Study On Fixed Point Problems In Menger Probability S-Metric Space

In 1942,Menger used distribution functions instead of nonnegative real num-ber as values of the metric and proposed the Menger probability metric space(for short,Menger PM-space).Since then,many scholars have begun to study the fixed point problems in Menger PM-space,and introduce Menger PGM-space and Menger PbM-space based on the notions of Menger PM-space and other generalized metric spaces such as G-metric space and b-metric space.In this pa-per,we defined the notion of a generalized probabilistic space as a generalization of a Menger PM-space and a S-metric space,which is called Menger probability S-metric space,and investigate its topological properties.Then,we obtain some fixed point theorems(including common fixed point and coupled fixed point)un-der various types of new contractive conditions.Meantime,we present practical examples to illustrate the validity of the new resultsIn the first chapter,we mainly introduce the practical value of fixed point theory and the research at home and abroad.In the second chapter,we introduce a new class of generalized metric space,which is called Menger PSM-space.We prove the space is a Hausdorff space,and define the concepts of neighborhood,convergence sequence,Cauchy sequence,completeness,etc,in the space.Then,in the complete Menger PSM-space,in the third chapter,we prove several new fixed point theorems under probabilistic ?-contractions,and give several examples to support the new results.In the fourth chapter,some new common fixed point the-orems of two pairs of weakly compatible mappings are proved in the condition of algebraic sum(?)and one pair of mappings satisfying(E.A)property.Meantime,a specific example is provided to exemplify the new results.In the fifth chapter,we obtain some fixed point theorems by introducing a generalized ?-type contrac-tive mapping in Menger PSM-space.Meantime,an application to Volterra type integral equation is given to support the obtained results.In the sixth chapter,in partial-order Menger PSM-space,some new coupled coincidence points and cou-pled common fixed point theorems are proved,and a practical example is given to illustrate the validity of new results.