# Article

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Keywords:
singular differential equation with deviating arguments; the Valée-Poussin problem; existence theorem; uniqueness theorem
Summary:
For the differential equation $u^{(n)}(t)= f(t,u(\tau _{1}(t)),\dots ,u^{(n-1)}(\tau _{n}(t))),$ where the vector function $f:\ ]a,b[\,\times {R}^{kn} \rightarrow {R}^{k}$ has nonintegrable singularities with respect to the first argument, sufficient conditions for existence and uniqueness of the Vallée–Poussin problem are established.
References:
[1] Bessmertnych G. A.: On existence and uniqueness of solutions of multipoint Vallée-Poussin problem for nonlinear differential equaitions. Differentsial’nyje Uravnenija 6, No 2 (1970), 298-310 (in Russian). MR 0267183
[2] de la Vallée-Poussin Ch. J.: Sur l’equation differentielle lineaire de second ordre. Détermination d’une integrale par deux valeurs assignées. Extension aux équations d’ordre $n$. J. Math. pures et appl. 8, No 2 (1929), 125-144.
[3] Kiguradze I. T.: On a singular multi-point boundary value problem. Ann. Mat. Pura ed Appl. 86(1970), 367-399. MR 0271449 | Zbl 0251.34012
[4] Kiguradze I. T.: Some singular boundary value problems for ordinary differential equations. Tbilisi: Tbilisi University Press (1975) (in Russian). MR 0499402 | Zbl 0521.34019
[5] Kiguradze I. T.: On some singular boundary value problems for ordinary differential equations. Equadiff 5 Proc. 5 Czech. Conf. Diff. Equations and Appl. Leipzig: Teubner Verlagsgesselschaft (1982), 174-178. Zbl 0521.34019
[6] Kiguradze I. T.: On the solvability of the Valée-Poussin problem. Differentsial’nyje Uravnenija 21, No 3 (1985), 391-398. MR 0785447
[7] Kiguradze I. T.: On a boundary value problems for higher ordinary differential equations with singularities. Uspekhi Mat. Nauk 41, No. 4 (1986), 166-167 (in Russian).
[8] Kiguradze I., Tskhovrebadze G.: On the two-point boundary value problems for systems of higher order ordinary differential equations with singularities. Georgian Math. J. 1, No 1(1994), 31-45.
[9] Lasota A., Opial Z.: L’existence et l’unicité des solutions du probléme d’interpolation differentielle ordinaire d’ordre $n$. Ann. Polon. Math. 15, No 3(1964), 253-271. MR 0173804
[10] Levin A. J.: Nonoscillatory solutions of equation $x^{(n)}+ p_1(t) x^{(n-1)}+ \dots + p_n(t)x=0$. Uspekhi Mat. Nauk 24, No 2(1969), 43-96 (in Russian).
[11] Levin A. J.: On a multi-point boundary value problem. Nauchnie Dokl. Vis. Shkoli 5 (1958), 34-37 (in Russian).
[12] Opial Z.: Linear problems for systems of nonlinear differential equations. J. Diff. Equat. 3, No 4(1967), 580-594. MR 0216068 | Zbl 0161.06102
[13] Sansone G.: Equazioni Differenzialli nel campo reale. Bologna: Zanichelli (1948). MR 0026731
[14] Tskhovrebadze G. D.: On a multipoint boundary value problem for nonlinear ordinary differential equations with singularities. Arch. Math. 30, No 3(1994), 171-206. MR 1308353

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