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On a class of functional boundary value problems for second-order functional differential equations with parameter. (English) Zbl 0788.34069
The existence of a solution to the problem (1) \(y''(t)-Q[y,y'] (t) y(t)=F [y,y', \mu](t)\), (2) \(\alpha_ 1(y(t_ 1)-y(t) | J_ 1)=0\), \(y(t_ 2)=0\), \(\alpha_ 2(y(t_ 3)-y(t) | J_ 3)= 0\) is proved under some conditions where \(-\infty<t_ 1<t_ 2<t_ 3<\infty\), \(\alpha_ i\) are continuous functionals, \(y(t) | J_ i\) denotes the restriction of \(y\) to \(J_ i=[t_ i,t_{i+1}]\), \(J=[t_ 1,t_ 3]\), \(\mu\) is a parameter, \(Q:X \times X \to X\), \(F:X \times X \times I \to X\) are continuous operators and \(X=C^ 0([J,R])\), \(I=[a,b]\), \(Q[y,z](t)>0\) on \(J\) for all \([y,z] \in X \times X\).

MSC:
34K10 Boundary value problems for functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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References:
[1] S. Staněk: On a class of five-point boundary value problems in second-order functional differential equations with parameter. Acta Math. Hungar, to appear. · Zbl 0801.34064
[2] S. Staněk: Three-point boundary value problem for nonlinear second-order differential equations with parameter. Czech. Math. J. 42 (117) (1992), 241-256. · Zbl 0779.34017
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[5] S. Staněk: Three-point boundary value problem of retarded functional differential equation of the second order with parameter. Acta UPO, Fac. rer. Nat. 97 Math. XXIX. (1990), 107-121. · Zbl 0759.34048
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