A Fixed Point Approach to the Stability of a Cauchy‐Jensen Functional Equation

Bae, Jae-Hyeong; Park, Won-Gil

doi:10.1155/2012/205160

Cited by 13 publications

(9 citation statements)

References 10 publications

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“…that is, the equations which characterize biadditive, bi-Jensen and Cauchy-Jensen mappings, respectively. Therefore, our results generalize stability outcomes for these equations (see, e.g., [2][3][4][5]9,14,15]).…”

Section: Introductionsupporting

confidence: 80%

On Stability of a General Bilinear Functional Equation

Bahyrycz

Sikorska

2021

Results Math

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We prove the Hyers–Ulam stability of the functional equation $$\begin{aligned}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\nonumber \\ \nonumber \\&\quad +C_{2}f(x_1,y_2)+C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \end{aligned}$$ f ( a 1 x 1 + a 2 x 2 , b 1 y 1 + b 2 y 2 ) = C 1 f ( x 1 , y 1 ) + C 2 f ( x 1 , y 2 ) + C 3 f ( x 2 , y 1 ) + C 4 f ( x 2 , y 2 ) in the class of functions from a real or complex linear space into a Banach space over the same field. We also study, using the fixed point method, the generalized stability of $$(*)$$ ( ∗ ) in the same class of functions. Our results generalize some known outcomes.

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Section: Introductionsupporting

confidence: 80%

On Stability of a General Bilinear Functional Equation

Bahyrycz

Sikorska

2021

Results Math

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“…Consider the probabilistic Equation ( 7) associated with (8). Assume that ( 14) holds and η 2 < 1, where η 2 is defined in (18). Suppose that there is a nonempty subset C of S := {L ∈ T |L( ) ≤ } such that (C, • ) is a BS, where • is given in (6), and the mapping M from C to C defined for each L ∈ T by ( 19) is a self mapping.…”

Section: Resultsmentioning

confidence: 99%

“…Hence, it is essential to analyze the stability of a solution to the suggested functional (7). For the details of the HU and HUR stability, we refer to [15][16][17][18][19]. Theorem 6.…”

Section: Stability Analysis Of the Suggested Functional Equationmentioning

confidence: 99%

Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning

2021

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Probabilistic functional equations have been used to analyze various models in computational biology and learning theory. It is worth noting that they are linked to the symmetry of a system of functional equations’ transformation. Our objective is to propose a generic probabilistic functional equation that can cover most of the mathematical models addressed in the existing literature. The notable fixed-point tools are utilized to examine the existence, uniqueness, and stability of the suggested equation’s solution. Two examples are also given to emphasize the significance of our findings.

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“…To finish this introductory section let us finally mention that some results on the stability of Cauchy-Jensen mappings can be found in [6,7,[24][25][26][27]31,32].…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, a function g 0 : G k → W given by the formula g 0 (x) = f (x, 0) is multi-Jensen, since according to (7) and (8)…”

Section: Introductionmentioning

confidence: 99%

On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings

Bahyrycz

Olko

2018

Results Math

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Recently, functions of several variables satisfying, with respect to each variable, some functional equation (among them Cauchy's, Jensen's, quadratic and other ones) have been studied. We give a new characterization of multi-Cauchy-Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy-Jensen. Next, using a fixed point theorem, it is proved that a function which approximately satisfies (on restricted domain) the equation characterizing such functions is close (in some sense) to the solution of the equation. This result is a tool for obtaining a generalized Hyers-Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples.Mathematics Subject Classification. 39B52, 39B82, 39B72, 47H10.

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A Fixed Point Approach to the Stability of a Cauchy‐Jensen Functional Equation

Cited by 13 publications

References 10 publications

On Stability of a General Bilinear Functional Equation

On Stability of a General Bilinear Functional Equation

Existence, Uniqueness, and Stability Analysis of the Probabilistic Functional Equation Emerging in Mathematical Biology and the Theory of Learning

On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings

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