Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation
In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term. The fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. The integral equivalent equation with impuls...
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| Format: | Article |
| Language: | English |
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Wiley
2019-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2019/2523615 |
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| author | D. K. Igobi U. Abasiekwere |
| author_facet | D. K. Igobi U. Abasiekwere |
| author_sort | D. K. Igobi |
| collection | DOAJ |
| description | In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term. The fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions is embedded in the space of generalized ordinary differential equations (GODEs), and the correspondence between the generalized ordinary differential equation and the nonhomogeneous retarded equation coupled with impulsive term is established by the construction of a local flow by means of a topological dynamic satisfying certain technical conditions. The uniqueness of the equation solution is proved. The results obtained follow the primitive Riemann concept of integration from a simple understanding. |
| format | Article |
| id | doaj-art-45a5b55958824e0c8d84c4020fca8c1d |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-45a5b55958824e0c8d84c4020fca8c1d2025-02-03T05:59:36ZengWileyInternational Journal of Differential Equations1687-96431687-96512019-01-01201910.1155/2019/25236152523615Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential EquationD. K. Igobi0U. Abasiekwere1Department of Mathematics and Statistics, University of Uyo, P.M.B. 1017, NigeriaDepartment of Mathematics and Statistics, University of Uyo, P.M.B. 1017, NigeriaIn this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term. The fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions is embedded in the space of generalized ordinary differential equations (GODEs), and the correspondence between the generalized ordinary differential equation and the nonhomogeneous retarded equation coupled with impulsive term is established by the construction of a local flow by means of a topological dynamic satisfying certain technical conditions. The uniqueness of the equation solution is proved. The results obtained follow the primitive Riemann concept of integration from a simple understanding.http://dx.doi.org/10.1155/2019/2523615 |
| spellingShingle | D. K. Igobi U. Abasiekwere Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation International Journal of Differential Equations |
| title | Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation |
| title_full | Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation |
| title_fullStr | Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation |
| title_full_unstemmed | Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation |
| title_short | Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation |
| title_sort | results on uniqueness of solution of nonhomogeneous impulsive retarded equation using the generalized ordinary differential equation |
| url | http://dx.doi.org/10.1155/2019/2523615 |
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