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Keywords:
two-point boundary value problem; curvature bound set; Leray-Schauder theorem; Bernstein-Hartman condition
two-point boundary value problem; curvature bound set; Leray-Schauder theorem; Bernstein-Hartman condition
Summary:
The solvability of second order differential systems with the classical separated or periodic boundary conditions is considered. The proofs use special classes of curvature bound sets or bound sets together with the simplest version of the Leray-Schauder continuation theorem. The special cases where the bound set is a ball, a parallelotope or a bounded convex set are considered.
References:
[1] Amster, P., Haddad, J.: A Hartman-Nagumo type condition for a class of contractible domains. Topol. Methods Nonlinear Anal. 41 (2013), 287-304. MR 3114309 | Zbl 1306.34033
[2] Bass, R. W.: On non-linear repulsive forces. Contributions to the Theory of Nonlinear Oscillations, Volume IV. Annals of Mathematics Studies 41. Princeton University Press, Princeton (1958), 201-211 S. Lefschetz. DOI 10.1515/9781400881758-011 | MR 0100704 | Zbl 0083.31405
[3] Bebernes, J. W.: A simple alternative problem for finding periodic solutions of second order ordinary differential systems. Proc. Am. Math. Soc. 42 (1974), 121-127. DOI 10.2307/2039687 | MR 0330597 | Zbl 0286.34055
[4] Bernstein, S. N.: Sur les équations du calcul des variations. Ann. Sci. Éc. Norm. Supér., Sér. III. 29 (1912), 431-485 French \99999JFM99999 43.0460.01. DOI 10.24033/asens.651 | MR 1509153
[5] Coster, C. De, Habets, P.: Two-Point Boundary Value Problems: Lower and Upper Solutions. Mathematics in Science and Engineering 205. Elsevier, Amsterdam (2006). DOI 10.1016/s0076-5392(06)x8055-4 | MR 2225284 | Zbl 1330.34009
[6] Fabry, C.: Nagumo conditions for systems of second-order differential equations. J. Math. Anal. Appl. 107 (1985), 132-143. DOI 10.1016/0022-247x(85)90358-0 | MR 0786017 | Zbl 0604.34002
[7] Fewster-Young, N.: A singular Hartman inequality for existence of solutions to nonlinear systems of singular, second order boundary value problems. Int. J. Differ. Equ. Appl. 14 (2015), 195-228. DOI 10.12732/ijdea.v14i3.2215 | MR 3621238 | Zbl 1337.34024
[8] Fewster-Young, N.: A priori bounds and existence results for singular boundary value problems. Electron. J. Qual. Theory Differ. Equ. (2016), 1-15. DOI 10.14232/ejqtde.2016.1.17 | MR 3487646 | Zbl 1363.34067
[9] Frigon, M.: Boundary and periodic value problems for systems of nonlinear second order differential equations. Topol. Methods Nonlinear Anal. 1 (1993), 259-274. DOI 10.12775/tmna.1993.019 | MR 1233095 | Zbl 0790.34022
[10] Frigon, M.: Boundary and periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition. Differ. Integral Equ. 8 (1995), 1789-1804. MR 1347980 | Zbl 0831.34021
[11] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics 568. Springer, Cham (1977). DOI 10.1007/BFb0089537 | MR 0637067 | Zbl 0339.47031
[12] Hartman, P.: On boundary value problems for systems of ordinary, nonlinear, second order differential equations. Trans. Am. Math. Soc. 96 (1960), 493-509. DOI 10.2307/1993537 | MR 0124553 | Zbl 0098.06101
[13] Hartman, P.: Ordinary Differential Equations. John Wiley & Sons, New York (1964). MR 0171038 | Zbl 0125.32102
