Existence and Ulam type stability for nonlinear Riemann–Liouville fractional differential equations with constant delay

. A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are deﬁned and studied. Several sufﬁcient conditions are obtained. Some of the obtained results are illustrated on fractional biological models.


Introduction
Ulam type stability concept is quite significant in realistic problems in many applications in numerical analysis, optimization, biology and economics etc. This type of stability guarantees that there is a close exact solution. Recently, several authors extended and discussed Ulam type stability to fractional differential equations. The Ulam type stability is well studied recently for Caputo fractional differential equations. For example, about Caputo fractional differential equations we refer [8,12], for Caputo fractional differential equations with impulses [14], about Caputo fractional differential equations with delays see, for example, [2,13], for ψ-Hilfer fractional derivative and a constant delay [11]. Note that in the case of the Riemann-Louisville (RL) fractional derivative only the case without any delays is studied (see, for example, [3,7,13,16]).
In addition, many real world processes and phenomena are characterized by the influence of past values of the state variable on the recent one and this leads to the inclusion of delays in the models. The analysis of RL delay fractional differential equations is rather complex (analytical solution computation, controllability analysis, etc.) and a very small class of equations could be solved in explicit form. It requires theoretical proofs of methods guarantee existence of enough close function to the unknown solution. One of these types of method is Ulam type stability. According to our knowledge this type of stability is not studied for nonlinear RL fractional differential equations with delays.
The main goal of the paper is to study scalar nonlinear RL fractional differential equations with a constant delay, to obtain some sufficient conditions for uniqueness and existence and to study Ulam type stability. The present paper is organized as follows. In Section 2, some notations and results about fractional calculus are given. In Section 3, an existence result, based on the Banach contraction principle, for the studied problem is presented. In Section 4, we prove three types of Ulam-Hyers stability results for the given RL fractional differential equation with a constant delay. Finally, in the last section, we illustrate the application of some of th obtained results on two fractional biological models: fractional generalization of Lasota-Ważewska model and fractional generalization of the logistic equation with a biological delay depending on the mechanistic details of the model.

Preliminary notes on fractional derivatives and equations
In this section, we introduce notations, definitions, and preliminary facts which are used throughout the paper. Let T : 0 < T < ∞,J = [0, T], J = (0, T], τ > 0 be given (the delay).
In this paper we will use the following definitions for fractional derivatives and integrals: • Riemann-Liouville fractional integral of order q ∈ (0, 1) where Γ(·) is the Gamma function.
• Riemann-Liouville fractional derivative of order q ∈ (0, 1) We will give fractional integrals and RL fractional derivatives of some elementary functions which will be used later: ([5]). The following equalities are true: The definitions of the initial condition for fractional differential equations with RL derivatives are based on the following result: (a) If there exists a.e. a limit lim t→t 0 + [(t − t 0 ) q−1 m(t)] = c, then there also exists a limit In the case of a scalar linear RL fractional differential equation we have the following result: has the following form [9, formula (4.1.14)] is the Mittag-Leffler function with two parameters (see, for example, [9]).
From Proposition 2.3 and Proposition 2.2 (a) we obtain the following result for the weighted form of the initial condition: has the following form Proposition 2.6 ([9]). For q ∈ (0, 1) the following properties hold.

Proposition 2.7 ([17, Corollary 2])
. Let a(t) be a nondecreasing function on J, g(t) be a nonegative, nondecreasing continuous function on J, and

Statement of the problem
Consider the initial value problem (IVP) for a nonlinear system of fractional differential equations with constant delay and q ∈ (0, 1) where RL 0 D q t denotes the RL fractional derivative, a, b ∈ R are constants, τ > 0 is the constant delay, the functions f : J × R → R → R, g ∈ C 0 .
First, we will consider the partial linear case of (3.1) without a delay, i.e.
Lemma 3.1 ([9]). The linear initial value problem (3.2) has the following integral representation for a solution We consider also the integral presentation (see [1]) of a special case of (3.1), i.e. we will consider the non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay with the initial conditions for t ∈ (nτ, (n + 1)τ], n = 1, 2, . . . , N.
We will consider the assumptions: (A2) The function g ∈ C 0 .

