Second Order Functional Differential Equations With Boundary Value Problems

Boundary value problems associated with functional differential equations have arisen from problems of physics and variational problems of control theory. It is one of most active domains of functional analysis studies at present, because it can explain well variousthe natural phenomenon. Differential equations with retardation or anticipation have become the hot spot which has been discussed in recent years as it's dense connecting with the problem of control. Since then many authors investigated the existence of solutions for boundary value problems concerning functional differential equations (see [2-8]). Among them, [2] studied the. the existence of solutions for boundary value problems concerning two-order functional differential equations with retardation by the using of nonlinear alternativesof Leray-Sehauder. [3] studied the existence of solutions for boundary value problems concerning one-order functional differential equations involving retardation and anticipation by the using of monotone iterative technique together with coupled lower and upper solutions. [6] studied the existence of multiple positive solutions for Floquet boundaryvalue problems concerning one-order functional differential equations with retardation by the using of fixed point theorem.[4] studied the the existence of solutions for boundary value problems concerning two-order functional differential equations with retardationby the using of fixed point theorem. where exist M > 0; measurable functions u_k : [0,1]â†'R⁺; nondccrcasing function L_k : R⁺â†'R⁺ so that[5] studied the existence of solutions for boundary value problems concerning twoorder functional differential equations with retardation also by the using of fixed point theorem. This simplified the restriction of f:(?)H₁ > max{||φ||_[-r,0],A},for (?)t∈[0,1],v∈C_r⁺ :|| v ||_[-r,0]≤H₁ so that f(t,v)≤ε|| v ||_[-r,0], whichε> 0 and∫₀^TG(s.s)εH₁ds+A≤H₁.In this paper, we arc concerned with the existence of solutions for kinds of nonlocal boundary value problems concerning two-order functional differential equations with anticipationby the using of fixed point theorem on the base of [5]. We arc also interested in the singular functional differential equations in Section 6.In this paper, we prove the existence of positive solutions for kinds of boundary value problems concerning two-order functional differential equations .r"(t) + f(t.x_t) = 0.In Section 3, we studied the existence of positive solutions for two-order functional differential equations under three points boundary value problems:which 0 < k < 1.η∈[0,T]. We not only improve the results in [5] but also apply the main results to functional differential equations with anticipation.In Section 4, we studied the existence of positive solutions for two-order functional differential equations under m points boundary value problems:which 0≤(?)k_i≤1,η_i∈[0,T]. This had expanded the range of boundary from three points to m points.In Section 5, we studied the existence of positive solutions for two-order functional differential equations under the integral boundary value problems:which 0 <∫₀^Tm(s)ds < 1,m: (0.T)â†'[0,+∞) is continuous. This simplified the restrictionof f: (?)C₁ >||φ||_[T,T+r]. for (?)t∈[0,1], v∈C_r⁺ :||v||_[T,T+r]≤C₁ so that f(l,v)≤ε||v||_[T,T+r], which In Section 6, we studied the existence of positive solutions for two-order functional differential equations under the integral boundary value problems:on the base of Section 5 which m: (0, T)â†'[0. +∞) arc continuous and 0 <∫₀^T m(s)ds < 1; h: (0.T)â†'[0. +∞) are continuous which can be singular at t = 0 and t = T, 0 <∫₀^Th(s)ds < +∞.