Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces
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Research Article Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces Yeol Je Cho,1 Madjid Eshaghi Gordji,2 and Somaye Zolfaghari3 1
Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea 2 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran 3 Department of Mathematics, Urmia University, Urmia, Iran Correspondence should be addressed to Madjid Eshaghi Gordji, [email protected] Received 29 July 2010; Accepted 31 August 2010 Academic Editor: Vijay Gupta Copyright q 2010 Yeol Je Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We achieve the general solution and the generalized stability result for the following functional equation in random normed spaces in the sense of Sherstnev under arbitrary t-norms: fx ky fx − ky k 2 fx y fx − y 2k 2 − 1/k 2 k − 2fkx − k 3 − k 2 − k 1/2k − 2f2x f2y − 8fy, where fy : fy − f−y for any fixed integer k with k / 0, ±1, 2.
1. Introduction In 1940, the stability problem of functional equations originated from a question given by Ulam 1, that is, the stability of group homomorphisms. Let G1 , · be a group and G2 , ∗, d be a metric group with the metric d. For any > 0, does there exist δ > 0, such that, if a mapping h : G1 → G2 satisfies inequality dhx · y, hx ∗ hy < δ. For all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with dhx, Hx < for all x ∈ G1 ? In other words, under what condition, does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers 2 gave a first affirmative answer to the question of Ulam for Banach spaces.
2
Journal of Inequalities and Applications Let f be a mapping between Banach spaces E and E such that, for some δ > 0, f x y − fx − f y ≤ δ,
∀x, y ∈ E.
1.1
Then there exists a unique additive mapping T : E → E such that fx − T x ≤ δ,
∀x, y ∈ E.
1.2
Moreover, if ftx is continuous in t ∈ R for any fixed x ∈ E, then T is linear. In 1978, Rassias 3 provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda 4 answered the question for the case p > 1, which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias stability of functional equations see 5–17. The functional equation f x y f x − y 2fx 2f y
1.3
is related to symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation 1.3 is said to be a quadratic mapping. It is well known that
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