DOI: 10.24193/fpt-ro.2021.2.52

Authors: Adrian Petrușel, Ioan A. Rus and Marcel Adrian Șerban

Abstract: Let (X₁,→) and (X₂, ↬) be two L-spaces, U be a nonempty subset of X₁ × X₂ such that U_x1 := {x₂ ∈ X₂ | (x₁, x₂) ∈ U} is nonempty, for each x₁ ∈ X₁. Let T₁ : X₁ → X₁, T₂ : U → X₂ be two operators and T : U → X₁ × X₂ defined by T(x₁,x₂) := (T₁(x₁),T₂(x₁,x₂)). If we suppose that T(U) ⊂ U, F_T1 ≠ ∅ and F_{T2(x1, · )} ≠ ∅ for each x₁ ∈ X₁, the problem is in which additional conditions T is a weakly Picard operator? In this paper we study this problem in the case when the convergence structures on X₁ and X₂ are defined by metrics. Some applications to the fixed point equations on spaces of continuous functions are also given.

Key Words and Phrases: Triangular operator, fibre contraction, weakly Picard operator, generalized metric space, generalized contraction, well-posedness, Ostrowki property, Ulam-Hyers stability, Volterra operator, functional differential equation, functional integral equation.

2010 Mathematics Subject Classification: 47H10, 54H25, 47H09, 45N05, 34K28.

Published on-line: July 1st, 2021.

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