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This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main results.
In this paper we are interested in the solvability of solutions for the following impulsive fractional differential equations with derivative dependence
where with and denotes the right Riemann–Loiuville fractional derivative of order the operator is defined as where and is the right Riemann–Liouville fractional derivative of order is the left Caputo fractional derivatives of order Suppose that:
(C1) and are continuous functions, and there exist positive constants such that
Fractional calculus is a generalization of the traditional calculus to arbitrary noninteger order. Fractional differential equations (FDEs) have played an important role in various fields [1,2] such as electricity, biology, electrical networks, mechanics, chemistry, rheology and probability, etc., With the help of fractional calculus, the natural phenomena and mathematical model can be more accurately described. As a consequence there was a rapid development of the theory and application concern with fractional differential equations. In particular, the solvability, attractivity, and multiplicity of solutions for FDEs have been greatly discussed. We refer to the monographs of Podlubny , Kilbas et al. , Diethelm , Zhou , the papers [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein.
More recently, starting with the pioneering work of Jiao and Zhou , the variational methods have been applied to investigate the existence and multiplicity of solutions for fractional differential equations, which possess the variational structures in some suitable functional spaces under certain boundary conditions in many papers, see [21,22,23,24,25,26,27,28,29,30] and the references therein. For instance, Sun and Zhang  by establishing a variational structure and applying Mountain Pass theorem and iterative technique, investigated the solvability of solutions to the following nonlinear fractional differential equations
where is a continuous function, and denote left and right Riemann-Loiuville fractional integrals of order respectively. In case , Galewski and Molica Bisci in  by using variational methods, proved that the following fractional boundary problems
has at least a nontrivial solution under some suitable conditions.
On the other hand, boundary value problems for impulsive differential equations are intensively discussed. Such problems arising from the real world appear in mathematical models with sudden and discontinuous changes of their states in biology, population dynamics, physics, engineering, etc. [31,32]. For their significance, it is very important and interesting to discuss the solvability of solutions for impulsive differential equations. Recently, the existence and multiplicity of solutions for impulsive FDEs are treated by using topological methods, critical point theory and the coincidence degree theory, for example see [33,34,35,36,37,38,39,40,41,42,43] and the references therein. Taking an impulsive fractional Dirichlet problem as a model, Bonanno et al. , and Rodríguez-López and Tersian  by applying variational methods, investigated the existence results of at least one and three solutions for the following impulsive fractional boundary value problems
Motivated by [21,33,44], in this paper we shall deal with the solvability of solutions for the problem (1) by using the variational methods and iterative methods. The characteristic of problem (1) is the presence of fractional derivative in the nonlinearity term. To the best of our knowledge, there is no result concerned with the solvability of solutions for impulsive FDEs, such as problem (1), by applying the variational methods and iterative methods. We know, contrary to those equations in [33,34,39,40,42,43,45], the problem (1) is of no the variational structure and it cannot be studied by directly using the well-developed critical point theory. Furthermore, due to the appearance of left and right Riemann–Liouville fractional integral and impulsive effect, the calculation of problem (1) will be more complicated.
Throughout the paper, we assume that and satisfies the following conditions:
(C2) uniformly for all and and for and
(C3) There exists a constant such that uniformly for all and
(C4) There are constants and such that
(C5) There exists two constants such that
(I1) There is a positive constant such that
According to assumptions (C2) and (C3), it is easy to obtain that for given there exists a positive constant independent of such that
Due to the fact that problem (1) is not variational, according to the idea be borrowed from [21,44], we will deal with a family of impulsive fractional boundary value problem without the fractional derivative of the solution; that is, we consider the following problems:
For each where the space will be introduced in Section 2. Obviously, problem (2) is of the variational structure and can be solved by applying the variational methods. Hence, for any we can deduce a unique solution with some bounds. Furthermore, we can prove that there exists a solution for problem (1) via iterative methods. Now let us give the preliminary result of the present paper:
Let Suppose that the hypotheses (C1)–(C5) and (I1) are satisfied; then there exist positive constants and independent of ω such that problem (2) has at least one solution satisfying
We will established the main results of the paper by an iterative method which depends on the solvability of problem (2). To obtain the solvability of problem (1), we also need the following assumptions:
(C6) There exist constants and such that the function f satisfies the following Lipschitz conditions:
(I2) There exist constants such that
Suppose that the hypotheses of Theorem 1 are satisfied. In addition, if (C6) and (I2) hold with , we can obtain the solution of problem (2) is unique in where
Assume conditions (C1)–(C6) and (I1), (I2) hold. Then problem (1) has at least one nontrivial solution provided
The article is organized as follows. In Section 2, we shall give some definitions and lemmas that will be helpful to discuss our main results. In Section 3, we will prove the solvability of the problem (2) and the existence of at least one nontrivial solution to the problem (1).
