Terminal value problem for causal differential equations with a Caputofractional derivative

Abstract

In this paper, we have given new definitions and obtained the unique solution of a fractional causal terminal value problem by combining the technique of generalized quasilinearization in the sense of upper and lower solutions.

Terminal Value Problems with Causal Operators

“…This is the main advantage of causal differential equations [11] in comparison with traditional models [10]. There has been a growing interest in this new area to study the concept of causal systems in the qualitative behaviors [3,11,22,24]. The monotone iterative technique [4,9,14,16,17,[20][21][22][23] coupled with the method of upper and lower solutions offers monotone sequences that converge uniformly and monotonically to the extremal solutions of the given nonlinear causal differential equations.…”

“…The study of differential equations [1,9,10,21] with causal operators [2-5, 7, 11-13, 19, 22-24] has a rapid development in the recent years and some results are assembled in a recent monograph [11]. The term of causal operators is adopted from engineering literature and the theory these operators has the powerful quality of unifying namely the fractional differential equations [2,11,14,15,24], ordinary differential equations [9,10], integro-differential equations [18], differential equations with finite or infinite delay, Volterra integral equations and neutral functional equations [3,11].…”

“…This is the main advantage of causal dierential equations [11] in comparison with traditional models [10]. There has been a growing interest in this new area to study the concept of causal systems in the qualitative behaviors [3,11,22,24].…”