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Set-Valued Quadratic Functional Equations
Results in Mathematics, 2017zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Lee, Jung Rye +3 more
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The quadratic function and quadratic equations
1985The function f(x), where f(x) = ax2 + bx + c, and a, b, c are constants, a ≠ 0, is called a quadratic function, or sometimes a quadratic polynomial. From elementary algebra $${(x + d)^2} \equiv {x^2} + 2dx + {d^2}.$$ Using this, we write $$a{x^2} + bx + c \equiv a\left( {{x^2} + \frac{b}{a}x + \frac{c}{a}} \right) \equiv a\left[ {{{\left( {x
J. E. Hebborn, C. Plumpton
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On the connection of the quadratic Lienard equation with an equation for the elliptic functions
Regular and Chaotic Dynamics, 2015Consider the differential equation \[ {d^2y\over dx^2}+ g(y)\Biggl({dy\over dx}\Biggr)^2+ h(y)= 0.\tag{\(*\)} \] The authors prove that \((*)\) can be transformed into the differential equation \[ w{d^2w\over d\xi^2}-{1\over 2}\Biggl({dw\over d\xi}\Biggr)^2+ 4\omega^3=0\tag{\(**\)} \] by means of the nonlocal transformation \[ w=F(y),\quad d\xi= G(y ...
Kudryashov, Nikolay A. +1 more
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On the quadratic functional equation on groups
Publicationes Mathematicae Debrecen, 2006Given two groups \(G,H\), the functional equation \[ f(xy)+f(xy^{-1})=2f(x)+2f(y),\qquad x,y\in G \tag{(1)} \] is called the quadratic functional equation, where \(f:G\to H\) is considered as an unknown function. Assuming that \(G\) and \(H\) are abelian and \(H\) is uniquely 2-divisible, \textit{J. Aczél} [Period. Math.-Phys. Astron., II. Ser. 20, 65--
Friis, P.d.P., Stetkær, H.
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Quadratic Operators and Quadratic Functional Equation
2012In the first part of this paper, we consider some quadratic difference operators (e.g., Lobaczewski difference operators) and quadratic-linear difference operators (d’Alembert difference operators and quadratic difference operators) in some special function spaces X λ . We present results about boundedness and find the norms of such operators.
M. Adam, S. Czerwik
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Alienation of the Quadratic and Additive Functional Equations
Analysis Mathematica, 2019Let \(G\) and \(H\) be uniquely \(2\)-divisible abelian groups. The Pexider-type functional equation \(f(x+y) + f(x-y) + g(x+y) = 2f(x) + 2f(y) + g(x) + g(y)\) is constructed by summing up the quadratic functional equation and additive Cauchy functional equation side by side. Here \(f, g : G \to H\).
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Conditional equations for quadratic functions
Acta Mathematica Hungarica, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Boros, Z., Garda-Mátyás, E.
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Quadratic Functional Equations
2009Quadratic functional equations, bilinear forms equivalent to the quadratic equation, and some generalizations are treated in this chapter. Among the normed linear spaces (n.l.s.), inner product spaces (i.p.s.) play an important role. The interesting question when an n.l.s. is an i.p.s. led to several characterizations of i.p.s.
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The quadratic functional equation on groups
Publicationes Mathematicae Debrecen, 2005The quadratic functional equation \[ f(xy)+f(xy^{-1})=2f(x)+2f(y) \] is considered on free groups. The author presents the result on a general solution of the above equation defined on a free group with values in an abelian group. In the proof some results concerning the Jensen functional equation are utilized.
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Quadratic Functional Equations
2011So far, we have discussed the stability problems of functional equations in connection with additive or linear functions. In this chapter, the Hyers–Ulam–Rassias stability of quadratic functional equations will be proved. Most mathematicians may be interested in the study of the quadratic functional equation since the quadratic functions are applied to
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