Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

Abstract: In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

existence of iterative solutions for a class of fractional initial value problem with non-monotone term D α 0 + u(t) = f (t, u(t)), for each, t ∈ (0, h), 0 < h < +∞, Motivated by the work above, we focus our attention on the more general problem: where D α 0 + is the standard Riemann-Liouville fractional derivative, f : (0, T] × R × R → R is a continuous function and 0 < α < 1. In the literature, several different definitions of fractional integrals and derivatives are present. Some of them such as the Riemann-Liouville integral, the Caputo and the Riemann-Liouville derivatives are thoroughly studied and actually used in applied models. Hilfer has introduced a generalized form of the Riemann-Liouville fractional derivative. In short, Hilfer fractional derivative D α,β 0 + is an interpolation between the Riemann-Liouville and Caputo fractional derivatives. The present paper is organized as follows. In Section 2, some notations are introduced and we recall some concepts of preliminaries about fractional calculus and auxiliary results. In Section 3, two results for the problem (1)- (2) are presented: the first one is based on the Banach contraction principle, the second one on Schaefer's fixed point theorem. In Section 4, we present Ulam-Hyers stability result for the problem (1)- (2). Finally, in the last Section, we give an example to illustrate the applicability of our main results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let 0 < T, J = [0, T]. By C(J, R) we denote the Banach space of all continuous functions from J into R with the norm ‖y‖∞ = sup{|y(t)| : t ∈ J}.
And L 1 (J, R) is the space of Lebesgue-integrable functions w : J → R with the norm In what follows > 0. We consider the weighted space of continuous functions Clearly C (J) is a Banach space.
Corollary 2.5. [26] Under the hypotheses of Lemma 2.4, assume further thatã(t) is a nondecreasing function for t ∈ [a, b). Then where E β (·) is the one parameter Mittag-Le er function.
Definition 2.6. The equation (1) is Ulam-Hyers stable if there exists a real number c f > 0 such that for each ϵ > 0 and for each solution z ∈ C 1−α (J) of the inequality there exists a solution y ∈ C 1−α (J) of equation (1) with Definition 2.7. The equation (1) is generalized Ulam-Hyers stable if there exists ψ f ∈ C(R+, R+), ψ f (0) = 0, such that for each solution z ∈ C 1−α (J) of the inequality there exists a solution y ∈ C 1−α (J) of the equation (1) with Definition 2.8. The equation (1) is Ulam-Hyers-Rassias stable with respect to ϕ ∈ C(J, R+) if there exists a real number c f > 0 such that for each ϵ > 0 and for each solution z ∈ C 1−α (J) of the inequality there exists a solution y ∈ C 1−α (J) of equation (1) with there exists a solution y ∈ C 1−α (J) of equation (1) with

Remark 2.11. A function z ∈ C 1−α (J) is a solution of the inequality (4) if and only if there is σ
One can have similar remarks for inequalities (5) and (3).

Existence of solutions
To prove the existence of solutions to (1)-(2), we need the following auxiliary Lemma.

Lemma 3.1. [6] The linear initial value problem
where λ ≥ 0 is a constant and q ∈ L(0, T), has the following integral representation for a solution where Eα,α(t) is the two-parameter Mittag-Le er function. In particular, when λ = 0, then the initial value problem (6) has a unique solution defined by The following assumptions will be used in our main results: (H1) There exist constants K > 0 and 0 < L < 1 such that where g : (0, T] → R be a function satisfying the functional equation Proof. Firstly, we need to show that the operator N is well defined, i.e, for every y ∈ C 1−α (J) and t > 0; the integral .
That is to say that the integral exists and belongs to C 1−α (J). The above inequality implies that Combining with lim t→0 + t 1−α (Ny)(t) = y 0 . The above arguments combined with Lemma 3.1, imply that the fixedpoint of the operator N solves (1)-(2) and vice versa. The proof is complete.
then, there exists a unique solution for the problem (1)-(2) in the space C 1−α (J).
By (H1) we have Then Therefore, for each t ∈ (0, T] By Lemma 2.2, we have which implies that By (8), the operator T is a contraction. Hence, by Banach's contraction principle, N has a unique fixed point which is a unique solution of the problem (1)-(2).
Our second result is based on Schaefer's fixed point theorem.
where q * = sup t∈J q(t) then the problem (1)-(2) has at least one solution in the space C 1−α (J).
Proof. Let the operator N defined in (7). We shall use Schaefer's fixed point theorem to prove that N has a fixed point. The proof will be given in several steps.
Consequently, N is continuous.
Step 2: N maps bounded sets into bounded sets in C 1−α . Indeed, it is enough to show that for any η * > 0, there exists a positive constant ℓ such that for each we have ‖Ny‖ C1−α ≤ ℓ. For each t ∈ (0, T], we have where g ∈ C 1−α (J) be such that g(t) = f (t, y(t), g(t)).
Step 4: A priori bounds. Now it remains to show that the set E = {y ∈ C 1−α : y = λN(y) for some 0 < λ < 1}, is bounded. Let y ∈ E, then y = λNy for some 0 < λ < 1. Thus, for each t ∈ (0, T], we have And by (H2), we have for each t ∈ (0, T], Thus, we get This implies, by (13) and Lemma 2.2, that for each t ∈ (0; T] we have Therefore Finally, by (9) we have This shows that the set E is bounded. As a consequence of Schaefer's fixed point theorem, we deduce that N has at least a fixed point y * ∈ C 1−α (J) which is a solution of the problem (1)-(2).
Proof. Let z ∈ C 1−α (J) be a solution of the inequality (4). Denote by y the unique solution of the initial value problem: Using Theorem 3.3, we obtain for each t ∈ (0, T] where g : (0, T] → R be a function satisfying the functional equation g(t)).
Since z solution of the inequality (4) and by Remark 2.11, we have Clearly, the solution of the problem (14)-(15) is given by where h : (0, T] → R be a function satisfying the functional equation Hence for each t ∈ (0, T], it follows that where g, h ∈ C 1−α (J) such that . Then, Therefore, for each t ∈ (0, T] Applying Corollary 2.5, we get Thus, the equation (1) is Ulam-Hyers-Rassias stable with respect to ϕ. The proof is complete.
Next, we present the following Ulam-Hyers stable result.
Proof. Let z ∈ C 1−α (J) be a solution of the inequality (3). Denote by y the unique solution of the initial value problem: By the same way of the proof of Theorem 4.1, we can easily show that which completes the proof of the theorem. Moreover, if we set ψ f (ϵ) = cϵ; ψ f (0) = 0, then, the equation (1) is generalized Ulam-Hyers stable.

An example
Consider the following initial value problem   (16) is Ulam-Hyers stable.