Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions
We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some suffic...
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Wiley
2016-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2016/3910972 |
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| author | Douglas R. Anderson Christopher C. Tisdell |
| author_facet | Douglas R. Anderson Christopher C. Tisdell |
| author_sort | Douglas R. Anderson |
| collection | DOAJ |
| description | We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs. |
| format | Article |
| id | doaj-art-bd7b8ccca37d48c09b1a11666cc063ad |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-bd7b8ccca37d48c09b1a11666cc063ad2025-02-03T05:47:15ZengWileyAbstract and Applied Analysis1085-33751687-04092016-01-01201610.1155/2016/39109723910972Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of SolutionsDouglas R. Anderson0Christopher C. Tisdell1Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USASchool of Mathematics and Statistics, The University of New South Wales (UNSW), Sydney, NSW 2052, AustraliaWe investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs.http://dx.doi.org/10.1155/2016/3910972 |
| spellingShingle | Douglas R. Anderson Christopher C. Tisdell Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions Abstract and Applied Analysis |
| title | Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions |
| title_full | Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions |
| title_fullStr | Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions |
| title_full_unstemmed | Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions |
| title_short | Discrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutions |
| title_sort | discrete approaches to continuous boundary value problems existence and convergence of solutions |
| url | http://dx.doi.org/10.1155/2016/3910972 |
| work_keys_str_mv | AT douglasranderson discreteapproachestocontinuousboundaryvalueproblemsexistenceandconvergenceofsolutions AT christopherctisdell discreteapproachestocontinuousboundaryvalueproblemsexistenceandconvergenceofsolutions |