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

Abstract

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)