Abstract
In this paper, we investigate the existence and uniqueness of S-asymptotically ω-periodic solutions to fractional differential equations of order \(q\in(0, 1)\) with finite delay in a Banach space X. Existence and uniqueness theorems, which are new even in the case of \(X=\mathbf{R}^{n}\) or \(A=0\), are established. As examples of applications of our existence and uniqueness results, we obtain the S-asymptotically ω-periodic solutions for the fractional-order autonomous neural networks with delay.
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1 Introduction
As one important branch of the research on evolution equations, the study of fractional differential equations in Banach spaces is very active recently due to its strong background in physics, chemistry, engineering, biology, financial sciences, etc. (cf., e.g., [1–18] and the references therein). In this paper, we are concerned with the following fractional differential equation:
where \(\delta>0\), \(q\in(0, 1)\) and the fractional derivative is understood here in the Caputo sense, \(A: D(A)\subset X\rightarrow X\) is the generator of an analytic semigroup on a Banach space X, f is a given function, \(u_{t}: [-\delta, 0]\rightarrow X\) is defined by \(u_{t}(\theta)=u(t+\theta)\) for \(\theta\in[-\delta, 0]\) (cf., e.g., [19–23]), and \(\phi\in C([- \delta, 0] , X)\). Our main purpose is to establish existence and uniqueness theorems about the S-asymptotically ω-periodic solutions to the (1.1).
Actually, while the almost periodic, almost automorphic, and weighted pseudo almost periodic solutions to various evolution equations are investigated by many scholars (cf., e.g., [1, 22–29]), the S-asymptotically ω-periodic solutions to some evolution equations are also studied by some researchers. There have been several interesting contributions to the investigation of S-asymptotically ω-periodic solutions of differential equations and fractional differential equations in finite as well as infinite dimensional spaces (cf. [1–3, 5, 19, 28, 30]). We also note that some papers about the existence of S-asymptotically ω-periodic solutions of fractional differential equations focus on the order \(q\in(1, 2)\) ([2, 5] and references therein). Therefore, motivated by all this work, we pay attention in this paper to the study of the existence of S-asymptotically ω-periodic (mild) solutions for differential equation of fractional order of type (1.1) for the case \(q\in(0, 1)\).
The paper is organized as follows. In Section 2, we recall some basic notations and concepts. In Section 3, we discuss the existence and uniqueness of S-asymptotically ω-periodic mild solution, and as a special case of our result, we present the corresponding result in the case of \(A=0\) (Theorem 3.7). In Section 4, we apply our result to a study of the existence and uniqueness of S-asymptotically ω-periodic solution for the fractional-order neural network with finite delay.
2 Basic notations and concepts
Throughout this paper, \((X, \Vert \cdot \Vert )\) is a Banach space, \(C_{b}(\mathbf{R}_{+}, X)\) denotes the space of the continuous bounded functions from \([0, +\infty)\) to X, endowed with the norm
\(C([-\delta, 0], X)\) denotes the space of the continuous functions from \([-\delta, 0]\) to X with the norm
Definition 2.1
The fractional integral of order q with the lower limit zero for a function \(f \in L^{1}[0, \infty)\) is defined as
where \(\Gamma(\cdot)\) is the gamma function.
Definition 2.2
The Caputo derivative of order q for a function \(f \in C^{1}[0, \infty)\) can be written as
3 Existence and uniqueness theorems
In this section we discuss the existence and uniqueness of S-asymptotically ω-periodic solutions for problem (1.1).
Let A be the infinitesimal generator of a uniformly exponentially stable analytic semigroup of linear operators \(\{T(t)\}_{t\geq0}\) on X such that
where M, \(\mu>0\) are constants.
Based on the work in [6, 12], we define the mild solution for problem (1.1) as follows.
Definition 3.1
A function \(u \in C([-\delta, +\infty], X)\) satisfying the equation
is called a mild solution of problem (1.1), where
Here \(\xi_{q}\) is a probability density function defined on \((0, \infty)\) (see [12]) such that
In fact, we can see that \(\xi_{q}(\sigma)\) is the Wright type function in [11, 15]. For \(-1 < r < \infty\), the following conclusions hold.
