Solvability of Functional Integral-Differential Equations in the Sobolev space w^{k,infinity}(R^n)
Subject Areas : StatisticsMasoome Hosseini Farahi 1 , Mahmoud Hassani 2 , Reza Allahyari 3
1 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
2 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
3 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
Keywords: شرایط کاراتئودوری, اندازه های نافشردگی, قضیه نقطه ثابت داربو, معادلات انتگرال- دیفرانسیل, فضاهای سوبولوف,
Abstract :
In 1930, Kuratowski introduced the concept of measure of noncompactness. Later, Banas and Goebel generalized this concept axiomatically, which is more convenient in applications. The principal application of measures of noncompactness in fixed point theory is contained in the Darbo'sfixed point theorem. This is a tool to investigate the existence and behaviour of solutions of manyclasses of integral equations such as Volterra, Fredholm and Uryson types.The technique of measure of noncompactness is applicable in several branches of nonlinear analysis. In particular, it is a very useful tool for several types of integral and integral-differential equations. In addition, the measure of noncompactness is also used in functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory. The purpose of this article is to introduce a new measure of noncompactness in the Sobolev space W^(k,∞) (R^n). The results are obtained to solve integral-differential equations. Finally, by providing an example to show the efficiency of our results.
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