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

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

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Article ID: iaor1990596
Country: United States
Volume: 2
Start Page Number: 1
End Page Number: 7
Publication Date: May 1989
Journal: Journal of Applied Mathematics and Stochastic Analysis
Authors: ,
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, the authors 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, they prove the continuous dependence of the solutions on initial conditions and parameters. Finally, the authors prove a global existence result and present an example to illustrate the theory.

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