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

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.