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Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Zagorodniuk, S. (2001)

Serdica Mathematical Journal

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2...

Analytic solutions of a nonlinear two variables difference system whose eigenvalues are both 1

Mami Suzuki (2011)

Annales Polonici Mathematici

For nonlinear difference equations, it is difficult to obtain analytic solutions, especially when all the eigenvalues of the equation are of absolute value 1. We consider a second order nonlinear difference equation which can be transformed into the following simultaneous system of nonlinear difference equations: ⎧ x(t+1) = X(x(t),y(t)) ⎨ ⎩ y(t+1) = Y(x(t), y(t)) where X ( x , y ) = λ ₁ x + μ y + ∑ i + j ≥ 2 c i j x i y j , Y ( x , y ) = λ ₂ y + ∑ i + j ≥ 2 d i j x i y j satisfy some conditions. For these equations, we have obtained analytic solutions in the cases "|λ₁| ≠ 1 or |λ₂| ≠ 1" or "μ...

Approximation of limit cycle of differential systems with variable coefficients

Masakazu Onitsuka (2023)

Archivum Mathematicum

The behavior of the approximate solutions of two-dimensional nonlinear differential systems with variable coefficients is considered. Using a property of the approximate solution, so called conditional Ulam stability of a generalized logistic equation, the behavior of the approximate solution of the system is investigated. The obtained result explicitly presents the error between the limit cycle and its approximation. Some examples are presented with numerical simulations.

Arithmetical aspects of certain functional equations

Lutz G. Lucht (1997)

Acta Arithmetica

The classical system of functional equations       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = n - s F ( x ) (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = ∑ d = 1 ∞ λ n ( d ) F ( d x ) (n ∈ ℕ) with complex valued sequences λ n . This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.

Currently displaying 101 – 120 of 1085