Fixed Point Theorems In Probabilistic Metric Space And Numerical Radius Of Lipschitz Dual Operator On Banach Space

There are two parts in this paper.In the first part we mainly study several fixed point theorems in Menger probabilistic metric spaces.In the second part we study the numerical range of Lipschitz operator in Banach spaces.Chapter one introduces the background and development of probabilistic metric space.It also intruduces some basic concepts,properties and topology of probabilistic metric space.The second chapter present an extension of the fixed point theorem in several ways : the probabilistic metric space is more general,the contractivity condition is better and the auxiliary functions belong to a wider class.Furthermore we prove the existence and uniqueness of fixed points in G complete and M complete Menger probability metric space.The greater progress is that we weaken the class of auxiliary functions ? and ? by assuming the continuity only at point t(28)0.The common fixed point theorem under generalized ? mappings is studied in this chapter.At the end of the chapter we introduce a Mengerprobability metric-like space as a natural way to extend Menger probability metric spaces.The third chapter mainly study the numerical radius of Lipschitz dual operator,which is denoted by Tl*.We study the distinction between linear dual operator and Lipschitz dual operator and get an important conclusion that if the Lipschitz operator T can attain its norm,then the numerical radius of Tl* is equal to its norm.Morever we give an example to show that the Lipschitz operator T attains its norm is not necessary to get that conclusion.