Attractivity for a k-dimensional system of fractional functional differential equations and global attractivity for a k-dimensional system of nonlinear fractional differential equations

In recent years, many researchers have been focused on investigation of fractional differential equations which has played an important role in different areas of science (see for example, [1–22] and the references therein). As you know, there are many practical applications of fractional differential equations in different fields of science such as economy, biology, and the study of forced van der Pol oscillators (see for example, [23–25] and the references therein). On the other hand, there are a few papers on the attractivity of solutions for fractional differential equations and fractional functional differential equations (see for example, [15] and [16]). For the details of basic notions of this paper such the standard Caputo fractional derivative, the standard Riemann-Liouville fractional derivative, and the fractional integral of order q for a function f see [18]. In 2011, Chen and Zhou reviewed the attractivity of solutions for the fractional functional differential equation ${}^{c}D^{\alpha }x(t)=f(t,x_t)$ for $t\in (t_0,\mathrm{\infty })$ via the boundary value condition $x(t)=\varphi (t)$ for $t_0-r\le t\le t_0$, where $t_0\ge 0$, $r>0$, $0<\alpha <1$, ^c D is the standard Caputo fractional derivative, $\varphi \in C([t_0-r,t_0],\mathbb{R})$ and $f:(t_0,\mathrm{\infty })\times C([-r,0],\mathbb{R})\to \mathbb{R}$ a function with some properties [16]. In 2012, Chen et al. reviewed the global attractivity of solutions for the nonlinear fractional differential equation $D^{\alpha }x(t)=g(t,x(t))$ for $t\in (t_0,\mathrm{\infty })$ via the boundary value problem ${[D^{\alpha -1}x(t)]}_{t=t_0}=x_0$, where $t_0\ge 0$, $0<\alpha <1$, $x_0$ is a constant, D is the standard Riemann-Liouville fractional derivative, and $g:(t_0,\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}$ a function with some properties [15]. Also, they investigated the global attractivity of solutions for the nonlinear fractional differential equation ${}^{c}D^{\alpha }x(t)=g(t,x(t))$ for $t\in (t_0,\mathrm{\infty })$ via the boundary value problem $x(t_0)=x_0$, where $t_0\ge 0$, $0<\alpha <1$, $x_0$ is a constant, ^c D is the standard Caputo fractional derivative, and $g:(t_0,\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}$ a function with some properties [15].

In this paper, we investigate the attractivity of solutions for a k-dimensional system of fractional differential equations. Also, we investigate the global attractivity of solutions for another k-dimensional system of nonlinear fractional differential equations.

via the boundary value problems ${[D^{{\alpha }_1-1}{x}_1(t)]}_{t=t_0}=x_1^0,{[D^{{\alpha }_2-1}{x}_2(t)]}_{t=t_0}=x_2^0,\dots ,{[D^{{\alpha }_k-1}{x}_k(t)]}_{t=t_0}=x_k^0$, where $0<{\alpha }_i<1$ for $i=1,2,\dots ,k$, $t_0\ge 0$, $t\in (t_0,\mathrm{\infty })$, D is the Riemann-Liouville fractional derivative, $J=(t_0,\mathrm{\infty })$, $x_1^0,\dots ,x_k^0$ are constants, and $g_i:J\times {\mathbb{R}}^n\times {\mathbb{R}}^n\times \cdots \times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is an integrable function satisfying some assumptions that will be specified later for $i=1,2,\dots ,k$. In fact, we say that the solution $(x_1(t),x_2(t),\dots ,x_k(t))$ of the problem (2.1) is attractive whenever if there exists a constant $b_i^0(t_0)>0$ such that $|{\varphi }_i(s)|\le b_i^0$ for all $i=1,2,\dots ,k$ and $s\in [t_0-r,t_0]$, then ${lim}_{t\to \mathrm{\infty }}{x}_i(t,t_0,{\varphi }_i)\to 0$. Also, the zero solution $x(t)$ of the problem (2.2) is said to be globally attractive whenever each solution tends to zero as $t\to \mathrm{\infty }$. Let $X=C(J,{\mathbb{R}}^n)$ be the Banach space of all continuous functions from J into ${\mathbb{R}}^n$ with the norm $\parallel x\parallel ={sup}_{t\in J}|x(t)|$, where $|\cdot |$ denotes a suitable complete norm on ${\mathbb{R}}^n$. It is clear that the product space is also a Banach space, where ${\parallel (x_1,x_2,\dots ,x_k)\parallel }_{\ast }=\parallel x_1\parallel +\parallel x_2\parallel +\cdots +\parallel x_k\parallel$. We need the following Schauder fixed-point theorem and improvement of a fixed-point theorem of Krasnoselskii due to Burton, which one can find in [14, 17] and [19].

