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

*Boundary Value Problems*
**volume 2009**, Article number: 978605 (2009)

## Abstract

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 second-order 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 second-order two-point 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 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

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.

## 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.

(H_{1}) for any and there exist and such that

uniformly for , and

(H_{2}) For any and countable bounded set , there exist such that

with

where .

(H_{3}) 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 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

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 three-point 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.

## References

- 1.
Guo DJ, Lakshmikantham V, Liu XZ:

*Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications*.*Volume 373*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341. - 2.
Deimling K:

*Ordinary Differential Equations in Banach Spaces, Lecture Notes in Mathematics*.*Volume 596*. Springer, Berlin, Germany; 1977:vi+137. - 3.
Lakshmikantham V, Leela S:

*Nonlinear Differential Equations in Abstract Spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications*.*Volume 2*. Pergamon Press, Oxford, UK; 1981:x+258. - 4.
Guo DJ, Lakshmikantham V:

*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275. - 5.
Liu Y: Boundary value problems for second order differential equations on unbounded domains in a Banach space.

*Applied Mathematics and Computation*2003, 135(2-3):569-583. 10.1016/S0096-3003(02)00070-X - 6.
Gupta CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.

*Journal of Mathematical Analysis and Applications*1992, 168(2):540-551. 10.1016/0022-247X(92)90179-H - 7.
Feng W, Webb JRL:Solvability of -point boundary value problems with nonlinear growth.

*Journal of Mathematical Analysis and Applications*1997, 212(2):467-480. 10.1006/jmaa.1997.5520 - 8.
Sun JX, Xu X, O'Regan D:Nodal solutions for -point boundary value problems using bifurcation methods.

*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(10):3034-3046. 10.1016/j.na.2007.02.043 - 9.
Xu X:Positive solutions for singular -point boundary value problems with positive parameter.

*Journal of Mathematical Analysis and Applications*2004, 291(1):352-367. 10.1016/j.jmaa.2003.11.009 - 10.
Ma R, Castaneda N:Existence of solutions of nonlinear -point boundary-value problems.

*Journal of Mathematical Analysis and Applications*2001, 256(2):556-567. 10.1006/jmaa.2000.7320 - 11.
Zhao J, Liu Z, Liu L: The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces.

*Journal of Computational and Applied Mathematics*2008, 222(2):524-530. 10.1016/j.cam.2007.11.020 - 12.
Zhang G, Sun JX:Positive solutions of -point boundary value problems.

*Journal of Mathematical Analysis and Applications*2004, 291(2):406-418. 10.1016/j.jmaa.2003.11.034 - 13.
Zhao Y-L, Chen H-B:Existence of multiple positive solutions for -point boundary value problems in Banach spaces.

*Journal of Computational and Applied Mathematics*2008, 215(1):79-90. 10.1016/j.cam.2007.03.025 - 14.
Liu B: Positive solutions of a nonlinear four-point boundary value problems in Banach spaces.

*Journal of Mathematical Analysis and Applications*2005, 305(1):253-276. 10.1016/j.jmaa.2004.11.037 - 15.
Guo DJ:Existence of positive solutions for th-order nonlinear impulsive singular integro-differential equations in Banach spaces.

*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(9):2727-2740. 10.1016/j.na.2007.02.019 - 16.
Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications.

*Nonlinear Analysis: Theory, Methods & Applications*1987, 11(5):623-632. 10.1016/0362-546X(87)90077-0

## 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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### Keywords

- Banach Space
- Fixed Point Theorem
- Infinite System
- Iterative Sequence
- Infinite Interval