Study On Some Fixed Point Theorems In Generalized Metric Spaces

Fixed point theory of operators is one of the main branches of functional analysis,which is widely used in finding the solutions of differential equations,integral equations,matrix equations and so on.As generalizations of a metric space,many concepts of generalized metric spaces are put forward,and fixed point problems in these new frameworks of spaces are deeply studied.In this thesis,we mainly study fixed point problems for operators in generalized metric spaces.Firstly,we discuss the existence and uniqueness of common fixed points and φ-common fixed points for operators in Gb-metric spaces.Secondly,using the fact that a Gp-metric can always induce a G-metric,we provide new methods to prove some fixed point theorems in Gp-metric spaces under various contractions.Finally,we establish some new fixed point theorems in bv(s)-metric spaces under three types of contractive conditions for a self-mapping.We provide some examples to show the validity of our main results.This thesis is divided into six chapters.In Chapter one,we introduce the background,current status and existing problems of generalized metric spaces and the fixed point theory in such kinds of spaces.Moreover,some related definitions and theorems are given.In Chapter two,we prove the common fixed point theorems under a class of contractive conditions for three self-mappings in Gb-metric spaces,and deduce that the corresponding results in G-metric spaces also hold.In Chapter three,we introduce the definition of φ-common fixed point and the function class H,and based on these,we prove the φ-common fixed point theorems for three self-mappings in Gb-metric spaces.As corollaries,we obtain the common fixed point results for one self-mapping in Gb-metric spaces.In Chapter four,using the G-metric induced by the Gp-metric,we discuss the relationships between the two kinds of spaces,and prove that fixed point theorems under various contractive conditions in G-metric spaces also hold in Gp-metric spaces under corresponding conditions.In Chapter five,we prove some fixed point theorems in bv(s)-metric spaces under three types of contractive conditionsIn Chapter six,we give some concluding remarks...