Takahiro Hayata

Stability of the Banach space valued Chebyshev differential equation

Abstract We prove that, if f is an approximate solution of the Banach space valued Chebyshev differential equation ( 1 − x 2 ) y ″ ( x ) − x y ′ ( x ) + n 2 y ( x ) = 0 for x ∈ ( − 1 , 1 ) , then f is near to an exact solution.

A Note on the Stability of an Integral Equation

Let p: ℝ → ℂ be a continuous function. We give a sufficient condition in order that the integral equation \(f(t) = f(0) +{ \int \nolimits \nolimits }_{0}^{\,t}p(s)f(s)\,\mathrm{d}s\) have the Hyers–Ulam stability. We also prove that if p has no zeros, then the sufficient condition is a necessary condition.

A reconsideration of Hua's inequality. Part II

We give a new interpretation of Huas inequality and its generalization. From this interpretation, we know the best possibility of those inequalities.

Zero Cells of the Siegel–Gottschling Fundamental Domain of Degree 2

Let be a fundamental domain of the Siegel upper half-space of degree n with respect to the Siegel modular group . According to Siegel himself, is determined by only finitely many polynomial inequalities. In case of degree n=2, Gottschling determined the minimal set of inequalities. The boundary of is of great concern in the literature not only from a homological point of view but also from the geometry of numbers. In this paper we compute the vertices of under the condition that the defining ideal is zero-dimensional (“0-cells”). We also discuss an equivalence relation among 0-cells.

On the Butler–Rassias Functional Equation and its Generalized Hyers–Ulam Stability

In this paper, we present a recent development of the Hyers–Ulam stability of the Butler–Rassias functional equation \(f(x + y) - f(x)f(y) = d\sin x\sin y\). We also show our most recent results on a Butler–Rassias type functional equation with complex variables.