Upper and Lower Solution Method for Nonlinear Differential Equations and Applications

Abstract

We have developed method of upper and lower solutions for nonlinear fractional newlinedifferential equations involving Riemann-Liouville fractional derivative and Caputo type newlinefractional derivative under different initial and boundary conditions. Upper and lower newlinesolution method together with fixed point theorem on cone is developed by using Greens newlinefunction. The basic idea of upper and lower solution method is to develop iterative newlinescheme based on notion of upper and lower solutions. Using upper and lower solutions newlineas initial iterations two monotone convergent sequences are constructed. They converge newlinemonotonically from above and below to maximal and minimal solutions respectively. newlineThe beauty of this method is that it is constructive and simple and hence powerful and newlineelegant. As an application of this upper and lower solution method, we obtain existence newlineand uniqueness results of solutions of nonlinear fractional differential equations under newlinedifferent conditions. newlineThe thesis comprises of eight chapters. The chapter one is introductory. The chapter two newlineis devoted to the development of the upper and lower solution method together with the newlinefixed point theorem on cone for Riemann-Liouville nonlinear fractional differential newlineequations with multipoint boundary value condition. The existence and uniqueness of newlinepositive solution in the closed set defined by the upper and lower solutions is obtained. newlineThe chapter three deals with the upper and lower solution method together with the fixed newlinepoint theorem on cone for nonlinear fractional differential equations with integral newlineboundary conditions. Upper and lower solution method is developed for this problem. As newlinean application of the upper and lower solution method existence and uniqueness of newlinesolutions of the problem are obtained. The chapter four is associated with the upper and newlinelower solution method Caputo nonlinear fractional differential equations even when the newlinefunctions on the right hand side are nonmonotone and existence and uniqueness results newlineare obtained by applying upper and lower sol