On Existence and Stability Results for Nonlinear Fractional Delay Differential Equations

abstract: We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, generalized Ulam–Hyers–Rassias stability, Eα–Ulam–Hyers stability and generalized Eα–Ulam–Hyers stability of fractional order delay differential equations.


Introduction
It is well known that many dynamical systems can be described more precisely by using fractional differential equations.Modeling of several physical phenomena appearing in science and engineering can suitably done via fractional differential equations.Hence the study of fractional differential equations have widespread interest.
Several researchers form mathematics community devoted to study existence, uniqueness and other qualitative properties for fractional delay differential equations (FDDEs) by various approaches.At many instances it is very difficult to obtain exact solution of FDDEs and in this case we are intended to obtain the approximate solution for such equations.The answer to the question, " Under 57 of successive approximation we obtain Ulam-Hyers stability, Ulam-Hyers-Rassias stability and E α -Ulam-Hyers stability results for delay differential equation (1.1).Finally, we provide examples to illustrate our obtained results.
Recently, Kucche et al. [15,16] established existence and various qualitative properties of solutions to nonlinear implicit fractional differential equations.

Preliminaries
In this section we present some basic definitions, notations and preliminaries.Basics of delay differential equations are considered from the monographs by Hale et al. [17] and Naito et al. [18].
Let R n is an n-dimensional linear vector space over the reals with the norm We use following results in our analysis.
where m(t) is a nondecreasing function, then For fundamentals of fractional calculus we refer the research monographs [19,20,21].Definition 2.1.Let g ∈ C[0, b] and α ≥ 0 then the Riemann-Liouville fractional integral of order α of a function g is defined as The one parameter Mittag-Leffler function is defined as , γ > 0.
Following lemmas play important role to obtain our main results.

Existence and Uniqueness Results
To obtain existence and uniqueness of solution to the initial value problem (1.1)-(1.2),we use the following Lemma.Lemma 3.1 ( [24], Modified version of contraction principle).Let X be a Banach Space and let D be an operator which maps the element of X into itself for which D r is a contraction, where r is a positive integer then D has a unique fixed point.
The proof of the following Lemma is close to the proof of Lemma 6.2 given in [21].
2) is equivalent to the following fractional Volterra integral equation Next theorem guarantee existence and uniqueness of solution to initial value problem (1.1)-(1.2).
be a continuous function that satisfies Lipschitz condition with respect to second variable Note that by definition of operator F , for any x, z ∈ B we have By using mathematical induction, for any x, z ∈ B and t ∈ [0, b] we prove that, By definition of operator F and using Lipschitz condition, we have for any x, z ∈ B and t For any t ∈ [0, b] and θ ∈ [−r, 0], we have −r ≤ t + θ ≤ b and hence Thus the inequality (3.2) holds for j = 1.Let us suppose that the inequality (3.2) holds for j = r ∈ N, hence we prove that (3.2) holds for j = r + 1.Let any x, z ∈ B and denote x * = F r x, z * = F r z.Then using definition of operator F and the Lipschitz condition of f , for any t ∈ [0, b] we get We write from (3.4), x An application of Lemma 2.2 gives, x − z B .
By using above inequality in (3.5) and then applying Lemma 2.4, we get Existence and Stability Results for FDDEs 61 Thus, We have proved that the inequality (3.2) holds for j = r + 1.By the principle of mathematical induction the proof of inequality (3.2) is completed.Combining (3.1) and (3.2), we obtain This gives, By definition of one parameter Mittag-Leffler function, we have .
Thus we can choose j ∈ N such that (Lb α ) j Γ(jα+1) < 1 so that F j is a contraction.Therefore by modified version of contraction principle, F has a unique fixed point x : [−r, b] → R n in B, which is the unique solution of the FDDEs (1.1)-(1.2). ✷

Ulam-Hyers Stability of FDDE
We adopt the definitions of Ulam-Hyers stability, generalized Ulam-Hyers stability and Ulam-Hyers-Rassias stability given in [2].Definition 4.1.We say that the equation (1.1) has Ulam-Hyers stability if there exists a real number then there exists a solution is Ulam-Hyers stable with initial conditions.Definition 4.2.We say that the equation (1.1) has generalized Ulam-Hyers stability if there exists then there exists a solution Definition 4.3.We say that the equation (1.1) has Ulam-Hyers-Rassias stability with respect to then there exists a solution Definition 4.4.We say that the equation (1.1) has generalized Ulam-Hyers-Rassias stability with respect to then there exists a solution In the following theorem by method of successive approximation we prove that FDDE (1.1) is Ulam-Hyers stable.
then there exists unique solution Existence and Stability Results for FDDEs

63
Proof: For every ǫ > 0, let y Then there exists a function σ y ∈ B (depending on y) such that, If y(t) satisfies (4.2) then in view of Lemma 3.2 it satisfies equivalent fractional integral equation Define , and consider the sequence x j ⊆ B defined by, Using mathematical induction firstly we prove that, By definition of successive approximations given above and (4.3) we have, which proves the inequality (4.4) for j = 1.Let us suppose that the inequality (4.4) hold for j = r ∈ N, we prove it also hold for j = r + 1 ∈ N. By using definition of successive approximations and Lipschitz condition of f , for any t ∈ [0, b] we obtain, Since (4.4) hold for j = r, we have Therefore by using Lemma 2.2 we get, Thus the inequality (4.5) reduces to Using Lemma 2.4, in the above inequality, we get, Therefore, which is the inequality (4.4) for j = r+1.Using principle of mathematical induction the proof of the inequality (4.4) is completed.Now using the estimation (4.4) for any

