Hyers–Ulam Stability of Two-Dimensional Flett’s Mean Value Points

: If a differentiable function f : [ a , b ] → R and a point η ∈ [ a , b ] satisfy f ( η ) − f ( a ) = f (cid:48) ( η )( η − a ) , then the point η is called a Flett’s mean value point of f in [ a , b ] . The concept of Flett’s mean value points can be generalized to the 2-dimensional Flett’s mean value points as follows: For the different points ˆ r and ˆ s of R × R , let L be the line segment joining ˆ r and ˆ s . If a partially differentiable function f : R × R → R and an intermediate point ˆ ω ∈ L satisfy f ( ˆ ω ) − f ( ˆ r ) = (cid:104) ˆ ω − ˆ r , f (cid:48) ( ˆ ω ) (cid:105) , then the point ˆ ω is called a 2-dimensional Flett’s mean value point of f in L . In this paper, we will prove the Hyers–Ulam stability of 2-dimensional Flett’s mean value points.


Introduction
In 1940, Ulam [1] raised an interesting question regarding the stability of group homomorphisms: Under what conditions is the approximate solution of an equation necessarily close to the exact solution of the equation?
In the following year, Hyers [2] positively solved the Ulam's question only in the case of additive functional equations under the additional assumption that G 1 and G 2 are all Banach spaces. Indeed, Hyers proved that every solution of inequality f (x + y) − f (x) − f (y) ≤ ε for all x and y, can be approximated by an exact solution (an additive function). In this case, the Cauchy additive functional equation, f (x + y) = f (x) + f (y), is said to satisfy the Hyers-Ulam stability, or it is called stable in the sense of Hyers and Ulam. Mean value theorem is one of the most important theorems in real analysis. According to Lagrange's mean value theorem, for any planar 'regular' curve between two endpoints, there exists at least one point between these endpoints at which the tangent to the curve is parallel to the secant through those endpoints. More precisely, if f : [a, b] → R is a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c ∈ (a, b) such that In 1958, Flett proved another mean value theorem which is a variant of the Lagrange's mean value theorem (see [3]): In particular, the point η in the Flett's mean value theorem is called a Flett's (mean value) point of f in [a, b].
In Section 3 of the present paper, the Hyers-Ulam stability of 2-dimensional Flett's mean value points will be proved (We may refer to Theorem 2 for the exact definition of 2-dimensional Flett's points). Theorem 3 of this paper is a generalization as well as an extension of ([4] Theorem 2.2) and at the same time it is a counterpart of Theorem 2 for the 2-dimensional Flett's points.

Preliminaries and Historical Backgrounds
We can apply the concept of Hyers-Ulam stability to other mathematical objects. Ulam and Hyers [5] appear to be the first ones who apply the Hyers-Ulam stability concept to differential expressions. The Ulam and Hyers' theorem is essential to establish the main result of the present paper. [5]) Assume that f : R → R is n times differentiable in a neighborhood N of a point t 0 , f (n) (t 0 ) = 0, and that f (n) (t) changes sign at t 0 . Then, for any given ε > 0, there is a δ > 0 with the property that if a function g : R → R is n times differentiable in N and g satisfies | f (t) − g(t)| < δ for t ∈ N, then there exists a point t 1 ∈ N satisfying g (n) (t 1 ) = 0 and |t 1 − t 0 | < ε.

Theorem 1. (Ulam and Hyers
One similar question, such as the question of Ulam, may be raised for various kinds of mean value points: If a function f has a mean value point η and g is a function very close to f , will g have a mean value point near η? Indeed, Das, Riedel and Sahoo studied Hyers-Ulam stability of Flett's points (see [6]). However, there was unfortunately some incompleteness in their proof. Thereafter, Lee, Xu and Ye [7] created a counterexample to show that the proof of [6] was incorrect, and succeeded in proving the Hyers-Ulam stability of the Sahoo-Riedel's points.

Main Theorem
According to the generalization of Lagrange's mean value theorem (see ([8] Theorem 4.1)), for every function f : R × R → R with continuous partial derivatives f x and f y and for all distinct pairs (p, q) and (u, v) in R × R, there exists an intermediate point (η, ξ) on the line segment connecting the points (p, q) and (u, v) that satisfies For any points (p, q) and (u, v) of R × R, the Euclidean inner product is denoted by (p, q), (u, v) and is defined as For notational simplicity, we denote ( f x , f y ) by f if f : R × R → R is a function of two variables with partial derivatives f x and f y . Using the above definition and notation, we can rewrite (1) as This can be further simplified to Now we present a generalization of Flett's mean value theorem for real valued functions of two variables (see ([8] Theorem 5.4)).
Theorem 2. (Two-dimensional Flett's mean value theorem) Given the different pointsr andŝ of R × R, let L be the line segment connectingr andŝ. For each function f : R × R → R that has continuous partial derivatives f x , f y and satisfies f (r) = f (ŝ), there exists an intermediate pointω ∈ L that satisfies Such a pointω will be called a 2-dimensional Flett's (mean value) point of f in L.
We define In ([9] Theorem 3.1), we proved the Hyers-Ulam stability of the 2-dimensional Lagrange's points by making use of Theorem 1. In a context similar to the preceding task, using a theorem of [6] and the proof of ([9] Theorem 3.1), we will prove a Hyers-Ulam type stability for the 2-dimensional Flett's points.

Theorem 3.
Given the different pointsr andŝ of R × R, let L be the line segment connectingr andŝ. Assume that f belongs to Γ andω 0 is the unique 2-dimensional Flett's point of f in L. For any given ε > 0, there is a δ > 0 with the property that if g belongs to Γ and g satisfies the inequality | f (x) − g(x) − f (r) + g(r)| < δ for allx = (x, y) ∈ L, then g has a 2-dimensional Flett's pointω 3 in L satisfying |ω 0 −ω 3 | < ε.
Proof. Letr = (p, q) andŝ = (u, v) be two different points of R × R. By putting h = u − p and k = v − q, the coordinates of every point on L are expressed byω = (p + ht, q + kt) for some t ∈ [0, 1]. An auxiliary function F : R → R will be defined by F(t) = f (ω) = f (p + ht, q + kt), and we calculate the derivative of this function as Since f belongs to Γ, we obtain We will now define another auxiliary function F : R → R by It then follows from (5) that for all t ∈ (0, 1]. It is clear that F is continuous on [0, 1] and continuously differentiable on (0, 1]. We now assert that there is a t 0 ∈ (0, 1) satisfying From (5), we know that F (0) = F (0). If F (1) = F (0), then by Rolle's theorem, there exists a t 0 ∈ (0, 1) such that F (t 0 ) = 0 and our assertion is established.
With reference to Table 1, Theorem 1 can be translated into Statement (8).
Indeed, referring to Table 1, Theorem 1 can be translated into: For any givenε > 0, there exists aδ > 0 with the property that if a function G : R → R is differentiable in N and satisfies |F (t) − G(t)| <δ for each t ∈ N, then there is a t 3 ∈ N satisfying G (t 3 ) = 0 and |t 3 − t 0 | <ε.