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
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.
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
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
and in the anti-periodic case it suffices that
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
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
We finish this introduction with a lemma
Lemma 1.3. Let
be defined by
Whereand are real constants such that
Then exists and is continuous and defined by
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
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
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
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)
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).
Similarly if is no ndecreasing then we have
Using (2.5), (2.6) and Step 2, we haveand But
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
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.
Example 3.1: Consider the nonlinear coupled boundary value system (BVS for short) with nonlinear CBCs
Letandare the coupled lower and upper solutions of the BVS (3.1)-(3.2). Consequently
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