In this paper we need the following definitions and properties of the fractional calculus. Let and be the left and right fractional integrals of order as follows
(see [2,4]).Let f be a function defined on Then the left and right Riemann–Liouville fractional derivatives of order γ for function f denoted by and are represented by
Note that if is such that and by (i) of Lemma 1, we can consider with the following norm
which is equivalent to (4) and we still denote by for short.
(, Proposition 5.6).Assume that and the sequence converges weakly to u in i.e., Then in that is, as
A function is called a solution of problem (1), if
the limits exist and satisfy the following impulsive condition
u satisfies the Equation (1) a.e. on and the boundary condition
A function is said to be a weak solution of problem (1), if
Associated to the boundary value problem (2) for given we have the functional defined by
where and Obviously, using the hypothesis (C1) we deduce that is continuous, differentiable and
for any Moreover, the critical point of is a solution of the problem (2).
(see ).Let E be a real Banach space. If any sequence for which is bounded and as possesses a convergent subsequence of . Then we say Φ satisfies Palais-Smale(PS) condition in
(see ).Let E be a real Banach space and satisfy the (PS) condition. Suppose that and
there exist constants such that and
there exists an such that
Then, Φ possesses a critical value Moreover, c can be characterized as
where is an open ball in E of radius ρ centered at 0 and
3. Proof of Theorems 1–3
The proof will be divided into four steps. We prove that the energy functional has the mountain pass geometric structure, that it is satisfies the (PS)-condition and finally that the obtained solutions have the uniform bounds.
For we show that there exist positive numbers ρ and such that for uniformly for
In fact, By (C2), (C3) and Remark 1, we have for any
Thanks to (I1), one has
Thus for any by (10), (11) and Lemma 1, one has
Choosing and let We may take sufficiently small such that
Hence This implies that satisfies assumption (i) of Lemma 3.
Fix We show that there exists such that and where ρ is given in (I).
Using (I2), we obtain that there is such that the following inequalities
hold. In fact, for any and set then
which implies that
So we have
where This implies (13) is satisfied.
From (C5) and (13), we obtain that for and
where are positive constants independent of Choosing with Since (14) implies that there is large enough such that and if we take So satisfies assumption (ii) of Lemma 3. The energy functional has the mountain pass geometric structure.
Fix We prove that satisfies the Palais-Smale condition on the space
For any sequence such that is a bounded sequence and as Then, there are two positive constants such that for n sufficiently large
Thus, it follows from (C4) and (I1) that
where is a positive constant independent of and Therefore, is bounded in
Since is a reflexive Banach space. It follows from Lemma 1 and Proposition 2 that is bounded in and Hence, we can assume that there exists some such that the sequence in and in and
which implies the second term of (15)
as According to Remark 1, we get
as Thus, the third term of (15)
as That is, is a Cauchy sequence in This implies that has a convergent sequence in Thus satisfies (PS) condition.
Obviously, Therefore, applying Lemma 3, we deduce that admits a nontrivial critical points in with
where and has been given in (II). So problem (2) has at least one weak solution for any
Fix We prove that there exist positive constants and independent of ω such that
Since is the solution of problem (2), then one has
By Remark 1, (I1) and Lemma 1, we have
for any So
Combined with by choosing small enough such that , we obtain
Notice that satisfying (16), then taking a special pass we have
where denote positive constants. Let
Since , then the function can achieve its maximum at some and the value can be taken as Obviously it is independent of Then (18) implies that there exists independent of such that Therefore, this completes the proof of Theorem 1. □
It follows from Theorem 1 that there exists at least one weak solution of problem (2). Next, fix we show that the solution of problem (2) is unique. In fact, if there are two different solutions and satisfying the first equation in problem (2) a.e. Then
Combining with the condition (C6), (I2) and Lemma 1, we have
Since we can deduce that and . This ends the proof of Theorem 2. □
According to Theorem 1, We can construct a iterative sequence as solutions of the following problem
Obtained by the Mountain Pass theorem, starting with an arbitrary
According to (IV) of Theorem 1, we have It follows from (6) that
So by (9), we have
Hence, by (C6), (I2), and the Hölder inequality, we get
which implies that
According to the condition of Theorem 3, The we know that is a Cauchy sequence in Therefore the sequence strongly converges in to some Theorem 1 guarantees
By (C6), we have, for any ,
as which implies that u is the solution of problem (1). Hence, we obtain a nontrivial solution of problem (1). This completes the proof. □
Finally, in this paper, we present an explicit example to illustrate our main result.
