content_image/296117.jpg content_image/296116.jpg
satisfies 

on [ p, q $_{]}$. Next, consider the operator Υ : C ([ p, q ] , R ) → C ([ p, q ] , R ) given by 

<img src='content_image/296117.jpg'>

<img src='content_image/296116.jpg'>

Then it is nondecreasing and by [14, p. 14], it is contractive. So it has a unique fixed point u$_{∗}$ ∈ C ([ p, q ] , R $_{)}$, i.e., u$_{∗}$ = Υ $_{u$_{∗}$}$. Since u ≤ Υ u and u$_{4}$ ≥ Υ $_{u$_{4}$}$, by standard arguments (see [14, Theorem 2.2]), we get u ≤ u$_{∗}$ ≤ $_{u$_{4}$}$. The proof is finished. □ 

## Acknowledgements 

This work was supported by the Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0153/16, by the Slo- vak Research and Development Agency under the contract No. APVV-14-0378, and by the National Scholarship Programme of the Slovak Republic. 

## References 

[1] Akhmet, M.U.: Li-Yorke chaos in the impact system. J. Math. Anal. Appl. $_{351}$, 804–810 (2009). 

[2] Akhmet, M. U.: Creating a chaos in a system with relay. Int. J. Qualit. Th. Diff. Eqs. Appl. $_{3}$, 3–7 (2009). 

[3] Akhmet, M.U.: Dynamical synthesis of quasi-minimal sets. Int. J. Bifur. Chaos $_{19}$, 2423–2427 (2009). 

[4] Akhmet, M.U.: Devaney’s chaos of a relay system. Commun. Nonlinear Sci. Numer. Simulat. $_{14}$, 1486–1493 (2009). 

[5] Akhmet, M.U., Fen, M.O.: Chaotic period-doubling and OGY control for the forced Duffing equa- tion. Commun. Nonlinear Sci. Numer. Simulat. $_{17}$, 1929–1946 (2012). 

[6] Akhmet, M.U., Fen, M.O.: Chaos generation in hyperbolic systems. Discontinuity, Nonlinearity, and Complexity $_{1}$, 367–386 (2012). 

[7] Akhmet, M.U., Fen, M.O.: Replication of chaos, Commun. Nonlinear Sci. Numer. Simul. $_{18}$, 2626–2666 (2013). 

[8] Akhmet, M.U., Fen, M.O.: Entrainment by chaos. J. Nonlinear Sci. $_{24}$, 411–439 (2014). 

