Quasi-Triangular Spaces, Pompeiu-Hausdorff Quasi-Distances, and Periodic and Fixed Point Theorems of Banach and Nadler Types

Let , -index set. A quasi-triangular space is a set with family satisfying . For any , a left (right) family generated by is defined to be , where and furthermore the property holds whenever two sequences and in satisfy and and . In , using the left (right) families generated by ( is a special case of ), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on ; for each type we construct of left (right) set-valued quasi-contraction , and we prove the convergence, existence, and periodic point theorem for such quasi-contractions. We also construct two types of left (right) single-valued quasi-contractions and we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi-contractions. ( ) generalize ultra quasi-triangular and partiall quasi-triangular spaces (in particular, generalize metric, ultra metric, quasi-metric, ultra quasi-metric, -metric, partial metric, partial -metric, pseudometric, quasi-pseudometric, ultra quasi-pseudometric, partial quasi-pseudometric, topological, uniform, quasi-uniform, gauge, ultra gauge, partial gauge, quasi-gauge, ultra quasi-gauge, and partial quasi-gauge spaces).