Nonlinear caputo fractional impulsive differential equations and generalized comparison results

摘要

It is known that Caputo fractional differential equations play an important role in modeling many physical situation. The models represented by Caputo fractional differential equation in general are better and efficient models than its counterpart with integer derivative models. In this work, we consider nonlinear Caputo impulsive fractional differential equations with initial conditions. Further, the impulses occur in the nonhomogeneous term. Initially, we have computed the solution of the linear Caputo impulsive fractional differential equation explicitly using the method of mathematical induction. We have developed comparison results in terms of coupled lower and upper solutions when the nonlinear terms are sums of an increasing and decreasing functions of the unknown function. Finally, we have developed generalized monotone method for the Nonlinear Caputo Impulsive Fractional Differential Equations with initial conditions. This proves the existence coupled minimal and maximal solutions of the nonlinear problem. Finally, under uniqueness condition, we prove the existence of the unique solution of the nonlinear Caputo fractional im-pulsive differential equation with initial condition. Further, the interval of existence is guaranteed by the upper and lower solutions. In this work, we have obtained the basic tools to enable us to develop the generalized iterative method for the nonlinear Caputo fractional impulsive differential with initial conditions. The basic tools developed are the explicit solution of the corresponding linear Caputo fractional impulsive differential equations with initial condition. We have achieved this by applying Laplace transform method. Laplace transform method is the most suitable method since the Caputo fractional derivative is a convolution integral. This explicit form is useful in establishing the uniqueness of the solution of the linear Caputo fractional impulsive differential conditions. We have developed two comparison theorems wh ich are useful in proving the monotonicity of the linear iterates that will arise in the generalized monotone method and the uniqueness of the solution of the nonlinear Caputo fractional impulsive differential equation. We have also presented some numerical results. See [8, 19] for results on generalized iterative method for Caputo fractional differential equations without impulses.