THE EXISTENCE OF SOLUTIONS FOR A NONLOCAL PROBLEM OF AN IMPLICIT FRACTIONAL-ORDER DIFFERENTIAL EQUATION

Fractional calculus which involves integro differential singular operators has lately caught attracted many engineers, physicists and certainly paying attention mathematician owing to their extensive applications in a multiplicity of fields such as dynamical systems, solid mechanics, viscoelasticity, control etc., see for instance the monographs by: Baleanu et al. [1], Coimbra et al. [2], Coimbra [3], Dalir and Bashour [4], Diethelm [5], Glockle and Nonnenmacher [6], Hilfer [7], Ingman and Suzdalnitsky [8], Kilbas et al. [9], Machado et al. [10], Metzler et al. [11] Rossikhin and Shitikova [12], Sabatier et al. [13], Samko and Marichev [14], Sweilam and AL-Mrawm [15], Yajima and Yamasaki [16] and the references in that. Numerous researchers form mathematics group fond on investigating existence, stability, uniqueness and additional properties for implicit fractional differential problems (IFDPs) by assorted formulas of fractional differential equations with different formulae of fractional derivative operators. The researchers investigate the case of implicit functions, they considered the nonlinear function f depends on the fractional derivative of the unknown function, see for example, Abbas et al. [17], Benavides [18], Benchohra et al. [19][23], El-Sayed and Bin-Taher [24][26], Guezane-Lakoud and Khaldi [27], Nieto et al. [28], Vityuk and Mykhailenko [29] and references therein. The papers on integrable solutions for fractional differential equations is extremely constrained, see papers by: Benchohra et al. [19, 20,22,23], El-Sayed and Abd El-Salam [30–32], El-Sayed and Hashem [33] and references therein. Motivated by the above works, In this paper, we study the existence of at least one integrable solution for the implicit fractional order differential problem (IFDP):