Author

# Gian Luigi Forti

Bio: Gian Luigi Forti is an academic researcher from University of Milan. The author has contributed to research in topic(s): Cauchy's equation & Functional equation. The author has an hindex of 8, co-authored 25 publication(s) receiving 671 citation(s).

##### Papers

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TL;DR: In this paper, a survey about Hyers-Ulam stability of functional equations and systems in several variables is presented, with a focus on the stability of the Ulam model.

Abstract: The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.

464 citations

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TL;DR: In this paper, it was shown that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(y, y)), exhibit phenomena impossible in the one-dimensional case.

Abstract: We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

46 citations

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TL;DR: In this article, a parametric class of triangular maps on Q × I is defined, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2.

Abstract: The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

44 citations

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TL;DR: In this paper, the authors present and compare some of the most used notions of chaos for discrete dynamical systems, and compare them with a brief survey of chaos notions for discrete systems.

Abstract: In this brief survey we present and compare some of the most used notions of chaos for discrete dynamical systems.

29 citations

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TL;DR: In this article, the authors consider the problem of finding all functions in an abelian group such that for every (x, y) ∈ G ×G the Cauchy difference belongs to the set {a, a+b, a +2b,...,a+Mb}.

Abstract: We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, b∈B,b≠0,M a positive integer; find all functionsf:G →B such that for every (x, y) ∈G ×G the Cauchy differencef(x+y)−f(x)−f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.

17 citations

##### Cited by

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TL;DR: In this article, the stability of functional equations has been studied from both pure and applied viewpoints, and both classical results and current research are presented in a unified and self-contained fashion.

Abstract: In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

601 citations

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Abstract: The paper is devoted to some results on the problem of S. M. Ulam for the stability of functional equations in Banach spaces. The problem was posed by Ulam 60 years ago.

515 citations

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01 Oct 1992TL;DR: A survey of ideas and results stemming from the stability problem of Ulam and Ulam's stability problem can be found in this article, where the necessity for the commutativity of the Abelian semigroup and the sequential completeness of the topological vector space is considered.

Abstract: We present a survey of ideas and results stemming from the following stability problem of S. M. Ulam. Given a groupG1, a metric groupG2 and e > 0, find δ > 0 such that, iff: G1 →G2 satisfiesd(f(xy),f(x)f(y)) ⩽ δ for allx, y ∈G1, then there exists a homomorphismg: G1 →G2 such thatd(f(x),g(x))⩽e for allx ∈ Gl. For Banach spaces the problem was solved by D. Hyers (1941) with δ = e and
$$g(x) = \mathop {\lim }\limits_{n \to \infty } f(2^n x)/2^n .$$
Section 2 deals with the case whereG1 is replaced by an Abelian semigroupS andG2 by a sequentially complete locally convex topological vector spaceE. The necessity for the commutativity ofS and the sequential completeness ofE are also considered.

469 citations

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TL;DR: In this paper, a survey about Hyers-Ulam stability of functional equations and systems in several variables is presented, with a focus on the stability of the Ulam model.

Abstract: The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.

464 citations

01 Jan 2003

TL;DR: In this article, it was shown that the theorems of Hyers, Rassias and Gajda concerning the stability of the Cauchy's functional equation in Banach spaces are direct consequences of the alternative of fixed point.

Abstract: In this paper, we show that the theorems of Hyers, Rassias and Gajda concerning the stability of the Cauchy's functional equation in Banach spaces, are direct consequences of the alternative of fixed point.

400 citations