-
(A1)
\(\int_{0}^{\infty} \sigma^{r} \xi_{q}(\sigma)\,d\sigma=\frac{\Gamma(1+r)}{ \Gamma(1+qr)}\);
-
(A2)
\(\int_{0}^{\infty}\xi_{q}(\sigma)e^{-z \sigma}\,d\sigma=E_{q}(-z)\), \(z \in\mathbf{C}\);
-
(A3)
\(\int_{0}^{\infty}q\sigma\xi_{q}(\sigma)e^{-z \sigma}\,d\sigma=E _{q,q}(-z)\), \(z\in\mathbf{C}\),
where \(E_{q}(\cdot)\) (\(E_{q,q}(\cdot)\)) is the Mittag-Leffler function (the generalized Mittag-Leffler function) (cf., e.g., [11, 15]).
Remark 3.2
-
(i)
Noting that \(\int_{0}^{\infty} \xi_{q}(\sigma)\,d\sigma=1\), we get
$$\begin{aligned} \bigl\Vert Q(t)\bigr\Vert \leq M \quad \text{and}\quad \lim _{t\rightarrow \infty }\bigl\Vert Q(t)\bigr\Vert =0. \end{aligned}$$(3.2) -
(ii)
In view of (A1), we have
$$\begin{aligned}& \bigl\Vert R(t)\bigr\Vert \leq \frac{M}{\Gamma(q)},\quad t\geq0, \end{aligned}$$(3.3)$$\begin{aligned}& \begin{aligned}[b] \int_{0}^{t}(t-s)^{q-1}\bigl\Vert R(t-s) \bigr\Vert \,ds &\leq q M \int_{0}^{t} \int_{0} ^{\infty} \sigma\xi_{q}(\sigma) (t-s)^{q-1}e^{-\mu(t-s)^{q} \sigma }\,d\sigma \,ds \\ & \leq M \int_{0}^{\infty}\xi_{q}(\sigma)\,d\sigma \int_{0}^{\infty } e^{-\mu\tau}\,d\tau \\ & = \frac{M}{\mu}. \end{aligned} \end{aligned}$$(3.4) -
(iii)
If \(A\in\mathbf{R}^{n\times n}\) is a constant matrix, then A generates a bounded operator semigroup \(T(t)=e^{A t}\) on X. Hence
$$\begin{aligned}& Q(t) = \int_{0}^{\infty} \xi_{q}(\sigma) e^{A t^{q}\sigma}\,d\sigma =E_{q}\bigl(A t^{q}\bigr), \\& R(t) = q \int_{0}^{\infty} \sigma\xi_{q}( \sigma)e^{A t^{q}\sigma }\,d\sigma=E_{q,q}\bigl(A t^{q}\bigr). \end{aligned}$$It follows from Definition 3.1 that
$$\begin{aligned} u(t)=\textstyle\begin{cases} \phi(t), & t\in[-\delta, 0], \\ {E_{q}(A t^{q})\phi(0)+ {\int_{0}^{t}(t-s)^{q-1}E_{q,q}(A (t-s)^{q})f(s, u_{s})\,ds}}, & t>0, \end{cases}\displaystyle \end{aligned}$$is a mild solution of the following problem:
$$\begin{aligned} \textstyle\begin{cases} {}^{c} D_{t}^{q} u(t)= \operatorname{Au}(t) +f(t, u_{t}),& t\geq0, \\ u(t)=\phi(t), & t\in[-\delta, 0]. \end{cases}\displaystyle \end{aligned}$$(3.5)It is not difficult to see that \(u(t)\) actually is the solution for the problem (3.5).
The following definition of S-asymptotically ω-periodic functions taking values in a Banach space X is from [28].
Definition 3.3
A function \(h \in C_{b}(\mathbf{R}_{+}, X)\) is called S-asymptotically ω-periodic if there exists \(\omega> 0\) such that \({\lim_{t\rightarrow\infty}(h(t+\omega)-h(t))=0}\). In this case, we say that ω is an asymptotic period of h.