.6)
Existence and Stability Results for FDDEs

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Hence the series, converges absolutely and uniformly on [0, b] with respect to the norm • .Let us suppose Then, is the r th partial sum of the series (4.7), therefore we can write, Further by definition of successive approximations we have, Clearly x ∈ B. We prove that this limit function is the solution of fractional integral equation Using definition of successive approximations for any t ∈ [0, b], we have Now for any t ∈ [0, b], we write from equations (4.7) and (4.8), Using inequality (4.4), we obtain Applying Lemma 2.2, we get Using (4.11) and (4.12) in (4.10), we obtain .

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Thus Since both the series on the right hand side of above inequality are convergent, by taking limit as j → ∞, we obtain This implies Further, from equations (4.6), (4.7) and (4.9), we have This proves that the equation (1.1) is Ulam-Hyers stable.Moreover as x (k) (0) = y (k) (0), k = 0, 1, ..., m − 1, the equation (1.1) has Ulam-Hyers stability with the initial conditions.It remains to prove the uniqueness of x(t).Assume x(t) is another solution of (1.1) with the initial conditions x(k By using Lipschitz condition we find that Using Lemma 2.2, An application of Lemma 2.3 to above inequality with u(t) = x t − xt C and a(t) = 0, we obtain ) is generalized Ulam-Hyers stable with initial conditions.Note 1.An example is given in last section to illustrate without initial condition x(t) is neither unique nor necessarily the best approximate solution to FDDE (1.1).
Next we obtain Ulam-Hyers-Rassias stability result for the equation (1.1) by method of successive approximation.
and λ > 0 is constant satisfying 0 < λL < 1, then there exists unique solution then there exists a function σ y ∈ B (depending on y) such that , Existence and Stability Results for FDDEs

69
Then y satisfies the fractional integral equation Define the sequence of approximation x j ⊆ B as in proof of Theorem 4.1 starting with zeroth order approximation x 0 (t) = y(t), t ∈ [−r, 0].By using mathematical induction we prove that, Using definition of successive approximations and (4.16) we have, which is the inequality (4.17) for j = 1.Let us suppose that the inequality (4.17) hold for j = r ∈ N. Then by definition of successive approximations and Lipschitz condition of f , for any t ∈ [0, b] we obtain, Since (4.17) hold for j = r, we have By an application of Lemma 2.2 to the above inequality gives Thus the inequality (4.18) reduces to which is the inequality (4.17) for j = r + 1.The proof of the inequality (4.17) is completed by principle of mathematical induction.Using the inequality (4.17) and the fact 0 Hence the series converges absolutely and uniformly on [0, b], say to x(t) in the norm • .Define, Proceeding as in the proof of Theorem 4.1 one can show that x(t) is a solution of (1.1) with x (k) (0) = y (k) (0), k = 0, 1, 2, ..., m − 1, that satisfies Therefore equation (1.1) is Ulam-Hyers-Rassias stable.✷

E α -Ulam-Hyers stability
We consider the following definitions of E α -Ulam-Hyers stabilities introduced by Wang and Li [3].Definition 5.1.We say that equation (1.1) has E α -Ulam-Hyers stability if there exists a real number K > 0 such that for each ǫ > 0, if y : then there exists a solution Definition 5.2.We say that equation (1.1) has generalized E α -Ulam-Hyers stability if there exists a nonnegative function there exists a solution For every ǫ > 0, if y : [−r, b] → R n in B satisfies, then there exists unique solution Proof: We define the sequence of approximations as in Theorem 4.1.Noting that x 0 (t) = y(t), we write from (4.4) (4.7) and (4.9)

Examples
We remark that the argument " without initial condition we cannot obtain best and unique approximate solution " of Huang et al. [13] about the best and unique approximate solution for integer order delay differential equations is also hold for fractional order delay differential equations.
To illustrate this we consider an example in the space R 1 .Example 6.1: Consider the fractional delay differential equation Then the function y Further note that y(t) is not a solution of equation (6.1) as for any t ∈ [0, 1], = 0.
By using successive approximations defined in Theorem 4.1 we obtain first approximate solution to (6.1) as Define, Then it is easy to verify that the function ψ(t) forms a solution of (6.1).Also we find Next, we see that the function ψ * (t) defined by Existence and Stability Results for FDDEs 73 is also a solution of equation (6.1) and is such that This shows that ψ * (t) is better approximate solution than ψ(t).

L
. Other stability results for (6.2) can be obtained similarly.
To illustrate existence and stability results for fractional delay differential equation obtained in this paper we give the following example.Since any two norms on a finite dimensional linear spaces are equivalent here we consider the example in R 2 with the norm Example 6.2: Consider the fractional delay differential equation of the form: , x 2 (t − 1) , t ∈ [0, 1], (6.4) Then for any φ, ψ ∈ C([−1, 0], R 2 ), we find Other stability results for the equation (6.4) can discussed similarly.