On The Attractivity Result for Some Nonlinear Functional Differential and Integral Equations of Fractional Order

Abstract

Nonlinear differential and integral equation of fractional order are an newlineimportant topic of the nonlinear analysis. Recently to investigations have many newlinephysical systems can be represented more than accurately through the fractional newlinederivative formulation.The study of nonlinear functional differential and integral newlineequations have been affecting the large area of literature by some of the authors all newlineover the world see ([43],[70],[71]). Some of the types of the problem may be newlineinvolving the discontinuous terms[107]. Therefore, it is of interest to study the newlinenonlinear differential and integral equation have attracted to attention of several newlinemathematician of the world since a long times work have been done to the concerning newlinethe various aspect of the solutions for nonlinear differential and integral equations newline(see. [8],[84],[93],[97]). newlineThe origin of the nonlinear integral equation in Banach Algebra lies in the newlineworks for famous Indian Physicists Chandrasekhar (1980) in his studies newlineThermodynamics [104].The method will be developed for existence solution of newlinenonlinear integral and differential equations in fractional order is very much newlinecumbrances and involves several technique [52].The established a general tools for newlinesolving nonlinear differential and integral equations, of integer order have been newlinestudied by some of the authors for their different aspects of solutions by using Banach newlinefixed point principle, Shauders fixed point and Kransnoselskii fixed point principle newline[25]. newlineFractional calculus is the branch of mathematical analysis to study which newlinecontracts with investigation and applications of integrals and derivatives of arbitrary newlineorder [69]. To the origin of the fractional calculus goes back to 1695. The famous newlinemathematician Leibniz. Introduced to derivatives of order half. The basic newlinemathematical ideas of fractional calculus were developed long ago of newlineMathematicians, G. b. Leibniz (1659), Euler s (1738), Laplace (1820), J. Fourier newline(1822), Abel (1823), Liouville (1834), Riemann (1847), Lagrange (1849), newlineGurnewland (1867), L