American Journal of Applied Mathematics
Volume 3, Issue 3-1, June 2015, Pages: 19-24

Positive Solutions of a Singular System with Two Point Coupled Boundary Conditions

Department of Mathematics, School of Science and Technology, University of Management and Technology, Lahore, Pakistan

(N. A. Asif)

Naseer Ahmad Asif. Positive Solutions of a Singular System with Two Point Coupled Boundary Conditions. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 19-24. doi: 10.11648/j.ajam.s.2015030301.14

1. Introduction

Coupled boundary conditions arises in the study of reaction-diffusion equations and Sturm-Liouvillie problems, see [3,4,11] and Chapter 13 of [19]. The study of elliptic systems with coupled boundary conditions was initiated by Agmon and coauthors [2]. In [2], the authors studied elliptic systems with the following type of coupled boundary conditions

(1.1)

where  and  are differential operators from  to  and  is a separable Hilbert space. Mehmeti [14], Mehmeti, Nicaise [15] and Nicaise [16] have been studied coupled boundary conditions in the study of interaction problems and elliptic operators on polygonal domains. Coupled boundary conditions have also some applications in Mathematical Biology. For example, Leung [11] studied the following reaction-diffusion system for prey-predator interaction:

(1.2)

subject to the coupled boundary conditions

(1.3)

where  are positive constants,  have Holder continuous partial derivatives up to second order in compact sets,  is a unit outward normal at  and  and  have Holder continuous first derivatives in compact subsets of . The functions  respectively represent the density of prey and predator at time  and at position . Similar coupled boundary conditions are also studied in [5] for biochemical system. All the above mention works are related to coupled boundary conditions for partial differential equations. To the best of our knowledge, we believe that coupled system of nonlinear ordinary differential equations subject to coupled boundary conditions have never been studied previously.

For the study of existence theory for boundary value problems, we refer the readers to the recent papers [6, 7, 9, 10, 12, 13, 17, 18]. An excellent resource on singular boundary value problems with an extensive bibliography was produced by Agarwal and O’Regan [1].

Recently, the author and coauthors [8] have studied the existence of positive solution to the following system under certain assumptions

(1.4)

The present paper is a generalization of [8] to a more general coupled singular system subject to coupled boundary conditions. We study existence of positive solutions to the following system of boundary value problems

(1.5)

where the nonlinearities  are continuous and are allowed to be singular at , ; ,  and  on . By singularity we mean that the functions ,  are allowed to be unbounded at . Here, we study existence of at least one -positive solution for the system (1.5) under more general and simple assumptions as compared to the assumptions in [8] for (1.4).

By a -positive solution of the system (1.5), we mean that ,  satisfies (1.5),   on  and  on

Throughout the paper, assume that the following conditions hold:

(A1)  on   and

(A2)  are continuous with  and  on

(A3)  and  where  are continuous and nonincreasing on , , ,  are continuous and nondecreasing on

(A4)

where

for

(A5)  and

(A6) for real constants  and  there exist continuous functions  and  defined on  and positive on  and constants  satisfying  such that  and  on

(A7)  and  for any real constant

Remark 1.1. Since ,  are continuous, , , , , and are monotone increasing. Hence,  and  are invertible. Moreover,  and  are also monotone increasing.

2. Main Result: Existence of at Least One Positive Solutions Introduction

Theorem 2.1. Assume that (A1) − (A7) hold. Then, the system (1.5) has at least one -positive solution.

Proof. In view of (A4), we can choose real constant M > 0 such that

(2.1)

where

From the continuity of  and  we choose  small enough such that

(2.2)

Choose a real constant  such that

(2.3)

Choose  such that  For each  define retractions  and  by

and

Consider the modified system of BVPs

(2.4)

Since ,  are continuous and bounded on , by Schauder’s fixed point theorem, it follows that the modified system of BVPs (2.4) has a solution  Using (2.4) and (A2), we obtain

which on integration from  to , use the boundary conditions (BCs), implies that

(2.5)

Integrating (2.5) from 0 to t, using the BCs and (2.5), we have

(2.6)

From (2.5) and (2.6), it follows that

(2.7)

Now, we show that

(2.8)

First, we prove  for  Suppose  for some  Using (2.4) and (A3), we have

which implies that

Integrating from  to , using the BCs, we obtain

which can also be written as

Using the increasing property of , we obtain

and using the increasing property of , leads to

Which is a contradiction to (2.3). Hence,  for

Similarly, we can show that  for

Now, we show that

(2.9)

Suppose  From (2.4), (2.5), (2.8) and (A3), it follows that

which implies that

Integrating from  to  using the BCs, we obtain

which can also be written as

The increasing property of  and  leads to

(2.10)

Integrating from  to  using the BCs and (2.10), we obtain

(2.11)

From (2.11) and (2.7), it follows that

which implies that

Thus,  is a solution of the following coupled system of BVPs

(2.12)

satisfying

(2.13)

We claim that

(2.14)

(2.15)

where

To prove (2.14), consider the following relation

(2.16)

which implies that

Using (A6) and (2.13), we obtain

which implies that

(2.17)

Similarly, using (A6) and (2.13), we obtain

(2.18)

Now, using (2.18) in (2.17), we have

Hence,

(2.19)

Similarly, using (2.17) in (2.18), we obtain

(2.20)

Now, from (2.16), it follows that

and using (A6), (2.13), (2.19) and (2.20), we obtain (2.14).

Similarly, we can prove (2.15).

Now, using (2.12), (A3), (2.13), (2.14) and (2.15), we have

(2.21)

In view of (2.13), (2.21), (A1) and (A7), it follows that the sequences  are uniformly bounded and equicontinuous on  Hence, by Arzela-Ascoli theorem, there exist subsequences  of  and  such that  converges uniformly to  on  Also,  Moreover, from (2.14) and (2.15), with  in place of  and taking  we have

which shows that  and  on ,  and  on  Further,  satisfy

Passing to the limit as  we obtain

which implies that

Hence,  is a -positive solution of the system (1.5).

Example 2.2. Consider the following coupled system of singular BVPs

(2.22)

where  satisfying  ,  and  such that

Choose , , , , , ,  and . Then, , ,  and .

Also, choose , . Then,

Clearly, (A1) - (A7) are satisfied. Hence, by Theorem 2.1, the system (2.22) has at least one -positive solution.

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