# Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

### Abstract

In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order \(\beta\in(1,2]\) subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous \(\beta\)-order fractional cosine family, while the nonlinear forcing term is of Carathéodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

### References

D. Araya, C. Lizama, “Almost automorphic mild solutions to fractional differential equations”, Nonlinear Analysis. vol. 69, no. 11, pp. 3692–3705, 2008.

D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012.

J. Bana´s, K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics. no. 60, Marcel Dekker Inc., New York, 1980.

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology, 2001.

S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag Berlin, Heidelberg, 2008.

D. Delbosco, L. Rodino, “Existence and uniqueness for a fractional differential equation”, J. Math. Anal. Appl., vol. 204, no. 2, pp. 609–625, 1996.

K. Ezzinbi, S. Ghnimi, and M.A. Taoudi, “Existence results for some nonlocal partial integrodifferential equations without compactness or equicontinuity”, J. Fixed Point Theory Appl., vol. 21 , no. 2, Art. 53, pp. 1–24, 2019.

V. Gafiychuk, B. Datsko, and V. V. Meleshko, “Mathematical modeling of time fractional reaction-diffusion systems”, J. Comput. Appl. Math., vol. 220, no. 1-2, pp. 215–225, 2019.

H. P. Heinz, “On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions”, Nonlinear Anal. vol. 7 , no. 12, pp. 1351–1371, 1983.

R. Hilfer, “Applications of Fractional Calculus in Physics”, World Scientific, Singapore, 2000.

S. Ji, G. Li, “Solutions to nonlocal fractional differential equations using a noncompact semi-group”, Electron. J. Differ. Equ., vol. 2013, no. 240, pp. 1–14, 2013.

M. Kamenskii, V. Obukhovskii, and P. Zekka, “Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces”, De Gruyter, Berlin, 2001.

K. Li, J. Peng, and J. Gao, “Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness”, Fract. Calc. Appl. Anal., vol. 15, no. 4 , pp. 591–610, 2012. DOI: 10.2478/s13540-012-0041-0.

K. Li, J. Peng, and J. Gao, “Nonlocal fractional semilinear differential equations in separable Banach spaces”, Electron. J. Differential Equations, no. 07, pp. 1–7, 2013.

K. Li, J. Peng, and J. Gao, “Controllability of nonlocal fractional differential systems of order α ∈ (1, 2] in Banach spaces,” Rep. Math. Phys. vol. 71, no. 1, pp. 33–43, 2013.

M.M. Matar, “Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions”, Electronic Journal of Differential Equations. no. 155, pp. 1–7, 2009.

H. Mönch, “Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces”, Nonlinear Anal., vol. 4, no. 5, pp. 985–999, 1980.

G. M. Mophou, “Almost automorphic solutions of some semilinear fractional differential equations”, Int. J. Evol. Equ. vol. 5, no. 1, pp. 109–115, 2010.

M. Muslim, “Existence and approximation of solutions to fractional differential equations”, Mathematical and Computer Modelling. vol. 49, no. 5-6, pp. 1164–1172, 2009.

H. Qin, X. Zuo, J. Liu, and L. Liu, “Approximate controllability and optimal controls of fractional dynamical systems of order 1 < q < 2 in Banach spaces”, Adv. Difference Equ., pp. 1–17, 2015.

X. B. Shu, Q. Wang, “The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < α < 2”, Comput. Math. Appl., vol. 64, no. 6, pp. 2100–2110, 2012.

V. E. Tarasov, “Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles”, Fields and Media, Springer, New York, 2010.

C. C. Travis, G.F. Webb, “Compactness, regularity, and uniform continuity properties of strongly continuous cosine families”, Houston J. Math. vol. 3, no. 4, pp. 555–567, 1977.

C. C. Travis, G.F. Webb, “Cosine families and abstract nonlinear second order differential equations”, Acta Math. Acad. Sci. Hungar. vol. 32, no. 1-2, pp. 75–96, 1978.

B. Zhu, L. Liu, and Y. Wu, “Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations”, Fract. Calc. Appl. Anal., vol. 20, no. 6, pp. 1338–1355, 2017. DOI: 10.1515/fca-2017-0071.

*CUBO, A Mathematical Journal*, vol. 22, no. 3, pp. 361–377, Dec. 2020.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.