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

Let C={Cα} α∈A∈[1;∞) A, A‐index set. A quasi‐triangular space (X,PC;A) is a set X with family PC;A={pα:X2→[0,∞), α∈A} satisfying ∀α∈A ∀u,v,w∈X {pα(u,w)≤Cα[pα(u,v)+pα(v,w)]}. For any PC;A, a left (right) family JC;A generated by PC;A is defined to be JC;A={Jα:X2→[0,∞), α∈A}, where ∀α∈A ∀u,v,w∈X {Jα(u,w)≤Cα[Jα(u,v)+Jα(v,w)]} and furthermore the property ∀α∈A {limm→∞pα(wm,um)=0} (∀α∈A {limm→∞pα(um,wm)=0}) holds whenever two sequences (um:m∈N) and (wm:m∈N) in X satisfy ∀α∈A {limm→∞supn>mJα(um,un)=0 and limm→∞Jα(wm, um) = 0} (∀α∈A {limm→∞supn>mJα(un,um)=0 and limm→∞Jα(um, wm) = 0}). In (X,PC;A), using the left (right) families JC;A generated by PC;A (PC;A is a special case of JC;A), we construct three types of Pompeiu‐Hausdorff left (right) quasi‐distances on 2X; for each type we construct of left (right) set‐valued quasi‐contraction T : X → 2X, 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 T : X → X and we prove the convergence, existence, approximation, uniqueness, periodic point, and fixed point theorem for such quasi‐contractions. (X,PC;A) generalize ultra quasi‐triangular and partiall quasi‐triangular spaces (in particular, generalize metric, ultra metric, quasi‐metric, ultra quasi‐metric, b‐metric, partial metric, partial b‐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).