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The solution of the two-point boundary value problem for a nonlinear differential equation of the third order. (English) Zbl 0627.34022
For the problem $$x\prime''+x=0$$, $$x^{(i)}(0)-x^{(i)}(2\pi /\sqrt{3})=0$$, $$i=0,1,2$$ Green’s function G(t,s) is constructed. It is shown that $$G(t,s)>0$$ on its domain. With its help the existence of a periodic solution to the nonlinear equation $$x\prime''+F(t,x,x',x'')+x=e(t)$$ is established.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
##### Keywords:
third order differential equation; Green’s function
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##### References:
 [1] COLLATZ L.: Funkcionální analýza a numerická matematika. Praha, 1970. [2] KOLMOGOROV A. N., FOMIN S. V.: Základy teorie funkcí a funkcionální analýzy. Praha, 1975. [3] REKTORYS K. a kol.: Přehled užité matematiky. Praha, 1963. · Zbl 0175.15801 [4] ŠEDA V.: An Application of Green’s Function in the Differential Equations. Proceedings of Equadiff II, Acta Fac. Rerum Natur. Univ. Comenian. Math. 17, 1967, 221–235. · Zbl 0186.14902
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