Abstract
Existence and multiplicity of solutions (with prescribed nodal properties) for the two-point boundary value problem associated to a second order ODE is proved. First, we consider an autonomous sublinear problem; secondly, we deal with a nonautonomous superlinear asymmetric one.
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D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at and superlinear at, Math. Z. 219 (1995), 499–513.
M. Arias and J. Campos, Exact number of solutions of a one-dimensional Dirichlet problem with jumping nonlinearities, Differential Equations and Dynamical Systems 5 (1997), 139–161.
A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Math. 152 (1984), 143–197.
A.K. Ben-Naoum and C. De Coster, On the existence and multiplicity of positive solutions of the p-Laplacian separated boundary value problem, Differential Integral Equations 10 (1997), 1093–11–12.
G.J. Butler, Periodic solutions of sublinear second order differential equations, J. Math. Anal. Appl. 62 (1978), 676–690.
A. Capietto and W. Dambrosio, Multiplicity results for some two-point superlinear asymmetric boundary value problem, Nonlinear Analysis TMA 38 (1999), 869–896.
A. Capietto and W. Dambrosio, Boundary value problems with sublinear conditions near zero, NoDEA 6 (1999), 149–172.
A. Capietto, W. Dambrosio and F. Zanolin, Infinitely many radial solutions to a boundary value problem in a ball, Quad. Dip. Mat. Univ. Torino, Annal. Mat. Pura Appl., to appear.
A. Castro and R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities, II, J. Math. Anal. Appl. 133 (1988), 509–528.
Y. Cheng, On the existence of radial solutions of a nonlinear elliptic bvp in an annulus, Math. Nachr. 165 (1994), 61–77.
D.G. Costa, D.G. De Figueiredo and P.N. Srikanth, The exact number of solutions for a class of ordinary differential equations through Morse index theory, J. Differential Equations 96 (1992), 185–199.
W. Dambrosio, Time-map techniques for some boundary value problem, Rocky Mountain J. Math. 28 (1998), 885–926.
W. Dambrosio, Boundary value problems for second order strongly nonlinear differential equations, Ph.D. thesis, 1998.
H. Dang, K. Schmitt and R. Shivaji, On the number of solutions of boundary value problems involving the p-Laplacian and similar nonlinear operators, Electr. J. Differential Equations 1 (1996), 1–9.
D.G. De Figueiredo and B. Ruf, On a superlinear Sturm-Liouville equation and a related bouncing problem, J. Reine Angew. Math. 421 (1991), 1–22.
G. Dinca and L. Sanchez, Multiple solutions of boundary value problems: an elementary approach via the shooting method, NoDEA 1 (1994), 163–178.
M.J. Esteban, Multiple solutions of semilinear elliptic problems in a ball, J. Differential Equations 57 (1985), 112–137.
C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Mat. 60 (1993), 266–276.
S. Fucik, Solvability of Nonlinear Equations and Boundary Value Problems, Reidel, Boston, 1980.
M. Garcia-Huidobro, R. Manasevich and F. Zanolin, Strongly nonlinear second order ODE’s with rapidly growing terms, J. Math. Anal. Appl. 202 (1996), 1–26.
M. Garcia-Huidobro, R. Manasevich and F. Zanolin, Infinitely many solutions for a Dirichlet problem with a non-homogeneous p-Laplacian like operator in a ball, Advances in Differential Equations 2 (1997), 203–230.
M. Garcia-Huidobro and P. Ubilla, Multiplicity of solutions for a class of nonlinear second-order equations, Nonlinear Analysis TMA 28 (1997), 1509–1520.
G. Harris and B. Zinner, Some remarks concerning exact solution numbers for a class of nonlinear boundary value problems, J. Math. Anal. Appl. 182 (1994), 571–588.
J. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1971), 983–996.
M.A. Krasnosel’skii, A.I. Perov, A.I. Povolotskii and P. P. Zabreiko, Plane Vector Fields, Academic Press, New York, 1966.
A.C. Lazer and P.J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 275–283.
J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Series, vol. 40, Amer. Math. Soc., Providence, RI, 1979.
J. Mawhin, C. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, preprint.
V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topological Methods in Nonlinear Analysis 10 (1997), 387–397.
Z. Opial, Sur les périodes des solutions de l’équation différentielle x” + g(x) = 0, Ann. Polon. Math. 10 (1961), 49–72.
R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, LNM 1458, Springer-Verlag, Berlin, 1990.
B.L. Shekhter, On existence and zeros of solutions of a nonlinear two-point boundary value problem, J. Math. Anal. Appl. 97 (1983), 1–20.
M. Struwe, Multiple solutions of anticoercive boundary value problems for a class of ordinary differential equations of second order, J. Differential Equations 37 (1980), 285–295.
B. Zinner, Multiplicity of solutions for a class of Superlinear Sturm-Liouville problems, J. Math. Anal. Appl. 176 (1993), 282–291.
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Capietto, A. (2001). On The Use of Time-Maps in Nonlinear Boundary Value Problems. In: Grossinho, M.R., Ramos, M., Rebelo, C., Sanchez, L. (eds) Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 43. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0191-5_14
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DOI: https://doi.org/10.1007/978-1-4612-0191-5_14
Publisher Name: Birkhäuser, Boston, MA
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Online ISBN: 978-1-4612-0191-5
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