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

Abstract EN

Let C = { C α } α ∈ A ∈ [ 1 ; ∞ ) A , A -index set.

A quasi-triangular space ( X , P C ; A ) is a set X with family P C ; A = { p α : X 2 → [ 0 , ∞ ) , α ∈ A } satisfying ∀ α ∈ A ∀ u , v , w ∈ X { p α ( u , w ) ≤ C α [ p α ( u , v ) + p α ( v , w ) ] } .

For any P C ; A , a left (right) family J C ; A generated by P C ; A is defined to be J C ; A = { J α : X 2 → [ 0 , ∞ ) , α ∈ A } , where ∀ α ∈ A ∀ u , v , w ∈ X { J α ( u , w ) ≤ C α [ J α ( u , v ) + J α ( v , w ) ] } and furthermore the property ∀ α ∈ A { lim m → ∞ p α ( w m , u m ) = 0 } ( ∀ α ∈ A { lim m → ∞ p α ( u m , w m ) = 0 } ) holds whenever two sequences ( u m : m ∈ N ) and ( w m : m ∈ N ) in X satisfy ∀ α ∈ A { lim m → ∞ sup n > m J α ( u m , u n ) = 0 and lim m → ∞ J α ( w m , u m ) = 0 } ( ∀ α ∈ A { lim m → ∞ sup n > m J α ( u n , u m ) = 0 and lim m → ∞ J α ( u m , w m ) = 0 } ) .

In ( X , P C ; A ) , using the left (right) families J C ; A generated by P C ; A ( P C ; A is a special case of J C ; A ), we construct three types of Pompeiu-Hausdorff left (right) quasi-distances on 2 X ; for each type we construct of left (right) set-valued quasi-contraction T : X → 2 X , 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 , P C ; 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).