Abstract
The present paper is concerned with the existence of solutions, in infinite- dimensional Banach spaces, for impulsive differential inclusions and equations of order \(q\in (1,2)\), with anti-periodic conditions and involving the Caputo derivative whether in the generalized sense (via the Riemann–Liouville fractional derivative,) or in the normal sense and whether the lower limit on each impulsive subinterval \( (t_{i} ,t_{i+1}],\, i=0,1,\ldots ,m\) is keep at zero or is set at \(t_{i}\). Using the technique of fixed point and the properties of the measure of noncompactness, existence results are obtained. Moreover, we derive in reflexive Banach spaces, by using a new version weakly convergent criteria in the space of piecewise continuous functions, an existence result of solutions without assuming any condition on the multivalued function in terms of measure of noncompactness. Some examples will be given to illustrate the obtained results.
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Agarwal, R.P., Hristova, S., O’Regan, D.: Aurvey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19(2), 290–318 (2016)
Agarwal, R.P., Ahmed, B.: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1200–1214 (2011)
Agarwal, R.P., Benchohra, M., Hamani, S.: Survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)
Agarwal, P., Choi, J.: Certain fractional integral inequalities associated with pathway fractional integral operators. Bull. Korean Math. Soc. 53(1), 181–193 (2016)
Agarwal, P., Choi, J.: Fractional calculus operators and their image formulas. J. Korean Math. Soc. 53(5), 1183–1210 (2016)
Ahmad, B., Nieto, J.J.: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 15(3), 981–993 (2011)
Ahmed, B.: Existence of solutions for fractional differential equations of order \(q\in (2,3]\) with anti-periodic boundary conditions. J. Appl. Math. Comput. 34(1–2), 385–391 (2010)
Alsaedi, A., Ahmed, B., Assolami, A.: On anti-periodic boundary value problems of higher-order fractional differential equations. Abstr. Appl. Anal. 2012 Article ID 325984,15 pages(2012)
Agur, Z., Cojocaru, L., Mazaur, G., Anderson, R.M., Danon, Y.L.: Pulse mass measles vaccination across age shorts. Proc. Natl. Acad. Sci. USA 90, 11698–11702 (1993)
Aubin, J.P., Frankoeska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Bajlekova, E.G.: Fractional Evolution Equations in Banach Spaces. Eindhoven University of Technology, Eindhoven (2001)
Belov, S.A., Chistyakov, V.V.: A selection principle for mappings of bounded variation. J. Math. Anal. Appl. 249, 351–366 (2000)
Benchohra, M., Henderson, J., Seba, D.: Boundary value problems for fractional differential inclusions in Banach spaces. Fract. Differ. Calc. 2(1), 99–108 (2012)
Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Carporation, New York (2006)
Ben Makhlouf, A., Hammami, M.A., Sioud, K.: Stability of fractional order nonlinear system depending on a parameter. Bull. Korean Math. Soc. 54(4), 1309–1321 (2017)
Bochner, S., Taylor, A.E.: Linear functionals on certain spaces of abstractly valued functions. Ann. Math. 39, 913–944 (1938)
Bothe, D.: Multivalued perturbation of m-accerative differential inclusions. Israel J. Math. 108, 109–138 (1998)
Cardinali, T., Rubbioni, P.: Impulsive mild solution for semilinear differential inclusions with nonlocal conditions in Banach spaces. Nonlinear Anal. 75, 871–879 (2012)
Cernea, A.: On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, 133–143 (2012)
Chen, Y., Nieto, J.J., O’Regan, D.: Anti-periodic solutions for fully nonlinear first-order differential equations. Math. Comput. Model. 46, 1183–1190 (2007)
Dunford, N., Schwartz, J.H.: Linear Operators. Wiley, New York (1976)
Fečkan, M., Zhou, Y., Wang, J.R.: On the concept and existence of solutions for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050–3060 (2012)
Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)
Ibrahim, A.G.: Fractional differential inclusions with anti-periodic boundary conditions in Banach spaces. Electron. J. Qual. Theory Differ. Equ. 2014(67), 1–32 (2014)
