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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:
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