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):

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): where f : (0, T ] × R 3 → R, k : [0, T ] × [0, T ] → R are given functions and D α is the Riemann-Liouville fractional-order derivative of order α ∈ (0, 1).Moreover, we will examine the uniqueness of the solution in L 1 (J, R), J = (0, T ] space and in the weighted space C 1−α (J, R).We provide examples to clarify our acquired outcomes.

Preliminaries
Let L 1 (J, R) denoted the space of all Lebesgue integrable functions on the interval J = (0, T ] with the standard norm Definition 2.1.( [1,14]) The Riemann-Liouville fractional integral of the function f ∈ L 1 (J, R) is known as The following theorems will be needed.Theorem 2.1.(Kolmogorov compactness criterion [34]) The superposition operator generated by f is identified as follows: For the existence of a solution to problem (1.1)-(1.2),we need to the following lemma.Lemma 3.1.The problem D α u(t) = y(t), under the condition is equivalence to the Volterra integral equation Proof.Letting D α u(t) = y(t), then DI 1−α u(t) = y(t), integrating both sides from 0 to t, we get operating by I α on both sides, we get differentiating both sides, we get (3.1).Conversely, let u(t) be a solution of (3.1), operating by I 1−α on it, then and where y(t) is the solution of the functional integral equation Proof.Letting y(t) be such that y(t) = D α u(t), substituting in equation (1.1), then we have using Lemma 3.1, substitute by the estimation of u(t) in (3.4), then we get (3.3).Theorem 3.1.Let the assumptions (h 1 )-(h 3 ) are satisfied. .
Evidently, B r is closed, bounded and also convex.Now, we will demonstrate that F B r ⊂ B r ; actually, from (3.5), (3.6) and from the assumptions (h 2 )-(h 3 ), let y be an arbitrary element in B r , we have which signifies that the operator F maps B r into itself.Assumption (h 1 ) guarantees that F is continuous.Now, we will demonstrate that F is compact, that is from assumptions (h 1 ) − (h 2 ) and Theorem 2.2 we get f ∈ L 1 (J, R), it follows that [34] 1 Hence, (F y) h → (F y) uniformly as h → 0.Then, by Theorem 2.1, we have got that F (Ω) is relative compact, i.e., F is a compact operator.As a consequence of Schauder's fixed point theorem [37], the operator F has a fixed point in B r , which demonstrates the existence of at least one solution y 2) has a unique solution u ∈ L 1 (J, R).
Proof.From condition (h 4 ) we can obtain, this show that the assumptions of Theorem 3.1 are satisfied.Let y 1 , y 2 ∈ L 1 (J, R) be two solutions of the functional integral equation (3.3), then Thus Example 3.1.Consider the following IFDP: where   Consider the following assumption: Let the assumptions (h 3 ), (h 4 ) and (h 6 ) be satisfied.If Proof.Define the operator F by (3.6), the operator Firstly, we evaluate f in the first term of the last inequality, from condition (h 4 ) we have Secondly, we evaluate the second term in the inequality (4.1), we have Now we return to equation (4.1), substituting from (4.2) and (4.3) into (4.1),we obtain ) then IFDP (1.1)-(1.2) has at least one solution u ∈ L 1 (J, R).Proof.Convert functional integral equation (3.3) into a fixed point problem.Consider the following operator F
R) of the functional integral equation (3.3), consequently from (3.2) we have u(t) has at least one solution, therefore IFDP (1.1)-(1.2) has at least one solution in B r .