## Sobolev Type Differential Equations

##### Abstract

An imbedding method for solving linear Fredholm integral equations was introduced by Sobolev [3] which involves the solution of the following differential equation with initial value for the resolvent kernel [see pdf for notation].
The differential equation in (1.1) is unusual in that the solution K(t,y,x) is evaluated at different combinations of the independent variables (t,y,x) . We will refer to any differential equation with this property as
a Sobolev type differential equation.
We introduce a Sobolev type differential equation which generalizes (1.1) and consider conditions for existence and uniqueness. A Picard type theorem is obtained, which by way of application, is used to obtain conditions for existence and uniqueness for the system (1.1).
Some further results are obtained for Sobolev type differential equations including solution of a linear constant coefficient equation, a Bellman-Gronwall type inequality and a variation of constants formula for solutions of perturbed linear equations. These results generalize in a natural way corresponding well known results as found in [2].
An application of Sobolev's imbedding technique to nonlinear Fredholm integral equations was obtained by Kagiwada and Kalaba [1]. The result obtained there involves a system of integro-differential equations of Sobolev type. Theory relating to this result is more complex and will be developed in a subsequent paper.