[14] Heinz, E.: On certain nonlinear elliptic differential equations and univalent mappings. J. Anal. Math. 5 (1956/1957), 197-272. DOI 10.1007/BF02937346 | MR 0136852 | Zbl 0085.08701
[15] Henderson, J., Sheng, Q., Tisdell, C. C.: Constructive existence results for solutions to systems of boundary value problems via general Lyapunov methods. Differ. Equ. Appl. 9 (2017), 57-68. DOI 10.7153/dea-09-05 | MR 3610874 | Zbl 1365.34043
[16] Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Fundamentals. Grundlehren der mathematischen Wissenschaften 305. Springer, Berlin (1993). DOI 10.1007/978-3-662-02796-7 | MR 1261420 | Zbl 0795.49001
[17] Knobloch, H.-W.: On the existence of periodic solutions for second order vector differential equations. J. Differ. Equations 9 (1971), 67-85. DOI 10.1016/0022-0396(70)90154-3 | MR 0277824 | Zbl 0211.11801
[18] Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Éc. Norm. Supér., Sér. III. 51 (1934), 45-78 French. DOI 10.24033/asens.836 | MR 1509338 | Zbl 0009.07301
[19] Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Regional Conference Series in Mathematics 40. AMS, Providence (1979). DOI 10.1090/cbms/040 | MR 0525202 | Zbl 0414.34025
[20] Mawhin, J.: Variations on Poincaré-Miranda's theorem. Adv. Nonlinear Stud. 13 (2013), 209-217. DOI 10.1515/ans-2013-0112 | MR 3058216 | Zbl 1278.55004
[21] Mawhin, J., Szymańska-Dębowska, K.: Convex sets and second order systems with nonlocal boundary conditions at resonance. Proc. Am. Math. Soc. 145 (2017), 2023-2032. DOI 10.1090/proc/13569 | MR 3611317 | Zbl 1393.34035
[22] Nagumo, M.: Über die Differentialgleichung $y'' = f(t, y, y')$. Proc. Phys.-Math. Soc. Japan, III. Ser. 19 (1937), 861-866 German. Zbl 0017.30801
[23] Opial, Z.: Sur la limitation des dérivées des solutions bornées d'un système d'équations différentielles du second ordre. Ann. Polon. Math. 10 (1961), 73-79 French. DOI 10.4064/ap-10-1-73-79 | MR 0126020 | Zbl 0097.29401
[24] Rouche, N., Mawhin, J.: Équations différentielles ordinaires. Tome I: Théorie générale. Tome II: Stabilité et solutions périodiques. Masson et Cie, Paris French (1973). MR 0481182 | Zbl 0289.34001
[25] Schmitt, K., Thompson, R.: Boundary value problems for infinite systems of second-order differential equations. J. Differ. Equations 18 (1975), 277-295. DOI 10.1016/0022-0396(75)90063-7 | MR 0374594 | Zbl 0302.34081
[26] Szymańska-Dębowska, K.: On a generalization of the Miranda Theorem and its application to boundary value problems. J. Differ. Equations 258 (2015), 2686-2700. DOI 10.1016/j.jde.2014.12.022 | MR 3312640 | Zbl 1336.47056
[27] Taddei, V.: Two-points boundary value problems for Carathéodory second order equations. Arch. Math., Brno 44 (2008), 93-103. MR 2432846 | Zbl 1212.34039
[28] Taddei, V., Zanolin, F.: Bound sets and two-point boundary value problems for second order differential equations. Georgian Math. J. 14 (2007), 385-402. DOI 10.1515/GMJ.2007.385 | MR 2341286 | Zbl 1133.34013
[29] Thorpe, J. A.: Elementary Topics in Differential Geometry. Undergraduate Texts in Mathematics. Springer, New York (1979). DOI 10.1007/978-1-4612-6153-7 | MR 0528129 | Zbl 0404.53001
[30] Tisdell, C. C., Tan, L. H.: On vector boundary value problems without growth restrictions. JIPAM, J. Inequal. Pure Appl. Math. 6 (2005), Article No. 137, 10 pages \99999MR99999 2191606 \filbreak. MR 2191606 | Zbl 1097.34018
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