Theorem 2.1 If U is a nonempty, closed, bounded, and convex subset of the Banach space X and $T:U\to U$ is completely continuous, then T has a fixed point.

Theorem 2.2 Let S be a nonempty, closed, convex, and bounded subset of the Banach space X, $A:X\to X$ a contraction with constant $l<1$, $B:S\to X$ a continuous map which $B(S)$ resides in a compact subset of X and $x=Ax+By$ and $y\in S$ implies $x\in S$. Then the operator equation $Ax+Bx=x$ has a solution in S.

for $i=1,2,\dots ,k$. It is easy to check that $(x_1(t),x_2(t),\dots ,x_k(t))$ is a solution of the problem (2.1) if and only if $(x_1(t),x_2(t),\dots ,x_k(t))$ is a fixed point of the operator T.

for all $t\in J$ and $f_i\in L^{\frac{1}{{\alpha }_{i1}}}(J\times C([-r,0],{\mathbb{R}}^n)\times C([-r,0],{\mathbb{R}}^n)\times \cdots \times C([-r,0],{\mathbb{R}}^n),{\mathbb{R}}^n)$. Then the problem (2.1) has at least one attractive solution $(x_1,x_2,\dots ,x_k)$ such that $x_i\in C([t_0-r,\mathrm{\infty }),{\mathbb{R}}^n)$ for all $i=1,2,\dots ,k$.

where $\tilde{t}$ is a constant. It is easy to check that $S_1$ is a closed, bounded, and convex subset of . We show that the operator T has a fixed point in $S_1$. This implies that the problem (2.1) has a solution. Note that $|T_i(x_1,x_2,\dots ,x_k)(t)|\le {(t-t_0)}^{-{\gamma }_{i1}}$ for all $i=1,2,\dots ,k$ and so $T(S_1)\subset S_1$. Now, we show that T is continuous. Let $(x_1^m,x_2^m,\dots ,x_k^m),(x_1,x_2,\dots ,x_k)\in S_1$ for all $m\ge 1$ and ${lim}_{m\to \mathrm{\infty }}|x_i^m(t)-x_i(t)|=0$ for all $i=1,2,\dots ,k$. Then, we have ${lim}_{m\to \mathrm{\infty }}{f}_i(t,x_{1_t}^m,x_{2_t}^m,\dots ,x_{k_t}^m)=f_i(t,x_{1_t},x_{2_t},\dots ,x_{k_t})$ for all $i=1,2,\dots ,k$ and $t>t_0$. Let $ϵ>0$ be given. Choose $\tilde{T}>t_0$ such that $t\ge \tilde{T}$ implies that ${(t-t_0)}^{-{\gamma }_{i1}}<\frac{ϵ}{2}$. Let ${\nu }_{i1}=\frac{{\alpha }_i-1}{1-{\alpha }_{i1}}$ and note that $1+{\nu }_{i1}>0$ for $i=1,2,\dots ,k$. Also, we have

$\begin{array}{c}|T_i(x_1^m,x_2^m,\dots ,x_k^m)(t)-T_i(x_1,x_2,\dots ,x_k)(t)|\hfill \\ \le \frac{1}{\mathrm{\Gamma }({\alpha }_i)}{\int }_{t_0}^t{(t-s)}^{{\alpha }_i-1}|f_i(s,x_{1_s}^m,x_{2_s}^m,\dots ,x_{k_s}^m)-f_i(s,x_{1_s},x_{2_s},\dots ,x_{k_s})|ds\hfill \\ \le \frac{1}{\mathrm{\Gamma }({\alpha }_i)}{\left\{{\int }_{t_0}^t{\left[{(t-s)}^{{\alpha }_i-1}\right]}^{\frac{1}{1-{\alpha }_{i1}}}ds\right\}}^{1-{\alpha }_{i1}}\hfill \\ \times {\left[{\int }_{t_0}^t\right|f_i(s,x_{1_s}^m,x_{2_s}^m,\dots ,x_{k_s}^m)-f_i(s,x_{1_s},x_{2_s},\dots ,x_{k_s}){|}^{\frac{1}{{\alpha }_{i1}}}ds]}^{{\alpha }_{i1}}\hfill \\ \le \frac{1}{\mathrm{\Gamma }({\alpha }_i)}{\left(\frac{1}{1+{\nu }_{i1}}{(t-t_0)}^{1+{\nu }_{i1}}\right)}^{1-{\alpha }_{i1}}\hfill \\ \times {\left[{\int }_{t_0}^{\tilde{T}}\right|f_i(s,x_{1_s}^m,x_{2_s}^m,\dots ,x_{k_s}^m)-f_i(s,x_{1_s},x_{2_s},\dots ,x_{k_s}){|}^{\frac{1}{{\alpha }_{i1}}}ds]}^{{\alpha }_{i1}}\hfill \\ \le \frac{1}{\mathrm{\Gamma }({\alpha }_i)}{\left(\frac{1}{1+{\nu }_{i1}}{(\tilde{T}-t_0)}^{1+{\nu }_{i1}}\right)}^{1-{\alpha }_{i1}}{(\tilde{T}-t_0)}^{{\alpha }_{i1}}\hfill \\ \times \underset{t_0\le s\le \tilde{T}}{sup}|f_i(s,x_{1_s}^m,x_{2_s}^m,\dots ,x_{k_s}^m)-f_i(s,x_{1_s},x_{2_s},\dots ,x_{k_s})|\hfill \end{array}$

