On the stability of generalized d'Alembert and Jensen functional equations

In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and GÂ·avruta for the quadratic functional equation:f(2x + y) + f(2x Â¡ y) = 2f(x + y) + 2f(x Â¡ y) + 4f(x) Â¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by thequadratic functional equation in normed spaces to 2-Banach spaces.

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Non-Archimedean hyperstability of Cauchy–Jensen functional equations on a restricted domain

Journal of Applied Analysis ◽

10.1515/jaa-2018-0015 ◽

2018 ◽

Vol 24 (2) ◽

pp. 155-165

Author(s):

Iz-iddine EL-Fassi

Keyword(s):

Fixed Point ◽

Functional Equations ◽

Metric Spaces ◽

Fixed Point Approach ◽

Nonlinear Anal ◽

Stability Of Functional Equations ◽

Jensen Functional ◽

Empty Subset ◽

The Stability ◽

Fixed Positive Integer

Abstract Let X be a normed space, {U\subset X\setminus\{0\}} a non-empty subset, and {(G,+)} a commutative group equipped with a complete ultrametric d that is invariant (i.e., {d(x+z,y+z)=d(x,y} ) for {x,y,z\in G} ). Under some weak natural assumptions on U and on the function {\gamma\colon U^{3}\to[0,\infty)} , we study the new generalized hyperstability results when {f\colon U\to G} satisfies the inequality d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)% \biggr{)}\leq\gamma(x,y,z) for all {x,y,z\in U} , where {\frac{x+y}{\alpha}+z\in U} and {\alpha\geq 2} is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.

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Stability of Bi-Additive Mappings and Bi-Jensen Mappings

Symmetry ◽

10.3390/sym13071180 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1180

Author(s):

Jae-Hyeong Bae ◽

Won-Gil Park

Keyword(s):

Functional Equation ◽

Stability Problem ◽

Additive Functional ◽

Cauchy Equation ◽

Self Similarity ◽

Additive Functional Equation ◽

Jensen Functional Equation ◽

Jensen Functional ◽

The Stability ◽

Additive Mappings

Symmetry is repetitive self-similarity. We proved the stability problem by replicating the well-known Cauchy equation and the well-known Jensen equation into two variables. In this paper, we proved the Hyers-Ulam stability of the bi-additive functional equation f(x+y,z+w)=f(x,z)+f(y,w) and the bi-Jensen functional equation 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w).

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On the stability of a class of cosine type functional equations

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v37i2.29563 ◽

2017 ◽

Vol 37 (2) ◽

pp. 35-49 ◽

Cited By ~ 1

Author(s):

John Michael Rassias ◽

Driss Zeglami ◽

Ahmed Charifi

Keyword(s):

Functional Equation ◽

Stability Problem ◽

Functional Equations ◽

Arbitrary Group ◽

The Stability ◽

Complex Valued

The aim of this paper is to investigate the stability problem for the pexiderized trigonometric functional equation f₁(xy)+f₂(xσ(y))=2g₁(x)g₂(y), x,y∈G, where G is an arbitrary group, f₁,f₂,g₁ and g₂ are complex valued functions on G and σ is an involution of G. Results of this paper also can be extended to the setting of monoids (that is, a semigroup with identity) that need not be abelian.

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The new investigation of the stability of mixed type additive-quartic functional equations in non-Archimedean spaces

Demonstratio Mathematica ◽

10.1515/dema-2020-0009 ◽

2020 ◽

Vol 53 (1) ◽

pp. 174-192

Author(s):

Anurak Thanyacharoen ◽

Wutiphol Sintunavarat

Keyword(s):

Functional Equation ◽

Normed Space ◽

Functional Equations ◽

Mixed Type ◽

Additive Group ◽

Ulam Stability ◽

Quartic Functional Equation ◽

The Stability

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

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The two-dimensional hydrodynamical theory of moving aerofoils. IV

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1947.0019 ◽

1947 ◽

Vol 188 (1015) ◽

pp. 439-463

Keyword(s):

Stability Problem ◽

Two Dimensional ◽

Centre Of Gravity ◽

First Approximation ◽

Von Karman ◽

Moving Cylinder ◽

Period Equation ◽

The Stability ◽

Von Kármán ◽

Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

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Some Structural Properties of Systolic Tree Automata

Fundamenta Informaticae ◽

10.3233/fi-1989-12409 ◽

1989 ◽

Vol 12 (4) ◽

pp. 571-585

Author(s):

E. Fachini ◽

A. Maggiolo Schettini ◽

G. Resta ◽

D. Sangiorgi

Keyword(s):

Structural Properties ◽

Stability Problem ◽

Sufficient Condition ◽

Tree Automata ◽

Necessary And Sufficient Condition ◽

The Stability ◽

Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

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Some remarks on the stability of the Cauchy equation and completeness

Aequationes Mathematicae ◽

10.1007/s00010-021-00804-y ◽

2021 ◽

Author(s):

Harald Fripertinger ◽

Jens Schwaiger

Keyword(s):

Normed Space ◽

Additive Function ◽

Functional Equations ◽

Constant Function ◽

Cauchy Equation ◽

Cauchy Difference ◽

International Conference ◽

The Difference ◽

The Stability ◽

The Given

AbstractIt was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

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