Existence of solution for first-order coupled system with nonlinear coupled boundary conditions
- Naseer Ahmad Asif^{1},
- Imran Talib^{1}Email authorView ORCID ID profile and
- Cemil Tunc^{2}
Received: 19 March 2015
Accepted: 19 July 2015
Published: 7 August 2015
Abstract
In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.
Keywords
1 Introduction
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 [1–3]. Moreover, while dealing with nonlinear ordinary differential systems (ODSs for short) mostly authors only focus attention on the differential systems with uncoupled boundary conditions [4–6]. But, on the other hand, very little research work is available where the differential systems are coupled not only in the differential systems but also through the boundary conditions [7, 8]. Our system (1)-(2) deals with the latter case.
Definition 1.1
In the same way, a supersolution is a function \((\beta_{1},\beta _{2} )\in C^{1}[0,1]\times C^{1}[0,1]\) that satisfies the reversed inequalities in (5). In what follows we shall assume that \((\alpha_{1},\alpha_{2} )\preceq (\beta_{1},\beta_{2} )\), if \(\alpha_{1}(t)\leq\beta_{1}(t)\) and \(\alpha_{2}(t)\leq\beta _{2}(t)\), for all \(t\in[0,1]\) or \((\alpha_{1},\alpha_{2} )\succeq (\beta_{1},\beta_{2} )\), if \(\alpha_{1}(t)\geq\beta _{1}(t)\) and \(\alpha_{2}(t)\geq\beta_{2}(t)\), for all \(t\in[0,1]\).
Lemma 1.2
Proof
2 Coupled lower and upper solutions
The following definition is very helpful to construct the statement of the main result (2.2), and also it covers different possibilities for the nonlinear function h.
Definition 2.1
Theorem 2.2
Proof
For the sake of simplicity we divide the proof into three steps.
Step 2: It remains to show that \((u,v)\in [\alpha_{1},\beta_{1} ]\times [\alpha_{2},\beta_{2} ]\).
We claim that \((u,v )\preceq (\beta_{1},\beta_{2} )\). If \((u,v )\npreceq (\beta_{1},\beta_{2} )\), then \(u\npreceq\beta_{1}\) and/or \(v\npreceq\beta_{2}\). If \(u\npreceq\beta _{1}\), then there exist some \(r_{0}\in[0,1]\), such that \(u-\beta_{1}\) attains a positive maximum at \(r_{0}\in[0,1]\). We shall consider three cases.
Case 3. Similarly \(h_{\beta}\) is monotone nondecreasing. We shall change the inequality (25) by \((u(0),v(0) )\preceq (\beta_{1}(0),\beta_{2}(0) )-h (\beta_{1}(0),\beta _{2}(0),\alpha_{1}(1),\alpha_{2}(1) )\) and again we get a contradiction. Consequently, \((u,v )\preceq (\beta_{1},\beta_{2} )\), for all \(t\in[0,1]\). Similarly, we can show that \((u,v )\succeq (\alpha_{1},\alpha_{2} )\), for all \(t\in[0,1]\).
Step 3: Now, it remains to show that \((u,v)\) satisfies the boundary condition (2).
Remark 2.3
Theorem 2.4
3 Examples
Example 3.1
Example 3.2
4 Conclusion
The new existence results are established for a nonlinear ordinary coupled system with nonlinear CBCs. The developed result unifies the treatment of many first-order problems [12–15]. Examples are included to verify the theoretical results. The existence results are also discussed when the lower and upper solutions are in reverse order \((\alpha_{1},\alpha_{2})\succeq(\beta_{1},\beta_{2})\).
Declarations
Acknowledgement
The authors are grateful for the referees careful reading and comments on this paper that led to the improvement of the original manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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