Abstract
Based on some previous works, an equivalent equations is obtained for the differential equations of fractional-orderq ∈(1, 2) with non-instantaneous impulses, which shows that there exists the general solution for this impulsive fractional-order systems. Next, an example is used to illustrate the conclusion.
1 Introduction
Fractional differential equations has gained much attention in literature because of its applications for description of hereditary properties in many fields, and some progresses were gotten in computation methods, controllability, existence etc. for fractional differential equations [1–6]. Moreover, impulsive fractional (partial) differential equations were widely studied [7–29] due to importance in description of some processes in which sudden, discontinuous jumps occur, and general solution has been discovered for several impulsive fractional order systems in [30–35].
However, Hernandez and O’Regan in [36] pointed out that the instantaneous impulses (considered in almost all papers about impulsive differential equations) cannot characterize some processes such as evolution processes in pharmacotherapy, and presented a kind of impulsive differential equations with non-instantaneous impulses. Moreover, the existence of solution is considered for some fractional order systems with non-instantaneous impulses in [37, 38].
Based on the above-works, we will study the following fractional order system with non- instantaneous impulses.
here
Next, let us introduce the concept of the fractional derivative and some conclusions in Section 2, and provide main result in section 3, and give an example to show the usefulness of the obtained result.
2 Preliminaries
[39]. The left-sided Riemann-Liouville fractional integral
where Γ(·) is the Gamma function.
[39]. The Caputo fractional derivative
whereD = d/dt and q ∈ (n - 1, n).
[39, 40]. If the function g(t, x) is continuous, then the initial value problem
is equivalent to the following nonlinear Volterra integral equation of the second kind,
[31]. Let ξ and ζ be two constants. The impulsive system
is equivalent to the integral equation
provided that the integral in (2.2) exists. Here J0 = [0, t1] and Jk = (tk, tk+1] (k = 1, 2,..., m).
3 Main result
For convenience, letf = f(τ, x(τ)) in this section. Consider condition (1.1a) in system (1.1) by using two different approches:
Next, substituting (i) into system (1.1), we get
That is,
In fact, x͂(t) satisfies conditions (1.1a)–(1.1c) in system (1.1). But, we will show that x͂(t) isn’t a solution of system (1.1). For system (1.1), we have
And system (3.4) is equivalent to
Moreover, letting
Therefore, if x͂(t) is a solution of system (1.1), then (3.6) is equivalent to (3.5). Thus,
Eq. (3.7) is an unfit equation, which means that x͂(t) isn’t a solution of system (1.1). Therefore, we will regard x͂(t) as an approximate solution to seek the exact solution of system (1.1).
Substituting (ii) into system (1.1), we obtain
By initial conditions x(sk) = gk(sk, x(sk)) and x′(sk) = g′k(sk, x(sk)) (here k = 1, 2, ..., N), we obtain B0 = 0, C0 = 0,
and
Substituting (3.9)-(3.10) into (3.8), we get
In fact, Eq. (3.11) satisfies conditions (1.1a)–(1.1c) and
Therefore, Eq. (3.11) satisfies all conditions of system (1.1), and it is a solution of system (1.1).
Let ξk and ζk (here k = 1, 2, ..., N) be some constants. System (1.1)isequivalent with the integral equation
provided that the integral in (3.12) exists.
‘Sufciency’; the solution of (1.1) for t ∈ (0, t1] satisfies
and x(t) = g1(t, x(t)) for t ∈ (t1, s1].
For t ∈ (s1, t2], the approximate solution of (1.1) is given by
Let e1(t) = x(t) – x͂(t) for t ∈ (s1, t2]. Moreover, by the particular solution (3.11), the exact solution x(t) of system (1.1) satisfies
Thus,
This means e1(t) is connected with lim
where χ(·, ·) is an undetermined function with χ(0, 0) = 1. Thus,
On the other hand, letting t1 → s1, we get
Using Lemma 2.4 for system (3.18), we get 1 – χ(y, z) = ξ1y + ζ1z for ∀y, z ∈ R, here ξ1 and ζ1 are two constants. Thus,
and x(t) = g2(t, x(t)) for t ∈ (t2, s2].
Next, for t ∈ (sk, tk+1] (here k ∈ {1, 2, ..., N}), the approximate solution of (1.1) is provided by
Let ek(t) = x(t) - x͂(t) for t ∈ (sk, tk+1]. Moreover, by the particular solution (3.11), the exact solution x(t) of system (1.1) satisfies
Thus,
Similarly to (3.16), suppose
where κ(·, ·) is an undetermined function with κ(0, 0) = 1. Thus,
Moreover, considering a special case
Using Lemma 2.4 and Eq. (3.23) for (3.24), we have 1- κ(y, z) = ξky + ζkz for ∀y, z ∈ R, here ξk and ζk are two constants. Thus,
‘Necessity’; taking the fractional derivative to Eq. (3.12) fort ∈ (sk, tk+1] (here k = 1, 2, ..., N), we get
Therefore, Eq. (3.12) satisfies the condition (1.1a). Next, Eq. (3.12) also satisfies the conditions (1.1b) and (1.1c). Furthermore, by Eq. (3.12), we obtain
Thus,
So, Eq. (3.12) satisfies all conditions of system (1.1).
