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Multiplicity results for second order nonlinear problems with maximum and minimum. (English) Zbl 0920.34058
Consider the functional boundary value problem $x''(t)= [Fx](t),\;t\in J= [a,b],\;\min\{x(t): t\in J\}= \alpha,\;\max\{x(t): t\in J\}= \beta,$ where $$F: C^1(I)\to L^1(I)$$ is an operator, and $$\alpha$$, $$\beta$$ are given real numbers.
A solution is a function $$x\in AC^1(I)$$ satisfying the equation for a.e. $$t\in J$$ and the boundary conditions. Sufficient conditions for the existence of at least two solutions are presented by using a lemma of Bihari and a Bernstein-Nagumo growth condition.

##### MSC:
 34K10 Boundary value problems for functional-differential equations
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##### References:
 [1] Ambrosetti, Ann. Math. Pura Appl. 93 pp 231– (1972) [2] Bihari, Acta Math. Acad. Sci. Hung. 7 pp 71– (1956) [3] Brüll, Arch. Mat. (Brno 24 pp 163– (1988) [4] Brykalov, Diff. Urav. 27 pp 2027– (1991) [5] Brykalov, Diff. Urav. 29 pp 938– (1993) [6] Brykalov, Proceedings Georgian Acad. Sci. Math. 1 pp 273– (1993) [7] Chiappinelli, J. Differential Equations 69 pp 422– (1987) [8] Ding, Differential and Integral Equations 1 pp 31– (1988) [9] , and : Existence and Multiplicity Results for Periodic Solutions of Semilinear Duffing Equations, SISS A Report 56191/M, Trieste, (1991) [10] Fabry, Bull. Londom Math. Soc. 18 pp 173– (1986) [11] Fečkan, J. Differential Equations 113 pp 189– (1994) [12] and : Integral Inequalities and Theory of Nonlinear Oscillations, Nauka, Moscow, 1976 (in Russian) [13] Gaete, J. Math. Anal. Appl. 134 pp 257– (1988) [14] Harris, J. Differential Equations 95 pp 75– (1992) [15] Harris, J. Math. Anal. Appl. 182 pp 571– (1994) [16] Hard, SIAM J. Math. Anal. 17 pp 1332– (1986) [17] : Ordinary Differential Equations, John Wiley and Sons, New York, 1964 [18] Mawhin, Z. Angew. Math. Phys. 38 pp 257– (1987) [19] Nkashama, J. Differential Equations 84 pp 148– (1990) [20] Nkashama, J. Math. Anal. Appl. 140 pp 381– (1989) [21] Rachńková, Nonlinear Analysis 18 pp 497– (1992) [22] Rachńková, Nonlinear Analysis 22 pp 1315– (1994) [23] Retzloft, J. Math. Anal. Appl. 185 pp 501– (1994) [24] Ruf, Nonlinear Analysis 10 pp 157– (1986) [25] Schaaf, Trans. Amer. Math. Soc. 306 pp 853– (1988) [26] Stanêk, Arch. Math. (Brno) 28 pp 57– (1992) [27] Struwe, J. Differential Equations 38 pp 285– (1980) [28] Šeda, Differential and Integral Equations 8 pp 19– (1995) [29] Šenkyřik, Math. Bohemica 119 pp 113– (1994) [30] Vidossich, J. Math. Anal. Appl. 127 pp 459– (1987)
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