Some Studies on Quadratic Functional Integral and Differential Equations of Fractional Order

Abstract

In recent years, the topic of nonlinear integral and differential equations in newlineBanach algebras is received the attention of several authors. Now a days, there is a newlineconsiderable literature available in this direction. While functional differential newlineequations have been a very active area of research for a long time, the study of newlinefunctional differential equations in Banach algebras is relatively new to the literature. newlineVery recently, the study of problems of this type by using fixed point theorems have newlinebegun to appear in the literature; for example, see Dhage [25,26,29] and Dhage and newlineO regan [38] and the references therein. newlineThe theory of fractional calculus (that is fractional order differential and newlineintegral equation) has newly received a lot of attention and establishes a meaningful newlinebranch of nonlinear analysis. Number of research monographs and research papers newlinehas appeared and devoted to integral and differential equations of fractional order. newlineFixed point theory is a fascinating subject, with an enormous number of newlineapplications in various fields of mathematics. Fixed point theory constitutes an newlineimportant and the core part of the subject of nonlinear functional analysis and is newlineuseful for proving the existence theorems for nonlinear differential and integral newlineequations. The fixed point theory is useful for proving the existence the solutions of newlinethe problems governed by nonlinear differential or integral equations [63, 94]. newlineDifferential and Integral equations are one of the most useful Mathematical newlinetools in both applied and pure mathematics. Moreover the theory of Differential and newlineIntegral equations is rapidly developing using the tools of Topology, Funct ional newlineAnalysis and Fixed point theory. This is particularly true of problems in the related newlinefields of Engineering, Mechanical Vibrations and Mathematical Physics. There are newlinenumerous applications of Differential equations and Integral equations of integer and newlinefractional orders in Electrochemistry, Viscoelasticity, Control theory, newlineElectromagnetism and Porous media etc. (s