By “Sufciency” and “Necessity”, system (1.1) is equivalent to Eq. (3.12). The proof is completed.
4 Example
Let us consider the general solution of the impulsive fractional system
By Theorem 3.1, system (4.1) has a general solution
where ξ and ζ are two constants.
Eq. (4.2) for
Therefore, Eq. (4.2) (for
Acknowledgments
The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant No. 21576033, 21636004, 61563023) and Jiujiang University Research Foundation (Grant No. 8400183).
References
[1] Yang X.J., Machado J.A.T., Baleanu D., Cattani C., On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016, 26(8), 110-118.10.1063/1.4960543Search in Google Scholar PubMed
[2] Yang X.J., Machado J.A.T., Hristov J., Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow, Nonlinear Dynamics, 2015, 84(1), 3-7.10.1007/s11071-015-2085-2Search in Google Scholar
[3] Yang X.J., Machado J.A.T., Srivastava H.M., A new numerical technique for solving the local fractional diffusion equation, Appl. Math. Comput.„ 2016, 274, 143-151.10.1016/j.amc.2015.10.072Search in Google Scholar
[4] Kailasavalli S., Baleanu D., Suganya S., Arjunan M. M., Exact controllability of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Analele Stiintifice ale Universitatii Ovidius Constanta-Seria Matematica, 2016, 24(1), 29-55.10.1515/auom-2016-0017Search in Google Scholar
[5] Suganya S., Baleanu D., Arjunan M.M., A note on fractional neutral integro-differential inclusions with state-dependent delay in Banach spaces, Journal of Computational Analysis and Applications, 2016, 20(7), 1302-1317.10.1016/j.camwa.2016.01.016Search in Google Scholar
[6] Suganya S., Baleanu D., Selvarasu S., Arjunan M.M., About the Existence Results of Fractional Neutral Integrodifferential Inclusions with State-Dependent Delay in Fréchet Spaces, Journal of Function Spaces, vol. 2016, Article ID 6165804, 9 pages, 2016.10.1155/2016/6165804Search in Google Scholar
[7] Yukunthorn W., Ntouyas S.K., Tariboon J., Impulsive Multiorders Riemann-Liouville Fractional Differential Equations, Discrete Dynamics in Nature and Society, vol. 2015, Article ID 603893, 9 pages, 2015.10.1155/2015/603893Search in Google Scholar
[8] Thaiprayoon C., Tariboon J., Ntouyas S.K., Impulsive fractional boundary-value problems with fractional integral jump conditions, Boundary Value Problems, vol. 2014, article 17, 16 pages, 2014.10.1186/1687-2770-2014-17Search in Google Scholar
[9] Zhang X., ZhangX., Liu Z., Ding W., Cao H., Shu T., On the general solution of impulsive systems with Hadamard fractional derivatives, Math. Prob. Eng., vol. 2016, Article ID 2814310, 12 pages, 2016.10.1155/2016/2814310Search in Google Scholar
[10] Yukunthorn W., Suantai S., Ntouyas S.K, Tariboon J., Boundary value problems for impulsive multi-order Hadamard fractional differential equations, Boundary Value Problems, vol. 2015, article 148, 13 pages, 2015.10.1186/s13661-015-0414-5Search in Google Scholar
[11] Fu X., Liu X., Lu B., On a new class of impulsive fractional evolution equations, Adv. Differ. Equ., vol. 2015, article 227, 16 pages, 2015.10.1186/s13662-015-0561-0Search in Google Scholar
[12] Yukunthorn W., Ahmad B., Ntouyas S.K., Tariboon J., On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal.: HS, 2016, 19, 77-92.10.1016/j.nahs.2015.08.001Search in Google Scholar
[13] Ahmad B., Sivasundaram S., Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal.: HS, 2009, 3, 251-258.10.1016/j.nahs.2009.01.008Search in Google Scholar
[14] Ahmad B., Sivasundaram S., Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal.: HS, 2010, 4, 134-141.10.1016/j.nahs.2009.09.002Search in Google Scholar
[15] Zhang X., Shu T., Liu Z., Ding W., Peng H., He J., On the concept of general solution for impulsive differential equations of fractional-order q ∈(2 ,3), Open math., 2016, 14, 452-473.10.1515/math-2016-0042Search in Google Scholar
[16] Ahmad B., Wang G., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Comput. Math. Appl., 2010, 59, 1341-1349.10.1016/j.camwa.2011.04.033Search in Google Scholar
[17] Tian Y., Bai Z., Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comput. Math. Appl., 2010, 59, 2601-2609.10.1016/j.camwa.2010.01.028Search in Google Scholar
