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Second order linear $q$ -difference equations: nonoscillation and asymptotics

Pavel Řehák (2011)

Czechoslovak Mathematical Journal

The paper can be understood as a completion of the $q$ -Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$ -difference equations. The $q$ -Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{ℕ_{0}} : = {q^{k} : k \in ℕ_{0}}$ with $q > 1$ . In addition to recalling the existing concepts of $q$ -regular variation and $q$ -rapid variation we introduce $q$ -regularly bounded functions and prove many related properties. The $q$ -Karamata theory is then...

Séries de $q$ -factorielles, opérateurs aux $q$ -différences et confluence

Anne Duval (2003)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Singular boundary value problem on infinite time scale.

Hao, Zhao-Cai, Liang, Jin, Xiao, Ti-Jun (2006)

Discrete Dynamics in Nature and Society

Some properties of solutions of complex q-shift difference equations

Hong-Yan Xu, Jin Tu, Xiu-Min Zheng (2013)

Annales Polonici Mathematici

Combining difference and q-difference equations, we study the properties of meromorphic solutions of q-shift difference equations from the point of view of value distribution. We obtain lower bounds for the Nevanlinna lower order for meromorphic solutions of such equations. Our results improve and extend previous theorems by Zheng and Chen and by Liu and Qi. Some examples are also given to illustrate our results.

Spectral theory from the second-order $q$ -difference operator.

Dhaouadi, Lazhar (2007)

International Journal of Mathematics and Mathematical Sciences

Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

Bilender P. Allahverdiev, Hüseyin Tuna (2020)

Communications in Mathematics

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Symmetries in connection preserving deformations.

Ormerod, Christopher M. (2011)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Systèmes aux $q$ -différences singuliers réguliers : classification, matrice de connexion et monodromie

Jacques Sauloy (2000)

Annales de l'institut Fourier

G.D. Birkhoff a posé, par analogie avec le cas classique des équations différentielles, le problème de Riemann-Hilbert pour les systèmes “fuchsiens” aux $q$ -différences linéaires, à coefficients rationnels. Il l’a résolu dans le cas générique: l’objet classifiant qu’il introduit est constitué de la matrice de connexion $P$ et des exposants en $0$ et $\infty$ . Nous reprenons sa méthode dans le cas général, mais en traitant symétriquement $0$ et $\infty$ et sans recours à des solutions à croissance “sauvage”. Lorsque $q$ ...

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