Stability has many names

Abstract

Abstract. A list of 22 definitions of stability of a functional equation and 5 definitions of stability of the alternation of two functional equations is given as well as some simple examples.Mathematics Subject Classification. 39B82, 39B62.

Stability of the Wave Equation with a Source

“…In general, it is somewhat tedious to estimate the upper bound of inequality (8). However, in view of (10) and (18), we can compute the upper bound less tediously:…”

“…When the above statement is true even if we replace and ( ) by ( ) and Φ( ), where , Φ : → [0, ∞) are functions not depending on and 0 explicitly, the corresponding differential equation (1) These terminologies will also be applied for other differential equations and partial differential equations. For more detailed definitions, we refer the reader to [1][2][3][4][5][6][7][8][9].…”

Fixed-point theorem in classes of function with values in a dq-metric space

“…The name of Ulam has been somehow connected with various definitions of stability (see, e.g., [1,12,16,24]), but roughly speaking, the following one describes our considerations in this paper (A B denotes the family of all functions mapping a nonempty set B into a nonempty set A, R stands for the set of all real numbers and R + :=[0, ∞)). Definition 1.…”

On fixed points of a linear operator of polynomial form of order 3

“…Later, that notion has been generalized in several various directions, which by now are very often collectively called the Ulam type stability. Numerous papers on this subject have been published so far and we refer to [1][2][3]6,13,[18][19][20]25] for more details, some discussions, recent results and further references. Let us also mention here that the problem of stability of functional equations is connected to the notions of shadowing (see [16,26,27]) and the theory of perturbation (see [11,23]).…”

“…Various definitions of that type of stability are possible for particular equations (see, e.g., [1,18,25]), but roughly speaking, the following one describes our considerations to some extent: given a metric space ( …”