Abstract
If a differentiable function f : [ a , b ] → R and a point η ∈ [ a , b ] satisfy f ( η ) − f ( a ) = f ′ ( 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 r ^ and s ^ of R × R , let L be the line segment joining r ^ and s ^ . If a partially differentiable function f : R × R → R and an intermediate point ω ^ ∈ L satisfy f ( ω ^ ) − f ( r ^ ) = ω ^ − r ^ , f ′ ( ω ^ ) , then the point 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.
Highlights
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?
Hyers proved that every solution of inequality k f ( x + y) − f ( x ) − f (y)k ≤ ε for all x and y, can be approximated by an exact solution
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
Summary
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?. Hyers proved that every solution of inequality k f ( x + y) − f ( x ) − f (y)k ≤ 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. 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
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