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In this paper we extend some results of V. Berinde, Şt. Măruşter and I.A. Rus (Saturated contraction principles for nonself operators, generalizations and applications, Filomat, 31:11(2017), 3391-3406), which were given in metric spaces, to Kasahara spaces. Some research directions are also presented.

In this paper we study the following problems: I. Let (M, d) be a complete metric space and f, g : M → M be two operators. We suppose that:

Several notions of abstract Volterra operators on spaces of functions of one variable are well known. In this paper, we introduce various notions of abstract Volterra operators in spaces of functions of several variables. Some fixed point equations with such abstract Volterra operators are also studied. The basic ingredient in the theory of step by step contraction is the notion of G-contraction. The relevance of step by step contraction principle is illustrated by applications in the theory of Darboux-Ionescu problem. [ABSTRACT FROM AUTHOR]

In this paper we give conditions in which the integral equation... [ABSTRACT FROM AUTHOR]

Let (M, d) be a metric space, .... In this paper, we will give sufficient conditions ensuring that a Frum-Ketkov operator is weakly Picard. Some generalized Frum-Ketkov operators are also studied. [ABSTRACT FROM AUTHOR]

The main aim of this note is to investigate empirically the relationship between the spectral radius of the derivative of a function f : Rm→Rm and the spectral radius of the derivatives of its iterates, which is done by means of some numerical experiments for mappings of two and more variables. In this way we give a partial answer to an open problem raised in [Rus, I. A., Remark on a La Salle conjecture on global asymptotic stability, Fixed Point Theory, 17 (2016), No. 1, 159-172] and [Rus, I. A., A conjecture on global asymptotic stability, communicated at the Workshop "Iterative Approximation of Fixed Points", SYNASC2017, Timis¸oara, 21-24 September 2017] and also illustrate graphically the importance and difficulty of this problem in the general context. An open problem regarding the domains of convergence is also proposed. [ABSTRACT FROM AUTHOR]

Let (X, d) be a metric space, f, fn : X → X, with Ff = ..., n ∈ N. For the fixed point equation (1) x = f(x) we consider the following iterative algorithm, (2) x ∈ X; x0 = x; xn+1(x) = fn(xn(x)); n ∈ N. By definition, the algorithm (2) is convergent if, xn(x) → x*(x) ∈ Ff as n → ∞, ∀ x ∈ X. In this paper we give some conditions on fn and f which imply the convergence of algorithm (2). In this way we improve some results given in [Rus, I. A., An abstract point of view on iterative approximation of fixed points. impact on the theory of fixed point equations, Fixed Point Theory, 13 (2012), No. 1, 179-192]. In our results, in general we do not suppose that, Ff ≠ ∅. Some research directions are formulated. [ABSTRACT FROM AUTHOR]

In this note, using the weakly Picard operator technique we will present some qualitative results for operators generated by an operator defined on a Cartesian product of a metric space with itself. The work is based on a recent paper of the authors. [ABSTRACT FROM AUTHOR]

In this paper we give a new proof of a result by S. Reich and A.J. Zaslavski (S. Reich and A.J. Zaslavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8(2007), No. 2, 303-307). Some new fixed point theorems for nonself generalized contractions are also given. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝp, p ∈ ℕ* be a nonempty subset and B(Ω) be the Banach lattice of all bounded real functions on Ω, equipped with sup norm. Let X ⊂ B(Ω) be a linear sublattice of B(Ω) and A: X → X be a positive linear operator with constant functions as the fixed point set. In this paper, using the weakly Picard operators techniques, we study the iterates of the operator A: Some relevant examples 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]

Inclusion relations between strongly demicontractive mappings, quasi (L,m)-contractions, and Kannan contractions are established. As corollaries, T-stability and strong convergence of Picard and Mann iterations for strongly demicontractive mappings are obtained. [ABSTRACT FROM AUTHOR]

Let X be a Banach lattice. In this paper we give conditions in which an increasing linear operator, A : X → X is weakly Picard operator (see I.A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58(2003), No. 1, 191-219). To do this we introduce the notion of "invariant linear partition of X with respect to A" and we use contraction principle and Maia's fixed point theorem. Some applications are also given. [ABSTRACT FROM AUTHOR]

In this paper we will present an abstract point of view on iterative approximation schemes of fixed points for multivalued operators. More precisely, we suppose that the algorithms are convergent and we will study the impact of this hypothesis in the theory of operatorial inclusiosns: data dependence, stability and Gronwall type lemmas. Some open problems are also presented. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following problem (see I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006, pp. 193-194): Let X be a set with structure and (X, S(X),M) be a large fixed point structure on X. Let U ∈ S(X). An operator p : U → U is by definition coincidence producing operator on (X, S(X),M) if for each f ∈ M(U) there exists u ∈ U such that f(u) = p(u). The problem is to study the existence and the properties of these class of operators. For the case of topological space with fixed point property with respect to continuous operators, the starting papers are given by W. Holsztynski (Une g'en'eralization du théorème de Brouwer sur les points invariants, Bull. Acad. Pol. Sc., 12(1964), 603-606) and by H. Schirmer (Coincidence producing maps onto trees, Canad. Math. Bull., 10(1967), 417-423) and for the case of metric spaces with fixed point property with respect to nonexpansive operators, by W.A. Kirk (Universal nonexpansive maps, 95-101, in Proc. 8th ICFPTA (2007), Yokohama Publ., 2008). [ABSTRACT FROM AUTHOR]