Qualitative analysis of nonlinear implicit neutral di ﬀ erential equation of fractional order

: In this paper, we discuss su ﬃ cient conditions for the existence of solutions for a class of Initial value problem for an neutral di ﬀ erential equation involving Caputo fractional derivatives. Also, we discuss some types of Ulam stability for this class of implicit fractional-order di ﬀ erential equation. Some applications and particular cases are presented. Finally, the existence of at least one mild solution for this class of implicit fractional-order di ﬀ erential equation on an inﬁnite interval by applying Schauder ﬁxed point theorem and the local attractivity of solutions are proved.

Benchohra et al. [13] established some types of Ulam-Hyers stability for an implicit fractional-order differential equation.
A. Baliki et al. [11] have given sufficient conditions for existence and attractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem.
Benchohra et al. [15] studied the existence of mild solutions for a class of impulsive semilinear fractional differential equations with infinite delay and non-instantaneous impulses in Banach spaces.This results are obtained using the technique of measures of noncompactness.
Motivated by these works, in this paper, we investigate the following initial value problem for an implicit fractional-order differential equation where C D α is the Caputo fractional derivative, h : J × R −→ R, g 1 : J × R × R −→ R and g 2 : J × R −→ R are given functions satisfy some conditions and J = [0, T ]. we give sufficient conditions for the existence of solutions for a class of initial value problem for an neutral differential equation involving Caputo fractional derivatives.Also, we establish some types of Ulam-Hyers stability for this class of implicit fractional-order differential equation and some applications and particular cases are presented.
Finally, existence of at least one mild solution for this class of implicit fractional-order differential equation on an infinite interval J = [0, +∞), by applying Schauder fixed point theorem and proving the attractivity of these mild solutions.
By a solution of the Eq (1.1) we mean that a function x satisfies the equation in (1.1).

Preliminaries
Definition 1. [23] The Riemann-Liouville fractional integral of the function f ∈ L 1 ([a, b]) of order α ∈ R + is defined by and when a = 0, we have For further properties of fractional operators (see [23,25,26]).

Main results
Consider the initial value problem for the implicit fractional-order differential Eq (1.1) under the following assumptions: is a continuous function and there exists a positive constant K 1 such that: is a continuous function and there exist two positive constants K, H such that: is a continuous function and there exists a positive constant K 2 such that: Lemma 3. Let assumptions (i)-(iii) be satisfied.If a function x ∈ C 2 (J, R) is a solution of initial value problem for implicit fractional-order differential equation (1.1), then it is a solution of the following nonlinear fractional integral equation Proof.Assume first that x is a solution of the initial value problem (1.1).From definition of Caputo derivative, we have Operating by I α−1 on both sides and using Lemma 2, we get Using initial conditions, we have Integrating both sides of (1.1), we obtain Then Conversely, assume that x satisfies the nonlinear integral Eq (3.1).Then operating by C D α on both sides of Eq (3.1) and using Lemma 2, we obtain Putting t = 0 in (3.1) and since g 1 is a continuous function, then we obtain Then we have Hence the equivalence between the initial value problem (1.1) and the integral Eq (3.1) is proved.Then the proof is completed.
Definition 3. The Eq (1.1) is Ulam-Hyers stable if there exists a real number c f > 0 such that for each > 0 and for each solution z ∈ C 2 (J, R) of the inequality Definition 4. The Eq (1.1) is generalized Ulam-Hyers stable if there exists
Definition 5.The Eq (1.1) 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 > 0 and for each solution z ∈ C 2 (J, R) of the inequality Definition 6.The Eq (1.1) is generalized Ulam-Hyers-Rassias stable with respect to ϕ ∈ C(J, R + ) if there exists a real number c f,ϕ > 0 such that for each solution z ∈ C 2 (J, R) of the inequality Now, our aim is to investigate the existence of unique solution for (1.1).This existence result will be based on the contraction mapping principle.
< 1, then there exists a unique solution for the nonlinear neutral differential equation of fractional order .
Proof.Define the operator N by:

In view of assumptions
Then < 1.It follows that N has a unique fixed point which is a solution of the initial value problem (1.1) in C 2 (J, R).
Proof.Let y ∈ C 2 (J, R) be a solution of the inequality Let x ∈ C 2 (J, R) be the unique solution of the initial value problem for implicit fractional-order differential Eq (1.1).By using Lemma 3, The Cauchy problem (1.1) is equivalent to Operating by I α−1 on both sides of (4.1) and then integrating, we get .
Also, we have thus the intial value problem (1.1) is Ulam-Heyers stable, and hence the proof is completed.