[18] Cao J., Chen H., Some results on impulsive boundary value problem for fractional differential inclusions, Electron. J. Qual. Theory Differ. Equ., 2010, 11, 1-24.10.14232/ejqtde.2011.1.11Search in Google Scholar
[19] Wang G., Ahmad B., Zhang L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal. Theory Methods Appl., 2011, 74, 792-804.10.1016/j.na.2010.09.030Search in Google Scholar
[20] Wang G., Ahmad B., Zhang L., Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl., 2010, 59, 1389-1397.10.1016/j.camwa.2011.04.004Search in Google Scholar
[21] Feckan M., Zhou Y., Wang J.R., On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 2012,17, 3050-3060.10.1016/j.cnsns.2011.11.017Search in Google Scholar
[22] Stamova I., Stamov G., Stability analysis of impulsive functional systems of fractional order, Commun. Nonlinear Sci. Numer. Simulat., 2014, 19, 702-709.10.1016/j.cnsns.2013.07.005Search in Google Scholar
[23] Zhang X., On impulsive partial differential equations with Caputo-Hadamard fractional derivatives, Adv. Differ. Equ., vol. 2016, article 281, 21pages, 2016.Search in Google Scholar
[24] Abbas S., Benchohra M., Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Anal. HS, 2010, 4, 406-413.10.1016/j.nahs.2009.10.004Search in Google Scholar
[25] Abbas S., Benchohra M., Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay, Fract. Calc. Appl. Anal., 2010, 13, 225-242.10.1515/dema-2013-0280Search in Google Scholar
[26] Abbas S., Agarwal R.P., Benchohra M., Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay, Nonlinear Anal. HS, 2010, 4, 818-829.10.1016/j.nahs.2010.06.001Search in Google Scholar
[27] Abbas S., Benchohra M., Gorniewicz L., Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative, Scientiae Mathematicae Japonicae, 2010, 72 (1), 49-60.Search in Google Scholar
[28] Benchohra M., Seba D., Impulsive partial hyperbolic fractional order differential equations in Banach spaces, J. Fract. Calc. Appl., 2011, 1 (4), 1-12.10.7153/fdc-02-07Search in Google Scholar
[29] Guo T., Zhang K., Impulsive fractional partial differential equations, Appl. Math. Comput., 2015, 257, 581-590.10.1016/j.amc.2014.05.101Search in Google Scholar
[30] Zhang X., Zhang X., Zhang M., On the concept of general solution for impulsive differential equations of fractional order q ∈ (0,1), Appl. Math. Comput., 2014, 247, 72-89.Search in Google Scholar
[31] Zhang X., On the concept of general solutions for impulsive differential equations of fractional order q ∈ (1, 2), Appl. Math. Comput., 2015, 268, 103-120.10.1016/j.amc.2015.05.123Search in Google Scholar
[32] Zhang X., The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect, Adv. Differ. Equ., vol. 2015, article 215, 16 pages, 2015.Search in Google Scholar
[33] Zhang X., Agarwal P., Liu Z., Peng H., The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1, 2), Open Math., 2015, 13, 908-930.10.1515/math-2015-0073Search in Google Scholar
[34] Zhang X., Shu T., Cao H., Liu Z., Ding W., The general solution for impulsive differential equations with Hadamard fractional derivative of order q∈(1, 2), Adv. Differ. Equ., vol. 2016, article 14, 36 pages, 2016.Search in Google Scholar
[35] Zhang X., Zhang X., Liu Z., Peng H., Shu T., Yang, S., The General Solution of Impulsive Systems with Caputo- Hadamard Fractional Derivative of Orderq ∈ C(ℜ(q) ∈ (1, 2)), Math. Prob. Eng., vol. 2016, Article ID 8101802, 20 pages, 2016.Search in Google Scholar
[36] Hernandez E., O’Regan D., On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 2013, 141, 1641-1649.10.1090/S0002-9939-2012-11613-2Search in Google Scholar
[37] Li P.L., Xu C.J., Mild solution of fractional order differential equations with not instantaneous impulses, Open Math., 2015, 13, 436-443.10.1515/math-2015-0042Search in Google Scholar
[38] Suganya S., Baleanu D., Kalamani P., Arjunan M.M., On fractional neutral integro-differential systems with state-dependent delay and non-instantaneous impulses, Adv. Differ. Equ., vol. 2015, article 372, 39 pages, 2015.Search in Google Scholar
[39] Kilbas A.A., Srivastava H.H., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).Search in Google Scholar
[40] Diethelm K., Ford N.J., Analysis of fractional differential equations, J. Math. Anal. Appl., 2002, 265, 229-248.10.1006/jmaa.2000.7194Search in Google Scholar
© 2016 Xianmin Zhang et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.