The Existence Of Solutions Of Differential Equations With Nonlinear Boundary Conditions

Nonlinear functional analysis is a research subject in applied mathematics that has both profound theory and widespread application. It takes the nonlinear problems ap-pearing in mathematics and natural sciences as background to establish some general theories and methods to handle nonlinear problems.The thesis is divided into three chapters, The chapter 1, we consider the existence of at least one positive solution to the following singular semipositone coupled system of nonlocal boundary value problems where f, g : (0,1) x [0,+?) ? [0,+?) are continuous and f, g may be singular at t = 0,1, q : (0,1) ? (-?,+?) is Lebesgue integrable and q(t) can have finitely many singularities in [0,1], Hi : R?R are continuous and Hi([0,+?)) C [0,+?), In par-ticular, realizing in the Stieltjes integral representation ?1(y) =J?[0,1] y(t)d?1(t),?2(x)??[0,1] x(t)d?2(t), with ?i : [0,1] ?R of bounded variation on [0,], so we no longer assume that ? is necessarily montonically increasing. By using the fixed point theorem,we gain the existence of solutions. Compared with the document [13] and [16], the equation (1.1.1)rather than only one equation and the nonlinearity f(t, x) may be singular at t=0,1 and q(t) can have finitely many singularities in [0,1]. Moreover, we do not assume Hi satisfies merely an asymptotic condition. Compared with the document [14], the equation (1.1.1)are also coupled equations, but our equations are singular semipositone and the f(t,u(t))need not have a lower bound and we do not assume Hi satisfies superlinearity conditions at t = 0 and t = +?.The chapter 2, we study the existence of at least one positive solution to the following singular semipositone coupled system of nonlocal boundary value problems where ?1,?2 > 0 are parameters, f, g : [0,1] × [0,+?)? [0,+?) are continuous and f,g may be singular at t=0, 1 and f, g may be singular at u=0, Hi : R?R are continuous and Hi([0,+?)) C [0,+?), In particular, realizing in the Stieltjes integral representation ?1(y) = ?[0,1]y(t)d?1(t),?2(x)=?[0,1] x(t)d?(t) with ?i:[0,1]? R of bounded variation on [0,1]. By using the fixed point theorem, we gain the existence of solutions. Compared with the document [24], the equation (2.1.1) have a more general integral boundary conditions and but also we have a parameter A and we don't need f,g nonincreasing in x, for each fixed t, we only need continuous. Compared with the document [14], we study coupled systems, rather than only one equation. And we have nonlinear, nonlocal boundary conditions rather than integral boundary value conditions.Compared with the document [16], we study coupled systems, rather than only one equa-tion. Although our f is also about the first and second argument is singular and we do not assume Hi (i = 1, 2) satisfies merely an asymptotic condition.The chapter 3, we study the existence of at least one positive solution to the following nonlocal boundary value problems where ? > 0 is a parameter,f : [0,1] ×R?R is continuous Hi: R? R are continuou and Hi([0,+?)) (?) [0,+?). In particular, realizing in the Stieltjes integral representa-tion ?1(y) = ?[0,1]y(t)d?1(t), ?2(x)= ?[0,1]x(t)d?2(t), with ?i: [0,1] ?R of bounded variation on [0,1]. By using the fixed point theorem, we gain the existence of solution-s. Compared with the document [13], the equation (3.1.1) have two nonlinear, nonlocal boundary conditions rather than one, and we have a parameter A. Compared with the document [25], the equation (3.1.1) not only have a more general integral boundary con-ditions, but also we study semipositone problems.