Stability of a Monomial Functional Equation on a Restricted Domain

In this paper, we prove the stability of the following functional equation ∑ i=0 nCi(−1) f (ix + y) − n! f (x) = 0 on a restricted domain by employing the direct method in the sense of Hyers.


Introduction
Let V and W be real vector spaces, X a real normed space, Y a real Banach space, n ∈ N (the set of natural numbers), and f : V → W a given mapping.Consider the functional equation for all x, y ∈ V, where n C i := n! i!(n−i)! .The functional Equation ( 1) is called an n-monomial functional equation and every solution of the functional Equation ( 1) is said to be a monomial mapping of degree n.The function f : R → R given by f (x) := ax n is a particular solution of the functional Equation (1).In particular, the functional Equation ( 1) is called an additive (quadratic, cubic, quartic, and quintic, respectively) functional equation for the case n = 1 (n = 2, n = 3, n = 4, and n = 5, respectively) and every solution of the functional Equation ( 1) is said to be an additive (quadratic, cubic, quartic, and quintic, respectively) mapping for the case n = 1 (n = 2, n = 3, n = 4, and n = 5, respectively).
A mapping A : V → W is said to be additive if A(x + y) = A(x) + A(y) for all x, y ∈ V.It is easy to see that A(rx) = rA(x) for all x ∈ V and all r ∈ Q (the set of rational numbers).A mapping A n : V n → W is called n-additive if it is additive in each of its variables.A mapping A n is called symmetric if A n (x 1 , x 2 , . . ., x n ) = A n (x π (1) , x π(2) , . . ., x π(n) ) for every permutation π : {1, 2, . . ., n} → {1, 2, . . ., n}.If A n (x 1 , x 2 , . . ., x n ) is an n-additive symmetric mapping, then A n (x) will denote the diagonal A n (x, x, . . ., x) for x ∈ V and note that A n (rx) = r n A n (x) whenever x ∈ V and r ∈ Q.Such a mapping A n (x) will be called a monomial mapping of degree n (assuming A n ≡ 0).Furthermore, the resulting mapping after substitution x 1 = x 2 = . . .= x l = x and x l+1 = x l+2 = . . .= x n = y in A n (x 1 , x 2 , . . ., x n ) will be denoted by A l,n−l (x, y).A mapping p : V → W is called a generalized polynomial (GP) mapping of degree n ∈ N provided that there exist A 0 (x) = A 0 ∈ W and i-additive symmetric mappings A i : , for all x ∈ V and A n ≡ 0. For f : V → W, let ∆ h be the difference operator defined as follows: for all n ∈ N and all h ∈ V.For any given n ∈ N, the functional equation ∆ n+1 h f (x) = 0 for all x, h ∈ V is well studied.In explicit form we can have The following theorem was proved by Mazur and Orlicz [1,2] and in greater generality by Djoković (see [3]).
Theorem 1.Let V and W be real vector spaces, n ∈ N and f : V → W, then the following are equivalent: In 2007, L. Cȃdariu and V. Radu [4] proved a stability of the monomial functional Equation (1) (see also [5][6][7]), in particular, the following result is given by the author in [6].
Theorem 2. Let p be a non-negative real number with p = n, let θ > 0, and let f : for all x, y ∈ X.Then there exist a positive real number K and a unique monomial function of degree n F : holds for all x ∈ X.The mapping F : X → Y is given by for all x ∈ X.
The concept of stability for the functional Equation (1) arises when we replace the functional Equation ( 1) by an inequality (2), which is regarded as a perturbation of the equation.Thus, the stability question of functional Equation (1) is whether there is an exact solution of (1) near each solution of inequality (2).If the answer is affirmative with inequality (3), we would say that the Equation ( 1) is stable.
