Integrability of the resolvent and the stability properties of the zero solution of linear Volterra integrodifferential systems are studied. In particular, it is shown that, the zero solution is uniformly stable if and only if the resolvent is integrable in some sense. It is also shown that, the zero solution is uniformly asymptotically stable if and only if the resolvent is integrable and an additional condition in terms of the resolvent and the kernel is satisfied. Finally, the integrability of the resolvent is obtained under an explicit condition.
In an Alexandrov space with curvature bound, we prove that a curvature takes the extreme value over some specially constructed surfaces if and only if each of the surfaces is totally geodesic and locally isometric to a surface with constant curvature.
A system of two nonlinear differential equations with an irregular type singularity not satisfying the Poincaré condition is studied. A two-parameter family of bounded solutions is constructed by the fixed point technique. The domain of holomorphy of the set of functions appearing in the fixed point technique is to be given by a family of the product of two circles over every point in a domain of independent variable. The radius of one circle depends on the argument of the independent variable only, while that of the other essentially depends on the independent variable itself.
Here, we compute the mean curvature of the geodesic sphere at any point in some symmetric spaces and determine the lower bound of the mean curvature of a closed hypersurface of constant mean curvature in it. With the Hessian Comparison Theorem, we also show that there is a lower bound for the mean curvature of any closed hypersurface of constant mean curvature in a manifold with a pole satisfying a curvature condition.
We prove a gradient estimate and Liouville type theorems for the solutions of the Poisson equation on a complete manifold whose Ricci curvature is suitably restricted.
The explicit Howe duality correspondence is partially solved in the case of irreducible type 2 dual reductive pairs defined over a non-Archimedean local field.
We calculate Donaldson's simple invariant for a surface of general type known as the Horikawa surface and for regular elliptic surfaces, by using Kronheimer's method. As a corollary, it is shown that there exist infinitely many differentiable structures on these surfaces and a torus sum of them.
We define a class of wavelet transforms as a continuous and microlocal version of the Littlewood-Paley decompositions. Hörmander's wave front sets as well as the Besov and Triebel-Lizorkin spaces may be characterized in terms of our wavelet transforms.
In this paper we offer very general Opial-type inequalities involving higher order r-derivatives. From these inequalities we then deduce extended and improved versions of several recent results. Some applications which dwell upon the importance of the obtained inequalities are also included.
Asymptotic behaviour of the heat kernels on some explicitly known quantum spaces are studied. Then the heat kernels are shown to be logarithmically divergent. These results suggest to us that the "dimensions" of these quantum spaces would not be zero but less than one so that these quantum spaces look almost like "discrete spaces".
We show that under some boundary conditions a hyperbolic fiber space with noncompact complete hyperbolic fibers has at most finitely many essentially nontrivial sections.