A fixed point theorem for a $c$-continuous mappings in the space $l_p (\mathbb{Z}, \mathbb{R}), 1 ≤ p ≤ ∞,$  are obtained.

[1] Mukhamadiev E. On the invertibility of functional operators in the space of functions bounded on an axis / / Mat. notes. - 1972.- 11, N3. - P. 269-274.

[2] Mukhamadiev E. Research on the theory of periodic and bounded solutions of differential equations: Diss. ... Doctor of Physical and Mathematical Sciences. Dushanbe. 1978. - 289 p.

[3] Slyusarchuk V.E. Invertibility of almost periodic $c$-continuous functional operators // Mat. sb. 1981. 116(158), N4(12).- P. 483-501.

[4] Slyusarchuk V.E. Integral representation of $c$-continuous linear operators // Reports of the Academy of Sciences of the Ukrainian SSR. - 1981. - series A, N8. - P. 34-37.

[5] Slyusarchuk V.E. Invertibility of non-autonomous differential-functional operators // Mat. sb. 1986. 130(172), N1(5).- P. 86-104.

[6] Slyusarchuk V.E. Necessary and sufficient conditions for the invertibility of non-autonomous functional-differential operators // Mat. notes. - 1987. - 42, N2. - P. 262-267.

[7] Slyusarchuk V.E. Necessary and sufficient conditions for the invertibility of uniformly $c$-continuous functionally differential operators // Ukr. mat. zhurn. - 1989. 41, N2. - P. 201-205.

[8] Kurbatov V.G. Linear differential-difference equations. - Voronezh: Publishing house of Voronezh. University, 1990. - 168 p.

[9] Chan Hiu Bong. Almost periodic and bounded solutions of linear functional-differential equations: Diss. ... Doctor of Physical and Mathematical Sciences. - Kyiv. 1993.- 255 p.

[10] Slyusarchuk V.E. Method of c-continuous operators in the theory of pulse systems // Abstracts of reports of the All-Union Conference on the Theory and Applications of Functional Differential Equations. - Dushanbe. 1987. - P. 102-103.

[11] Slyusarchuk V.E. Weakly nonlinear disturbances of pulse systems / / Mathematical physics and nonlinear mechanics. - 1991. Issue 15(49). - P. 32-35.

[12] Lyusternik L.A., Sobolev V.I. A short course in functional analysis. - M.: Higher School, 1982. - 272 p.

[13] Schauder J. Die Fixpunktsatz in Funktionalräume // Studia Math. - 1930. - 2 . - P. 171-180.

[14] Slyusarchuk V.E. Necessary and sufficient conditions for the Ipschitz invertibility of nonlinear difference operators in spaces $l_p (\mathbb{Z}, \mathbb{R}), 1 ≤ p ≤ ∞$ // Mat. notes.- 2000.- 68, N3.- P. 448-454.

[15] Slyusarchuk V.Yu. Necessary and sufficient conditions for the invertibility of nonlinear difference mappings in the space $l_∞ (\mathbb{Z}, \mathbb{R})$ / / Mat. studii. - 2000. - 13, N1. - P. 63-73.

[16] Slyusarchuk V.E. Bounded solutions of nonlinear elliptic equations // Uspekhi mat. sciences. - 1980.- 35, N1(211).- P. 215-216.

[17] Slyusarchuk V.E. Bounded solutions of pulse systems // Differential equations. - 1983. - 19, N4. - P.588-596.

[18] Slyusarchuk V.E. Weakly nonlinear perturbations of normally solvable functional-differential and discrete equations // Ukr. mat. zhurn. - 1987. - 39, N5. - P.660-662.

[19] Slyusarchuk V.E. $\mathcal{P}$-continuous operators and their application to solving problems of mathematical physics / / Integral transformations and their application to boundary value problems: 3b. scientific. pr. - K.: Institute of Mathematics of the NAS of Ukraine, 1997. - Issue 15. - P. 188-226.