Geodesic
On the Riemann--Hilbert problem for difference and $q$-difference systems
Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 326-343.

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We study an analog of the classical Riemann–Hilbert problem stated for the classes of difference and $q$-difference systems. We generalize Birkhoff's existence theorem. 
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     author = {I. V. Vyugin and R. I. Levin},
     title = {On the {Riemann--Hilbert} problem for difference and $q$-difference systems},
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I. V. Vyugin; R. I. Levin. On the Riemann--Hilbert problem for difference and $q$-difference systems. Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 326-343. http://geodesic.mathdoc.fr/item/TRSPY_2017_297_a17/

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[2] Birkhoff G. D., “General theory of linear difference equations”, Trans. Amer. Math. Soc., 12:2 (1911), 243–284 | DOI | MR | Zbl

[3] Birkhoff G. D., “The generalized Riemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations”, Proc. Amer. Acad. Arts Sci., 49:9 (1913), 521–568 | DOI | Zbl