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In this article, we prove the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: f(f(x) – f(y)) = f(x + y) + f(x – y) – f(x) – f(y), where f maps from a(β, p)-Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers-Ulam-Rassias stability for the composite s-functional inequality is discussed via our results.
Content available remote Stability of an additive-quadratic-quartic functional equation
In this paper, we investigate the stability of an additive-quadratic-quartic functional equation f(x+2y)+f(x-2y) - 2f(x+y) - 2f(-x-y) - 2f(x-y) - 2f(y-x)+4f(-x)+2f(x) - f(2y) - f(-2y)+4f(y)+4f(-y)=0 by the direct method in the sense of Găvruta.
We have proved the Hyers-Ulam stability and the hyperstability of the quadratic functional equation f(x+y+z) +f(x+y−z) +f(x−y+z) +f(−x+y+z) = 4[f(x) +f(y) +f(z) ] in the class of functions from an abelian group G into a Banach space.
In this paper, the author established the general solution and generalized Ulam–Hyers–Rassias stability ofn-dimensional additive functional equation (…) in generalized 2-normed space.
This work deals with the Feigenbaum's functional equation in the broad sense (…), where φ2 is the 2-fold iteration of φ, f(x) is a strictly increasing continuous function on [0, 1] and satisfies (...). Using constructive method, we discuss the existence of single-valley-extended continuous solutions of the above equation.
The continuous solutions of a composite functional equation are characterized. An applications to the problem of invariant curves is presented.
The paper contains connections between oscillation of solutions of iterative functional equations, difference equations and differential equations with advanced or delayed arguments. New oscillatory criteria for these equations are given.
Content available remote Exponent in one of the variables
A periodicity functional equation of one complex variable which characterizes the exponential function is discussed. This functional equation can be generalized to equation for functions depending on two complex variables. It is conjectured that the second functional equation also characterizes the exponent. Applications to representations of complex continuous elementary functions are discussed.
Content available remote Homogeneous non-symmetric means of to variables
Let f, g : I approaches R be given continuous functions on the interval I such that g is not equal to 0, and h := f/g is strictly monotonic (thus invertible) on I. Taking an increasing nonconstant functiong g ž on [0, 1].
Content available remote Solutions with big graph of the equation of invariant curves
We consider the functional equation of invariant curves [phi(f(x, phi(x))) = g(x, phi(x))] and we look for its solution which has a big graph. Such a graph is big from the point of view of topology and measure theory.
Content available remote Positivity of Schilling functions
We prove that every non-trivial [L^1]-solution of the Schilling problem is either positive or negative almost everywhere on its support.
Content available remote On iterative roots of homeomorphisms of the circle
In this paper we deal with the problem of existence and uniqueness of continuous iterative roots of homeomorphisms of the circle. Let F : [S^1 --> S^1] be a homeomorphism without periodic points. If the limit set of the orbit [F^k(z), k belongs to Z] equals [S^1], then F has exactly n iterative roots of n-th order. Otherwise F either has no iterative roots of n-th order or F has infinitely many iterative roots depending on an arbitrary function. In this case we determined all iterative roots of n-th order of F.
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