On Existence For Solutions Of Some Classes Of Nonlinear Differential Equation Boundary Value Problems

Nonlinear functional Analysis is an important branch of morderm mathmatics. The singular differential equation has developed on the basis of nonlinear functional analysis, because it can explain a lot of natural phenomenal, more and more mathematicans are devoting their times to the study of sigular differential equation.In the second chapter, we will exploit the skill of the fixed point theorem of cone expansion and compression to study a class of fourth ordered singular differential equation:where / satisfy:(H1)f ∈ C((0,1) x [0,+∞),(0,+∞)),f(t,u) ≤p(t)q(u). where q ∈C([0, +∞), [0, +∞)), f01 t(1 - t}p(t}dt ≤(H2} a. > 0, such that f01t (1 - t)p(t) max[ga(t),a] q(u)dt≤6a. where (H3)0<β<α, such thatwhere the defination of gp(t) is same as ga(t),we obtained the fowlling results:Theorem 1.1: Suppose^) (H-^) holds, then (3) have positive solution u 6 C2[0,1] n C4(0,1).Remark 1: Although (Hi] holds,/ is not monotone in it .Corallary 2.1: Suppose (Hi) holds, f(t,u) is increasing in u, there exists a > 0, such thatthen (3) has positive solution On the condition of f(t,u) is decreasing in u, we obtain the condition of problem (3) has positive solution.is nonincreas-ing in it.we obtained the fowlling results:Theorem 2.2: If (H3), (H4) holds, there exists , then problem (3) has at least one positive solution Remark 2: Throrem 2.2 claim that / is monotone in it, but the condition is very simple.In section three, we consider fourth order boundary value problemWe supposeWe notewe obtained the fowlling results:Theorem 3.1: Suppose (H1) holds, if f satisfy:then boundary value problem (12)has at least one positive solution. We supposeTheorem 3.2: Suppose (H1) holds, if one of the following two conditions holds:suppose (H2) holds, and there exist R0 > 0, such thatwhere then boundary value problem (12) has at least two positive solutions.In fourth section, we consider third-order four-point boundary value problemthe existence theory of solution.we make the following assumptions:is strict decreasing in u.(H2) For any a, then there exist , such thatWe obtain the following result:Theorem4.1: Assume (H1) and (H2) hold, there exist B(t), a(t), which is the upper and lower solutions of problem (38) satisfy a(t) < B(t), t [a, b],then for all real number :, problem(38) have solution u(t) andRemark3: Compared with citr [30], we have a new comparison result which is better than cite [30]. The method we use is different from that in cite [30].