The cone theory together with Mönch fixed point theorem and a monotone iterative technique is used to investigate the positive solutions for some boundary problems for systems of nonlinear secondorder differential equations with multipoint boundary value conditions on infinite intervals in Banach spaces. The conditions for the existence of positive solutions are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is also derived.
1. Introduction
In recent years, the theory of ordinary differential equations in Banach space has become a new important branch of investigation (see, e.g., [1–4] and references therein). By employing a fixed point theorem due to Sadovskii, Liu [5] investigated the existence of solutions for the following secondorder twopoint boundary value problems (BVP for short) on infinite intervals in a Banach space :
where On the other hand, the multipoint boundary value problems arising naturally from applied mathematics and physics have been studied so extensively in scalar case that there are many excellent results about the existence of positive solutions (see, i.e., [6–12] and references therein). However, to the best of our knowledge, only a few authors [5, 13, 14] have studied multipoint boundary value problems in Banach spaces and results for systems of secondorder differential equation are rarely seen. Motivated by above papers, we consider the following singular point boundary value problem on an infinite interval in a Banach space
where and with In this paper, nonlinear terms and may be singular at , and/or , where denotes the zero element of Banach space . By singularity, we mean that as or
Very recently, by using Shauder fixed point theorem, Guo [15] obtained the existence of positive solutions for a class of thorder nonlinear impulsive singular integrodifferential equations in a Banach space. Motivated by Guo's work, in this paper, we will use the cone theory and the Mönch fixed point theorem combined with a monotone iterative technique to investigate the positive solutions of BVP (1.2). The main features of the present paper are as follows. Firstly, compared with [5], the problem we discussed here is systems of multipoint boundary value problem and nonlinear term permits singularity not only at but also at . Secondly, compared with [15], the relative compact conditions we used are weaker. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is worked out to illustrate the main results.
2. Preliminaries and Several Lemmas
Let
Evidently, . It is easy to see that is a Banach space with norm
and is also a Banach space with norm
where
Let with norm
Then is also a Banach space. The basic space using in this paper is .
Let be a normal cone in with normal constant which defines a partial ordering in by . If and , we write . Let . So, if and only if . For details on cone theory, see [4].
In what follows, we always assume that . Let . Obviously, for any . When , we write , that is, . Let and . It is clear, are cones in and , respectively. A map is called a positive solution of BVP (1.2) if and satisfies (1.2).
Let denote the Kuratowski measure of noncompactness in and , respectively. For details on the definition and properties of the measure of noncompactness, the reader is referred to [1–4]. Let be all Lebesgue measurable functions from to . Denote
Let us list some conditions for convenience.
() for any and there exist and such that
uniformly for , and
() For any and countable bounded set , there exist such that
with
where .
() imply
In what follows, we write and . Evidently, , and are closed convex sets in and , respectively.
We will reduce BVP (1.2) to a system of integral equations in . To this end, we first consider operator defined by
where
Lemma 2.1.
If condition is satisfied, then operator defined by (2.12) is a continuous operator from into .
Proof.
Let
By virtue of condition , there exists an such that
where
Hence
Let , we have, by (2.19)
which together with condition implies the convergence of the infinite integral
Thus, we have
which together with (2.13) and implies that
Therefore, by (2.15) and (2.20), we get
Differentiating (2.13), we obtain
Hence,
It follows from (2.24) and (2.25) that
So, . On the other hand, it can be easily seen that
So, . In the same way, we can easily get that
where Thus, maps into and we get
where
Finally, we show that is continuous. Let . Then is a bounded subset of . Thus, there exists such that for and . Similar to (2.24) and (2.26), it is easy to have
It is clear,
and by (2.20),
It follows from (2.33) and (2.34) and the dominated convergence theorem that
It follows from (2.32) and (2.35) that as . By the same method, we have as . Therefore, the continuity of is proved.
Lemma 2.2.
If condition is satisfied, then is a solution of BVP (1.2) if and only if is a fixed point of operator .
Proof.
Suppose that is a solution of BVP (1.2). For integrating (1.2) from to , we have
Integrating (2.36) from 0 to , we get
Thus, we obtain
which together with the boundary value conditions imply that
Substituting (2.40) and (2.41) into (2.37) and (2.38), respectively, we have
It follows from Lemma 2.1 that the integral and the integral are convergent. Thus, is a fixed point of operator .
Conversely, if is fixed point of operator , then direct differentiation gives the proof.
Lemma 2.3.
Let be satisfied, is a bounded set. Then and are equicontinuous on any finite subinterval of and for any there exists such that
uniformly with respect to as
Proof.
We only give the proof for operator , the proof for operator can be given in a similar way. By (2.13), we have
For we obtain by (2.44)
Then, it is easy to see by (2.45) and that is equicontinuous on any finite subinterval of .
Since is bounded, there exists such that for any . By (2.25), we get
It follows from (2.46) and and the absolute continuity of Lebesgue integral that is equicontinuous on any finite subinterval of .
