Several Types Of Differential Equations And Their Applications

Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain all kinds of natural phenomena, more and more mathematics are devoting their time to it. Among them, the nonlinear 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. We apply theory of ordering, expansion and compression theory, upper and lower solution method and fixed point theorems to study some nonlinear BVPs. This paper is composed of four chapters.In chapter 1, is the introduction of this paper, which introduces the main contents of this paper.In chapter 2, the existence of positive solutions for singular fourth order three-point boundary value problemwas investigated . A necessary and sufficient condition for the existence of C²[0, 1],C³[0,1] positive solutions is obtained under the condition that nonlinear item are all superlinear or one is superlinear, another is sublinear by constructing with fixed point theorem of cone expansion and compression . Where 0<α<1,0<η<1 are constants, α,β,γ,δ are nonnegative and satisfy Δ= αγ + αδ + βγ > 0, f,g ∈ C((0,1) × [0,+oo),[0,+oo)) and f(t,1) > 0,g(t,1) > 0,(?) t∈ (0,1),f,g are singular at t = 0,1. The method; we applied are difference with reference[1] and [2], we also obtain two positive solutions under the condition that nonlinear item , one is superlinear, another is sublinear.In chapter 3, we investigate the existence of positive solutions for singular fourth-order three-point boundary value problemwith f∈((0,1)×[0,+∞),(0,+∞)) and f(t,dx(t)) ≤g(d)f(t,x(t)), g(d)/d² is integral in [1,+∞). A necessary and sufficient condition for the existence of C²[0,1] positive solutions as well as C³[0,1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem.In chapter 4, we apply the fixed point index theory to obtain the existence of two positive solutions established for Sturm-liouville boundary value problems of the formwhere J = [0,1],0 1< t₂ < … m< … <1/2,t_∞:= lim_{m→+∞}t_m = 1/2,α_i≥0,β_i≥0(i =1,2),ρ =α₂β₁ + α₁β₂ + α₁β₂∫₀¹ds/p(s)> 0. p(t)∈ C(J,R⁺),f ∈ C(J×R⁺,R⁺),I_k∈ C(R⁺,R⁺),R⁺ = [0,+∞), Δx'|t=t_k = x'(t_k⁺) -x'(t_k^-), and ∫₀¹ds/p(s) < +∞.