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

This article is concerned with the existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps due to Martelli.

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 four-point boundary value problem y''(t)+a(t)f(y(t)=θ, 0<t<1, y(0)=μy(ξ), y(1)=βy(η) in Banach space E, where θ is zero element of E, 0<ξ≤η< 1, 0<μ< 1/1−ξ, 0<β<1/η and μξ(1-β)+(1-μ)(1-βη)>0. As an application, we also give one example to demonstrate our results.

With the help of the coincidence degree continuation theorem, the xistence of periodic solutions of a nonlinear second-order differential equation with deviating argument x''(t)+ƒ1(x(t)x'(t+ƒ2(x(t))(x'(t))2+g(x(t-r(t))=0, under some assumptions are obtained.

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 generalize Sturm-Liouville four-point boundary value problem y''(t)+a(t)f(y (t))=θ, 0<t<1, αy(0)-βy1(0)=μ1y(ξ), γy(1)+δy'(1)=μ2y(η), in Banach space E, where θ is the zero element of E, 0<ξ<1, 0<η<1. As an application, we also give an example to demonstrate our results.