A Refinement of Quasilinearization Method for Caputo's Sense Fractional-Order Differential Equations

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions. 1. Introduction The well-known quasilinearization method [1, 2] in differential equation has been employed to obtain a sequence of lower and upper bounds which are the solutions of linear differential equations that converge quadratically. However, the convexity and concavity assumption that is demanded by the method of quasilinearization has been a stumbling block for further development of the theory. Recently, this method has been generalized, refined, and extended in several directions so as to be applicable to a much larger class of nonlinear problems by not demanding convexity and concavity property [1, 3–7]. Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications [3, 6, 7]. The theory of nonlinear fractional-order dynamic systems has been investigated depending on the development in the theory of fractional-order differential equations. In this context, generalized quasilinearization method has been reconsidered, and similar results parallel to classical theory of differential equations have been obtained [1, 2, 8]. In this work, the quasilinearization technique coupled with lower and upper solutions is employed to study Caputo’s fractional-order differential equation for which particular and general results that include several special cases are obtained. Moreover, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and the sequences converge to the solutions of the original nonlinear problems. Instead of imposing the convexity assumption on the function involved, we assume weaker conditions as well as for the concave functions. This is a definite advantage of this constructive technique. Furthermore, these monotone flows are shown to converge semiquadratically. Consider the following initial value problem: where and is Caputo’s sense fractional-order derivative. Let and be the lower and upper solutions of (1.1) satisfying the following inequalities (1.2) and (1.3), respectively, on : The corresponding Volterra fractional integral equation is Caputo’s sense fractional-order