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.

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.

We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.