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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 357651, 14 pages
First-Order Three-Point Boundary Value Problems at Resonance Part III
1Jabatan Matematik dan Statistik, Universiti Teknologi MARA Perlis, 02600 Arau, Malaysia
2Department of Mathematics, The University of Queensland, Brisbane, QLD 4072, Australia
3Fakulti kejuruteraan Awam, Universiti Teknologi MARA Perlis, 02600 Arau, Malaysia
Received 11 December 2011; Accepted 17 January 2012
Academic Editor: Yeong-Cheng Liou
Copyright © 2012 Mesliza Mohamed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The main purpose of this paper is to investigate the existence of solutions of BVPs for a very general case in which both the system of ordinary differential equations and the boundary conditions are nonlinear. By employing the implicit function theorem, sufficient conditions for the existence of three-point boundary value problems are established.
Consider where , , and are constant square matrices of order , is an matrix with continuous entries, is continuous, is a continuous function where , , , and is continuous.
Our existence theorem uses the implicit function theorem; see for example Nagle . Nagle  extended the alternative method considered by Hale  for handling the periodic case of non-self-adjoint problems subject to homogeneous boundary conditions. These results extend the work of Feng and Webb  and Gupta  of three-point BVPs with linear boundary conditions for and to nonlinear boundary conditions. Feng and Webb  studied the existence of solutions of the following BVPs (1.3) and (1.4): where , , is a continuous function, and is a function in . Both of the problems are resonance cases under the assumption for the problem (1.3), and for the problem (1.4). The problem for nonlinear boundary conditions for discrete systems has been studied by Rodriguez [7, 8]. Rodriguez  extended results of Halanay , who considered periodic boundary conditions and also extended those of Rodriguez  and Agarwal  who considered linear boundary conditions. To our knowledge there appears to be no research in the literature on multipoint BVPs for systems of first-order equations with nonlinear boundary conditions at resonance. The results of this paper fill this gap in the literature.
Our results are analogues for three-point boundary conditions of those periodic boundary conditions for perturbed systems of first-order equations at resonance considered by Coddington and Levinson  and Cronin [13, 14]. Moreover, our results extend the work of Urabe , Liu , and of Nagle , where he solved the two-point BVP using the Cesari-Hale alternative method.
Theorem 2.1 (see Coppel [17, Page 19]). (i)Let be a continuous function of for all points in an open set and all values near .(ii)Let be any noncontinuable solution of the differential equation If is defined on the interval and is unique, then is defined on for all sufficiently near and is a continuous function of its threefold arguments at any point .
Theorem 2.2 (see Coppel [17, Page 22]).
(i) Let be a continuous function of for all points in a domain and all values of the vector parameter near .
(ii)Let be a solution of the differential equation
defined on a compact interval .
(iii)Suppose that has continuous partial derivatives , at all points with .
Then for all sufficiently near the differential equation has a unique solution over that is close to the solution of . The continuous differentiability of with respect to and implies the additional property that the solution is differentiable with respect to for near .
We recall the following results of .
Lemma 2.3 (see ). Consider the system where is an matrix with continuous entries on the interval . Let be a fundamental matrix of (2.4). Then the solution of (2.4) which satisfies the initial condition is where is a constant -vector. Abbreviate to . Thus .
Lemma 2.4 (see ). Let be a fundamental matrix of (2.4). Then any solution of (1.1) and (2.5) can be written as The solution (1.1) satisfies the boundary conditions (1.2) if and only if where , = + − , + , and is the solution of (1.1) given .
Next we suppose that is a singular matrix. This is sometimes called the resonance case or degenerate case. Now we consider the case rank , . Let denote the null space of , and let denote the complement in of ; that is, Let be a basis for such that is a basis for and a basis for .
Let be the matrix projection onto Ker , and , where is the identity matrix. Thus is a projection onto the complementary space of , and Without loss of generality, we may assume We will identify with and with whenever it is convenient to do so.
Let be a nonsingular matrix satisfying Matrix can be computed easily. The nature of the solutions of the branching equations depends heavily on the rank of the matrix .
Next we give a necessary and sufficient condition for the existence of solutions of of three-point BVPs for such that the solution satisfies where for suitable .
We need to solve (2.7) for when is sufficiently small. The problem of finding solutions to (1.1) and (1.2) is reduced to that of solving the branching equations (2.7) for as function of for . So consider (2.7) which is equivalent to Multiplying (2.7) by the matrix and using (2.11), we have where = + − and = + .