In 1998, A. Gilányi dealt with the stability of monomial functional equation for the case p = 0 (see [18,19]) and he proved for the case when p is a real constant (see [20]).Thereafter, C.-K. Choi proved stability theorems for many kinds of restricted domains, but his theorems are mainly connected with the case of p = 0.If p < 0 in (2), then the inequality (2) cannot hold for all x ∈ X, so we have to restrict the domain by excluding 0 from X.
The main purpose of this paper is to generalize our previous result (Theorem 2) by replacing the real normed space X with a restricted domain S of a real vector space V and by replacing the control function θ(||x|| p + ||y|| p ) with a more general function ϕ : S 2 → [0, ∞).

Stability of the Functional Equation (1) on a Restricted Domain
In this section, for a given mapping f : V → W, we use the following abbreviation and Proof.From the equalities we get the equality for all x ∈ R. Since the coefficient of the term x 2i of the left-hand side in ( 6) is n C i (−1) n−i and the coefficient of the term x 2i of the right-hand side in ( 6) is We easily know that the coefficient of the term x 2i−1 of the left-hand side in ( 6) is 0 and the coefficient of the term x 2i−1 of the right-hand side in (6 So we get the Equality (5).
We rewrite a refinement of the result given in [6].
holds for all x, y ∈ V.In particular, if D n f (x, y) = 0 for all x, y ∈ V, then Proof.By (4), (5), and the equality ∑ n j=0 n C j = 2 n , we get the equalities for all x, y ∈ V.
Lemma 3. If f satisfies the functional equation D n f (x, y) = 0 for all x, y ∈ V\{0} with f (0) = 0, then f satisfies the functional equation D n f (x, y) = 0 for all x, y ∈ V.
We rewrite a refinement of the result given in [7].
Theorem 3. (Corollary 4 in [7]) A mapping f : V → W is a solution of the functional Equation (1) if and only if f is of the form f (x) = A n (x) for all x ∈ V, where A n is the diagonal of the n-additive symmetric mapping A n : V n → W.
Proof.Assume that f satisfies the functional Equation (1).We get the equation x ∈ V, where A 0 (x) = A 0 is an arbitrary element of W and A i (x)(i = 1, 2, . . ., n) is the diagonal of an i-additive symmetric mapping A i : V i → W. On the other hand, f (2x) = 2 n f (x) holds for all x ∈ V by Lemma 2, and so f (x) = A n (x).
Conversely, assume that f (x) = A n (x) for all x ∈ V, where A n (x) is the diagonal of the n-additive symmetric mapping we see that f satisfies (1), which completes the proof of this theorem.
Theorem 4. Let S be a subset of a real vector space V and Y a real Banach space.Suppose that for each x ∈ V\{0} there exists a real number r x > 0 such that rx ∈ S for all r ≥ r x .Let ϕ : for all x, y ∈ S. If the mapping f : S → Y satisfies the inequality for all x, y ∈ S, then there exists a unique monomial mapping of degree n F : for all x ∈ S, where In particular, F is represented by for all x ∈ V.
Proof.Let x ∈ V\{0} and m be an integer such that 2 m ≥ r x .It follows from (7) in Lemma 2 and (10) that From the above inequality, we get the following inequalities and for all m ∈ N.So the sequence { f (2 m x) 2 mn } m∈N is a Cauchy sequence for all x ∈ V\{0}.Since lim m→∞ f (2 m 0) 2 mn = 0 and Y is a real Banach space, we can define a mapping F : V → Y by for all x ∈ V.By putting m = 0 and letting m → ∞ in the inequality (12), we obtain the inequality (11) if x ∈ S.
From the inequality (10), we get for all x, y ∈ V\{0}, where 2 m ≥ r x , r y .Since the right-hand side in the above equality tends to zero as m → ∞, we obtain that F satisfies the inequality (1) for all x, y ∈ V\{0}.By Lemma 3 and F(0) = 0, F satisfies the Equality (1) for all x, y ∈ V. To prove the uniqueness of F, assume that F is another monomial mapping of degree n satisfying the inequality (11) for all x ∈ S. The equality F (x) = F (2 m x) follows from the Equality (8) in Lemma 2 for all x ∈ V\{0} and m ∈ N. Thus we can obtain the inequalities ) for all x ∈ V, i.e., F(x) = F (x) for all x ∈ V.This completes the proof of the theorem.
