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In the following 2 theorems, we prove the modified Hyers–Ulam stability of the quadratic functional equation for the cases p < 2 and p > 2 separately

# On the stability of functional equations and a problem of Ulam

Acta Applicandae Mathematica, no. 1 (2001): 23-130

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 ar...更多

• Šemrl [RS1] introduced a simple counterexample to Theorem 1.3 for p = 1 as follows: The continuous real-valued function defined by f (x) = x log2(x + 1) for x 0, x log2 |x − 1| for x < 0 satisfies the inequality (1.7) with θ = 1 and |f (x) − cx|/|x| → ∞, as x → ∞, for any real number c.
• It can be proved that if G is a Banach space and if f is continuous in t for every fixed x ∈ G A is a linear function in the same way as in Theorem 1.3.

• Ulam [Ul1], for very general functional equations, one can ask the following question: When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation? if we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality lie near the solutions of the strict equation?
• Ulam gave a wide ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the following question concerning the stability of homomorphisms: Let G1 be a group and let G2 be a metric group with a metric d(·, ·)
• Ulam [Ul1] raised the following question concerning the stability of homomorphisms: Let G1 be a group and let G2 be a metric group with a metric d(·, ·)
• In the following 2 theorems, we prove the modified Hyers–Ulam stability of the quadratic functional equation (3.1) for the cases p < 2 and p > 2 separately
• We introduce a Hyers–Ulam stability result which was presented by J

• The authors have seen that the condition that a function f satisfies one of the inequalities (1.2), (1.5), (1.10), (1.12), (1.15) and (1.18) on the whole space, assures them of the existence of a unique additive function which approximates f within a given distance.
• F. Skof [Sk2] proved the following theorem and applied the result to the study of an asymptotic behavior of additive functions.
• According to Theorem 1.1, there is a unique additive function A: R → E such that the inequality (1.23) holds for any x in R.
• According to Theorem 1.15, there exists a unique additive function An: E1 → E2 satisfying g(x) − An(x) 9δn (b) for all x in E1.
• The authors look at the so-called Hammerstein equation: x = λAf (x) subject to the following conditions: (H1) (E, K) and (F, P ) are real Banach spaces ordered by the cones K and P , respectively, and the cone K is normal and solid; (H2) The function f : E → F is continuous and the operator A: F → E is linear and compact; (H3) A is strongly positive, that is y − x ∈ P \{0} implies that A(y) − A(x) ∈ int(K).

• Let conditions (H1), (H2), (H3) hold and suppose that the followings are true:  f (K) ⊂ P ;  f is ψ-additive;  limn→∞ 2−nf (2nx) > 0 for all x ∈ K \{0}.
• Assume that the mapping E is RN+-asymptotically equivalent to a ψ-additive mapping H : RN → RN and the following hypotheses are satisfied:  ψ satisfies assumptions (i), (ii), (iii);  H is off-diagonal negative with respect to RN+;  the operator T (x) = limn→∞ 2−nH (2nx) is nonsingular on RN+ (i.e., T (x) =

• Šemrl [RS1] introduced a simple counterexample to Theorem 1.3 for p = 1 as follows: The continuous real-valued function defined by f (x) = x log2(x + 1) for x 0, x log2 |x − 1| for x < 0 satisfies the inequality (1.7) with θ = 1 and |f (x) − cx|/|x| → ∞, as x → ∞, for any real number c.
• It can be proved that if G is a Banach space and if f is continuous in t for every fixed x ∈ G A is a linear function in the same way as in Theorem 1.3.
• The authors have seen that the condition that a function f satisfies one of the inequalities (1.2), (1.5), (1.10), (1.12), (1.15) and (1.18) on the whole space, assures them of the existence of a unique additive function which approximates f within a given distance.
• F. Skof [Sk2] proved the following theorem and applied the result to the study of an asymptotic behavior of additive functions.
• According to Theorem 1.1, there is a unique additive function A: R → E such that the inequality (1.23) holds for any x in R.
• According to Theorem 1.15, there exists a unique additive function An: E1 → E2 satisfying g(x) − An(x) 9δn (b) for all x in E1.
• The authors look at the so-called Hammerstein equation: x = λAf (x) subject to the following conditions: (H1) (E, K) and (F, P ) are real Banach spaces ordered by the cones K and P , respectively, and the cone K is normal and solid; (H2) The function f : E → F is continuous and the operator A: F → E is linear and compact; (H3) A is strongly positive, that is y − x ∈ P \{0} implies that A(y) − A(x) ∈ int(K).
• Let conditions (H1), (H2), (H3) hold and suppose that the followings are true:  f (K) ⊂ P ;  f is ψ-additive;  limn→∞ 2−nf (2nx) > 0 for all x ∈ K \{0}.
• Assume that the mapping E is RN+-asymptotically equivalent to a ψ-additive mapping H : RN → RN and the following hypotheses are satisfied:  ψ satisfies assumptions (i), (ii), (iii);  H is off-diagonal negative with respect to RN+;  the operator T (x) = limn→∞ 2−nH (2nx) is nonsingular on RN+ (i.e., T (x) = 0 