doi: 10.17706/ijapm.2015.5.2.115-125
Existence of Extremal Solutions of Differential Equations with Delay in Banach Spaces
- 1 School of Mathematical Sciences, Shandong Normal University, Ji'nan, 250014, Shandong, China.
- 2 the Department of Mathematics, College of Arts and Science faculty, Abu Dhabi University, UAE.
Abstract
In this paper, by using the monotone iterative method and Daher's fixed point theorem and an inequality of noncompact measure of Monch and Von Harten, we study the existence of maximum solution and minimal solution and the iterative approximation of the following initial value problem for O.D.E with delay in Banach spaces <span id="cke_bm_89S" style="display: none;"> </span><span id="cke_bm_86S" style="display: none;"> </span><span id="cke_bm_84S" style="display: none;"> </span><em>u</em><span id="cke_bm_89E" style="display: none;"> </span><span id="cke_bm_86E" style="display: none;"> </span><span id="cke_bm_84E" style="display: none;"> </span>’ = <span id="cke_bm_90S" style="display: none;"> </span><span id="cke_bm_87S" style="display: none;"> </span><em>f</em><span id="cke_bm_90E" style="display: none;"> </span><span id="cke_bm_87E" style="display: none;"> </span>(t, <span id="cke_bm_91S" style="display: none;"> </span><em>u</em><span id="cke_bm_91E" style="display: none;"> </span>(<em>t</em>), <span id="cke_bm_92E" style="display: none;"> </span><span id="cke_bm_88E" style="display: none;"> </span><em>ut</em>), <em>ut</em>0 = <em>ψ</em>0 and give a sufficient condition of the existence of solutions of the above problem. The obtained results improve and generalize the corresponding results.<span id="cke_bm_85E" style="display: none;"> </span>
Keywords
- Banach space
- measure of noncompactness
- differential equation
- normal cone
- 2000 Mathematics Subject Classification 34G20
- 47H07
How to Cite
Jinjun Fan, Xiaodi Li, and Haydar Akca, "Existence of Extremal Solutions of Differential Equations with Delay in Banach Spaces," International Journal of Applied Physics and Mathematics, vol. 5, no. 2, pp. 115-125, 2015. https://doi.org/ 10.17706/ijapm.2015.5.2.115-125
Copyright & License
Copyright © 2015 by the authors. 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 (CC BY 4.0).