Solutions of nonstandard initial value problems for a first order ordinary differential equation. (English) Zbl 0717.34007

Venkatesulu, M.; Srinivasu, P. D. N.

J. Appl. Math. Simulation 2, No. 4, 225-238 (1989).

Summary: 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 \(y'=f(t,y,y')\), \(y(t_ 0)=y_ 0\) 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.

MSC:

34A12

Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

nonstandard initial value problems; scalar-valued connected continuous dependence; functional equation; global existence; example