Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces

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 second-order two-point boundary value problems (BVP for short) on infinite intervals in a Banach space :

(1.1)

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 second-order 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

(1.2)

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 th-order nonlinear impulsive singular integro-differential 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.

Let

(2.1)

Evidently, . It is easy to see that is a Banach space with norm

(2.2)

and is also a Banach space with norm

(2.3)

where

(2.4)

Let with norm

(2.5)

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

(2.6)

Let us list some conditions for convenience.

(H₁) for any and there exist and such that

(2.7)

uniformly for , and

(2.8)

(H₂) For any and countable bounded set , there exist such that

(2.9)

with

(2.10)

where .

(H₃) imply

(2.11)

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

(2.12)

where

(2.13)

(2.14)

Lemma 2.1.

If condition is satisfied, then operator defined by (2.12) is a continuous operator from into .

Proof.

Let

(2.15)

(2.16)

By virtue of condition , there exists an such that

(2.17)

where

(2.18)

Hence

(2.19)

Let , we have, by (2.19)

(2.20)

which together with condition implies the convergence of the infinite integral

(2.21)

Thus, we have

(2.22)

which together with (2.13) and implies that

(2.23)

Therefore, by (2.15) and (2.20), we get

(2.24)

Differentiating (2.13), we obtain

(2.25)

Hence,

(2.26)

It follows from (2.24) and (2.25) that

(2.27)

So, . On the other hand, it can be easily seen that

(2.28)

So, . In the same way, we can easily get that

(2.29)

where Thus, maps into and we get

(2.30)

where

(2.31)

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

(2.32)

It is clear,

(2.33)

and by (2.20),

(2.34)

It follows from (2.33) and (2.34) and the dominated convergence theorem that

(2.35)

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

(2.36)

Integrating (2.36) from 0 to , we get

(2.37)

(2.38)

Thus, we obtain

(2.39)

which together with the boundary value conditions imply that

(2.40)

(2.41)

Substituting (2.40) and (2.41) into (2.37) and (2.38), respectively, we have

(2.42)

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

(2.43)

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

(2.44)

For we obtain by (2.44)

(2.45)

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

(2.46)

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

(2.47)

uniformly with respect to as

Combining with (2.45), we need only to show that for any there exists sufficiently large such that

(2.48)

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

(2.49)

Proof.

The proof is similar to that of Lemma in [5], we omit it.

Lemma 2.5 (see [1, 2]).

Mönch Fixed-Point 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

(2.50)

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 .

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),

(3.1)

By Lemma 2.4, we have

(3.2)

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

(3.3)

Then, by [1, Theorem ], (2.44), (3.1), (3.3), , and Lemma 2.7, we obtain

(3.4)

It follows from (3.2) and (3.4) that

(3.5)

In the same way, we get

(3.6)

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

(3.7)

(3.8)

(3.9)

(3.10)

Proof.

From (3.7), one can see that and

(3.11)

By (3.7) and (3.11), we have that and

(3.12)

which imply that . Similarly, we have . Thus, . It follows from (2.13) and (3.9) that

(3.13)

By Lemma 2.1, we get and

(3.14)

By Lemma 2.6 and (3.13), we have

(3.15)

It follows from (3.14), by induction, that

(3.16)

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,

(3.17)

By and (3.16), we have

(3.18)

Noticing (3.17) and (3.18) and taking limit as in (3.9), we obtain

(3.19)

In the same way, taking limit as in (3.10), we get

(3.20)

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

(3.21)

Hence, by (3.17), we obtain

(3.22)

Similarly, we have

(3.23)

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

(3.24)

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.

Consider the infinite system of scalar singular second order three-point boundary value problems:

(4.1)

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

(4.2)

Let . Then for . It is clear, for any . Notice that for , by (4.2), we get

(4.3)

which imply is satisfied for and

(4.4)

Let , where

(4.5)

(4.6)

(4.7)

(4.8)

Let be given, and be any sequence in , where . By (4.5), we have

(4.9)

So, is bounded and by the diagonal method together with the method of constructing subsequence, we can choose a subsequence such that

(4.10)

which implies by virtue of (4.9)

(4.11)

Hence It is easy to see from (4.9)–(4.11) that

(4.12)

Thus, we have proved that is relatively compact in

For any , we have by (4.6)

(4.13)

where is between and . By (4.13), we get

(4.14)

In the same way, we can prove that is relatively compact in , and we can also get

(4.15)

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.