On the Stability of the Linear Mapping in Banach Spaces

Rassias, Themistocles M.

doi:10.2307/2042795

Cited by 392 publications

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“…Ulam in a talk at a conference at the Wisconsin University in 1940 and it represents the starting point of the Hyers-Ulam stability theory of functional equations (see [8,9]). The subject was later strongly developed by many authors, see for example: [1,4,11,12,16]. An interesting connection between the stability of the Cauchy equation and subadditive set-valued functions was established by Smajdor [17] and Gajda and Ger [6].…”

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confidence: 99%

The properties of functional inclusions and Hyers–Ulam stability

Piszczek

2012

Aequat. Math.

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confidence: 99%

The properties of functional inclusions and Hyers–Ulam stability

Piszczek

2012

Aequat. Math.

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“…Be that as it may, later, Rassias [81] considered mappings with unbounded Cauchy differences and proved the next Theorem 1.4 for 0 ≤ p < 1 thus generalizing Hyers' result. Rassias [81] observed that the same proof also worked for p < 0 and asked about the possibility of extending the result for p ≥ 1; the affirmative answer for p > 1 was provided by Gajda [37]. Thus one has the following starting result.…”

Section: A(x + Y) − A(x) − A(y)mentioning

confidence: 54%

“…For more general results in this direction the reader can consult Rassias [81,84], Gajda [37], Isac and Rassias [53], Rassias andŠemrl [86], Gȃvruţa [40], Jung [59] and the surveys of Hyers and Rassias [52] and Forti [36]. It should be noted that most of these results are particular cases of a general result of Forti [33] about stability of functional equations of the form g(F (x, y)) = G(f (x), f (y)).…”

Section: A(x + Y) − A(x) − A(y)mentioning

confidence: 99%

Banach space techniques underpinning a theory for nearly additive mappings

Sánchez

Castillo

2002

Dissertationes Math.

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Ulam's stability problem for additive maps and the theory of extensions of Banach spaces have remained so far unaware of each other. To support this assertion it is enough to peruse the survey articles [36,52] In fact, both worlds constitute two faces of the same coin. In this spirit, the purpose of this paper is to progress towards a global understanding of the problem, and we shall do so by developing a theory of vector-valued nearly additive mappings on semigroups. While a unified theory seems to be out of reach by now, the part of the theory involving quasi-normed groups and maps with values in Banach spaces seems to be ripe with these pages. We hasten to alert the reader that this paper can by no means be considered a survey, although it contains occasional historical comments; two excellent survey papers are [36,52].

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“…In this regard Hyers [8] was the first mathematician who answered the Ulam's question for the additive mapping in complete normed spaces. Latter on, from 1982 to 1998, Rassias [19] developed the conditions under which linear and nonlinear mappings are Hyers-Ulam stable. Jung [11], established Hyers-Ulam stability for nonlinear mapping on a restricted domains.…”

Section: Introductionmentioning

confidence: 99%

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

Ali¹,

Shah²,

Khan³

2017

J. Nonlinear Sci. Appl.

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This paper is concerned with developing some conditions that reveal existing and stability analysis for solutions to a class of differential equations with fractional order. The required conditions are obtained by applying the technique of degree theory of topological type. The concerned problem is converted to the integral equation and then to operator equation, where the operator is defined by T : C[0, 1] → C[0, 1]. It should be noted that the assumptions on nonlinear function f(t, u(t)) does not usually ascertain that the operator T being compact. Moreover, in this paper we also establish some conditions under which the solution of the considered class is Hyers-Ulam stable and also satisfies the conditions of Hyers-Ulam-Rassias and generalized Hyers-Ulam stability. Proper example is provided for the illustration of main results.

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On the Stability of the Linear Mapping in Banach Spaces

Cited by 392 publications

References 0 publications

The properties of functional inclusions and Hyers–Ulam stability

The properties of functional inclusions and Hyers–Ulam stability

Banach space techniques underpinning a theory for nearly additive mappings

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

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