Existence Of Positive Solutions For Boundary Value Problems Of Nonlinear Differential Equations

Nonlinear functional analysis is a research field of mathematics with profound theories and extensive applications. It constructs many general theories and methods to deal with nonlinear problems on the basis of the study of the nonlinear problems which appeared in mathematics and the natural sciences. Because it can explain all kinds of natural phenomena commendably, its rich theory and advanced method have provided the effective theory tool for solving the nonlinear problem which emerges incessantly in the technical domain one after another. At present, nonlinear functional analysis mainly covers topology degree theory, critical point theory, partial order method, analysis method, monotone mapping theory, and soon.The theory of differential equations in Banach spaces is an important branch of nonlin-ear analysis, which is applied to many fields, such as engineering and control theory. The research on boundary value probelms of nonlinear differential equations is an enduring topic, in recent years, the existenee of solutions to boundary value problems for nonlinear differen-tial equations attracts many researchers’attention, and a large number of results have been obtained. After three centuries development, fractional calculus has been applied in more and more fields. Compared with the integer calculus, it can describe the processes with memory and hereditary in biology, physics, chemistry, financial and so on, and these processes can be described by fractional differential equation. However, the properties of nonlocal and sin-gular of the fractional differential operator make it difficult in theory research. Thus, it has important theoretical and practical significance to research on the boundary value problems for fractional differential equation.This paper studies the existence, uniqueness and multiplicity of positive solutions to sev-eral kinds of singular semipositone boundary value problems for nonlinear integer (fraction-al) differential equation (system). Through the application of nonlinear functional analysis has been employed in the present paper, such as cone theory, fixed point theory, Krasnosel-skii’s fixed point theorem and monotone iterative technique, some new results have been obtained, most of which are published on the important academic journals in foreign coun-tries, such as 《Appl. Math. Comput.》(SCI), I Adv. Difference Equ.》 (SCI), 《Abstr. Appl. Anal.》, ect.The dissertation is divided into six chapters. In Chapter I, the background of nonlin- ear functional analysis has been introduced, some preliminary definitions and properties of nonlinear funetion alanalysis are given, also several lemmas on the existence of fixed point, which play an important role in the next chapters.In Chapter Ⅱ, by using the fixed point theory on a cone with a special norm and space, we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals. The work improves many known results including singular and non-singular casesIn Chapter Ⅲ, we study the extremal solutions of a class of fractional integro-differential equation with integral conditions on infinite intervals involving the p-Laplacian operator. By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. At last, an example is given to confirm our main results.In Chapter IV, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HTV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.In Chapter V, we obtain the existence of positive solutions for fractional differential sys-tem with coupled boundary conditions. In 5.1, we study the positive solutions of a (n-1, 1)-type fractional differential system with coupled integral boundary conditions. The con-ditions for the existence of positive solutions to the system are established. In addition, we derive explicit formulae for the estimation of the positive solutions, and obtain the u-nique positive solution when certain additional conditions hold. An example is then given to demonstrate the validity of our main results. In 5.2, we study the existence of a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. By using the properties of the Green’s function and the Kxasnosel’skii fixed point theorem, we obtain the existence results of positive solutions under some conditions concerning the nonlinear functions. The method of this paper is a uni-fied method for establishing the existence of positive solutions for the nonlinear differential equations with coupled boundary conditions.In Chapter VI, we investigate the existence of positive solutions for a class of fractional differential systems with integral boundary conditions, and the nonlinear source terms which may be singular with respect to both time and space variables.