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Solvability for second-order nonlocal boundary value problems with
Boundary Value Problems volume 2014, Article number: 255 (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.
In this paper, we consider the following second-order nonlinear differential equation with integral boundary conditions:
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 ;
(H3) let , and there exist nonnegative measurable functions α, β and γ such that and
(H4) there exists a function such that and
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.
A continuous operator is said to be quasi-linear if
is a closed subset of Y;
is linearly homeomorphic to for some .
Let be a quasi-linear operator. Let and be an open and bounded set with the origin . Then , is said to be M-compact in if , is a continuous operator, and there exist a vector subspace of Y satisfying and an operator being continuous and compact such that, for ,
, ⇔ ;
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.
Let Z be the space of all bounded continuous 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 constant such that (respectively, ) for all (respectively, ) and all .
3 Main result
Let X be the set of the functions such that
where w is the function in the assumption (H3). Then X is a Banach space equipped with a norm , where
Let Y denote the Banach space equipped with a usual norm
It is well known that, for any and ,
Thus, for all , where .
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 .
Define by , where
Then is continuous. Let Ω be an open bounded subset of X such that . For , define by . By (H3) and the Lebesgue dominated convergence theorem, is continuous. Denote by N. Then problem (1) is equivalent to , . Define by
Then are continuous.
Assume that (H1) and (H2) hold. Then the operatoris quasi-linear. Moreover, and.
Clearly, , and it is linearly homeomorphic to . Next, we show that
Let . Then there exists such that
Thus . In a similar manner, .
On the other hand, let satisfying . Take
Then , and . Thus, . Since are continuous, ImM is closed in Y. Consequently, M is a quasi-linear operator.
Let be linear operators which are defined as follows:
where () are the constants in the assumption in (H4). Then, by direct calculations,
Define the bounded linear operators and by
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.
Let . Then is a complement space of in X, i.e., . Define , for , by
Since Ω is bounded, there exists a constant such that for any . For , and for almost all , by (H3)
which implies that
Since and , for ,
First, we prove that is compact by using Theorem 2.4. Let with the usual sup norm. For ,
Here is the constant in Remark 3.1(1). Thus and are bounded in Z.
Let and let be given. First, for any with , we have
and since and w are uniformly continuous on a compact interval in , there exists such that if with , then
In a similar manner, there exists such that if with , then
Letting , if with , then
Consequently, and are equicontinuous on any compact intervals in .
For , by L’Hôspital’s rule,
In a similar manner,
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.
Finally, we show that (i)-(iv) hold in Definition 2.2. Since , . For , , and . Consequently, . Since for any ,
and . For , . Then, for and ,
On the other hand, for and ,
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;
(H6)there exists a positive constant C such that ifor, then either
Then problem (1) has at least one solution in X provided that
Here, D is the constant defined in (5).
We divide the proof into three steps.
Step 1. Let
We will prove that is bounded. For , .
By (H5), there exist and such that
which imply that
and . Then we have
On the other hand, by (4),
It follows that
By (7), is bounded.
Step 2. Define a homeomorphism by
Assume (H6)(1) holds, i.e., there exists a positive constant C such that if or , then . Let
Let . Then for some . If , . Since J is homeomorphism, . By (H6), we obtain and . If , then .
For , by , we obtain
If or , then, by (H6)(1), we obtain
which is a contradiction. Thus, is bounded.
In the case that (H6)(2) holds, we take
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
Let . By Step 2, we know that , for every . Thus, by the homotopy property of the degree, we obtain
By Theorem 2.3, has at least one solution in , and consequently problem (1) has at least one solution in X. □
Consider the following second-order nonlinear differential equation:
Define by , where
where . Since for and , , and thus (H1), (H2), and (H3) hold.
Take , then
Thus, (H4) holds.
Take and . If , for , then . If for , then for , and
Thus, (H5) holds.
For any , if or , then
Thus, (H6)(2) is satisfied.
Since , , , and , we have
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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).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final draft.