American Journal of Applied Mathematics
Volume 3, Issue 3-1, June 2015, Pages: 54-59

Existence of Solutions to a Second Order Coupled System with Nonlinear Coupled Boundary Conditions

Naseer Ahmad Asif*, Imran Talib

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

Email address:

(N. A. Asif)
(I. Talib)

To cite this article:

Naseer Ahmad Asif, Imran Talib. Existence of Solutions to a Second Order Coupled System with Nonlinear 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. 54-59. doi: 10.11648/j.ajam.s.2015030301.19


Abstract: We study existence of solution in the presence of upper and lower solutions of some second-order nonlinear coupled ordinary differential system (ODS for short) depending on first order derivatives with nonlinear coupled boundary conditions (CBCs for short). Our method for nonlinear coupled system with nonlinear CBCs is new and it unifies the treatment of many different second order problems. Nagumo condition is used to define bound for the derivative of the solution. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder's fixed point theorem play an important role in establishing the arguments.

Keywords: Lower and Upper Solutions, Coupled System, Coupled Boundary Conditions, Arzela-Ascoli Theorem, Schauder's Fixed Point Theorem


1. Introduction

In this paper, we study existence of solution in the presence of upper and lower solutions of some second-order nonlinear coupled ODS with nonlinear CBCs of the type

(1.1)

(1.2)

where and  are continuous functions. A significant motivation factor for the study of the above system has been the applications of the nonlinear differential equations to the areas of mechanics; population dynamics; optimal control; ecology; biotechnology; harvesting; and physics [11, 15, 16]. Moreover, while dealing with nonlinear ODS mostly authors only focus on attention to the differential systems with uncoupled boundary conditions [1, 4, 6].

But, on the other hand, very few research work is available where the differential systems are coupled not only in the differential systems but also through the boundary conditions [2, 9, 12, 14]. Our considered System (1.1)-(1.2) deals with the latter case.

The other productive aspect of the article is the generalization of the classical concepts that had been discussed in [3, 5, 7, 8, 10, 13,17]. We mean to say if and  then (1.2) implies the periodic boundary conditions (BCs for short). Also if and  then (1.2) implies the anti-periodic BCs. In order to obtain a solution satisfying some initial or BCs and lying between a subsolution and supersolution, we need additional conditions. For example, in the periodic case it suffices that

(1.3)

and in the anti-periodic case it suffices that

(1.4)

so to generalize the classical results (1.3) and (1.4), the concept of coupled lower and upper solutions is defined in Section 2 that allows us to obtain a solution in the sector Also (2.1) implies (1.3) and (1.4).

Definition 1.1. We say that a function  is a sub solution of (1.1) if

(1.5)

In the same way, a super solution is a function that satisfies the same inequalities in reverse order. For we define the set

Definition 1.2. We say that and satisfies Nagumo condition relative to the intervalsand respectively, if for

there exists a constant  such that

and a continuous function such that

(1.6)

and

We finish this introduction with a lemma

Lemma 1.3. Let

  be defined by

Whereand are real constants such that

and here

Then exists and is continuous and defined by

where

and

2. Coupled Lower and Upper Solutions

To cover different possibilities for the nonlinear boundary functions  and  we introduce the following concept.

Definition 2.1. We say that are coupled lower and upper solutions for the problem (1.1) and (1.2) if  is a sub solution and  is asupersolution for the system (1.1),and

(2.1)

Theorem 2.2. Assume that  are coupled lower and upper solutions for the problem (1.1)-(1.2). Also assume that and satisfies a Nagumo condition relative to the intervals  and  respectively. Suppose that  is no ndecreasing in the third and fourth arguments. In addition suppose that the function  in  is monotone and the functions

have got the same kind of monotonocity as  then there exists at least one solution  of the problem (1.1)-(1.2). Furthermore,

Proof. Let  and consider the modified system

(2.2)

with

and

Note that if  is a solution of (2.2), then  is a solution of (1.1)-(1.2).

For the sake of simplicity we divide the proof in three steps:

Step 1: We define the mappings

by

and

Clearly  is continuous and compact by the direct application of Arzela-Ascoli theorem. Also from Lemma 1.3 with and  exists and is continuous.

On the other hand, solving (2.2) is equivalent to find a fixed point of

Now, Schauder's fixed point theorem guarantees the existence of at least one fixed point sinceis continuous and compact.

Step 2: If  is a solution of (2.2), then By definition of  we see that Thus, if is no ndecreasing, we have by condition (2.1)

(2.3)

Similarly, if  is no nincreasing, then (2.3) holds. Hence  Now, it remains to show that  for We claim  If then either  and/or If then there exist some  such that  So,  attains apositive maximum at  Thus  and But,

a contradiction. Similarly one can show that  Hence

Step 3: If  is a solution of (2.2) then  satisfies (1.2).

We claim

(2.4)

If

then

(2.5)

Similarly if is no ndecreasing then we have

(2.6)

Using (2.5), (2.6) and Step 2, we haveand But

(2.7)

a contradiction. Similarly if is no nincreasing we get same contradiction. Consequently, (2.4) holds. By definition of and Step 2, the second boundary condition is obvious. Consequently, satisfies (1.2).

Step 3: If is a solution of (2.2) then

We claimIf then either and/or Ifthen there exists such that Moreover using the Lagrange Theorem there exists with So,

Now consider an interval or such that and with

or

In the first situation we obtain form (1.2) that

Using (1.6), Step 2 andfor all  we get a contradiction.

Similarly in the second situation we get a contradiction. HenceThe proof of the other inequality is similar.

3. Example

Example 3.1: Consider the nonlinear coupled boundary value system (BVS for short) with nonlinear CBCs

(3.1)

(3.2)

Letandare the coupled lower and upper solutions of the BVS (3.1)-(3.2). Consequently

and

Furthermore the coupled lower and upper solutions satisfies the system (2.1). And the functions

satisfies the Nagumo condition (1.6) with and respectively. Hence by Theorem (2.2), BVS (3.1)-(3.2) has at least one solution


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