On moment convergence for some order statistics

1.   Introduction

Fractional differential equations rise in many fields, such as biology, physics and engineering. There are many results about the existence of solutions and control problems (see ^{[1,2,3,4,5,6]}).

It is well known that the nonexistence of nonconstant periodic solutions of fractional differential equations was shown in ^[7,8,11] and the existence of asymptotically periodic solutions was derived in ^[8,9,10,11]. Thus it gives rise to study the periodic solutions of fractional differential equations with periodic impulses.

Recently, Fečkan and Wang ^[12] studied the existence of periodic solutions of fractional ordinary differential equations with impulses periodic condition and obtained many existence and asymptotic stability results for the Caputo's fractional derivative with fixed and varying lower limits. In this paper, we study the Caputo's fractional evolution equations with varying lower limits and we prove the existence of periodic mild solutions to this problem with the case of general periodic impulses as well as small equidistant and shifted impulses. We also study the Caputo's fractional evolution equations with fixed lower limits and small nonlinearities and derive the existence of its periodic mild solutions. The current results extend some results in ^[12].

2.   Caputo derivatives with varying lower limits

Set $\xi_{q}(\theta) = \frac{1}{q}\theta^{-1-\frac{1}{q}}\varpi_{q}(\theta^{-\frac{1}{q}})\geq0,$ $\varpi_{q}(\theta) = \frac{1}{\pi}\sum_{n = 1}^{\infty}(-1)^{n-1}\theta^{-nq-1}\frac{\Gamma(nq+1)}{n!}\sin(n\pi q), \ \theta\in(0, \infty).$ Note that $\xi_{q}(\theta)$ is a probability density function defined on $(0, \infty)$, namely $\xi_{q}(\theta)\geq0, \ \theta \in (0, \infty)$ and $\int_{0}^{\infty}\xi_{q}(\theta)d\theta = 1.$

Define $\mathscr{T}: X\to X$ and $\mathscr{S}: X\to X$ given by

Lemma 2.1. ([13,Lemmas 3.2,3.3]) The operators $\mathscr{T}(t)$ and $\mathscr{S}(t), t\geq0$ have following properties:

(1) Suppose that $\sup_{t\geq 0}\|S(t)\|\leq M$. For any fixed $t\geq0$, $\mathscr{T}(\cdot)$ and $\mathscr{S}(\cdot)$ are linear and bounded operators, i.e., for any $u\in X$,

(2) $\{\mathscr{T}(t), t\geq0\}$ and $\{\mathscr{S}(t), t\geq0\}$ are strongly continuous.

(3) $\{\mathscr{T}(t), t > 0\}$ and $\{\mathscr{S}(t), t > 0\}$ are compact, if $\{S(t), t > 0\}$ is compact.

2.1.   General impulses

Let $\mathbb{N}_0 = \{0, 1, \cdots, \infty\}$. We consider the following impulsive fractional equations

where ${}^{c}D_{t_k, t}^{q}$ denotes the Caputo's fractional time derivative of order $q$ with the lower limit at $t_k$, $A:D(A)\subseteq X \to X$ is the generator of a $C_0$-semigroup $\{S(t), t\geq0\}$ on a Banach space $X$, $f: \mathbb{R}\times X\to X$ satisfies some assumptions. We suppose the following conditions:

(Ⅰ) $f$ is continuous and $T$-periodic in $t$.

(Ⅱ) There exist constants $a > 0$, $b_k > 0$ such that

(Ⅲ) There exists $N\in \mathbb{N}$ such that $T = t_{N+1}, t_{k+N+1} = t_k+T$ and $\Delta_{k+N+1} = \Delta_k$ for any $k\in \mathbb{N}$.

It is well known ^[3] that (2.1) has a unique solution on $\mathbb{R}_+$ if the conditions (Ⅰ) and (Ⅱ) hold. So we can consider the Poincaré mapping

By [14,Lemma 2.2] we know that the fixed points of $P$ determine $T$-periodic mild solutions of (2.1).

Theorem 2.2. Assume that (I)-(III) hold. Let $\Xi: = \prod_{k = 0}^{N}Mb_kE_q(Ma(t_{k+1}-t_k)^q)$, where $E_q$ is the Mittag-Leffler function (see [3, p.40]), then there holds

If $\Xi < 1$, then (2.1) has a unique $T$-periodic mild solution, which is also asymptotically stable.

