On the center-stable manifolds for some fractional differential equations of Caputo type ∗

Abstract. This paper is devoted to study the existence of center-stable manifolds for some planar fractional differential equations of Caputo type with relaxation factor. After giving some necessary estimation for Mittag–Leffler functions, some existence results for center-stable manifolds are established under the mild conditions by virtue of a suitable Lyapunov–Perron operator. Moreover, an explicit example is given to illustrate the above result. Finally, high-dimensional case is considered.


Introduction
Fractional calculus (FC) has a long history almost as well as the one of standard integer calculus.Thereafter, fractional differential equations (FDEs) have been recognized as one of the best tools to be applied in interdisciplinary field such as viscoelastic materials and electromagnetic problems; see [3,4,7,12,13,16,18,19,[25][26][27][28][29] and references therein.In particular, existence and stability results for some FDEs involving two different Caputo derivatives have been studied extensively, one can refer to [1, 2, 5, 8-10, 14, 15, 17, 20-22, 24].Very recently, an interesting local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations is given in [6], where the fixed point of Lyapunov-Perron operator describes the set of all solutions near the fixed point tend to zero is called stable manifold of hyperbolic fixed point.However, stable manifolds results for FDEs of different type are still not enough.
Here we would like to emphasize that the methods used to deal with (2) are much different from (1).Concerning on (2), we cannot apply a known formula on Mittag-Leffler functions to simplify the form of solution to (2).In order to obtain the existence of the stability of the solution, we simply our problem and set x(0) = x = (0, x 2 ) T .Even in this special case, we also have to overcome expatiatory computation from the estimation on the possible integral of Mittag-Leffler functions, more precisely, we have to study the asymptotic behavior of Mittag-Leffler functions E α , E α,2 , and E α,α for α ∈ (1,2).
By a center-stable manifold of (2) we mean the set of all small x and x for which the solution of ( 2) is bounded on R + when the time variable tends infinite.
To achieve our aim, we adopt the same idea in [6] and construct a suitable Lyapunov-Perron operator as follows: Then we need show that the center-stable manifold of (2) can be characterized as a fixed point of the above Lyapunov-Perron operator F and the fixed point is bounded.This paper is organized as follows.In Section 2, we give some fundamental estimation related to Mittag-Leffler functions.In Section 3, we give the main result of this paper about center-stable manifolds.An example is given to demonstrate the application of our main result.In the final section, we extend the previous stable manifold result for planar fractional order relaxation differential equations involving the order α ∈ (0, 1) to highdimensional case.

A sequence of integral estimation related to Mittag-Leffler functions
The following explicit estimation of Mittag-Leffler functions is useful in the sequel, which has been reported in one of our submitted paper.To give some results for the asymptotic behavior of Mittag-Leffler functions E α , E α,2 , and E α,α for α ∈ (1, 2), we recall the following.
We give the estimation for Mittag-Leffler functions.
Lemma 5.For λ > 0, we define Then, for any function g ∈ X ∞ (R + , R), the following statements hold for all t ∈ [0, 1]: ) and the fact that Mittag-Leffler functions are increasing functions on [0, ∞), we have (ii) Like above, we have The proof is complete.
Remark 4. Lemma 5 presents some explicit integral bound estimation by using the asymptotic behavior of Mittag-Leffler functions, which will be used to derive the stable manifold for the proposed fractional systems.
By Lemmas 3 and 4 we can define finite constants: Now we can prove the following lemma.
https://www.mii.vu.lt/NALemma 6.For λ > 0, we define Then, for any function g ∈ X ∞ (R + , R) with g := sup t 0 |g(t)/ (t)| < ∞, the following statements hold for all t > 1: On the other hand, we get Consequently, we get (ii) Similarly, we get (iii) Like above, we get On the other hand, Furthermore, we obtain Consequently, we get The proof is finished.

Existence of stable manifolds theorem
By Lemmas 5 and 6 the operator F in (4) is well defined.In what follows, we state and prove some fundamental properties of F , which are used later to prove the existence of stable manifolds.
Let V ⊂ U ⊂ R 2 and W ⊂ R 2 be open neighborhoods of zero.Define a center-stable manifold By the above results we know that φ 2 (•, x, x) = F 2 (φ 2 (•, x, x)).
Nonlinear Anal.Model.Control, 23 (5):642-663 Now we are ready to state and prove the main result on stable manifolds.
To end this section, we give an example for which we can compute explicitly its stable manifold.

High-dimensional case
In this section, we extend the result for planar fractional differential equations in [6] to high-dimensional case: where x−y , x , y r with f (0) = 0, lim r→0 l f (r) = 0.The solution of ( 8) is formulated by Consider the Lyapunov-Perron operator T : where K, M are defined in [6, pp. 161-162].
(ii) Since ξ(t) is a fixed point of T , we can get It is easy to get ξ j (0) = x j , and thus we get which is a solution of (8).
(ii) It is easy to see that The proof is completed.