On Carlson Type Inequalities and Fixed Point Theory

• Main Authors: Jen-Chun Fang, 方仁駿
• Other Authors: Gou-Sheng Yang, Ph.D.
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/82297338451455325695

博士 === 淡江大學 === 數學學系 === 87 === This dissertation is divided into two parts. The first part deals with generalization of Carlson type inequalities, while the second part focues on the fixed point theorem in hyperspace $\mbf {CC}({\bf X})$, which equipped with the weak topology $\mbf{\cal T}_w$. Suppose $\{a_n\}_{n=1}^\infty$ is a sequence of nonnegative real numbers and $f$ is a nonnegative function on $[0,\infty)$. In 1934, Carlson proved that the following inequalities $\sum\limits _{n=1}^\infty a_n\!\le\!\sqrt\pi \Big(\sum\limits _{n=1}^\infty a^2_{n}\Big)^{{1\over 4}} \Big(\sum\limits _{n=1}^\infty n^2a^2_{n}\Big)^{{1\over 4}}$ and $\int\limits _0^\infty f(x)dx\!\le\!\sqrt \pi (\int\limits _0^\infty f^2(x)dx)^{{1\over 4}} (\int\limits _0^\infty x^2f^2(x)dx)^{{1\over 4}}$. After Carlson rendered inequalities in 1934, some subsequent theories concerning the generalization and application of Carlson type inequalities have been published. Carlson type inequalities and its generalizations have several applications in momenta problems, in the theory of interpolation, in the homogenous weights and in optimal reconstruction of a sampling signal. The first part of this paper is to expand on the Carlson type inequalities, basing on the academic researches done in recent years. Next part is about the fixed point theorem in the hyperspace $(\mbf{CC}({\bf X}),\mbf{\cal T}_w)$. Let ${\bf X}$ be a Banach space, ${\bf X}^*$ its topological dual, and $\mbf {CC}({\bf X})$ the collection of all non-empty, compact, convex subsets of ${\bf X}$, and $h$ the natural Hausdorff metric on $\mbf {CC}({\bf X})$. Let ${\bf Z}$ denote the complex plane, $\mbf{CC}({\bf Z})$ denote the collection of all non-empty, compact, convex subsets of ${\bf Z}$, and $h$ the natural Hausdorff metric on $\mbf {CC}({\bf Z})$. Hu and Huang proved that, $h(x^*(A), x^*(B))\le \|x^*\|h(A,B)$, for each $x^*\in {\bf X}$, therefore, they define $\mbf{\cal T}_w$ to be the weakest topology on the hyperspace $\mbf{CC}({\bf X})$ such that each $x^*:(\mbf{CC}({\bf X}), \mbf{\cal T}_w)\rightarrow (\mbf{CC}({\bf Z}),h)$ is continuous. In 1965, Kirk proved that if $K$ is a non-empty, weakly compact, convex subset of a Banach space ${\bf X}$, with normal structure, then every nonexpansive mapping of $K$ into $K$ has a fixed point in $K$. It is the main purpose of the second part to have the Kirk''s theorem extended to the hyperspace $\mbf{CC}({\bf X})$ equipped with the weak topology $\mbf{\cal T}_w$.