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On a theorem of L. Lefton. (English) Zbl 0756.34028
The author reconsiders the two-point boundary-value problem \(Ly+y^ 3=f(x)\), \(x\in(a,b)\), \(M_ 1(y)=M_ 2(y)=0\) with \(Ly\equiv y''+p(x)y'+q(x)y\), \(M_ 1(y)=\alpha_ 1y(a)+\alpha_ 2y(b)+\alpha_ 3y'(a)+\alpha_ 4y'(b)\), \(M_ 2(y)=\beta_ 1y(a)+\beta_ 2y(b)+\beta_ 3y'(a)+\beta_ 4y'(b)\), where \(p(x)\) and \(q(x)\) are given integrable functions while \(\alpha_ 1\) and \(\beta_ 1\) are given real constants. The known function \(f(x)\) is supposed to be ”small”. His aim is to give a new and simple proof for the existence theorem previously stated by L. Lefton [J. Differ. Equations 85, No. 1, 171-185 (1990; Zbl 0699.34020)]. To this end he follows the Lyapunov-Schmidt method.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
Citations:
Zbl 0699.34020
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References:
[1] LEFTON L.: Existence of small solutions to a resonant boundary value problem with large nonlinearity. J. Differential Equations 85 (1990), 171-185. · Zbl 0699.34020
[2] LALOUX B., MAWHIN J.: Coincidence index and multiplicity. Trans. Amer. Math. Soc. 217 (1976), 143-162. · Zbl 0334.47041
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