Proof. By the mild solution of (2.1), we mean that $u\in C((t_k, t_{k+1}), X)$ satisfying

Let $u$ and $v$ be two solutions of (2.3) with $u(0) = u_0$ and $v(0) = v_0$, respectively. By (2.3) and $(II)$, we can derive

Applying Gronwall inequality [15, Corollary 2] to (2.4), we derive

which implies

By (2.6) and (Ⅱ), we derive

which implies that (2.2) is satisfied. Thus $P:X\to X$ is a contraction if $\Xi < 1$. Using Banach fixed point theorem, we obtain that $P$ has a unique fixed point $u_0$ if $\Xi < 1$. In addition, since

we get that the corresponding periodic mild solution is asymptotically stable.

2.2.   Small equidistant and shifted impulses

We study

where $h > 0$, $\bar{\Delta}\in X$, and $f:X\to X$ is Lipschitz. We know ^[3] that under above assumptions, (2.8) has a unique mild solution $u(u_0, t)$ on $\mathbb{R}_+$, which is continuous in $u_0\in X$, $t\in \mathbb{R}_+ \setminus\{kh|k\in \mathbb{N}\}$ and left continuous in $t$ ant impulsive points $\{kh|k\in \mathbb{N}\}$. We can consider the Poincaré mapping

Theorem 2.3. Let $w(t)$ be a solution of following equations

Then there exists a mild solution $u(u_0, t)$ of (2.8) on $[0, T]$, satisfying

If $w(t)$ is a stable periodic solution, then there exists a stable invariant curve of Poincaré mapping of (2.8) in a neighborhood of $w(t)$. Note that $h$ is sufficiently small.

Proof. For any $t\in(kh, (k+1)h), k\in \mathbb{N}_0$, the mild solution of (2.8) is equivalent to

So

and

Inserting

into (2.10), we obtain

since

where $L_{loc}$ is a local Lipschitz constant of $f$. Thus we get

and (2.12) gives

So (2.11) becomes

Since $\mathscr{T}(t)$ and $\mathscr{S}(t)$ are strongly continuous,

Thus (2.14) leads to its approximation

which is the Euler numerical approximation of

Note that (2.10) implies

Applying (2.15), (2.16) and the already known results about Euler approximation method in ^[16], we obtain the result of Theorem 2.3.

Corollary 2.4. We can extend (2.8) for periodic impulses of following form

where $\bar{\Delta}_k\in X$ satisfy $\bar{\Delta}_{k+N+1} = \bar{\Delta}_k$ for any $k\in \mathbb{N}$. Then Theorem 2.3 can directly extend to (2.17) with

instead of (2.9).

Proof. We can consider the Poincaré mapping

with a form of

where

By (2.13), we can derive

Then we get

By (2.15), we obtain that $P_h(u_0)$ leads to its approximation

Moreover, equations

has the Euler numerical approximation

with the step size $h^q$, and its approximation of $N+1$ iteration is (2.19), the approximation of $P_h$. Thus Theorem 2.3 can directly extend to (2.17) with (2.18).

3.   Caputo derivatives with fixed lower limits and weak nonlinearities

Now we consider following equations with small nonlinearities of the form

where $\epsilon$ is a small parameter, ${}^{c}D_{0}^{q}$ is the generalized Caputo fractional derivative with lower limit at $0$. Then (3.1) has a unique mild solution $u(\epsilon, t)$. Give the Poincaré mapping

Assume that

(H1) $f$ and $\Delta_k$ are $C^2$-smooth.

Then $P(\epsilon, u_0)$ is also $C^2$-smooth. In addition, we have

where $\omega(t)$ satisfies

and

Thus we derive

Theorem 3.1. Suppose that (I), (III) and (H1) hold.

1). If $(\mathscr{T}(T)-I)$ has a continuous inverse, i.e. $(\mathscr{T}(T)-I)^{-1}$ exists and continuous, then (3.1) has a unique $T$-periodic mild solution located near $0$ for any $\epsilon\neq0$ small.

2). If $(\mathscr{T}(T)-I)$ is not invertible, we suppose that $\ker(\mathscr{T}(T)-I) = [u_1, \cdots, u_k]$ and $X = im(\mathscr{T}(T)-I)\oplus X_1$ for a closed subspace $X_1$ with $\dim X_1 = k$. If there is $v_0\in[u_1, \cdots, u_k]$ such that $B(0, v_0) = 0$ (see (3.7)) and the $k\times k$-matrix $DB(0, v_0)$ is invertible, then (3.1) has a unique $T$-periodic mild solution located near $\mathscr{T}(t)v_0$ for any $\epsilon\neq0$ small.