Kamenskii, M., Obukhowskii, V., Zecca, V.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter Saur. Nonlinear Anal. Appl., vol. 7. Walter, Berlin (2001)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)
Kim, K.H., Lim, S.: Asymptotic behaviors of fundemental solutions and its derivatives to fractional diffusion wave equations. J. Korean Math. Soc. 53(4), 929–967 (2016)
Lan, Q.A., Lin, W.: Positive solutions of systems of Caputo fractional differential equations. Commun. Appl. Anal. 17(1), 61–86 (2013)
Liu, Z., Zeng, B.: Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type. Appl. Math. Comput. 257, 178–189 (2015)
Luo, Z., Shen, J., Nieto, J.J.: Anti-periodic boundary value problem for first-order impulsive ordinary differential equations. Comput. Math. Appl. 49, 253–261 (2005)
Mahmudov, N.I., Unul, S.: Existence of solutions of order fractional three-point boundary value problem with integral boundary conditions. Abstract Appl. Anal. 2014 ID198632 (2014)
Nakao, M.: Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. J. Math. Anal. Appl. 204, 754–764 (1996)
O’Regan, D.: Fixed point theorems for weakly sequentially closed maps. Arch. Math. 36, 61–70 (2000)
Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. TMA 69, 3877–3896 (2008)
Rida, S.Z., El-Sherbiny, H.M., Arafa, A.A.M.: On the solution of the fractional nonlinear Schrődinger equation. Phys. Lett. A. 372, 553–558 (2008)
Shao, J.: Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys. Lett. A. 372, 5011–5016 (2008)
Shu, X.B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. TMA 74, 2003–2011 (2011)
Shu, X.B., Shi, Y.: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465–467 (2016)
Shu, X.B., Wang, Q.: The existence of mild solutions for fractional differential equations with nonlocal conditions of order\(1<\alpha <2\). Comput. Math. Appl. 64, 2100–2110 (2012)
Wang, X.: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383–2391 (2011)
Wang, J.R., Ibrahim, A.G., Fečkan, M.: Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl. Math. Comput. 257, 103–118 (2015)
Wang, J.R., Fečkan, M., zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 194, 806–831 (2016)
Wang, J.R., Zhou, Y., Fečkan, M.: On recent development in the theory of boundary value problems fractional differential equations. Comput. Math. Appl. 64, 3008–3020 (2012)
Wang, J.R., Zhou, Y., Fečkan, M.: Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions. Bull. Sci. math. 141, 727–746 (2017)
Wang, J.R., Ibrahim, A.G., Fečkan, M.: Anti-periodic solutions for differential inclusions of arbitrary fractional order in Banach spaces and involving the generalized Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 34, 1–22 (2016)
Wang, J.R., Ibrahim, A.G., O’Regan, D.: Controllability of fractional evolution inclusions with noninstantaneous impulses. Int. J. Nonlinear Sci. Numer. Simul. (2017). https://doi.org/10.1515/IJNSNS-2017-0090
Wang, J.R., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal. Real World Appl. 12, 3642–3653 (2011)
Zavalishchin, S.T., Sesekin, A.N.: Dynamic Impulse Systems, Theory and Applications. Kluwer Academic Publishers Group, Dordrecht (1997)
Acknowledgements
The author gratefully acknowledges the Deanship of Scientific Research, King Faisal University of Saudi Arabia, for their financial support this research under Grant No. 170060. Also, we would like to thank the referee for their valuable and useful comments.
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Shangjiang Guo.
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Ibrahim, A.G. Differential Equations and Inclusions of Fractional Order with Impulse Effects in Banach Spaces. Bull. Malays. Math. Sci. Soc. 43, 69–109 (2020). https://doi.org/10.1007/s40840-018-0665-2
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DOI: https://doi.org/10.1007/s40840-018-0665-2
Keywords
- Boundary value problems
- Fractional derivative
- Impulsive differential inclusions
- Measure of noncompactness