Class Of Nonlinear Boundary Value Problems And Its Applications

Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain all kinds of natural phenomenal, more and more mathematics are devoting their time to it. Among them, the nonhnear boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analysis mathematics. The present paper employs the cone theory, fixed point index theory, and Krasnoselskii fixed point theorem and so on, to investigate the existence of positive solutions of several classes of differential equations singular boundary value problem. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. Most results of this paper are published or to appear in important journals of the world or China, for example, 《 Journal of Qufu Normal University 》 , 《 Journal of Mathematical Study 》 , etc.This paper is divided into three chapters according to contents.In the first chapter,by using fixed point index theorem we shall study a class of second order third points singular boundary value problem:where η∈ (0,1) is a constant, a∈((0,1), [0, ∞)), f ∈ C([0,∞), [0,∞)).In this paper,f satisfies more extensively limits as following: whereWe can obtain the existence of positive solution, where m,M as follow:For the convenience, we make the following assumptions:(H₃) f∈C([0, ∞), [0, ∞)), satisfies one of following conditions:(i)(ii) We can obtain the following result:Theorem 1.1.1 Assume conditions (H₁), (H₂), (H₃)(i) or (H₁), (H₂), (H₃)(ii)hold.Then, the boundary value problem (1.1.1) has at least a C[0,l] positive solution.Remark 1.1.1 The proof of the Theorem 1.1.1 in this paper is based on looking for the operator B. It is noticeable that it is difficult to prove the existence of positive solution by using the theory of the cone expansion and compression.Remark 1.1.2 In this paper, Theorem 1.1.1 extend and improve the result of the paper [11] in essence.In the second chapter , we exploit the fixed point theory of cone expansion and compression to study the positive solutions of a class fourth-order semipositive boundary value problem:where f satisfies the Caratheodory condition .We can obtain the following result:Theorem 2.1.2 If the following conditions hold:(H₁) f : [0,1] × [0, ∞) â†' R satisfies the condition Caratheodory and exist the function M(x) ∈ L¹(0,1),M(x) > 0, such that fora.e.x ∈ [0,1] and y ≥0,f(x,y) ≥ -M(x);(H2) lim^oo ^LL = oo in compact interval [a, f3) C [0,1] almost uniformly holds. Then, the boundary value problem(2.1.1) has a positive solution for enough small A > 0.In the third chapter , we exploit the fixed point index theory of cone and the fixed point theory of cone expansion and compression to study the positive solutions of two classes boundary value problems with p-Laplacian.In the first section ,we exploit the fixed point index theory of cone to study the positive solutions of two classes boundary value problems with p-Laplacian:0p(u')' + a{t)f(u) + b(t)g{u) = 0, 0 < t < 1, u'(0) = u(l) = 0,where 2s, (0p)¹(s) = l.For the convenience, we make the following assumptions:(HO /,q I (a(r) + b(r))dr J ds < oo;(H3) 0 < /0+ < M , m < f < oo , 0 < g+ < M , m < < oo;(H4) 0 < /+ < M , m < fo < oo , 0 < y+ < M , m < p^ < oo;(H5) 0 < /0+ < M , m < fa < oo , 0 < g^ < M , m< g^ < oo;(H6) 0 < /+ < M , m < /0- < oo , 0 < g+ < M , m < g^ < oo;(H7) f(u),g(u) is monotonously increase,and 3 Mi > 0 such thatwherevP-\ J9i JO-16i(l — 6i) / q{!\a{r) + b{r))dr)di \Jo JoWe can obtain the following results:Theorem 3.1.2 Assume conditions (Hj), (H2), (H3) or (Hi), (H2), (H4) hold. Then, the boundary value problem(3.1.1) has at least a positive solution .Theorem 3.1.3 Assume conditions (Hi), (H2), (H5), (H7) or(Hi), (H2), (He), (H7) hold. Then, the boundary value problem(3.1.1) has at least two positive solutions.Remark 3.1.1 In this paper, we discuss the existence of positive solutions of second order singular boundary value problem with p-Laplacian.As 6 = 0, the results in this paper is the same as the Theorem 1 in the paper [23]. But compared with Theorem 1 in the paper[23],our results extend and improve the results from the paper [23] because the limits in the paper [23] are substituted by super or infer limits in this paper. Furthermore, the limits from the paper[23] can only take 0 or 00, while the limits in this paper take any value in some interval including 0 or 00. Meanwhile, our method of the proof is different from the paper[23] in essence.Remark 3.1.2 As p = 2, the problem in this paper is the same as the problem from the paper[24], but compared with Theorem 1 in the paper[24], the limits in the paper [24] are substituted by super or infer limits in this paper. Furthermore, the limits from the paper[24] can only take 0 or 00, while the limits in this paper take any value in some interval including 0 oroo. Meanwhile, our method of the proof is different from the paper[24] in essence. So we extend and improve the Theorem 1 in the paper [24].Remark 3.1.3 We can obtain the same results by using the condition m < fe < oo, m < g^ < oo or 0 < /0+ < M,0 p{u'{\)) = 0,where p(s) is p-Laplacian operator,i.e.