Abstract:

Recently [2] a class of new differential equations called Sobolev type differential equations was introduced and Picard type existence theorem for such equations was proved among other results like variation of constants formula and Gronwall type inequalities for such equations. However Peano's type existence result was open for such equations since it was not clear how to proceed. Special cases of such equations occur in imbedding method for solving Fredholm integral equations [5].
In this paper we consider Volterra integral equations of Sobolev type which include the foregoing class of differential equations as special cases. We consider Peano's type existence result, discuss integral inequalities, prove existence of extremal solutions and derive a comparison theorem. For future use, we state, as a corollary, Peano's existence theorem for differential equations of Sobolev type, thus
2
solving an open problem. Our results naturally include as a special case the results for Volterra integral inequalities of usual type [1,3,4].
Consider the following system of integral equations
[see pdf for notations] where [see pdf for notations]
R and ^ is an open subset of Rn. For convenience, we list below needed assumptions.
[see pdf for notations]
for every set [see pdf for notations] and for every interval [see pdf for notations].
We also use [see pdf for notations] to denote the ball in Rn of radius e centered at u(t0,x0). 