We consider multipoint and two-point BVPs for second order ordinary differential equations with a Carathéodory right hand side. We prove the existence of solutions provided there exist upper and lower solutions of the BVP and the upper solution is less than the lower one.
@article{bwmeta1.element.bwnjournal-article-apmv65z3p235bwm,
author = {Irena Rach\r unkov\'a},
title = {Upper and lower solutions satisfying the inverse inequality},
journal = {Annales Polonici Mathematici},
volume = {66},
year = {1997},
pages = {235-244},
zbl = {0868.34014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv65z3p235bwm}
}
Irena Rachůnková. Upper and lower solutions satisfying the inverse inequality. Annales Polonici Mathematici, Tome 66 (1997) pp. 235-244. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv65z3p235bwm/
[000] [1] S. R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974. | Zbl 0286.34018
[001] [2] E. N. Dancer, On the ranges of certain damped nonlinear differential equations, Ann. Mat. Pura Appl. (4) 119 (1979), 281-295. | Zbl 0416.34029
[002] [3] R. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, Berlin, 1977. | Zbl 0339.47031
[003] [4] W. Gao and J. Wang, On a nonlinear second order periodic boundary value problem with Carathéodory functions, Ann. Polon. Math. 62 (1995), 283-291. | Zbl 0839.34031
[004] [5] I. T. Kiguradze, Some Singular Boundary Value Problems for Ordinary Differential Equations, Univ. Press, Tbilisi, 1975 (in Russian).
[005] [6] H. W. Knobloch, Eine neue Methode zur Approximation periodischer Lösungen nicht-linearer Differentialgleichungen zweiter Ordnung, Math. Z. 82 (1963), 177-197. | Zbl 0117.05404
[006] [7] A. Lepin and L. Lepin, Boundary Value Problems for Ordinary Differential Equations of the Second Order, Zinatne, Riga, 1988 (in Russian). | Zbl 0661.34014
[007] [8] J. Mawhin, Topological Degree Methods in Nonlinear BVPs, CBMS Regional Conf. Ser. in Math. 40, Providence, R.I., 1979.
[008] [9] J. Mawhin and J. R. Ward Jr. Periodic solutions of some forced Liénard differential equations at resonance, Arch. Math. (Basel) 41 (1983), 337-351. | Zbl 0537.34037
[009] [10] J. J. Nieto, Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl. 130 (1988), 22-29. | Zbl 0678.34022
[010] [11] P. Omari, Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations, Rend. Inst. Mat. Univ. Trieste 20 (1988), 54-64. | Zbl 0719.34078
[011] [12] I. Rachůnková, The first kind periodic solutions of differential equations of the second order, Math. Slovaca 39 (1989), 407-415. | Zbl 0753.34028
[012] [13] I. Rachůnková, An existence theorem of the Leray-Schauder type for four-point boundary value problems, Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 100, Math. 30 (1991), 49-59. | Zbl 0752.34016
[013] [14] I. Rachůnková, On a transmission problem, Acta Univ. Palack. Olomuc. Fac. Rerum Nat. 105, Math. 31 (1992), 45-59. | Zbl 0807.34018
[014] [15] I. Rachůnková, On resonance problems for the second order differential equations, preprint.
[015] [16] R. Reissig, Schwingungssätze für die verallgemeinerte Liénardsche Differentialgleichung, Abh. Math. Sem. Univ. Hamburg 44 (1975), 45-51. | Zbl 0323.34033
[016] [17] N. I. Vasil'ev and Yu. A. Klokov, Foundations of the Theory of Boundary Value Problems for Ordinary Differential Equations, Zinatne, Riga, 1978 (in Russian).
[017] [18] F. Zanolin, Periodic solutions for differential systems of Rayleigh type, Rend. Inst. Mat. Univ. Trieste 12 (1980), 69-77. | Zbl 0467.34030