On antiperiodic boundary value problem for a semilinear differential inclusion of a fractional order 2 < q < 3
On antiperiodic boundary value problem for a semilinear differential inclusion of a fractional order q. The investigation of control systems with nonlinear units forms a complicated and very important part of contemporary mathematical control theory and harmonic analysis, which has numerous applications and attracts the attention of a number of researchers around the world. In turn, the development of the theory of differential inclusions is associated with the fact that they provide a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics, which are studied in various branches of the optimal control theory, mathematical physics, radio physics, acoustics etc. One of the best approaches to the study of this kind of problems is provided by the methods of multivalued and nonlinear analysis, which are distinguished as very powerful, effective and useful. However, the solving of these problems within the frameworks of existing theories is often a very difficult problem, since many of them find sufficiently adequate description in terms of differential equations and inclusions with fractional derivatives. The theory of differential equations of fractional order originates from the ideas of Leibniz and Euler, but only by the end of the XX century, interest in this topic increased significantly. In the 70s - 80s, this direction was greatly developed by the works of A.A. Kilbas, S.G. Samko, O.I. Marichev, I. Podlubny, K.S. Miller, B. Ross, R. Hilfer, F. Mainardi, H. M. Srivastava. Notice that the research in this direction will open up prospects and new opportunities for the studying of non-standard systems that specialists encounter while describing the development of physical and chemical processes in porous, rarefied and fractal media. It is known that a periodic boundary value problem is one of the classical boundary value problems of differential equations and inclusions. At the same time, in recent years, along with periodic boundary value problems, antiperiodic boundary value problems are of great interest due to their applications in physics and interpolation problems.
In this paper, we study an antiperiodic boundary value problem for semilinear differential inclusions with Caputo fractional derivative of order $q$ in Banach spaces. We assume that the nonlinear part is a multivalued map obeying the conditions of the Caratheodory type, boundedness on bounded sets, and the regularity condition expressed in terms of measures of noncompactness. In the first section, we present a necessary information from fractional analysis, Mittag -- Leffler function, theory of measures of noncompactness, and multivalued condensing maps. In the second section, we construct the Green's function for the given problem, then, we introduce into consideration a resolving multivalued integral operator in the space of continuous functions. The solutions to the boundary value problem are fixed points of the resolving multioperator. Therefore, we use a generalization of Sadovskii type theorem to prove their existence. Then, we first prove that the resolving multioperator is upper semicontinuous and condensing with respect to the two-component measure of noncompactness in the space of continuous functions. In a proof of a main theorem of the paper, we show that a resolving multioperator transforms a closed ball into itself. Thus, we obtain that the resolving multioperator obeys all the conditions of the fixed point theorem and we prove the existence of solutions to the antiperiodic boundary value problem.