Let \(\operatorname{SAP}_{\omega}(X)\) represent the space of all the X-valued S-asymptotically ω-periodic functions endowed with the uniform convergence norm denoted by \(\Vert \cdot \Vert _{\infty}\). Then, by virtue of [28], Proposition 3.5, \(\operatorname{SAP}_{\omega}(X)\) is a Banach space.
Set
Clearly, \(\operatorname{SAP}_{\omega, 0}(X)\) is a closed subspace of \(\operatorname{SAP}_{\omega }(X)\).
Lemma 3.4
Let \(u: [-\delta, +\infty)\rightarrow X\) be a function with \(u_{0} \in C([-\delta, 0], X)\) and \(u\vert_{[0, +\infty)}\in \operatorname{SAP}_{ \omega}(X)\). Then the function \(t \rightarrow u_{t}\) belongs to \(\operatorname{SAP}_{\omega}(C([-\delta, 0], X))\).
Proof
Since \(u_{t}\) is continuous on \([-\delta, 0]\), we see that there exists \(\overline{\theta}\in[-\delta, 0]\) such that
Setting \(\tau=t+\overline{\theta}\), we obtain
□
Set
For the function \(f:\mathbf{R}_{+} \times C([-\delta, 0], X)\rightarrow X \), we write
-
(H1)
there exists a function \(s\rightarrow L_{f}(s)\in L^{1}([0, t], \mathbf{R}_{+})\) such that
$$\bigl\Vert f(t, \psi_{1})-f(t, \psi_{2})\bigr\Vert \leq L_{f}(t)\Vert \psi _{1}-\psi_{2}\Vert _{[-\delta, 0]}, \quad \text{for all } t\geq0, \psi_{1}, \psi _{2} \in C\bigl([-\delta, 0], X\bigr), $$the function \(s\rightarrow\frac{L_{f}(s)}{(t-s)^{1-q}}\) belongs to \(L^{1}([0,t],\mathbf{R}_{+})\) and
$$\begin{aligned} \Lambda:=\sup_{t\geq0} \int_{0}^{t} \frac{L_{f}(s)}{(t-s)^{1-q}}\,ds< \frac{\Gamma(q)}{M}; \end{aligned}$$(3.6) -
(H2)
\({K:=\sup_{t\geq0}\int_{0}^{t}\frac{\Vert f(s, 0)\Vert }{(t-s)^{1-q}}\,ds< \infty}\);
-
(H3)
there exists \(\omega>0\), for all \(\varphi\in C([-\delta, 0], X)\), \({\lim_{t\rightarrow\infty} \Vert f(t+\omega, \varphi)-f(t,\varphi)\Vert =0}\).
Theorem 3.5
Assume that (H1)-(H3) hold. Then the problem (1.1) has a unique S-asymptotically ω-periodic mild solution.
Proof
For every \(\phi\in C([-\delta,0],X)\), we define the function \(y(t)=\phi(t)\) for \(t\in[-\delta, 0]\), \(y(t)=Q(t)\phi(0)\) for \(t\geq0\). Then \(y\in C([-\delta,\infty), X)\). Set
It is obvious that u satisfies (3.1) if and only if x satisfies \(x_{0}=0\) and for \(t\geq0\),
For each \(x\in\widetilde{C}_{b}(X)\), we write \(C_{1}=\Vert x\Vert _{\infty}+M \Vert \phi(0)\Vert +\Vert \phi \Vert _{[-\delta, 0]}\). Then
Hence
We consider the operator \(\mathcal{F}:\widetilde{C}_{b}(X) \rightarrow \widetilde{C}_{b}(X)\) as follows:
In view of (3.3), (3.7), (3.6) and (H2), we have
So, the operator \(\mathcal{F}\) is well defined.
It is clear that the fixed points of \(\mathcal{F}\) are mild solutions to problem (1.1).
Now, we show that \(\mathcal{F}\) is \(\operatorname{SAP}_{\omega, 0}(X)\)-valued.
For each \(x\in \operatorname{SAP}_{\omega, 0}(X)\), (3.2) implies that \(y\vert_{[0, \infty)}\in \operatorname{SAP}_{\omega}(X)\). It follows from Lemma 3.4 that the function \(t\rightarrow y_{t}\) belongs to \(\operatorname{SAP}_{\omega}(C([- \delta, 0], X))\).
