satisfies on [ p, q $_{]}$. Next, consider the operator Υ : C ([ p, q ] , R ) → C ([ p, q ] , R ) given by 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. □ 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. 20 [8] Akhmet, M.U., Fen, M.O.: Entrainment by chaos. J. Nonlinear Sci. $_{24}$, 411–439 (2014). [7] Akhmet, M.U., Fen, M.O.: Replication of chaos, Commun. Nonlinear Sci. Numer. Simul. $_{18}$, 2626–2666 (2013). [6] Akhmet, M.U., Fen, M.O.: Chaos generation in hyperbolic systems. Discontinuity, Nonlinearity, and Complexity $_{1}$, 367–386 (2012). [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). [4] Akhmet, M.U.: Devaney’s chaos of a relay system. Commun. Nonlinear Sci. Numer. Simulat. $_{14}$, 1486–1493 (2009). [3] Akhmet, M.U.: Dynamical synthesis of quasi-minimal sets. Int. J. Bifur. Chaos $_{19}$, 2423–2427 (2009). [2] Akhmet, M. U.: Creating a chaos in a system with relay. Int. J. Qualit. Th. Diff. Eqs. Appl. $_{3}$, 3–7 (2009). [1] Akhmet, M.U.: Li-Yorke chaos in the impact system. J. Math. Anal. Appl. $_{351}$, 804–810 (2009). References Acknowledgements Υ u ( t ) = a + b ( e − γ ( t − p ) + e − γ ( q − t ) ) + c ∫ q p e − γ | t − s $^{|}$u ( s ) ds. u$_{4}$ ( t ) ≥ a + b ( e − γ ( t − p ) + e − γ ( q − t ) ) + c ∫ q p e − γ | t − s $^{|}$u$_{4}$ ( s ) ds