By means of backward stochastic differential equations, the existence and uniqueness of the mild solution are obtained for the nonlinear Kolmogorov equations associated with stochastic delay evolution equations. Applications to optimal control are also given.

In this paper, we consider the following second order ordinary differential equation x''=ƒ(t, x (t),x'(t))+e(t), t∈(0, 1), (1.1) subject to one of the following boundary value conditions: x(0)=m-2∑i=1/αix(ξi), x(1)=n-2∑j=1/βjx(ηj), (1.2) x'(0)=m-2∑i=1/αix1 (ξi), x'(1)=n-2∑j=1/βjx1 (ηj), (1.3) x'(0)=m-2∑i=1/αix1 (ξi), x'(1)=n-2∑j=1/βjx (ηj), (1.4) where αi (1≤i≤m-2), βj (1≤j≤n-2) ∈ R, 0<ξ1<ξ2<……<ξm-2<1, 0<η1<η2<……<ηn-2< 1. When all the ais have no the same sign and all the bjs have no the same sign, some existence results are given for (1.1) with boundary conditions (1.2)-(1.4) at resonance case. We also give some examples to demonstrate our results.

In the present paper,we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions,the continuity and differentiability of the value function are proved.We also define a new type of Hamilton–Jacobi–Bellman(HJB) equation and prove that the value function is a viscosity solution of this HJB equation also defineanewtypeofHamilton–Jacobi–Bellman(HJB)equationandprovethatthevalue

We consider the singular three-point boundary value problems (Φp (y'))'+a(t)ƒ(y (t))=0, 0<t<1, y' (0)=0, y (1)=βy (n), where Φp (s)=|s|p-2s, p>2, 0<β< 1, 0<η< 1, f ∈ C ((0,+∞), (0,+∞)), a: (0, 1)～(0,+∞), and has countably many singularities in (0, 1/2). We show that there exist countably many positive solutions by using the fixed-point index theory.

In this paper, by using the fixed points of strict-set contractions, we study the existence of at least one or two positive solutions to the second-order three-point boundary value problem y''(t)+a(t)ƒ(y(t))=θ, 0<t<1, y(0)=θ, y(1)=βy(η) in Banach space E, where θ is the zero element of E, 0<β<1, 0<η<1 and a (t) is allowed to change sign on [0, 1]. As an application, we also give one example to demonstrate our results.