The Well-Posedness of Fractional Systems with Affine-Periodic Boundary Conditions

Abstract

This paper is devoted to study the existence and uniqueness of solutions for a class of nonlinear fractional dynamical systems with affine-periodic boundary conditions. We can show that there exists a solution for an $\alpha$-fractional system via the homotopy invariance of Brouwer degree, where $0<\alpha \le 1$. Furthermore, using Gronwall–Bellman inequality, we can prove the uniqueness of the solution if the nonlinearity satisfies the Lipschitz continuity. We apply the main theorem to the fractional kinetic equation and fractional oscillator with constant coefficients subject to affine-periodic boundary conditions. And in appendix, we give the proof of the nonexistence of affine-periodic solution to a given $(\alpha ,Q,T)$-affine-periodic system in the sense of Riemann–Liouville fractional integral and Caputo derivative for $0<\alpha <1$.

Appendix

Proof of Lemma 2.4

Proof

Case 1 If $\beta =n$, then

$$\begin{aligned} ^{RL}_{~~~a}\mathrm{I}^{\beta }_{t}{^{C}_{a}}\mathrm{D}^{\beta }_{t}f(\tau )&= \underbrace{\int _{a}^{t}\cdots \int _{a}^{t}}_{n-times} f^{(n)}(\tau ) \mathrm{d} \tau =\underbrace{\int _{a}^{t}\cdots \int _{a}^{t}}_{(n-1)-times}(f^{(n-1)} (\tau )-a_{1}^{(0)}) \mathrm{d}\tau \\&=\underbrace{\int _{a}^{t}\cdots \int _{a}^{t}}_{(n-2)-times}(f^{(n-2)}(\tau ) -a_{2}^{(0)}-a_{2}^{(1)}t) \mathrm{d}\tau \\&=\cdots \\&=\int _{a}^{t}(f^{'}(\tau )-a_{n-1}^{(0)}-a_{n-1}^{(1)}t- \cdots -a_{n-1}^{(n-2)}t^{n-2}) \mathrm{d}\tau \\&=f(t)-a_{n}^{(0)}-a_{n}^{(1)}t-\cdots -a_{n}^{(n-1)}t^{n-1}\\&\triangleq f(t)-\sum _{k=0}^{n-1}c_{k}t^{k}, \end{aligned}$$

where $c_{k}=a_{n-1}^{(k)}\in {\mathbb {R}}^{1}$.

Case 2 If $n-1<\beta <n$, then

$$\begin{aligned} \begin{array}{ll} ^{RL}_{~~~a}\mathrm{I}^{\beta }_{t}{^{C}_{a}}\mathrm{D}^{\beta }_{t}f(\tau )={^{RL}_{~~~a}}\mathrm{I}^{\beta }_{t}{^{RL}_{~~~a}}\mathrm{I}^{n-\beta }_{t}{^{C}_{a}}\mathrm{D}^{n}_{t}f(\tau )={^{RL}_{~~~a}}\mathrm{I}^{n}_{t}{^{C}_{a}}\mathrm{D}^{n}_{t}f(\tau ), \end{array} \end{aligned}$$

and by the proof of Case 1, we have

$$\begin{aligned} ^{RL}_{~~~a}\mathrm{I}^{\beta }_{t}{^{C}_{a}}\mathrm{D}^{\beta }_{t}f(\tau )=f(t)-\sum _{k=0}^{n-1}c_{k}t^{k}. \end{aligned}$$

$\square$

A Proof of the Nonexistence of $(\alpha ,Q,T)$-Affine-Periodic Solution for a Given $(\alpha ,Q,T)$-Affine-Periodic System when $0<\alpha <1$

Proof

Consider the $(\alpha ,Q,T)$-affine-periodic system

$$\begin{aligned} ^{C}_{a} \mathrm{D}_{t}^{\alpha }{\varvec{x}}={\varvec{f}}(t,{\varvec{x}}), \quad (t,{\varvec{x}})\in [0,T]\times {\mathbb {R}}^{n}, \end{aligned}$$

(6.1)

where $0<\alpha <1, {Q}\in GL_{n}({\mathbb {R}}), T>0, a\in {\mathbb {R}}.$

If ${\varvec{x}}={\varvec{x}}(t)$ is an $(\alpha ,Q,t)$-affine-periodic solution of (6.1), then ${\varvec{x}}(t+T)=Q{\varvec{x}}(t), \forall t\in [0,T].$

Set

$$\begin{aligned} {\varvec{y}}(t)\triangleq {\varvec{x}}(t)=Q^{-1}{\varvec{x}}(t+T), \end{aligned}$$

then y should satisfy $^{C}_{a} \mathrm{D}_{t}^{\alpha }{\varvec{y}}={\varvec{f}}(t,{\varvec{y}}).$ But in fact, it follows the definitions of Riemann–Liouville fractional integral and Caputo fractional derivative that

$$\begin{aligned} {^{C}_{a} \mathrm{D}_{t}^{\alpha }}{\varvec{y}}(t)&={^{RL}_{a} \mathrm{I}_{t}^{1-\alpha }}{^{C}_{a} \mathrm{D}_{t}^{1}}{\varvec{y}}(t)\\&={^{RL}_{a} \mathrm{I}_{t}^{1-\alpha }}{{\varvec{y}}^{'}}(t)\\&=\frac{1}{\Gamma (1-\alpha )}\int _{a}^{t}(t-\tau )^{-\alpha }{\varvec{y}}^{'}(\tau )\mathrm{d}\tau \\&=\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{t}(t-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau +T)\mathrm{d}\tau \\&=\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a+T}^{t+T}(t+T-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau )\mathrm{d}\tau \\&=\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{t+T}(t+T-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau )\mathrm{d}\tau \\&\quad -\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{a+T}(t+T-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau )\mathrm{d}\tau \\&=Q^{-1}f(t+T,{\varvec{x}}(t+T))-\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{a+T}(t+T-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau )\mathrm{d}\tau \\&=f(t,Q^{-1}{\varvec{x}}(t+T))-\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{a+T}(t+T-\tau )^{-\alpha }{\varvec{x}}^{'}(\tau )\mathrm{d}\tau \\&=f(t,{\varvec{y}}(t))-\frac{Q^{-1}}{\Gamma (1-\alpha )}\int _{a}^{a+T}(t+T-\tau )^{-\alpha }{\varvec{y}}^{'}(\tau )\mathrm{d}\tau . \end{aligned}$$

When $a=-\infty ,$

$$\begin{aligned} {^{C}_{a} \mathrm{D}_{t}^{\alpha }}{\varvec{y}}=f(t,{\varvec{y}}), \end{aligned}$$

which is reasonable and this case is also in the sense of Weyl fractional integral, but we don’t know what is the placement in infinity. When $a>-\infty ,$ that is to say, the lower limit of integral is finite or there is a “truncation” destroys the shift-invariant, which shows the nonexistence of the $(\alpha ,Q,T)$-affine-periodic solution for a given $(\alpha ,Q,T)$-affine-periodic system. And some similar results about periodicity or quasi-periodicity for fractional integrals and derivatives of periodic functions can be found in Nieto et al.’s work, see [3,4,5]. $\square$

Remark 6.1

The above conclusion shows that in order to obtain the existence and uniqueness of affine-periodic solutions for fractional affine-periodic dynamical systems, the fractional differential operators must keep shift-invariant apart from Weyl fractional differential operators. Thus, one direction for the future research is to modify the definition of fractional derivative or to seek “quasi-periodic solutions”. This is our ongoing work and will be reported elsewhere.