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The article is devoted to investigation of applications of infinite systemsof functional equations for modeling of functions with complicated localstructure, that are defined in terms of the nega-$\tilde Q$-representation. Theinfinite system of functional equations $$ f\left(\hat\varphi^k(x)\right)=\tilde \beta_{i_{k+1},k+1}+\tildep_{i_{k+1},k+1}f\left(\hat \varphi^{k+1}(x)\right), $$ where $k=0,1,...$, $\hat\varphi$ is a shift operator of the $\tilde Q$-expansion, $x=\Delta^{-\tilde Q}_{i_1(x)i_2(x)...i_n(x)...}$, are investigated. It is proved, that the systemhas the unique solution in the class of determined and bounded on $[0;1]$functions and continuity of the solution. His analytical presentation isfounded. Conditions of its monotonicity and nonmonotonicity, differential,integral properties are studied. Conditions under which the solution of thefunctional equations system is a distribution function of random variable$\eta=\Delta^{\tilde Q} _{\xi_1\xi_2...\xi_n...}$ with independent $\tildeQ$-symbols are discovered.  The results of the article was represented in Fourth all-Ukrainian ScientificConference of Young Scientists on Mathematics and Physics, Kyiv, April 23-25,2015 (https://www.researchgate.net/publication/301765100).  The investigation was represented at the seminar on fractal analysis ofInstitute of Mathematics of the National Academy of Sciences of Ukraine on,October 30, 2014(http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1).

tatra mountains mathematical publications

On one class of functions with complicated local structure that the  solutions of infinite systems of functional equations (On one application of  infinite systems of functional equations in the functions theory)