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Sobolev Type Differential Equations

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Sobolev Type Differential Equations

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dc.contributor.author Lord, M. E. en
dc.contributor.author Lakshmikantham, V. en
dc.date.accessioned 2010-06-03T16:14:09Z en
dc.date.available 2010-06-03T16:14:09Z en
dc.date.issued 1977-06 en
dc.identifier.uri http://hdl.handle.net/10106/2303 en
dc.description.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. en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;58 en
dc.subject Integro-differential equations en
dc.subject Bell-Gronwall inequality en
dc.subject Differential equation en
dc.subject Linear constant coefficient equation en
dc.subject Sobolev type en
dc.subject.lcsh Mathematics Research en
dc.title Sobolev Type Differential Equations en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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