Let and Consider the following fractional boundary value problem:
Compared with problem (1), and By taking and and all Then by simple computation, it is easy to show that the function f satisfies the assumptions (C1)-(C5) and the function satisfies the hypotheses (I1).
For the conditions (C6) and (I2), for all it follows that
Thus, we can choose and where In this case, it suffices to verify that
From (19), we estimate the value of where is dependent of Since
Then we may choose where such that with By direct computation via Mathematica, we have and
According to the arbitrariness of and we may take enough small such that Then we obtain
Then all conditions in Theorem 1 are satisfied. Consequently the problem (21) admits at least one nontrivial solution.
In this work, we studied a class of impulsive fractional boundary value problems with nonlinear derivative dependence. Due to the fact that the studied problem (1) is of no the variational structure and it cannot be studied by directly using the well-developed critical point theory. First, we considered a family of impulsive fractional boundary value problem without the fractional derivative of the solution. Second, we give sufficient conditions of the existence of at least one nontrivial solution for problems (1). The used technical approach is based on variational methods and iterative methods. In future work, it is worth investigating multiplicity of solutions for the problem (1), and the existence of solutions to impulsive fractional differential equations involving p-Laplacian.
Y.Z. and J.X. contributed equally in writing this article; supervision, H.C. All authors read and approved the final manuscript.
The research was supported by Hunan Provincial Natural Science Foundation of China (2019JJ40068), and by National Natural Science Foundation of China (11601048 and 11671403).
The authors thank the anonymous referees for their careful reading and insightful comments.
Conflicts of Interest
The authors declare no conflict of interest.
Podlubny, I. Fractional Differential Equations, Mathematics in Science and Engineering; Academic Press: New York, NY, USA, 1999; Volume 198. [Google Scholar]
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier Science: Amsterdam, The Netherlands, 2006. [Google Scholar]
Diethelm, K. The Analysis of Fractional Differential Equation; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
Zhou, Y. Basic Theory of Fractional Differential Equations; World Scientific: Singapore, 2014. [Google Scholar]
Agarwal, R.P.; Benchohra, M.; Hamani, S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math.2010, 109, 973–1033. [Google Scholar] [CrossRef]
Ahmad, B.; Sivasundaram, S. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput.2010, 217, 480–487. [Google Scholar] [CrossRef]
Zhao, Y.L.; Chen, H.B.; Huang, L. Existence of positive solutions for nonlinear fractional functional differential equation. Comput. Math. Appl.2012, 64, 3456–3467. [Google Scholar] [CrossRef][Green Version]
Fečkan, M.; Zhou, Y.; Wang, J.R. On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul.2012, 17, 3050–3060. [Google Scholar] [CrossRef]
Wang, F.; Cui, Y.; Zhou, H. Solvability for an infinite system of fractional order boundary value problems. Ann. Funct. Anal.2019, 10, 395–411. [Google Scholar] [CrossRef]
Zhang, Y. Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance. J. Inequal. Appl.2018, 2018, 198. [Google Scholar] [CrossRef][Green Version]
Zhou, Y. Attractivity for fractional differential equations in Banach space. Appl. Math. Lett.2018, 75, 1–6. [Google Scholar] [CrossRef]
Cui, Y.J.; Ma, W.J.; Sun, Q.; Su, X.W. New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal. Model. Control2018, 23, 31–39. [Google Scholar] [CrossRef]
Zou, Y.M.; He, G.P. On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett.2017, 74, 68–73. [Google Scholar] [CrossRef]
Cui, Y.J. Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett.2016, 51, 48–54. [Google Scholar] [CrossRef]
Fu, Z.D.; Bai, S.K.; O’Regan, D.; Xu, J.F. Nontrivial solutions for an integral boundary value problem involving Riemann–Liouville fractional derivatives. J. Inequal. Appl.2019, 2019, 104. [Google Scholar] [CrossRef]
