Monotone iterative technique for nonlinear differential equation of fractional order

Abstract

In this paper, we mainly study the existence of solution of fractional differential equations. Firstly, the existence of the maxmum solution and minmum solution of the differential equation are proved by using the fixed point theorem and the monotone iteration method. Secondly, the existence of the solution of the original equation is proved by using the newly constructed differential equation. Finally, the application of the monotone iteration method is given through an example.

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

“…are a couple of monotone sequences of ordered coupled of lower and upper quasi-solutions of ( 7) and (8).…”

“…is pair of coupled lower and upper quasi-solutions of ( 7) and (8). Now, we shall prove that k v and k w are ordered.…”

“…Fractional differential equations and systems are appearing in a variety of scientific and engineering branches being mathematical modelling in many fields, namely, in Physics, electromagnetic, acoustic, viscoelasticity, electrochemistry, economics, signal and image processing, control theory, etc. For details, one can see [1]- [8]. Many works treated the problems concerning fractional differential equations or systems by using several methods.…”

“…Now, let us prove that, for each 1 k ≥ , the pair k v and k w are an ordered coupled lower and upper quasi-solutions of ( 7) and (8). By adding ( ) , , ,…”

“…So v * satisfies ( 9) and (10), in the same manner, we prove that w * satisfies (11) and (12). Therefore, v * and w * are coupled quasi-solutions of ( 7) and (8)…”