We can give a generalization of Theorems 2 and 5 in [6] as the following corollary.
Corollary 1.Let p and r be real numbers with p < n and r > 0, let X be a normed space, ε > 0, and f : X → Y be a mapping such that D n f (x, y) ≤ ε( x p + y p ) for all x, y ∈ X with x , y > r.Then there exists a unique monomial mapping of degree n F : x p (for kx > r).
In particular, if p < 0, then f is a monomial mapping of degree n itself.
Proof.If we set ϕ(x, y) := ε( x p + y p ) and S = {x ∈ X| x > r}, then there exists a unique monomial mapping of degree n F : for all x ∈ X with x > r by Theorem 4. Notice that if F is a monomial mapping of degree n, then D n F((k + 1)x, −kx) = 0 for all x ∈ X and k ∈ R. Hence the equality holds for all x ∈ X and k ∈ R.So if F : X → Y is the monomial mapping of degree n satisfying ( 14), then F : X → Y satisfies the inequality with a real number k for all kx > r.
for all x ∈ X\{0} by (15).Since lim k→∞ k p = 0 and the inequality holds for any fixed x ∈ X\{0} with x > r and all natural numbers k, we get f (0) = F(0).
Theorem 5. Let S be a subset of a real vector space V and Y a real Banach space.Suppose that for each x ∈ V there exists a real number r x > 0 such that rx ∈ S for all r ≤ r x .Let ϕ : for all x, y ∈ S. Suppose that a mapping f : V → Y satisfies the inequality (10) for all x, y ∈ S, where ix + y ∈ S for all i = 0, 1, . . ., n.Then there exists a unique monomial mapping of degree n F : for all x with nx ∈ S, where Φ(x) is defined as in Theorem 4. In particular, F is represented by for all x ∈ V.
Proof.Let x ∈ V and m be an integer such that 2 −m (n + 1) ≤ r x .It follows from (7) in Lemma 2 and (10) that for all x ∈ V. From the above inequality, we get the inequality for all x ∈ V and m ∈ N.So the sequence {2 nm f x 2 m } m∈N is a Cauchy sequence by the inequality (16).From the completeness of Y, we can define a mapping F : V → Y by for all x ∈ V.Moreover, by putting m = 0 and letting m → ∞ in (18), we get the inequality (17) for all x ∈ S with (n + 1)x ∈ S. From the inequality (10), if m is a positive integer such that ix+y 2 m ∈ S for all i = 0, 1, . . ., n, then we get for all x, y ∈ V. Since the right-hand side in this inequality tends to zero as m → ∞, we obtain that F is a monomial mapping of degree n.To prove the uniqueness of F assume that F is another monomial mapping of degree n satisfying the inequality (17) for all x ∈ S with (n + 1)x ∈ S.So the equality F (x) = 2 nm F x 2 m holds for all x ∈ V by (8) in Lemma 2. Thus, we can infer that for all positive integers m, where 2 m for all x ∈ V.This completes the proof of the theorem.
We can give a generalization of Theorem 3 in [6] as the following corollary.
From the definition of f , the inequality (22), and the inequality (23), we obtain that Therefore, f satisfies (21) for all x, y ∈ R. Now, we claim that the functional Equation ( 1) is not stable for p = n in Corollarys 1 and 2. Suppose on the contrary that there exists a monomial mapping of degree n F : R → R and constant d > 0 such that | f (x) − F(x)| ≤ d|x| n for all x ∈ R. Notice that F(x) = x n F(1) for all rational numbers x.So we obtain that

Conclusions
The advantage of this paper is that we do not need to prove the stability of additive quadratic, cubic, and quartic functional equations separately.Instead we can apply our main theorem to prove the stability of those functional equations simultaneously.