Two Types Of Boundary Value Problems Of Second Order Singular Nonlinear Differential Equations And Their Applications

Nonlinear functional analysis is an important branch of modern analytical mathematics, because it can explain many kinds of natural phenomena, more and more mathematicians are devoting their time into it. Among them, the nonlinear boundary value problem comes from a lot of branches of applied mathematics and physics, it is one of the most active fields that is studied in analytical mathematics. This paper considered the existence of solutions to boundary value problem of several kinds of nonlinear systems of differential equations by the cone theory, fixed point index theory and so on. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three sections according to contents.In chapter 1, apply the fixed point index theorem on cone concerning the first eigenvalue corresponding to the relevant linear operator, we obtained existence results of positive solutions of the following nonlinear singular two-point boundary value problems for second-order impulsive differential equationwhereα,β,γ,γ≥0,ρ=βγ+αγ+αδ> 0, J = (0,1), 0 < t₁ < t₂ <…< t_m <1, J' = J\{t₁, t₂,…,t_m}, J|- = [0,1], J₀ = (0,t₁], J₁ = (t₁,t₂],…, J_m =(t_m, 1), f∈C(J|-×R⁺ , R⁺ ). I_k, (I|-)_k∈C(R⁺,R⁺), R⁺ = [0, +∞), u'|_t=tk=u'(t_k⁺ ) - u'(t_k^-), u|_t=tk= u(t_k⁺ ) - u(t_k^- ), u'(t_k⁺), u(t_k⁺ ), (u'(t_k^- ), u(t_k^-)) denote the right (left) limit of u'(t) and u(t) at t = t_k respectively, h{t)∈C(J,R⁺ ) and may be singular at t = 0,1.In Chapter 2, by computing the fixed point indexes m K×K which is the Cartesian product of two cones in space C[0,1], we establish the existence of at least one and two positive solutions of the following nonlinear two-point boundary value problems of second-order impulsive differential system:whereα,β,γ,δ≥0,ρ=βγ+αγ+αδ> 0, J = (0,1), 0 < t₁ < t₂ <…m <1, J' = J\{t₁,t₂,…,t_m}, f_i∈C(J|-×R⁺×R⁺, R⁺), I_{i, k}∈C(R⁺, R⁺), h_i(t)∈C(J, (0, +∞)) (i = 1,2) and may be singular at t = 0,1, R⁺ = [0, +∞).In Chapter 3, by computing the fixed point indexes in K₁×K₂, we get the existence of one or two positive solutions for the following differential systemwhere J = [0,1], J' = (0,1),η_i∈J',α_i > 0, 1 -α_iη_i> 0, g_i : [0,∞)×[0,∞)�'[0,∞) is continuous, a_i : J'�'[0,∞) is continuous, does not vanish identically on any subinterval and may be singular at t = 0,1, h_i : J'�'(0,1], i=1,2.