The Research Of Solutions For Initial Value Problems Of A Class Of Fractional Nonlinear Differential Equations And Timoshenko Beam Lattice System

This paper consists of two parts. The first one is related the existence and unique-ness of solutions for initial value problems of a class of nonlinear differential equationsof fractional order. The second one is concerned with the lattice system of Timoshenkobeam, including the existence and uniqueness of solutions of fractional Timoshenkobeam lattice system, and the existence of exponential attractor for Timoshenko beamlattice system of integer order. This paper is organized as follows:In chapter1, the background and research value of fractional differential equa-tions and Timoshenko beam lattice system are introduced, and the inequalities, basicdefinitions and theories of this paper are given.In chapter2, the solutions for initial value problems of a class of caputo nonlineardifferential equations of fractional order are considered. Firstly, the differential equa-tions of fractional order are transformed into equivalent integral equations based onbasic properties of the Caputo operator. Further, by using continuous compact map-ping theorem and the Banach contraction principle, the existence and uniqueness ofsolutions are proved and two sufficient conditions for existence of solutions are given.In chapter3, the existence and uniqueness of solutions of Timoshenko beam latticesystem of fractional order are considered. First of all, the spatial variables of initialvalue problem for Timoshenko beam equations is discretized, and the corresponding fractional lattice system are obtained. Further, by using Schauder’s fixed point theoremand Banach contraction mapping principle，the existence of solutions of the latticesystem are studied, and the sufficient conditions for unique solutions are obtained.In chapter4, for Timoshenko beam lattice system of integer order, according tothe sufficient conditions for the existence of exponential attractors for general latticedynamical systems, the existence of exponential attractor for the lattice system is dis-cussed.In chapter5, this paper is summarized and looked ahead.