Abstract
In the present paper, we deal with the existence and multiplicity of solutions for the following impulsive fractional boundary value problem
where \(\alpha \in (1/p, 1]\), \(1<p<\infty \), \(0 = t_0<t_1< t_2< \cdots< t_n < t_{n+1} = T\), \(f:[0,T]\times \mathbb {R} \rightarrow \mathbb {R}\) and \(I_j : \mathbb {R} \rightarrow \mathbb {R}\), \(j = 1, \ldots , n\), are continuous functions, \(a\in C[0,T]\) and
By using variational methods and critical point theory, we give some criteria to guarantee that the above-mentioned impulsive problems have at least one weak solution and a sequences of weak solutions.
Similar content being viewed by others
References
Ahmad, B., Nieto, J.: Existence of solutions for impulsive anti-periodic boundary value problem of fractional order. Taiwan J. Math. 15(3), 981–993 (2011)
Anguraj, A., Karthikeyan, P.: Anti-periodic boundary value problem for impulsive fractional integro differential equations. Acta Math. Hung. 13(3), 281–293 (2010)
Belmekki, M., Nieto, J., Rodríguez-López, R.: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl., Art. ID 324561 (2009)
Benchohra, M., Cabada, A., Seba, D.: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. Article ID 628916 (2009)
Bogun, I.: Existence of weak solutions for impulsive \(p\)-Laplacian problem with superlinear impulses. Nonlinear Anal. RWA 13, 2701–2707 (2012)
Bonanno, G., Rodríguez-López, R., Tersian, S.: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(3), 717–744 (2014)
Cao, J., Chen, H.: Impulsive fractional differential equations with nonlinear boundary conditions. Math. Comput. Model. 55, 303–311 (2012)
Dai, B., Su, H., Hu, D.: Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse. Nonlinear Anal. 70, 126–134 (2009)
El-Sayed, A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33, 181–186 (1998)
Georescu, P., Morosanu, G.: Pest regulation by means of impulsive controls. Appl. Math. Comput. 190, 790–803 (2007)
George, P., Nandakumaran, A., Arapostathis, A.: A note on controllability of impulsive systems. J. Math. Anal. Appl. 241, 276–283 (2000)
Jiao, F., Zhou, Y.: Existence of solution for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)
Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos 22(4), 1–17 (2012)
Kilbas, A., Trujillo, J.: Differential equations of fractional order: methods, results and problems I. Appl. Anal. 78, 153–192 (2001)
Kilbas, A., Trujillo, J.: Differential equations of fractional order: methods, results and problems II. Appl. Anal. 81, 435–493 (2002)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Klimek, M.: On solutions of linear fractional differential equations of a variational type. The Publishing Office of Czestochowa University of Technology, Czestochowa (2009)
Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6. World Scientific, Teaneck (1989)
Liu, Z., Lu, L., Szántó, I.: Existence of solutions for fractional impulsive differential equations with \(p\)-Laplacian operator. Acta Math. Hung. 141(3), 203–219 (2013)
Mawhin, J., Willen, M.: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74. Springer, Berlin (1989)
Mendez, A., Torres, C.: Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivarives. Fract. Calc. Appl. Anal. 18(4), 875–890 (2015)
Nyamoradi, N.: Infinitely many solutions for a class of fractional boundary value problems with Dirichlet boundary conditions. Mediterr. J. Math. 11(1), 75–87 (2014)
Nyamoradi, N., Rodrígues-López, R.: On boundary value problems for impulsive fractional differential equations. Appl. Math. Comput. 271, 874–892 (2015)
Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)
Rabinowitz, P.: Minimax method in critical point theory with applications to differential equations, CBMS American Mathematical Society, vol. 65 (1986)
Rivero, M., Trujillo, J., Vázquez, L., Velasco, M.: Fractional dynamics of populations. Appl. Math. Comput. 218, 1089–1095 (2011)
Rodríguez-López, R., Tersian, S.: Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(4), 1016–1038 (2014)
Samoilenko, A., Perestyuk, N.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Samko, S., Kilbas, A., Marichev, O.: Fractional integrals and derivatives: Theory and applications. Gordon and Breach, New York (1993)
Schechter, M.: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999)
Shen, J., Li, J.: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal. 10, 227–243 (2009)
Szulkin, A.: Ljusternik–Schnirelmann theory on \(C^1\)-manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 119–139 (1988)
Torres, C., Nyamoradi, N.: Existence and multiplicity result for a fractional p-Laplacian equation with combined fractional derivatives (preprint)
Torres, C.: Boundary value problem with fractional \(p\)-Laplacian operator. Adv. Nonlinear Anal. 5(2), 133–146 (2016)
Torres, C.: Existence of solution for fractional Hamiltonian systems. Electron. J. Differ. Equ. 2013(259), 1–12 (2013)
Torres, C.: Mountain pass solution for a fractional boundary value problem. J. Fract. Calc. Appl. 5(1), 1–10 (2014)
Torres, C.: Existence of a solution for fractional forced pendulum. J. Appl. Math. Comput. Mech. 13(1), 125–142 (2014)
Torres, C.: Ground state solution for a class of differential equations with left and right fractional derivatives. Math. Methods Appl. Sci. 38, 5063–5073 (2015)
Torres, C.: Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simulat. 27, 314–327 (2015)
Xu, J., Wei, Z., Ding, Y.: Existence of weak solutions for \(p\)-Laplacian problem with impulsive effects. Taiwan J. Math. 17(2), 501–515 (2013)
Zeidler, E.: Nonlinear Functional Analysis and It’s Applications III Variational Methods and Optimization. Springer, New York (1985)
Zhang, X., Zhu, C., Wu, Z.: Solvability for a coupled system of fractional differential equations with impulses at resonance. Bound. Value. Probl. 80, 23 (2013). doi:10.1186/1687-2770-2013-80
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Hackensack (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Torres Ledesma, C.E., Nyamoradi, N. Impulsive fractional boundary value problem with p-Laplace operator. J. Appl. Math. Comput. 55, 257–278 (2017). https://doi.org/10.1007/s12190-016-1035-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-016-1035-6