Applying the topological transversality method of Granas and the a priori bounds technique we prove some existence results for systems of differential inclusions of the form y'' ∈ F(t,y,y'), where F is a Carathéodory multifunction and y satisfies some nonlinear boundary conditions.
@article{bwmeta1.element.bwnjournal-article-apmv54z3p195bwm,
author = {L. H. Erbe and W. Krawcewicz},
title = {Nonlinear boundary value problems for differential inclusions y'' [?] F(t,y,y')},
journal = {Annales Polonici Mathematici},
volume = {55},
year = {1991},
pages = {195-226},
zbl = {0731.34078},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv54z3p195bwm}
}
L. H. Erbe; W. Krawcewicz. Nonlinear boundary value problems for differential inclusions y'' ∈ F(t,y,y'). Annales Polonici Mathematici, Tome 55 (1991) pp. 195-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv54z3p195bwm/
[000] [1] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984.
[001] [2] J. W. Bebernes and K. Schmitt, Periodic boundary value problems for systems of second order differential equations, J. Differential Equations 13 (1973), 33-47. | Zbl 0253.34020
[002] [3] J.-M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris 265 (1967), 333-336. | Zbl 0164.16803
[003] [4] H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, Paris 1982.
[004] [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, 1977.
[005] [6] M. Cecchi, M. Marini and P. Zecca, Asymptotic properties of the solutions of nonlinear equations with dichotomies and applications, Boll. Un. Mat. Ital. (6) 1-C (1982), 209-234. | Zbl 0511.34039
[006] [7] M. Cecchi, M. Marini and P. Zecca, Linear boundary value problems for systems of ordinary differential equations on non-compact intervals, I, II, Ann. Mat. Pura Appl. (4) 123 (1980), 267-285; 134 (1980), 367-379. | Zbl 0442.34016
[007] [8] A. Cellina and A. Lasota, A new approach to the definition of topological degree for multivalued mappings, Rend. Accad. Naz. Lincei 47 (1969), 434-440.
[008] [9] K. C. Chang, The obstacle problems and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), 117-146. | Zbl 0405.35074
[009] [10] J. Dugundji and A. Granas, Fixed Point Theory, Vol. 1, PWN, Warszawa 1982.
[010] [11] L. H. Erbe and H. W. Knobloch, Boundary value problems for systems of second order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 61-76. | Zbl 0582.34024
[011] [12] L. H. Erbe, W. Krawcewicz and S. Chen, Some existence results for solutions of differential inclusions with retardations, this issue, 227-239. | Zbl 0731.34079
[012] [13] L. H. Erbe and P. K. Palamides, Boundary value problems for second-order differential systems, J. Math. Anal. Appl. 127 (1) (1987), 80-92. | Zbl 0635.34017
[013] [14] L. H. Erbe and K. Schmitt, On solvability of boundary value problems for systems of differential equations, J. Appl. Math. Phys. 38 (1987), 184-192. | Zbl 0635.34016
[014] [15] C. Fabry, Nagumo conditions for systems of second order differential equations, J. Math. Anal. Appl. 107 (1985), 132-143. | Zbl 0604.34002
[015] [16] C. Fabry and P. Habets, The Picard boundary value problem for nonlinear second order vector differential equations, J. Differential Equations 42 (2) (1987), 186-198. | Zbl 0439.34018
[016] [17] M. Frigon, Application de la théorie de la transversalité topologique à des problèmes non linéaires pour des équations différentielles ordinaires, Dissertationes Math. 296 (1990). | Zbl 0728.34017
[017] [18] R. E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. 568, Springer, 1977. | Zbl 0339.47031
[018] [19] K. Gęba, A. Granas, T. Kaczyński et W. Krawcewicz, Homotopie et équations non linéaires dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 300 (10) (1985), 303-306. | Zbl 0585.47049
[019] [20] A. Granas, Homotopy extension theorem in Banach spaces and some of its applications to the theory of non-linear equations, Bull. Acad. Polon. Sci. 7 (1959), 387-394. | Zbl 0092.32302
[020] [21] A. Granas, Sur la méthode de continuité de Poincaré, C. R. Acad. Sci. Paris 287 (1976), 983-985.
[021] [22] A. Granas et Zine el Abdine Guennoun, Quelques résultats dans la théorie de Bernstein-Carathéodory de l'équation y'' = f(t,y,y'), C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 703-706.
[022] [23] A. Granas, R. B. Guenther and J. W. Lee, On a theorem of S. Bernstein, Pacific. J. Math. 74 (1978), 78-82.
[023] [24] A. Granas, R. B. Guenther and J. W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math. 10 (1980), 35-58. | Zbl 0476.34017
[024] [25] A. Granas, R. B. Guenther and J. W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. 244 (1981).
[025] [26] A. Granas, R. B. Guenther and J. W. Lee, Topological transversality II; Applications to the Neumann problem for y'' = f(t,y,y'), Pacific J. Math. 104 (1983), 95-109. | Zbl 0534.34006
[026] [27] A. Granas, R. B. Guenther, J. W. Lee and D. O'Regan, Topological transversality III; Applications to the boundary value problems on infinite intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), 335-348. | Zbl 0594.34019
[027] [28] J. Haddad and J. M. Lasry, Periodic solutions of functional differential inclusions and fixed points of G-selectionable correspondences, J. Math. Anal. Appl. 110 (1983), 295-312. | Zbl 0539.34031
[028] [29] P. Hartman, Ordinary Differential Equations, Wiley, New York 1964. | Zbl 0125.32102
[029] [30] T. Kaczyński, Topological transversality and nonlinear equations in locally convex spaces, 1987, preprint.
[030] [31] H. W. Knobloch, Boundary value problems for systems of nonlinear differential equations, in: Proc. Equadiff IV 1977, Lecture Notes in Math. 703, Springer, 1978, 197-204.
[031] [32] H. W. Knobloch and K. Schmitt, Nonlinear boundary value problems for systems of differential equations, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977), 139-159. | Zbl 0368.34009
[032] [33] W. Krawcewicz, Contribution à la théorie des équations non linéaires dans les espaces de Banach, Dissertationes Math. 273 (1988). | Zbl 0677.47038
[033] [34] J. Mawhin and K. Schmitt, Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 199-207. | Zbl 0573.35029
[034] [35] T. Pruszko, Topological degree methods in multivalued boundary value problems, Nonlinear Anal. 5 (9) (1981), 953-973.
[035] [36] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math. 229 (1984). | Zbl 0543.34008
[036] [37] K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. 2 (1978), 263-309. | Zbl 0378.35022
[037] [38] C. A. Stuart, Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Anal. 63 (1976), 59-75. | Zbl 0393.34010
[038] [39] P. Zecca and P. L. Zecca, Nonlinear boundary value problems in Banach space for multivalued differential equations on a noncompact interval, Nonlinear Anal. 3 (1979), 347-352. | Zbl 0443.34060