3). If $r_\sigma (D_{u_0}M(\epsilon, u_0)) < 0$, then the $T$-periodic mild solution is asymptotically stable. If $r_\sigma (D_{u_0}M(\epsilon, u_0))\cap(0, +\infty)\neq \emptyset$, then the $T$-periodic mild solution is unstable.

Proof. The fixed point $u_0$ of $P(\epsilon, x_0)$ determines the $T$-periodic mild solution of (3.1), which is equivalent to

Note that $M(0, u_0) = (\mathscr{T}(T)-I)u_0$. If $(\mathscr{T}(T)-I)$ has a continuous inverse, then (3.3) can be solved by the implicit function theorem to get its solution $u_0(\epsilon)$ with $u_0(0) = 0$.

If $(\mathscr{T}(T)-I)$ is not invertible, then we take a decomposition $u_0 = v+w$, $v\in[u_1, \cdots, u_k]$, take bounded projections $Q_1 : X\to im(\mathscr{T}(T)-I)$, $Q_2 : X\to X_1$, $I = Q_1+Q_2$ and decompose (3.3) to

and

Now $Q_1M(0, v+w) = (\mathscr{T}(T)-I)w$, so we can solve by implicit function theorem from (3.4), $w = w(\epsilon, v)$ with $w(0, v) = 0$. Inserting this solution into (3.5), we get

So

Consequently we get, if there is $v_0\in[u_1, \cdots, u_k]$ such that $B(0, v_0) = 0$ and the $k\times k$-matrix $DB(0, v_0)$ is invertible, then (3.1) has a unique $T$-periodic mild solution located near $\mathscr{T}(t)v_0$ for any $\epsilon\neq0$ small.

In addition, $D_{u_0}P(\epsilon, u_0(\epsilon)) = I+D_{u_0}M(\epsilon, u_0)+O(\epsilon^2)$. Thus we can directly derive the stability and instability results by the arguments in ^[17].

4.   An example

In this section, we give an example to demonstrate Theorem 2.2.

Example 4.1. Consider the following impulsive fractional partial differential equation:

for $\xi \in \mathbb{R}$, $t_k = \frac k3$. Let $X = L^{2}[0, \pi]$. Define the operator $A:D(A)\subseteq X\rightarrow X$ by $Au = \frac{d^{2}u}{dy^{2}}$ with the domain

Then $A$ is the infinitesimal generator of a $C_{0}$-semigroup $\{S(t), t\geq0\}$ on $X$ and $\|S(t)\| \leq M = 1$ for any $t\geq0$. Denote $u(\cdot, y) = u(\cdot)(y)$ and define $f:[0, \infty)\times X\rightarrow X$ by

Set $T = t_3 = 1$, $t_{k+3} = t_k+1$, $\Delta_{k+3} = \Delta_k$, $a = 1$, $b_k = |1+\xi|$. Obviously, conditions (I)-(III) hold. Note that

Letting $\Xi < 1$, we get $-E_{\frac12}(\frac{1}{\sqrt{3}})-1 < \xi < E_{\frac12}(\frac{1}{\sqrt{3}})-1$. Now all assumptions of Theorem 2.2 hold. Hence, if $-E_{\frac12}(\frac{1}{\sqrt{3}})-1 < \xi < E_{\frac12}(\frac{1}{\sqrt{3}})-1$, (4.1) has a unique $1$-periodic mild solution, which is also asymptotically stable.

5.   Conclusion

This paper deals with the existence and stability of periodic solutions of impulsive fractional evolution equations with the case of varying lower limits and fixed lower limits. Although, Fečkan and Wang ^[12] prove the existence of periodic solutions of impulsive fractional ordinary differential equations in finite dimensional Euclidean space, we extend some results to impulsive fractional evolution equation on Banach space by involving operator semigroup theory. Our results can be applied to some impulsive fractional partial differential equations and the proposed approach can be extended to study the similar problem for periodic impulsive fractional evolution inclusions.

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. This research is supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Foundation of Postgraduate of Guizhou Province (YJSCXJH[2019]031), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17.

Conflict of interest

All authors declare no conflicts of interest in this paper.