Differential Equation And Integral Differential Equations, The Existence Of Mild Solution,

The theory of neutral differential equations and integrodifferential equations is arising in the physical science,biology and other applied subjects.And with the development of the other subjects it has been a tremendous development.The nonlocal condition, which is a generalization of zhe classical initial condition, was motivated by physcial problems, the pioneering work on nonlocal conditions is due to Byszewski and it has been studied extensively.Nonlocal problems have better effects in applications than the tranditional Cauchy problems with initial condition. The nonlocal condition is used to escribe the diffusion phenomenon of a small amount of gas in a transparent tube . So more information is aviable.As the further research of differential equations ,neutral differential equations with nonlocal conditions have deep physical back ground because of all the stucture of its emergence. So the research on neutral differential equations with nonlocal conditions is very meaningful.Equations with delay are often more realistic to describe natural phenomena than those without delay. Therefore it is significance to study the existence of a class of partial neutral functional differential equations and integrodifferential equations with unbounded delay.The purpose of this paper is to prove the existence of mild solutions for a class of semilinear neutral differential equations with nonlocal conditions in a Banach space and the existence of mild solutions for a class of partial neutral functional differential equations and integrodifferential equations with unbounded delay.Neither the semigroup {T ( t ): t≥0}nor the function f is needed to be compact in our result.The results obtained are presented as follows:In the first chapter we give the existence of mild solutions for a class of semilinear neutral differential equations with nonlocal conditions in a Banachspace .We establish the existence of mild solutions by using the conditions in respect of the Hausdorff 's measure of noncompactness and the theory of Darbo -Sadovskii . And the compactness of T ( t ) and f are the special case of our conditions.Hence we generalize and improve some related results.In the second chapter,we prove the existence of mild solutions for a partial neutral functional differential equation with unbounded delay .We establish the existence of mild solutions by using the conditions in respect of the Hausdorff 's measure of noncompactneness and the theory of M ?n ch. Hence we generalize and improve some related results.Then we prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay by using the conditions in respect of the Hausdorff 's measure of noncompactneness and the theory of M ?n ch. And the compactness of T ( t ) and f are the special case of our conditions . Hence we generalize and improve some related results.