Ulam-Hyers-Rassias stability
Theorem 3. Let assumptions of Theorem 1 be satisfied, there exists an increasing function ϕ ∈ C(J, R) and there exists λ ϕ > 0 such that for any t ∈ J, we have then the Eq (1.1) is Ulam-Heyers-Rassias stable.
Proof.Let y ∈ C 2 (J, R) be a solution of the inequality Let x ∈ C 2 (J, R) be the unique solution of the initial value problem for implicit fractional-order differential Eq (1.1).By using Lemma 3, The Cauchy problem (1.1) is equivalent to Operating by I α−1 on both sides of (4.2) and then integrating, we get Also, we have then the initial problem (1.1) is Ulam-Heyers-Rassias stable, and hence the proof is completed.

Existence and attractivity of solutions on half line
In this section, we prove some results on the existence of mild solutions and attractivity for the neutral fractional differential equation (1.1) by applying Schauder fixed point theorem.Denote BC = BC(J), J = [0, +∞) and consider the following assumptions: (I) h : J × R −→ R is a continuous function and there exists a continuous function K h (t) such that: (II) g 1 : J × R × R −→ R satisfies Carathéodory condition and there exist an integrable function a 1 : R + −→ R + and a positive constant b such that: (III) g 2 : J × R −→ R satisfies Carathéodory condition and there exists an integrable function a 2 : R + −→ R + such that: By a mild solution of the Eq (1.1) we mean that a function x ∈ C(J, R) such that x satisfies the equation in (3.1).
Theorem 4. Let assumptions (I)-(IV) be satisfied.Then there exists at least one mild solution for the nonlinear implicit neutral differential equation of fractional order (1.1).Moreover, mild solutions of IVP (1.1) are locally attractive.
Proof.For any x ∈ BC, define the operator A by The operator A is well defined and maps BC into BC.Obviously, the map A(x) is continuous on J for any x ∈ BC and for each t ∈ J, we have (5.1) Thus A(x) ∈ BC.This clarifies that operator A maps BC into itself.
Finding the solutions of IVP (1.1) is reduced to find solutions of the operator equation A(x) = x.Eq (5.1) implies that A maps the ball B M := B(0, M) = {x ∈ BC : ||x(t)|| BC ≤ M} into itself.Now, our proof will be established in the following steps: Step 1: A is continuous.Let {x n } n∈N be a sequence such that x n → x in B M .Then, for each t ∈ J, we have Assumptions (II) and (III) implies that: Using Lebesgue dominated convergence theorem, we have Step 2: A(B M ) is uniformly bounded.It is obvious since A(B M ) ⊂ B M and B M is bounded.
Step 3: A(B M ) is equicontinuous on every compact subset [0, T ] of J, T > 0 and t 1 , t 2 ∈ [0, T ], t 2 > t 1 (without loss of generality), we get Thus, for a i = sup t∈[0,T ] a i , i = 1, 2 and from the continuity of the functions a i we obtain Continuity of h implies that Step 4: A(B M ) is equiconvergent.
Let t ∈ J and x ∈ B M then we have In view of assumptions (I) and (IV), we obtain Then A has a fixed point x which is a solution of IVP (1.1) on J.
Step 5: Local attactivity of mild solutions.Let x * be a mild solution of IVP (1.1).Taking x ∈ B(x * , 2M), we have Hence A is a continuous function such that A(B(x * , 2M)) ⊂ B(x * , 2M).Moreover, if x is a mild solution of IVP (1.1), then In view of assumption of (IV) and estimation (5.2), we get Then, all mild solutions of IVP (1.1) are locally attractive.

Conclusions
Sufficient conditions for the existence of solutions for a class of neutral integro-differential equations of fractional order (1.1) are discussed which involved many key functional differential equations that appear in applications of nonlinear analysis.Also, some types of Ulam stability for this class of implicit fractional differential equation are established.Some applications and particular cases are presented.Finally, the existence of at least one mild solution for this class of equations on an infinite interval by applying Schauder fixed point theorem and the local attractivity of solutions are proved.
as a particular case of Theorem 1 we can deduce an existence result for the initial value problem for implicit second-order differe-integral equation