Funkcialaj Ekvacioj

A Simplified Approach in the Study of Elliptic Differential Equations in UMD Spaces and New Applications

Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, A ...

2008 年 51 巻 2 号 p. 165-187
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.165

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This paper is devoted to provide some new clarifications in the study of complete abstract second order differential equations of elliptic type given in our recent papers [8], [9] and [10]. We improve the situation in the case of the space L^p(0,1;X) with a UMD Banach space X by considering an approach generalizing those used in [10]. On the other hand, we give some new examples to which our theory applies.

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Energy Dependent Inverse Scattering

Yutaka Kamimura

2008 年 51 巻 2 号 p. 189-219
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.189

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The paper discusses the inverse scattering problem to recover the potentials of an energy dependent Schrödinger equation from the scattering data. A necessary and sufficient condition for a given function to be the scattering data is obtained. The potential can be recovered by a simple formula from the solution of a Marchenko type equation with a function determined from the scattering data.

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Algebraic Transformations of ₃F₂

Mitsuo Kato

2008 年 51 巻 2 号 p. 221-243
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.221

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Let E(x) be the third order differential equation ₃F₂ satisfied by a generalized hypergeometric function ₃F₂ (a₀, a₁, a₂; b₁, b₂; x). Under some assumptions which guarantee that E(x) is irreducible and has no logarithmic solutions, we determine all the algebraic transformations E(x) = θ(x)E′(φ(x)), where E′(z) is a Fuchsian differential equation on P¹, φ(x) a rational function, and θ(x) a finite product of complex powers of rational functions. We find E′(z) also turns out to be a ₃E₂.

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Infinitely Many Periodic Solutions of Nonlinear Equations of Suspended Strings

Masaru Yamaguchi

2008 年 51 巻 2 号 p. 245-267
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.245

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We shall deal with a nonlinear equation of a suspended string with a special density to which a nonlinear time-independent external force is applied. We shall show the existence of infinitely many time-periodic solutions, near time-periodic solutions which are independent of position. The set of the periods is constructed by using algebraic numbers of degree 2 and the continued fractions.

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Non-Existence of Positive Radially Symmetric Solutions for the p-Laplacian Boundary Value Problem on Annular Domains

Fu-Hsiang Wong, Chin-Chen Chou, Wei-Cheng Lian, Shiueh-Ling Yu

2008 年 51 巻 2 号 p. 269-279
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.269

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In this paper we propose some criterion concerning the non-existence of positive solutions for the two-point boundary value problem of ordinary differential equations of p-Laplacian type. As an application, we also study the non-existence of positive radially symmetric solutions for the p-Laplacian equation on annular domain.

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Affine Schwarz Map for the Hypergeometric Differential Equation

Ryoichi Kobayashi, Tatsuya Nishizaka, Shoji Shinzato, Masaaki Yoshida

2008 年 51 巻 2 号 p. 281-305
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.281

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We propose an affine version of the Schwarz map for the hypergeometric differential equation, and study its image when the monodromy group is finite.

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Discrete Bernoulli Polynomials and the Best Constant of the Discrete Sobolev Inequality

Atsushi Nagai, Yoshinori Kametaka, Hiroyuki Yamagishi, Kazuo Takemura, ...

2008 年 51 巻 2 号 p. 307-327
発行日: 2008年
公開日: 2008/08/01

DOIhttps://doi.org/10.1619/fesi.51.307

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A discrete version of the Sobolev inequalty in the Hilbert space
${\\bf C}_0^N=\\biggl\\{{\\bf u}={ }^t(u(0),\\cdots,u(N-1)) \\in {\\bf C}^N \\,\\biggr|\\, \\displaystyle\\sum_{i=0}^{N-1}u(i)=0\\biggr\\},$
which is equipped with a suitable inner product, is derived. The best constant and best function of the discrete Sobolev inequality are also obtained from the theory of reproducing kernels, and are expressed by means of discrete analogues of the well-known Bernoulli polynomials. Some interesting properties of these discrete Bernoulli polynomials are also discussed.

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