Cheng, W.; Xu, J.F.; Cui, Y.J.; Ge, Q. Positive solutions for a class of fractional difference systems with coupled boundary conditions. Adv. Differ. Equ.2019, 2019, 249. [Google Scholar] [CrossRef]
Zhang, K.; Fu, Z.D. Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity. J. Funct. Spaces2019, 2019, 9046472. [Google Scholar] [CrossRef]
Qi, T.T.; Liu, Y.S.; Zou, Y.M. Existence result for a class of coupled fractional differential systems with integral boundary value conditions. J. Nonlinear Sci. Appl.2017, 10, 4034–4045. [Google Scholar] [CrossRef][Green Version]
Jiao, F.; Zhou, Y. Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos2012, 22, 1250086. [Google Scholar] [CrossRef]
Sun, H.R.; Zhang, Q.G. Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique. Comput. Math. Appl.2012, 64, 3436–3443. [Google Scholar] [CrossRef][Green Version]
Galewski, M.; Bisci, G.M. Existence results for one-dimensional fractional equations. Math. Meth. Appl. Sci.2016, 39, 1480–1492. [Google Scholar] [CrossRef]
Li, Y.N.; Sun, H.R.; Zhang, Q.G. Existence of solutions to fractional boundary-value problems with a parameter. Electron. J. Differ. Equ.2013, 141, 1–12. [Google Scholar]
Zhao, Y.L.; Chen, H.B.; Qin, B. Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods. Appl. Math. Comput.2015, 257, 417–427. [Google Scholar] [CrossRef]
Klimek, M.; Odzijewicz, T.; Malinowska, A.B. Variational methods for the fractional Sturm-Liouville problem. J. Math. Anal. Appl.2014, 416, 402–426. [Google Scholar] [CrossRef]
Chen, L.; Chen, C.; Yang, H.; Song, H. Infinite radial solutions for the fractional Kirchhoff equation. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.2019, 113, 2309–2318. [Google Scholar] [CrossRef]
Zhao, Y.L.; Shi, X.Y.; Tang, L. Multiple positive solutions for perturbed nonlinear fractional differential system with two control parameters. Adv. Differ. Equ.2019, 2019, 341. [Google Scholar] [CrossRef]
Zhang, X.G.; Liu, L.S.; Wu, Y.H. Variational structure and multiple solutions for a fractional advection-dispersion equation. Comput. Math. Appl.2014, 68, 1794–1805. [Google Scholar] [CrossRef]
Torres, C. Boundary value problem with fractional p-Laplacian operator. Adv. Nonlinear Anal.2016, 5, 133–146. [Google Scholar]
Zhao, Y.L.; Chen, H.B.; Zhang, Q.M. Infinitely many solutions for fractional differential system via variational method. J. Appl. Math. Comput.2016, 50, 589–609. [Google Scholar] [CrossRef]
Benchohra, M.; Henderson, J.; Ntouyas, S. Theory of Impulsive Differential Equations, Contemporary Mathematics and Its Applications; Hindawi Publishing Corporation: New York, NY, USA, 2006. [Google Scholar]
Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S. Theory of Impulsive Differential Equations; World Scientific Publishing Co. Inc.: Singapore, 1989; Volume 6. [Google Scholar]
Bonanno, G.; Rodríguez-López, R.; Tersian, S. Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal.2014, 17, 717–744. [Google Scholar] [CrossRef]
Rodríguez-López, R.; Tersian, S. Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal.2014, 17, 1016–1038. [Google Scholar] [CrossRef]
Bai, Z.; Dong, X.; Yin, C. Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl.2016, 2016, 63. [Google Scholar] [CrossRef]
Zuo, M.; Hao, X.; Liu, L.; Cui, Y. Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl.2017, 2017, 161. [Google Scholar] [CrossRef][Green Version]
D’Aguì, G.; Di Bella, B.; Tersian, S. Multiplicity results for superlinear boundary value problems with impulsive effects. Math. Methods Appl. Sci.2016, 39, 1060–1068. [Google Scholar] [CrossRef]
Wang, J.R.; Fečkan, M.; Zhou, Y. A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal.2016, 19, 806–831. [Google Scholar] [CrossRef]
Heidarkhani, S.; Zhao, Y.L.; Caristi, G.; Afrouz, G.A.; Moradi, S. Infinitely many solutions for perturbed impulsive fractional differential systems. Appl. Anal.2017, 96, 1401–1424. [Google Scholar] [CrossRef]
Zhao, Y.L.; Tang, L. Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods. Bound. Value Probl.2017, 2017, 123. [Google Scholar] [CrossRef][Green Version]
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