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In this paper we present some suggestions for unifying the terminology of the mappings appearing in fixed point iterative methods for the case when the setting is a metric space. We consider the following concepts: contraction type mapping, contractive type mapping and nonexpansive type mapping, for which some problems are formulated. [ABSTRACT FROM AUTHOR]

In the book, Fixed Point Structure Theory (I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006) there are studied fixed point structures on a set with structure. In this paper we introduce the notion of the set-mapping pair (U;M) (i.e., U:= a class of sets with the same type structure and for X; Y 2 U a set of mappings, M(X; Y), from X to Y is given) and the notion of fixed point structure (f.p.s.) on a such pair. After some examples of f.p.s. we study the preserving of the fixed point property by, (U;M)-bijections, retractions, cartesian product and exponential. We give some fixed point results in terms of a f.p.s. and we consider some special f.p.s.: f.p.s. with common fixed point property, with coincidence point property and with coincidence producing mappings. Some open problems are also formulated. [ABSTRACT FROM AUTHOR]

Let a < c < b real numbers, (B, |·|) a (real or complex) Banach space, H ∈ C([a, b] x [a, c] x B,B), K ∈ C([a,b]² x B,B), g ∈ C([a,b],B), A : C([a,c],B) → C([a,c],B) and B : C([a,b],B) → C([a, b],B). In this paper we study the following functional integral equation, x(t) = ∫ac H(t,s,A(x)(s))ds + ∫at, K(t,s B(x)(s))ds + g(t), t ∈ [a,b]. By a new variant of fibre contraction principle (A. Petrusel, I.A. Rus, M.A. Serban, Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations, Fixed Point Theory, 22 (2021), no. 2, 795-808) we give existence, uniqueness and convergence of successive approximations results for this equation. In the case of ordered Banach space B, Gronwall-type and comparison-type results are also given. [ABSTRACT FROM AUTHOR]

Let (X1,→) and (X2, ↬) be two L-spaces, U be a nonempty subset of X1 × X2 such that UX1:= {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, FT1 ≠ ∅ and FT2(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. [ABSTRACT FROM AUTHOR]

In this paper we present some remarks on the following problem: Let X be a (real or complex) Banach space, Ω ⊂ X be an open convex subset and f: Ω → Ω be an operator. We suppose that: (i) f ∈ C¹ (X, X); (ii) the differential of f at x, df(x): X → X is a Picard operator for all x ∈ Ω; (iii) the fixed point set of f, Ff ≠ ∅. The problem is in which conditions f is a Picard operator? In the case X:= Rm or X:= Cm, this problem is in connection with a LaSalle Conjecture (J.P. LaSalle, The Stability of Dynamical Systems, SIAM, No. 25, 1976) and with the Belitskii-Lyubich Conjecture (G.R. Belitskii and Yu.I. Lyubich, Matrix Norms and their Applications, Birkhäuser, 1988). We also formulate the following conjecture: Let X be a Banach space, Ω ⊂ X be an open convex subset and f: Ω → Ω be an operator. We suppose that: (i) f ∈ C¹ (Ω, X); (ii) dfk (x) is a Picard operator, ∀ x ∈ Ω, ∀ k ∈ N∗; (iii) Ff ≠ ∅. Then f is a Picard operator. Some research directions are also presented. [ABSTRACT FROM AUTHOR]