On approximate solutions of the generalized Volterra integral equation

Bahyrycz, Anna; Brzdęk, Janusz; Leśniak, Zbigniew

doi:10.1016/j.nonrwa.2014.04.004

Cited by 16 publications

(14 citation statements)

References 7 publications

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“…Such a method has been used in, e.g., [4][5][6]10,26,30,33,34]. Moreover, the results that we provide correspond to the outcomes in [3,9,13,16,18,21,24,25,31,32] (for more details see, e.g., [8,19,22]) and complement [4,Corollary 6]. …”

Section: This Equation Is a Generalization Of The Fréchet Functional supporting

confidence: 61%

On the generalized Fréchet functional equation with constant coefficients and its stability

2018

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Abstract. We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to show the existence of an exact solution of the equation that is close to a given approximate solution.Mathematics Subject Classification. 39B52, 39B82, 47H10.

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Section: This Equation Is a Generalization Of The Fréchet Functional supporting

confidence: 61%

On the generalized Fréchet functional equation with constant coefficients and its stability

2018

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“…(1) in Banach spaces) to Ulam's question was given by Hyers (see [24]). After his result a great number of papers (see for instance [7,8,[11][12][13][14]21,26,27,29,30] and the references given there) on the subject has been published, generalizing Ulam's problem and Hyers's theorem in various directions and to other (not only functional, see for example [4,37]) equations.…”

Section: Introductionmentioning

confidence: 99%

On an equation characterizing multi-additive-quadratic mappings and its Hyers–Ulam stability

Bahyrycz

Ciepliński

Olko

2015

Applied Mathematics and Computation

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“…[1, 3, 14, 44, 46, 49-65, 69-80, 82-85, 87-90, 95-99, 101-107, 109, 116-118, 120, 122, 124, 125, 129-134, 136, 137, 140-149, 151-153, 156-158, 160-162, 164, 173-179, 187, 189, 190, 195-201, 207, 208, 212, 214, 217, 219, 221-223, 226, 228, 230, 236, 239-241, 262, 266, 268, 269, 275-288, 296-303, 309, 311-322, 324, 330, 331] and also very recent survey papers [42,60,61]) have extensively studied stability theorems for several kinds of functional equations in various spaces, for example, Banach spaces, 2-Banach spaces, Banach n-Lie algebras, quasi-Banach spaces, Banach ternary algebras, non-Archimedean normed and Banach spaces, metric and ultra metric spaces, Menger probabilistic normed spaces, probabilistic normed space, p-2-normed spaces, C -algebras, Cternary algebras, Banach ternary algebras, Banach modules, inner product spaces, Heisenberg groups and others. Further, we have to pay attention to applications of the Hyers-Ulam-Rassias stability problems, for example, (partial) differential equations, Fréchet functional equations, Riccati differential equations, Volterra integral equations, group and ring theory and some kinds of equations (see [66, 142, 150, 154, 159, 176, 185, 186, 192-194, 259-261, 327, 329]).…”

Section: Remarkmentioning

confidence: 98%