In the following, we are in position to show that for any there exists such that
uniformly with respect to as
Combining with (2.45), we need only to show that for any there exists sufficiently large such that
for all as The rest part of the proof is very similar to Lemma in [5], we omit the details.
Lemma 2.4.
Let be a bounded set in . Assume that holds. Then
Proof.
The proof is similar to that of Lemma in [5], we omit it.
Mönch FixedPoint Theorem. Let be a closed convex set of and Assume that the continuous operator has the following property: countable, is relatively compact. Then has a fixed point in .
Lemma 2.6.
If is satisfied, then for imply that
Proof.
It is easy to see that this lemma follows from (2.13), (2.25), and condition . The proof is obvious.
Lemma 2.7 (see [16]).
Let and are bounded sets in , then
where and denote the Kuratowski measure of noncompactness in and , respectively.
Lemma 2.8 (see [16]).
Let be normal (fully regular) in , then is normal (fully regular) in .
3. Main Results
Theorem 3.1.
If conditions and are satisfied, then BVP (1.2) has a positive solution satisfying for
Proof.
By Lemma 2.1, operator defined by (2.13) is a continuous operator from into , and, by Lemma 2.2, we need only to show that has a fixed point in . Choose and let . Obviously, is a bounded closed convex set in space . It is easy to see that is not empty since . It follows from (2.27) and (3.6) that implies , that is, maps into . Let satisfying for some . Then We have, by (2.13) and (2.25),
By Lemma 2.4, we have
where , and .
By (2.21), we know that the infinite integral is convergent uniformly for So, for any we can choose a sufficiently large such that
Then, by [1, Theorem ], (2.44), (3.1), (3.3), , and Lemma 2.7, we obtain
It follows from (3.2) and (3.4) that
In the same way, we get
On the other hand, . Then, (3.5), (3.6), , and Lemma 2.7 imply that is, is relatively compact in Hence, the Mönch fixed point theorem guarantees that has a fixed point in . Thus, Theorem 3.1 is proved.
Theorem 3.2.
Let cone be normal and conditions be satisfied. Then BVP (1.2) has a positive solution which is minimal in the sense that for any positive solution of BVP (1.2). Moreover, and there exists a monotone iterative sequence such that as uniformly on and as for any where
Proof.
From (3.7), one can see that and
By (3.7) and (3.11), we have that and
which imply that . Similarly, we have . Thus, . It follows from (2.13) and (3.9) that
By Lemma 2.1, we get and
By Lemma 2.6 and (3.13), we have
It follows from (3.14), by induction, that
Let Then, is a bounded closed convex set in space and operator maps into . Clearly, is not empty since Let Obviously, and Similar to above proof of Theorem 3.1, we can obtain that is, is relatively compact in So, there exists an and a subsequence such that converges to uniformly on Since that is normal and is nondecreasing, it is easy to see that the entire sequence converges to uniformly on Since and are closed convex sets in space , we have It is clear,
By and (3.16), we have
Noticing (3.17) and (3.18) and taking limit as in (3.9), we obtain
In the same way, taking limit as in (3.10), we get
which together with (3.19) and Lemma 2.2 implies that and is a positive solution of BVP (1.2). Differentiating (3.9) twice, we get
Hence, by (3.17), we obtain
Similarly, we have
Let be any positive solution of BVP (1.2). By Lemma 2.2, we have and for It is clear that for any So, by Lemma 2.6, we have for any Assume that for Then, it follows from Lemma 2.6 that for that is, for Hence, by induction, we get
Now, taking limits in (3.24), we get for and the theorem is proved.
Theorem 3.3.
Let cone be fully regular and conditions and be satisfied. Then the conclusion of Theorem 3.2 holds.
Proof.
The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition , the conclusion is implied directly by (3.15) and (3.16), the full regularity of and Lemma 2.4.
4. An Example
Consider the infinite system of scalar singular second order threepoint boundary value problems:
Proposition 4.1.
Infinite system (4.1) has a minimal positive solution satisfying for
Proof.
Let with the norm . Obviously, is a real Banach space. Choose . It is easy to verify that is a normal cone in with normal constants 1. Now we consider infinite system (4.1), which can be regarded as a BVP of form (1.2) in with . In this situation, in which
Let . Then for . It is clear, for any . Notice that for , by (4.2), we get
which imply is satisfied for and
Let , where
Let be given, and be any sequence in , where . By (4.5), we have
So, is bounded and by the diagonal method together with the method of constructing subsequence, we can choose a subsequence such that
which implies by virtue of (4.9)
Hence It is easy to see from (4.9)–(4.11) that
Thus, we have proved that is relatively compact in
For any , we have by (4.6)
where is between and . By (4.13), we get
In the same way, we can prove that is relatively compact in , and we can also get
Thus, by (4.14) and (4.15), it is easy to see that holds for . Thus, our conclusion follows from Theorem 3.1. This completes the proof.
Acknowledgment
The project is supported financially by the National Natural Science Foundation of China (10671167) and the Natural Science Foundation of Liaocheng University (31805).
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