In this paper, we first prove that if a linear neutral functional differential equation is positive then it must degrade into a linear functional differential equation of retarded type. Then, we give some explicit criteria for positive linear functional differential equations. Consequently, we obtain a novel criterion for exponential stability of positive linear functional differential equations.

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We will write down the higher genus algebraic curves supporting icosahedral solutions of the sixth Painlevé equation, including the largest (genus seven) curve.

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We study the regularity of weak solutions for a class of second order semi-linear infinitely degenerate elliptic equations. We get the regularity of weak solutions up to the boundary for Dirichlet problem, by noting the logarithmic regularity estimate for a linear principal part. In relation to this linear part, we also show the controllability and strong maximum principle for second order hypoelliptic operators even in the case where they degenerate infinitely. Model equations naturally come from some variational problems, if one replace the Laplace operator in such as Yamabe problems by degenerate elliptic operators. In the infinitely degenerate case, a permissible nonlinear term is not fractional power, compared with elliptic or subelliptic case. To treat this nonlinear term, the nonlinear microlocal analysis is developed in the logarithmic Sobolev space.

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Our purpose is to show the uniqueness of positive solutions for a nonlinear elliptic equation with a harmonic potential in some energy space. This problem arises in the study on the stability of standing waves for a nonlinear Schrödinger equation, and in [3], the existence of positive solutions to the ellipitc equation has been shown by the standard variational method. Moreover, in [5] we have derived the uniqueness under n ≥ 3, where n denotes the space dimension. In this paper, we give the proof of the uniqueness for n = 1 and n = 2.

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In this paper we obtain some existence and nonexistence results of a problem involving the pseudobilaplacian operator. By the sub and supersolutions techniques we prove that the problem admits a positive solution when it is subhomogeneous. In the case where it is superhomogeneous, we prove the existence of nontrivial solutions. The nonexistence result follows from a Pohozaev-type identity.

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In this paper, we study the oscillatory behavior of an even order differential equation with impulses. Oscillation critera are established. In addition, examples show that impulses play an important role in the oscillations of the solutions.

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For abstract functional differential equations (FDE) and Volterra difference equations (VDE) in a Banach space, the local existence and smoothness of invariant manifolds, such as stable/unstable manifolds, center-stable/center-unstable manifolds and center manifolds, are established by means of the variation of constants formula in the phase space in [18] and [12]. Also, it is shown that in a neighborhood of the zero solution the behavior of solutions of FDE (resp. VDE) is described, in some sense, by a certain ordinary differential equation (resp. first order difference equation) in a finite dimensional space. As a corollaly, the principle of linearized stability is derived.

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