Research On The Existence Of Solutions For Some Classes Of Nonlinear Differential Equations In Banach Space

In this thesis, we investigated the existence of solutions for some classes of nonlinear differential equations in abstract spaces. The thesis consists of seven chapters.Chapter1is introduction. We present the historical background of the problems and the significance of this thesis. Boundary value problem occurs in various fields of natural science and is an important part of differential equation. The existence of solutions of boundary value problems in Banch space attracts much attention of many famous mathematicians all around the world. The fractional differential equations have existed in the literature for a long time, however, the preliminary research proceeds very slowly. There has been a significant development in ordinary and partial differential equations involving both Riemann-Liouville and Caputo fractional derivatives in recent years. Development on the existence of solutions of nonlinear differential equations with integral or fractional order in this thesis is given. We also introduce the background and main results of our works. At last, some preliminary knowledge needed in this thesis is summarized.In Chapter2, using the well known fixed point theorem on cone, the theory of fixed point index, the theory of Kuratovski noncompactness measure and strict-set-contraction operator, and some techniques of analysis, we discussed the existence and multiplicity of positive solutions for two kinds of nonlinear singular integral-differential equations in abstract spaces, and obtained some results which generalize and improve some known results in literatures.In Chapter3, we studied the existence and multiplicity of positive solutions of a class of multi-point system of boundary value differential equations in abstract spaces by using the theory of strict-set-contraction operator, fixed point theorem again. We obtained some new results.In Chapter4, firstly, by means of new comparison result coupled with the lower or upper solutions method, we discussed the existence and the estimating the error of the iterative positive solution for a kind of generalized Sturm-Liouville boundary value problem in abstract spaces. Our results are without making any compactness type assumption. Secon- dly, using the technique of monotone iterative on the regular cone, we investigated the existence of solutions for a class of impulsive fractional order differential equations with nonlinear boundary conditions in abstract spaces. Our nonlinear boundary conditions unify the treatment of initial and final value problem, anti-periodic boundary value problem, and general nonlinear boundary value problem.In Chapter5, by using the regularization, sequential techniques, fixed point theorem and diagonalization methods, we investigated the existence of solutions for a class of singular multi-point boundary value problem for fractional differential equation with more singular term on the half-line in abstract spaces. We obtain some results which extend the related known works in literatures.In Chapter6, by means of the classical fixed point theorems combined with the theory of resolvent operators for integral equations, we investigated the existence, uniqueness and continuous dependence of mild solutions for fractional neutral functional differential equation with nonlocal initial conditions and infinite delay in abstract spaces. Some new global results are obtained.In Chapter7, we investigated the nonempty of the solution set under both convexity and nonconvexity conditions on the multivalued right-hand side, measurable version Filippov's theorem and the corresponding relaxation theorem of solutions for fractional impulsive differential inclusions. The main tools are the theory of multivalued analysis, the fixed point theorem of multivalued operator, the fractional calculus and the technique of sequential.