Moreover, we have
Noting that \({t+\omega-s\geq\frac{t+\omega}{\omega}(\omega-s)}\), we have
which implies that
By (3.7), we get
By (H3), we can see that, for each \(\varepsilon>0\), there is a positive constant \(L_{1}\) such that
Noting that (3.4) and \({t-s\geq\frac{t}{L_{1}}(L_{1}-s)}\), we deduce that
Since
we know that there is a positive constant \(L_{2}>0\) such that
then
Thus,
So
Moreover, for \(x, \widetilde{x}\in \operatorname{SAP}_{\omega, 0}(X)\), we have
Hence
which means that \(\mathcal{F}\) is a contraction mapping. Then the proof now can be finished by using the contraction mapping principle. □
Remark 3.6
In the proof of Theorem 3.5, we can see the result is true when (H3) changes to
because now we can directly obtain \(\lim_{t\rightarrow\infty }I_{2}(t)=0\).
In the case of \(A\equiv0\), (1.1) takes the form of
It is well known that (3.10) is equivalent to the following integral equation:
Thus, we study (3.10) just like the case of
in (3.1). Clearly, we just need to revise (3.8) and (3.9), that is, if we replace (H2), (H3) by
and (H3′), respectively, then we can obtain the corresponding result of problem (3.10), hence we have the following result.
Theorem 3.7
Assume that (H1) (\(M\equiv1\)), (H2′) and (H3′) hold. Then the following fractional differential equation:
has a unique S-asymptotically ω-periodic solution.
4 Applications
Example 4.1
Let \(X=\mathbf{R}^{2}\). For the vector \(x=(x_{1}, x_{2})^{T} \in\mathbf{R}^{2}\), we define
For the matrix \(A=(a_{ij})_{2\times2}\), we define
We consider the following fractional-order neural network model with finite delay (FNND) on X:
Problem (4.1) can be written in the vector form as follows:
where
It is well known ([31]) that B generates a bounded operator semigroup
and
(i.e. \(M=1\), \(\mu=2\)). Moreover, (3.1) is now the solution of (4.1) (Remark 3.2(iii)).
For \(\varphi, \widetilde{\varphi} \in C([-1, 0], X)\), it is easy to see that
Since
we deduce that \((t+t^{\frac{5}{2}})^{-\frac{1}{3}}\in L^{1}(0, t)\) and
Hence
Moreover,
and
Therefore, the conditions in Theorem 3.5 are satisfied. Thus, by virtue of Theorem 3.5, the problem (4.1) has a unique S-asymptotically 1-periodic solution. We refer the reader to Figure 1 below for a numerical solution of (4.1).
Example 4.2
Let \(X=\mathbf{R}\). We consider the following fractional-order model on X:
Problem (4.2) can be written in the form as follows:
where
For any \(\varphi, \widetilde{\varphi} \in C([-1, 0], X)\), we have
Noting that
we get
So
Moreover,
In view of
being Fresnel integrals and (4.3), we obtain
Clearly,
Hence, for each \(\varepsilon>0\), there is a positive constant \(L>0\) such that
Therefore, noting that \(t-L>\frac{t}{L}(L-s)\), we obtain
i.e. (H3′) is satisfied. Thus, the conditions in Theorem 3.7 are fulfilled. Hence, by Theorem 3.7, the problem (4.2) has a unique S-asymptotically 1-periodic solution. We refer the reader to Figure 2 above for a numerical solution of (4.2).
