Abstract:
We study the existence of solutions for the periodic boundary value problem for some second order integro-differential equations with a general kernel. Also we develop the monotone method to approximate the extremal solutions of the problem.

We proposed a kind of doubly periodic Riemann boundary value problem on
two parallel curves. By using the method of complex functions, we investigated
the method for solving this kind of doubly periodic Riemann boundary value
problem of normal type and gave the general solutions and the solvable
conditions for it.

Abstract:
In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it.

Abstract:
In this paper, we investigate the positive solutions of periodic boundary value problem. By using critical point theory the existence of multiple positive solutions is obtained.

Abstract:
We obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper (Zhang and Wang, 2003). An example is also given to illustrate our result.

Abstract:
Fourth order boundary value problems arise in the study of the equilibrium of an elastaic beam under an external load. The author earlier investigated the existence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary value problems that arise in the equilibrium of an elastic beam depending on how the ends of the beam are supported. This paper concerns the existence and uniqueness of solutions of the fourth order boundary value problems with periodic boundary conditions.

Abstract:
This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.

Abstract:
Using a capacity approach and the theory of the measure’s perturbation of the Dirichlet forms, we give the probabilistic representation of the general Robin boundary value problems on an arbitrary domain ？, involving smooth measures, which give rise to a new process obtained by killing the general reflecting Brownian motion at a random time. We obtain some properties of the semigroup directly from its probabilistic representation, some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary. 1. Introduction The classical Robin boundary conditions on a smooth domain of ( ) is giving by where is the outward normal vector field on the boundary and a positive bounded Borel measurable function defined on . The probabilistic treatment of Robin boundary value problems has been considered by many authors [1–4]. The first two authors considered bounded -domains since the third considered bounded domains with Lipschitz boundary, and the study of [4] was concerned with -domains but with smooth measures instead of . If one wants to generalize the probabilistic treatment to a general domain, a difficulty arise when we try to get a diffusion process representing Neumann’s boundary conditions. In fact, the Robin boundary conditions (1.1) are nothing but a perturbation of , which represent Neumann’s boundary conditions, by the measure , where is the surface measure. Consequently, the associated diffusion process is the reflecting Brownian motion killed by a certain additive functional, and the semigroup generated by the Laplacian with classical Robin boundary conditions is then giving by where is a reflecting Brownian motion (RBM) and is the boundary local time, which corresponds to by Revuz correspondence. It is clear that the smoothness of the domain in classical Robin boundary value problem follows the smoothness of the domains where RBM is constructed (see [5–10] and references therein for more details about RBM). In [6], the RBM is defined to be the Hunt process associated with the form defined on by where is assumed to be bounded with Lipschitz boundary so that the Dirichlet form is regular. If is an arbitrary domain, then the Dirichlet form needs not to be regular, and to not to lose the generality we consider , the closure of in . The domain is so defined to insure the Dirichlet form to be regular. Now, if we perturb the Neumann boundary conditions by Borel’s positive measure [11–13], we get the Dirichlet form defined on by In the case of ( bounded with Lipschitz boundary), (1.4) is the form associated with

Abstract:
In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.