Since the matrix is nonsingular, solving (2.7) for is equivalent to solving (2.13) for . The following theorem due to Cronin [13, 14] gives a necessary condition for the existence of solutions to the BVP (1.1) and (1.2).
If is a nonsingular matrix then the implicit function theorem is applicable to solve (2.7) uniquely for as a function of in a neighborhood of the initial solution (see Cronin ). The implicit function theorem may be stated as in Voxman and Goetschel [18, page 222].
Theorem 2.7 (the implicit function theorem). Let be an open set, and let be function of class . Suppose . Assume that where . Then there are open sets and , with and , and a unique function such that for all with . Furthermore, is of class .
3. Main Results
Definition 3.1 (see ). Let denote the null space of , and let denote the complement in of . Let be the matrix projection onto , and , where is the identity matrix. Thus is a projection onto the complementary space of . If is properly contained in , then is an -dimensional vector space where . If , let and , then define a continuous mapping , given by
where is a differentiable function of and . By abuse of notation we will identify and when convenient and where the meaning is clear from the context so that in defining above from the context we interpreted as . Similarly we will sometimes identify and . Setting , we have
where ; note that from the context is interpreted as .
If and , then . Since , it follows that the matrix is the identity matrix. Thus define a continuous mapping , given by . Setting , we have .
The following theorem is the main theorem of this paper and gives sufficient conditions for the existence of solutions of (1.1), (1.2) for , for some . The existence theorem can be established using the implicit function theorem; see Theorem 2.7.
Theorem 3.2. If , let . Let the conditions , , and of Theorem 2.2 hold, and let , and be small enough so that (1.1) has a unique -vector defined on . Let , given by where is a differentiable function of and , and for . If and for some , then there is , , and such that (1.1), (1.2) has a unique solution for all such that and .
Proof. The existence and uniqueness of a solution for with follows directly from conditions (i), (ii), and (iii) of Theorem 2.2. Now for some , thus it follows from the implicit function theorem that there is , such that (3.3) has a unique solution , with , for all , . From this it follows that is a unique solution of the BVP (1.1), (1.2) which satisfies the initial value and and , where .
Theorem 3.4. Let the conditions , , and of Theorem 2.2 hold, and let , and be small enough so that (1.1) has a unique solution defined on . If , , and then there is , , and such that (1.1), (1.2) has a unique solution for all such that and .
Proof. If and , then . This implies . Since , it follows that , the identity matrix.
The existence and uniqueness of a solution for with follows directly from conditions (i), (ii) and (iii) of Theorem 2.2. Now If , for some ; thus it follows from the implicit function theorem that there is , such that (3.8) has a unique solution , with , for all , . From this it follows that is a unique solution of the BVP (1.1), (1.2) which satisfies the initial values for all , such that and .
4. Some Examples
Example 4.1. , rank , for .
Consider the BVP where , , . Then the BVP (4.1) is equivalent to where By Lemma 2.4, we find : The resonance happens if ; that is the case where . For , rank ; that is, Let denote the null space of . Thus is a basis for , and . Let be the matrix projection onto . . . Set = so that . In system (4.2), (4.3) let , , and let = . We need to show that which is a necessary condition in order to apply Theorem 2.6: Since , it follows that . From the boundary condition (4.3), we have . Then, by the variation of constants formula, we obtain Thus the BVP (4.2), (4.3) has a solution if , ; namely, , , , . Setting , thus and . Hence If or , then and
Hence by Theorem 3.2 there is , and such that the BVP (4.2), (4.3) has a unique solution which satisfies the initial values for all such that and .
Example 4.2. Rank .
Consider the BVP where , , , , By Lemma 2.4, the problem of solving (4.11), (4.12) is reduced to that of solving for provided solutions of initial value problems exist on for each . Thus we find : Since rank , it follows that the matrix is singular. Let denote the null space of . Thus is a basis for , and . Let be the matrix projection onto . . So . Set so that . Since , it follows that . Thus a necessary condition of Theorem 2.6 holds. We also have . To obtain we must first calculate ; that is the solution of . By Lemma 2.3, and boundary condition (4.12), has a solution with . We note that at , , where and . Hence and . Thus Thus the BVP (4.11), (4.12) has a solution if ; namely, and , and thus , , , and : where Thus , where and . Setting , we have , where . Writing out the components and setting , we obtain and . Hence where , , , and . Let , and . Hence If , then and Hence by Theorem 3.2 there is , and such that the BVP (4.11), (4.12) has a unique solution which satisfies the initial values for all such that and .
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