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A remark on the existence of small solutions to a fourth order boundary value problem with large nonlinearity. (English) Zbl 0779.34022
The existence of at least one small solution to the nonlinear boundary value problem $$y^{(4)}+(m^ 2+n^ 2)y''+m^ 2 n^ 2 y+\eta y^{2l}=f$$, $$y^{(i)}(0)=y^{(i)}(2\pi)$$, $$i=0,1,2,3$$, where $$0<m<n$$, $$l\geq 4$$, $$m,n\in N$$, $$\eta=\pm 1$$, under certain assumptions on $$f$$, is proved.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
small solution; nonlinear boundary value problem
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##### References:
 [1] DEIMLING K.: Nichtlineare Gleichungen and Abbildungsgrade. Springer-Verlag, Berlin-Heidelberg-New York, 1974. · Zbl 0281.47033 [2] DING S. H., MAWHIN J.: A multiplicity result for periodic solutions of higher order ordinary differential equations. Differential Integral Equations 1 (1988), 31-39. · Zbl 0715.34086 [3] GREGUŠ M., ŠVEC M., ŠEDA V.: Ordinary Differential Equations. (Slovak), Alfa, Bratislava, 1985. [4] LALOLIX B., MAWHIN J.: Coincidence index and multiplicity. Trans. Amer. Math. Soc. 217 (1976), 143-162. · Zbl 0334.47041 [5] LEFTON L.: Existence of small solutions to a resonant boundary value problem with large nonlineanty. J. Differential Equations 85 (1990), 171-185. · Zbl 0699.34020 [6] REKTORYS K. ETAL.: Survey of Applied Mathematics. (Czech), SNTL - Nakladatelství technické literatury, Praha, 1963. [7] SCHIJUR J. D.: Perturbation at resonance for a fourth order ordinary differential equation. J. Math. Anal. Appl. 65 (1978), 20-25. · Zbl 0406.34026 [8] ŠEDA V.: Some remarks to coincidence theory. Czechoslovak Math. J. 38 (113) (1988), 554-572. · Zbl 0721.47048
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