Existence Results for Non-instantaneous Impulsive Nonlinear Fractional Differential Equation Via Variational Methods

Benchohra, M., Henderson, J., Ntouyas, S.: Theory of Impulsive Differential Equations, Contemporary Mathematics and Its Applications. Hindawi Publishing Corporation, New York (2006)

Agarwal, R., Hristova, S., O’Regan, D.: Non-instantaneous Impulses in Differential Equations. Springer, Berlin (2017)

Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, New York (1999)

Diethelm, K.: The Analysis of Fractional Differential Equation. Spring, Heidelberg (2010)

Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

Wang, J., Fečkan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19(4), 806–831 (2016)

Stamova, I., Stamov, G.: Applied Impulsive Mathematical Models. Springer International Publishing, Berlin (2016)

Wang, J., Fečkan, M., Zhou, Y.: Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions. Bull. des Sci. Math. 141, 727–746 (2017)

Shah, K., Khalil, H., Ali Khan, R.: Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos, Solitons Fractals 77, 240–246 (2015)

Wang, G., Ahmad, B., Zhang, L., Nieto, J.J.: Comments on the concept of existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 19(3), 401–403 (2014)

D’Aguì, G., Di Bella, B., Tersian, S.: Multiplicity results for superlinear boundary value problems with impulsive effects. Math. Methods Appl. Sci. 39, 1060–1068 (2016)

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), 717–44 (2014)

Zhao, Y., Chen, H., Xu, C.: Nontrivial solutions for impulsive fractional differential equations via Morse theory. Appl. Math. Comput. 307, 170–179 (2017)

Torres, C., Nyamoradia, N.: Impulsive fractional boundary value problem with p-Laplace operator. J. Appl. Math. Comput. 55, 257–278 (2017)

Zhao, Y., Tang, L.: Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods, Bound. Value Probl. 2017, Article ID 123 (2017)

Heidarkhani, S., Zhao, Y., Caristi, G., Afrouz, G.A., Moradi, S.: Infinitely many solutions for perturbed impulsive fractional differential systems. Appl. Anal. 96(8), 1401–1424 (2017)

Li, P., Wang, H., Li, Z.: Solutions for impulsive fractional differential equations via variational methods, J. Funct. Spaces., Article ID 2941368 (2016)

Hernádez, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)

Pierri, M., ORegan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219, 6743–6749 (2013)

Muslim, M., Kumar, A., Fečkan, M.: Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. J. King Saud Univ. 30, 204–213 (2018)

Bai, L., Nieto, J.J.: Variational approach to differential equations with not instantaneous impulses. Appl. Math. Lett. 73, 44–48 (2017)

Bai, L., Nieto, J.J., Wang, X.: Variational approach to non-instantaneous impulsive nonlinear differential equations. J. Nonlinear Sci. Appl. 10, 2440–2448 (2017)

Hernández, E., Pierri, M., O’Regan, D.: On abstract differential equations with non instantaneous impulses. Topol. Methods Nonlinear Anal. 46, 1067–1085 (2015)

Agarwal, R., O’Regan, D., Hristova, S.: Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses. J. Appl. Math. Comput. 53, 147–168 (2017)

Gautam, G.R., Dabas, J.: Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses. Appl. Math. Comput. 259, 480–489 (2015)

Yang, D., Wang, J., O’Regan, D.: Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations. J. Frankl. Inst. 354, 6978–7011 (2017)

Wang, J., Fečkan, M., Tian, Y.: Stability analysis for a general class of non-instantaneous impulsive differential equations. Mediter. J. Math. 14, 1–21 (2017)

Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos 22, 1250086 (2012)

Khaliq, A., Rehman, M.: On variational methods to non-instantaneous impulsive fractional differential equation. Appl. Math. Lett. 83, 95–102 (2018)

Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Mat. Soc. Providence (1986)

Zeidler, E.: Nonlinear Functional Analysis and Its Applications, III, Variational Methods and optimization (Translated from the German by Leo F. Boron). Springer, New York (1985)

Bai, L., Dai, B., Nieto, J.J.: Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential. Electron. J. Qual. Theory Differ. Equ. 1, 1–13 (2018)