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Start Over Subject functional equation

1. Asymptotic behaviour of solutions of some linear delay differential equations.

Creator:
Čermák, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
asymptotic behaviour, differential equation, delayed argument, and functional equation
Language:
English
Description:
In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y'(x)=a(x)y(\tau(x))+b(x)y(x),\qquad x\in I=[x_0,\infty). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z'(x)=b(x)z(x),\qquad x\in I and a solution of the functional equation |a(x)|\varphi(\tau(x))=|b(x)|\varphi(x),\qquad x\in I.
Rights:
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2. Equation $f(p(x)) = q(f(x))$ for given real functions $p$, $q$

Creator:
Kopeček, Oldřich
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
homomorphism of mono-unary algebras, functional equation, and strictly increasing continuous real functions
Language:
English
Description:
We investigate functional equations $f(p(x)) = q(f(x))$ where $p$ and $q$ are given real functions defined on the set ${\Bbb R}$ of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions $p, q$ which are strictly increasing and continuous on ${\Bbb R}$. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction of any solution of this equation if some exists. This construction is demonstrated in detail and discussed by means of an example.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Equivariant mappings from vector product into G-space of vectors and ε-vectors with G = O(n, 1, ℝ)

Creator:
Glanc, Barbara, Misiak, Aleksander, and Stepień, Zofia
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
G-space, equivariant map, pseudo-Euclidean geometry, and functional equation
Language:
English
Description:
In this note all vectors and ε-vectors of a system of m ≤ n linearly independent contravariant vectors in the n-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation F(Au 1 , Au 2 , . . . , Au m ) = (det A) λ · A · F(u 1 , u 2 , . . . , u m ) with λ = 0 and λ = 1, for an arbitrary pseudo-orthogonal matrix A of index one and given vectors u 1 , u 2 , . . . , u m .
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. On the mean value of the generalized Dirichlet $L$-functions

Creator:
Ma, Rong, Yi, Yuan, and Zhang, Yulong
Type:
model:article and TEXT
Subject:
generalized Dirichlet $L$-functions, mean value properties, functional equation, and asymptotic formula
Language:
English
Description:
Let $q\ge 3$ be an integer, let $\chi $ denote a Dirichlet character modulo $q.$ For any real number $a\ge 0$ we define the generalized Dirichlet $L$-functions $$ L(s,\chi ,a)=\sum _{n=1}^{\infty }\frac {\chi (n)}{(n+a)^s}, $$ where $s=\sigma +{\rm i} t$ with $\sigma >1$ and $t$ both real. They can be extended to all $s$ by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet $L$-functions especially for $s=1$ and $s=\frac 12+{\rm i} t$, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

5. The impact of unbounded swings of the forcing term on the asymptotic behavior of functional equations

Creator:
Singh, Bhagat
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
oscillatory, nonoscillatory, exterior domain, elliptic, and functional equation
Language:
English
Description:
Necessary and sufficient conditions have been found to force all solutions of the equation \[ (r(t)y^{\prime }(t))^{(n-1)} + a(t)h(y(g(t))) = f(t), \] to behave in peculiar ways. These results are then extended to the elliptic equation \[ |x|^{p-1} \Delta y(|x|) + a(|x|)h(y(g(|x|))) = f(|x|) \] where $ \Delta $ is the Laplace operator and $p \ge 3$ is an integer.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

6. Why λ-additive (fuzzy) measures?

Creator:
Chiţescu, Ion
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
generalized measure (probabiity), λ-additive measure, and functional equation
Language:
English
Description:
The paper is concerned with generalized (i. e. monotone and possibly non-additive) measures. A discussion concerning the classification of these measures, according to the type and amount of non-additivity, is done. It is proved that λ-additive measures appear naturally as solutions of functional equations generated by the idea of (possible) non additivity.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public