The stability of d’Alembert’s functional equation

On the stability of a class of cosine type functional equations

“…Submitted January 21, 2015. Published March 13, 2017 The cosine functional equation (C) is generalized to the following functional equations f (x + y) + f (x − y) = 2f (x)g(y), x, y ∈ G, f (x + y) + f (x − y) = 2g(x)f (y), x, y ∈ G, f (x + y) + f (x − y) = 2g(x)g(y), x, y ∈ G, and their stabilities are explored by Kannappan and Kim ( [9], [12], [13]) and Tyrala [23]. The superstability problem for the pexiderized cosine type functional equation f 1 (x + y) + f 2 (x − y) = 2g 1 (x)g 2 (y), x, y ∈ G, on the abelian group (G, +) is investigated by Kim [13] and Kusollerschariya and Nakmahachalasint [14].…”

Stability of Trigonometric Functional Equations in Generalized Functions

“…This problem was solved affirmatively by Hyers 2 under the assumption that G 2 is a Banach space. In 1949-1950, this result was generalized by the authors Bourgin 3, 4 and Aoki 5 and since then stability problems of many other functional equations have been investigated 2,[6][7][8][8][9][10][11][12][13][14][15][16][17][18][19] . In 1990, Székelyhidi 18 has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations.…”

On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

“…Unfortunately, there was no use of these results until 1978 when Rassias [7] treated with the inequality of Aoki [3]. Following Rassias' result, a great number of papers on the subject have been published concerning numerous functional equations in various directions [6][7][8][9][10][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In 1990 Székelyhidi [24] has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations.…”

“…In 1990 Székelyhidi [24] has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations. We refer the reader to [9,10,18,19,25] for Hyers-Ulam stability of functional equations of trigonometric type. In this paper, following the method of Székelyhidi [24] we consider a distributional analogue of the Hyers-…”