Some Generalized B-Metric Spaces And Fixed Point Theorems

Fixed point theory is an important part of nonlinear functional analysis,which has been widely applied in different areas,such as economics,computational science,biology,chemistry and engineering,etc.As one of the earliest fixed point results,Banach contraction principle has been studied and extended in various spaces by numerous researchers to obtain abundant fixed point results,such as Kannan type,Chatterjea type,Reich type,Ciric type,Geraghty type and other nonlinear fixed point theorems.The purpose of this paper is to study fixed point theorems with corresponding applications in b-metric spaces and some generalized b-metric spaces.All the results given by us consist of five parts as follows.(1)In the framework of metric spaces,we introduce the concept of Geraghty-Ciric type contractions and establish the fixed point theorem for such mappings.By constructing two examples,we prove that these results improve the theorems obtained by Geraghty and Ciric.In addition,the unified form of Geraghty-Ciric type fixed point theorem was extended to b-metric spaces and applied in the periodic boundary value problems of first-order ordinary differential equations.(2)We introduce the notion of Kaleva-Seikkala’s type fuzzy b-metric spaces and give an example to illustrate that the fuzzy b-metric space is a generalization of the b-metric space and the fuzzy metric space.In such spaces,we establish Banach type,Reich type and Chatterjea type fixed point theorems with the contraction constants[0,1).Finally,various corollaries are presented to testify the fact that our main theorems extend the cases of b-metric spaces.(3)By a new lemma concerning Cauchy sequence,we prove the fixed point theorem for Ciric type quasi-contractions in rectangular b-metric spaces.This result generalizes Chatterjea type,Reich type and Hardy-Rogers type fixed point theorems,which gives a complete positive answer to the open question proposed by Geroge et al.After that,an example is presented to illustrate our main result.(4)In the context of b-metric spaces,the range of q-set-valued quasi-contraction constant is extended from[0,1/(s+s2))to(?)(1≤s<1+(?)),[0,1/s)(s>1+(?))with a new technical lemma.In addition,we establish the second result which extends the theorem concerning set-valued quasi-contraction type multifunctions from metric spaces to b-metric spaces.Finally,we give an unified result to improve the recent three fixed point theorems for set-valued mappings in b-metric spaces.(5)Applying a new lemma to prove that a Picard is a Cauchy sequence,we establish the theorem for graphical Banach type(G,G*)-contractions in graphical b-metric spaces,where the contraction constant is fully extended from[0,1/s2)to[0,1).By introducing a new property,we generalize the concepts of Kannan type and Reich type contractions to graphical b-metric spaces and establish the corresponding theorems.After that,we present two theorems concerning graphical Edelstein type and Meir-Keeler type(G,G*)-contractions in the setting of graphical b-metric spaces,which partially answer an open question in the literature.Finally,a counterexample is proposed to prove that the theorem for graphical Meir-Keeler type(G,G*)-contraction does not hold for s=1.