Positive Solutions of a Singular System with Two Point Coupled Boundary Conditions
Naseer Ahmad Asif
Department of Mathematics, School of Science and Technology, University of Management and Technology, Lahore, Pakistan
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To cite this article:
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
a contradiction to (2.2). Hence,
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.
References