Some Fixed Theorems And Application To Coupled Integral Equations On B-Metric Spaces

With the development of technology and the actual need,nonlinear problems be come increasingly one of hot spot,which people are researching.However,fixed point theory is regarded as a powerful tool to resolve nonlinear problems and have been used widely in more fields.Such as variational inequalities,integral equations,differential equations,economics,game theory and equilibrium theory and so on.however,in recently years,some scholars provided coupled fixed point theorem and coupled coincident point theorem,which obtain long-term development.Many people mainly generalized three concept in three directions:(1)Promote the fixed theory to a broader space.such as,partially order metric space,complete b-metric space,complete G-metric space,fuzzy metric space,quasi-partial metric space(2)Generalize the mapping to two dimensional,three dimensional,four dimensional and even N dimensional,obtain high dimensional coupled fixed theorems and coincident point theorems in partially order metric space.(3)Get relevant fixed point theorems by changing contraction condition.The paper mainly study the fixed point theory on complete b-metric spaces and its applications.The first chapter introduces the research background and current situation of the fixed point theory.The second chapter first introduces the concepts of binary relation and order,and then the related knowledge of metric spaces and the fixed point theorem are also given.The third chapter obtain the equivalence theorem of the fixed point theorem and coupled coincidence point theory in partially order metric spaces,and a specific example is used to verify the correctness of the above theory.The four chapter inspired of the predecessors,in order to expand the developing space of the fixed point theorem.The first section,we will generalize the coupled fixed point theorem to complete b-metric space,also get the equivalence theorem.Further,in the second quarter,we generalize the coupled fixed point theorem to coupled coincidence point theorems on complete b-metric space.It is the promotion of the fixed point theorem.In the paper,the fixed point theorems are based on coherent structure.we know that by adding order relation to metric space,we notice that the contraction conditions of the fixed point theorems become weak,and be able to settle the wider problems.Finally,we end up with several examples to verify the correctness and effectiveness of the theory.