we get ${lim}_{t_2\to t_1}|T_i(x_1,x_2,\dots ,x_k)(t_2)-T_i(x_1,x_2,\dots ,x_k)(t_1)|=0$ in all cases. This implies that the set $T(S_1)$ is equi-continuous. Since $T(S_1)\subset S_1$ is uniformly bounded, $T(S_1)$ is relatively compact. Now by using Theorem 2.1, T has a fixed point in $S_1$ which is a solution of the problem (2.1). Since $x(t)=(x_1(t),x_2(t),\dots ,x_k(t))\in S_1$, ${lim}_{t\to \mathrm{\infty }}x(t)=0$. Thus, $x(t)$ is an attractive solution for the problem (2.1). □

for all $i=1,2,\dots ,k$, $t\in J$ and $x_i\in C([t_0-r,\mathrm{\infty }),{\mathbb{R}}^n)$. Then the problem (2.1) has at least one attractive solution $(x_1,x_2,\dots ,x_k)$ such that $x_i\in C([t_0-r,\mathrm{\infty }),{\mathbb{R}}^n)$ for all $i=1,2,\dots ,k$.

where $\tilde{t}$ is a constant. By using a similar techniques and proof in Theorem 3.1, one can show that $T(S_2)\subset S_2$, T is continuous and $T(S_2)$ is relatively compact. Now by using Theorem 2.1, T has a fixed point in $S_2$ which is a solution of the problem (2.1). Since $x(t)=(x_1(t),x_2(t),\dots ,x_k(t))\in S_2$, ${lim}_{t\to \mathrm{\infty }}x(t)=0$. Thus, $x(t)$ is an attractive solution for the problem (2.1). □

for all $t\in J$. Then the problem (2.1) has at least one attractive solution $(x_1,x_2,\dots ,x_k)$ such that $x_i\in C([t_0-r,\mathrm{\infty }),{\mathbb{R}}^n)$ for all $i=1,2,\dots ,k$.

where $\tilde{t}$ is a constant. By using a similar techniques and proof in Theorem 3.1, one can show that $T(S_3)\subset S_3$, T is continuous and $T(S_3)$ is relatively compact. Now, by using Theorem 2.1, T has a fixed point in $S_3$ which is a solution of the problem (2.1). Since $x(t)=(x_1(t),x_2(t),\dots ,x_k(t))\in S_3$, ${lim}_{t\to \mathrm{\infty }}x(t)=0$. Thus, $x(t)$ is an attractive solution for the problem (2.1). □

where $B_i(x_1,x_2,\dots ,x_k)(t)=\frac{1}{\mathrm{\Gamma }({\alpha }_i)}{\int }_{t_0}^t{(t-s)}^{{\alpha }_i-1}{g}_i(s,x_1(s),x_2(s),\dots ,x_k(s))ds$ for all $i=1,2,\dots ,k$. It is easy to check that $(x_1(t),x_2(t),\dots ,x_k(t))$ is a solution of the problem (2.2) if and only if it is a fixed point of the operator T. Note that A is a contraction with constant 0.

Theorem 3.4 Suppose that for each $i\in \{1,2,\dots ,k\}$ there exist ${\alpha }_i<{\beta }_{i1}^{\prime }<1$ and $M_i\ge 0$ such that $|g_i(t,x_1(t),x_2(t),\dots ,x_k(t))|\le M_i{(t-t_0)}^{-{\beta }_{i1}^{\prime }}$ for all $t\in J$ and $x_1,\dots ,x_k\in C((t_0,\mathrm{\infty }),{\mathbb{R}}^n)$. Then the zero solution of the problem (2.2) is globally attractive.