On the stability of the linear mapping in Banach spaces

1978/11/01 by Themistocles M. Rassias · 23 citations
Computer Science · Mathematics · #Optimization and Variational Analysis #Functional Equations Stability Results #advanced mathematical theories

paper · pdf · doi:10.1090/s0002-9939-1978-0507327-1

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1 comma upper E 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E₁,E₂</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be two Banach spaces, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper E 1 right-arrow upper E 2"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f:E₁ → E₂</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a mapping, that is “approximately linear". S. M. Ulam posed the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam’s problem.

Cited by

• A characterization of Hyers–Ulam stability of first order linear differential operators
• Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces
• On the generalized quadratic mappings in quasi-Banach modules over a C^*--algebra
• On approximate ternary m-derivations and σ-homomorphisms
• Stability of ternary Jordan homomorphisms and derivations associated to the generalized Jensen equation
• Fuzzy stability of ring homomorphisms and ring derivations on fuzzy Banach algebras
• On perturbations of the multiadditive mappings via the concept of generalized Z -numbers
• The stability of a Cosine-Sine Functional Equation on abelian groups
• On Caputo-tempered fractional coupled systems with three-point boundary conditions via the vector degree of nondensifiability
• Multivariable mappings associated with the Drygas functional equation and a fixed point approach
• Jordan derivations on C^*-ternary algebras for a Cauchy-Jensen functional equation
• Continuous ∗-g-Frame in Hilbert C^∗-Modules
• Bi-additive s-functional inequalities and biderivations in modular spaces
• Stability of a mixed type quadratic, cubic and quartic functional equation
• Stability of a Volterra Integral Equation on Time Scales
• Stability of generalized mixed type additive-quadratic-cubic functional equation in non-Archimedean spaces
• Hyers-Ulam-Rassias stability of functional equations with parameters
• The stability of a quadratic type functional equation with the fixed point alternative
• Solvability of Infinite Systems of Nonlinear Caputo Fractional Differential equations in generalized Hahn Sequence Space
• Ulam stability of Jensen functional inequality in groups
• On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces
• Stability of a functional equation deriving from cubic and quartic functions
• On approximate cubic homomorphisms

Related

• Banach spaces with the Daugavet property
• Cell-like mappings between CS sets
• Irregular sets, the 𝛽-transformation and the almost specification property
• One dimensional 𝖱𝖢𝖣 spaces always satisfy the regular Weyl’s law
• A relation between pointwise convergence of functions and convergence of functionals