Jamuna, R.

Pinelas, Sandra

Rassias, J. M.

Ravi, K.

English

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INTERNATIONALPUBLICATIONS USAPanAmerican Mathematical JournalVolume 24(2014), Number 2, 61–84A Fixed Point Approach to the Stabilityof a Quadratic Quartic FunctionalEquation in Paranormed SpacesK. RaviSacred Heart CollegeDepartment of MathematicsTirupattur - 635601, Tamilnadu, Indiaschkravi@yahoo.co.inJ.M. RassiasNational and Capodistrain University of AthensPedagagical Department E.E.Section of Mathematics and Informatics4, Agamemnonos str, Aghia ParaskeviAthens 15342, Greecejrassias@primedu.uoa.grhttp://users.uoa.gr/∼jrassias/Sandra PinelasAcademia MilitarDepartamento de Cieˆncias Exactas e NaturaisAv. Conde Castro Guimara˜es2720-113 Amadora, Portugalsandra.pinelas@gmail.comR. JamunaR.M.K. College of Engineering and TechnologyDepartment of MathematicsR.S.M. Nagar, Puduvoyal, Gummidipoondi TalukTiruvallur Dist., Tamilnadu, India 601206rjamche31@gmail.comCommunicated by Ram U Verma(Received December 1, 2013; Revised Version Accepted April 26, 2014)6162 K. Ravi, J.M. Rassias, Sandra Pinelas, R. JamunaAbstractIn this paper, using fixed point method we prove the generalized Hyers-Ulam stabilityof a quadratic quartic functional equation for fixed integers k with k 6= 0, ±1 in para-normed spaces.AMS (MOS) Subject Classification: Primary 39B, Secondary 47H10Key words: Paranormed space, generalized Hyers-Ulam stability, fixed point, quadraticquartic functional equation.1 Introduction and PreliminariesA basic question in the theory of functional equations arises as follows: When is it true thata function, which approximately satisfies a functional equation, must be close to an exactsolution of the equation?If the problem accepts a unique solution, we say the equation is stable. The first stabilityproblem concerning group homomorphisms is related to a question of Ulam [30] in 1940.“Let G be a group and G′ be a metric group with metric d(·, ·). Given  > 0 does thereexists a δ > 0 such that if a function f : G→ G′ satisfies the inequality d(f(xy), f(x)f(y)) <δ for all x, y ∈ G, then there exists homomorphism H : G→ G′ with d(f(x),H(x)) <  forall x ∈ G?”In 1941 D.H. Hyers [11] gave the first affirmative partial answer to the question of Ulamfor Banach spaces. He proved the following celebrated theorem.Theorem 1. ([11]) Let X,Y be Banach spaces and let f : X → Y be a mapping satisfying‖f(x + y) − f(x) − f(y)‖ ≤  (1)for all x, y ∈ X. Then the limita(x) = limn→∞f(2nx)2n(2)exists for all x ∈ X and a : X → Y is the unique additive mapping satisfying‖f(x)− a(x)‖ ≤  (3)for all x ∈ X.Aoki [2] generalized Hyers theorem for additive mappings. In 1978, a generalized versionof the theorem of Hyers for approximately linear mappings was given by Th.M. Rassias [24].He proved the following:

A fixed point approach to the stability of a quadratic quartic functional equation in Paranormed spaces

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A fixed point approach to the stability of a quadratic quartic functional equation in Paranormed spaces

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