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. | Turk J Math (2017) 41: 1042 – 1052 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Terminal value problem for causal differential equations with a Caputo fractional derivative Co¸skun YAKAR, Mehmet ARSLAN∗ Department of Mathematics, Faculty of Fundamental Sciences, Gebze Technical University, Gebze, Turkey Received: • Accepted/Published Online: • Final Version: 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. Key words: Causal operator, terminal value problem, Caputo fractional derivative, quasilinearization method, quadratic convergence, upper and lower solutions 1. Introduction Recently, the study of differential equations [6] with causal operators [13] has rapidly developed and some results are assembled in [8,13,19,26]. The theory of causal operators is a powerful tool unifying the fractional order differential equations [4,16,25,27], ordinary differential equations [1,8,11,28], integro-differential equations [23], differential equations with finite or infinite delay, Volterra integral equations [23], and neutral functional equations [8,13,20]. There has been rapidly growing interest in the study of fractional differential equations [2,4,5,13,16,18,21,22,25,27] because recent investigations in science and engineering have indicated that the dynamics of many systems can be described more accurately by using differential equations of a noninteger order. It has recently been shown that causal differential equations [2,8,9,13,19,20,26] provide excellent models for real world problems [8] and its real time applications in a variety of disciplines. This is not only the main advantage of causal differential equations in comparison with the traditional models [12] and there
