated by using an extension of 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 medium, theory of drain flows and plasma physics. For an extensive collection of results to boundary value problems on unbounded domains, we refer the reader to a monograph by Agarwal and O’Regan . The study of nonlocal elliptic boundary value problems was investigated by Bicadze and Samarskiĭ , and later continued by Il’in and Moiseev  and Gupta . Since then, the existence of solutions for nonlocal boundary value problems has received a great deal of attention in the literature. For more recent results, we refer the reader to – and the references therein.
where , , 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 assumptions hold:
(H1) satisfy ;
(H2) is a continuous function which satisfy ;
where , , , , and will be defined in Section 3.
A boundary value problem is called 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. However, it is not suitable for the case L is nonlinear. Recently, Ge and Ren  extended Mawhin’s continuation theorem from the case of linear L to the case of quasi-linear L. The purpose of this paper is to establish the sufficient conditions for the existence of solutions to the problem (1) on the real line at resonance with by using an extension of Mawhin’s continuation theorem .
In this section, we recall some definitions and theorems. Let X and Y be two Banach spaces with the norms and , respectively.
is a closed subset of Y;
is linearly homeomorphic to for some .
, ⇔ ;
is the zero operator and , where ;
Here, is a complement space of in X, is the origin of Y and , are projections.
Now, we give an extension of Mawhin’s continuation theorem .
Letbe an open and bounded set with. Suppose thatis a quasi-linear operator and, is M-compact. In addition, if the following conditions hold:
(A2), whereis a homeomorphism with,
then the abstract equationhas at least one solution in.
Finally, we give a theorem which is useful to show the compactness of operators defined on an infinite interval.
S is bounded in Z;
S is equicontinuous on any compact interval of ;
S is equiconvergent at ±∞, that is, given , there exists a constant such that (respectively, ) for all (respectively, ) and all .
3 Main result
- (1)It is well known that, for any and ,
Since , then w is a continuous function which satisfies and .
For any continuous functions , we can choose a function which satisfies . For example, put , then .
Then are continuous.
Assume that (H1) and (H2) hold. Then the operatoris quasi-linear. Moreover, and.
Thus . In a similar manner, .
Then , and . Thus, . Since are continuous, ImM is closed in Y. Consequently, M is a quasi-linear operator.
where and . Then , are projections, and . By (H4), , and it follows from Lemma 3.2 that . □
Assume that (H1)-(H4) hold. Assume that Ω is an open bounded subset of X such that. Then, is M-compact on.
Here is the constant in Remark 3.1(1). Thus and are bounded in Z.
Consequently, and are equicontinuous on any compact intervals in .
By (6), we conclude that and are equiconvergent at ±∞. Thus, is compact in view of Theorem 2.4.
Next, we prove that is continuous. Let be a sequence in such that in X and in as . Then is bounded in X and pointwise as . Since R is compact, there exists a subsequence of such that in X as . By the Lebesgue dominated convergence theorem, as . Thus, . By a standard argument, is continuous.
Thus, is M-compact on . □
Now, we give the main result in this paper.
Assume that (H1)-(H4) hold. Assume also that the following hold:
(H5)there exist positive constants A and B such that iffor everyorfor every, then, i.e., eitheror;
Here, D is the constant defined in (5).
We divide the proof into three steps.
We will prove that is bounded. For , .
By (7), is bounded.
Let . Then for some . If , . Since J is homeomorphism, . By (H6), we obtain and . If , then .
which is a contradiction. Thus, is bounded.
and it follows that is bounded in a similar manner.
Step 3. Take an open bounded set in X. By Step 1,
(A1) for every .
Now we will show that
By Theorem 2.3, has at least one solution in , and consequently problem (1) has at least one solution in X. □
where . Since for and , , and thus (H1), (H2), and (H3) hold.
Thus, (H4) holds.
Thus, (H5) holds.
Thus, (H6)(2) is satisfied.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A1011225).
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