References
Agarwal, RP, de Andrade, B, Cuevas, C: On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differ. Equ. 2010, Article ID 179750 (2010)
Cuevas, C, de Souza, JC: S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22, 865-870 (2009)
Cuevas, C, Pierri, M, Sepulveda, A: Weighted S-asymptotically ω-periodic solutions of a class of fractional differential equations. Adv. Differ. Equ. 2011, Article ID 584874 (2011)
Diagana, T, Mophou, GM, N’Guérékata, G: On the existence of mild solutions to some semilinear fractional integro-differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 58 (2010)
dos Santos, JPC, Cuevas, C: Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations. Appl. Math. Lett. 23, 960-965 (2010)
El-Borai, MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433-440 (2002)
Li, F, Liang, J, Lu, TT, Zhu, H: A nonlocal Cauchy problem for fractional integrodifferential equations. J. Appl. Math. 2012, Article ID 901942 (2012)
Li, F, Liang, J, Xu, HK: Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, 510-525 (2012)
Liu, KW, Jiang, W: Stability of nonlinear Caputo fractional differential equations. Appl. Math. Model. 40(5-6), 3919-3924 (2016)
Lv, ZW, Liang, J, Xiao, TJ: Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order. Comput. Math. Appl. 62, 1303-1311 (2011)
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Mainardi, F, Paradisi, P, Gorenflo, R: Probability distributions generated by fractional diffusion equations. In: Kertesz, J, Kondor, I (eds.) Econophysics: An Emerging Science. Kluwer, Dordrecht (2000)
Mophou, GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72(3-4), 1604-1615 (2010)
Mophou, GM, N’Guérékata, G: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79, 315-322 (2009)
Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999)
Shu, X, Lai, Y, Chen, Y: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74(5), 2003-2011 (2011)
Shu, X, Wang, Q: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order \(1< \alpha<2 \). Comput. Math. Appl. 64(6), 2100-2110 (2012)
Wang, DJ, Xia, ZN: Pseudo almost automorphic solution of semilinear fractional differential equations with the Caputo derivatives. Fract. Calc. Appl. Anal. 18(4), 951-971 (2015)
Dimbour, W, Mophou, G, N’Guérékata, GM: S-asymptotically periodic solutions for partial differential equations with finite delay. Electron. J. Differ. Equ. 2011, 117 (2011)
Liang, J, Xiao, TJ: Solutions to nonautonomous abstract functional equations with infinite delay. Taiwan. J. Math. 10, 163-172 (2006)
Liang, J, Xiao, TJ: Solvability of the Cauchy problem for infinite delay equations. Nonlinear Anal. 58, 271-297 (2004)
Liu, JH: Bounded and periodic solutions of finite delay evolution equations. Nonlinear Anal. 34, 101-111 (1998)
Liu, JH: Periodic solutions of infinite delay evolution equations. J. Math. Anal. Appl. 247, 27-644 (2000)
Diagana, T: Almost periodic solutions to some second-order nonautonomous differential equations. Proc. Am. Math. Soc. 140, 279-289 (2012)
Diagana, T: Pseudo-almost periodic solutions for some classes of nonautonomous partial evolution equations. J. Franklin Inst. 348, 2082-2098 (2011)
Liang, J, Xaio, TJ, Yang, H: About periodicity of impulsive evolution equations through fixed point theory. Fixed Point Theory Appl. 2015, 1 (2015)
Liu, JH: Bounded and periodic solutions of semi-linear evolution equations. Dyn. Syst. Appl. 4, 341-350 (1995)
Henríquez, HR, Pierre, M, T’aboas, P: On S-asymptotically ω-periodic function on Banach spaces and applications. J. Math. Anal. Appl. 343, 1119-1130 (2008)
Hilfer, H: Applications of Fractional Calculus in Physics. World Scientific, Singapure (2000)
Dimbour, W, N’Guérékata, GM: S-asymptotically ω-periodic solutions to some classes of partial evolution equations. Appl. Math. Comput. 218, 7622-7628 (2012)
Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Acknowledgements
The work was supported partly by the NSF of China (11561077, 11571229, 11201413) and the NSF of Yunnan Province (2013FB034) and The Reserve Talents of Young and Middle-Aged Academic and Technical Leaders of the Yunnan Province. The authors thank the referees very much for their suggestions and comments.
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Li, F., Liang, J. & Wang, H. S-asymptotically ω-periodic solution for fractional differential equations of order \(q\in(0, 1)\) with finite delay. Adv Differ Equ 2017, 83 (2017). https://doi.org/10.1186/s13662-017-1137-y
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DOI: https://doi.org/10.1186/s13662-017-1137-y