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Journal of Applied Mathematics and Simulation
Volume 2, Issue 4, Pages 225-237
http://dx.doi.org/10.1155/S1048953389000183

Solutions of nonstandard initial value problems for a first order ordinary differential equation

Department of Mathematics, Sri Sathya Sai Institute of Higher Learning, Prasanthinilayam, A.P. 515 134, India

Received 21 April 1989; Revised 1 July 1989

Copyright © 1989 Hindawi Publishing Corporation. 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.

Abstract

Differential equations of the form y=f(t,y,y), where f is not necessarily linear in its arguments, represent certain physical phenomena and have been known to mathematicians for quite a long time. But a fairly general existence theory for solutions of the above type of problems does not exist because the (nonstandard) initial value problem y=f(t,y,y), y(t0)=y0 does not permit an equivalent integral equation of the conventional form. Hence, our aim here is to present a systematic study of solutions of the NSTD IVPs mentioned above.

First, we establish the equivalence of the NSTD IVP with a functional equation and prove the local existence of a unique solution of the NSTD IVP via the functional equation. Secondly, we prove the continuous dependence of the solutions on initial conditions and parameters. Finally, we prove a global existence result and present an example to illustrate the theory.