Existence and integral presentation of the solution
Now we will study the existence of the solution of (3.1) and its presentation, based on Lemma 3.2. We will use the Banach contraction principle. Then the operator Ω : Let t ∈ (0, τ] then according to assumption (A1) and Proposition 2.1 with β = q − 1 we get Let t > τ. Then according to assumption (A1), equality From inequalities (4.2) and (4.3) it follows Therefore, the integrals exist and Ωy(t) ∈ C 1−q (J).
Proof. (i) Let the function y ∈ C 1−q (J) be a solution of IVP (3.1). We will use an induction to prove the function y is a fixed point of the operator Ω.
By induction it follows the solution y is a fixed point of the operator Ω.
The proof follows from Theorem 4.5 and Proposition 2.6. In the case of an equation without a delay we obtain the following corollary.

Corollary 4.7.
Let τ = 0, a = b = 0, q ∈ (0, 1), the function f ∈ C(J × R, R) and there exists a Then the reduced initial value problem has a unique solution y ∈ C 1−q (J).
Note in the case without a delay it follows from the proof of Theorem 4.5 that we do not need the restriction q ∈ [0.5, 1).
there exists a solution x ∈ C 1−q (J) of the problem (3.1) with

Definition 5.3 ([10]). The problem (3.1) is generalized Ulam-Hyers-Rassias stable with respect
to Φ if there exists a real number c f > 0 such that for each solution y ∈ C 1−q (J) of the inequality there exists a solution x ∈ C 1−q (J) of the problem (3.1) with

Remark 5.4.
If the function f ∈ C(J × R 2 , R) then the function y ∈ C 1−q (J) is a solution of the inequality (5.1) iff there exist a function G ∈ C 1−q (J) which depends on y such that

Note we have a similar remark for inequality (5.3).
Based on Remark 5.4 and Definition 5.1 we get the following result. Lemma 5.6. Let the conditions of Theorem 4.5 be satisfied. If y ∈ C 1−q (J) is a solution of inequalities (5.1) then it satisfies the following integral-algebraic inequalities Proof. Let y ∈ C 1−q (J) be a solution of inequalities (5.1). According to Remark 5.5 it satisfies the IVP C 0 D q t y(t) = ay(t) + by(t − τ) + f (t, y(t), y(t − τ)) + G(t) for t ∈ J, y(t) = g(t) for t ∈ [−τ, 0], lim t→0+ t 1−q x(t) = g(0). Let t ∈ (0, τ]. Apply the inequalities |G(t)| ≤ ε and (2.2) and obtain The proof for t ∈ (τ, T] is similar and we omit it. Now we will study Ulam type stability of problem (3.1).

Theorem 5.7 (Stability results).
Assume the conditions of Theorem 4.5 are satisfied.
Proof. According to Theorem 4.5 the problem (3.1) has an unique solution x ∈ C 1−q (J) for which the integral presentation (4.7) holds.
(ii) The proof is similar to the one in (i).

Applications to some biological models
In this section we will apply the obtained results to some biological models and their fractional generalizations.

Model 1.
The investigation of blood cell dynamics is connected with formulating and studyig mathematical methods and models, numerical results, schemes to estimate parameters and prognosticate optimal treatments to particular diseases. In order to describe the survival of red blood cells in animals, Ważewska-Czyżewska and Lasota proposed in [15] the following delayed equation x (t) = −γx(t) + βe −αx(t−τ) where x(t) represents the number of red blood cells at time t, γ > 0 is the death probability for a red blood cell, a and β are positive constants related to the production of red blood cells per unit time and τ is the time delay between the production of immature red blood cells and their maturation for release in circulating blood stream. The well known Lasota-Ważewska model was extended and generalized by many authors. Now we will consider one fractional generalization.
Consider the following fractional generalization of the mentioned above model: with the initial conditions (3.4), (3.5), where q ∈ [0.5, 1), x is the number of red blood cells, β > 0 is the demand for oxygen, τ > 0 is the time required for erythrocytes to attain maturity, γ > 0 is the cell destruction rate.