Abstract
The method of lower and upper solutions deals mainly with existence results for boundary value problems. In this presentation, we will restrict attention to second order ODE problems with separated boundary conditions. Although some of the ideas can be traced back to E. Picard [16], the method of lower and upper solutions was firmly established by G. Scorza Dragoni [20]. This 1931 paper considered upper and lower solutions which are C2; in 1938, the same author extended his method to the L1;-Carathéodory case [21]. Upper and lower solutions with corners were considered by M. Nagumo in 1954 [13]. Since then a multitude of variants have been introduced. The Definitions 2.1 and 3.1 we present here tend to be general enough for applications and simple enough to model the geometric intuition built into the concept.
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References
K. Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45–62.
H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346–384.
H. Amann, A Ambrosetti and G. Mancini, Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities, Math. Z. 158 (1978), 179–194.
S. Bernstein, Sur certaines équations différentielles ordinaires du second ordre, C.R.A.S. Paris 138 (1904), 950–951.
C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations (F. Zanolin, ed.), C.I.S.M. Courses and Lectures 371, Springer-Verlag, New York (1996), 1–79.
C. De Coster and M. Henrard, Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. Differential Equations 145 (1998), 420–452.
P. Habets and F. Zanolin, Positive solutions for a class of singular boundary value problem, Boll. U.M.I. 9-A (1995), 273–286.
I.T. Kiguradze, A priori estimates for derivatives of bounded functions satisfying second-order differential inequalities, Differentsial’nye Uravneniya 3 (1967), 1043–1052.
I.T. Kiguradze, Some singular boundary value problems for ordinary nonlinear second order differential equations, Differentsial’nye Uravneniya 4 (1968), 1753–1773.
Y.S. Kolesov, Periodic solutions of quasilinear parabolic equations of second order, Trans. Moscow Math. Soc. 21 (1970), 114–146.
W. Mlak, Parabolic differential inequalities and Chaplighin’s method, Ann. Polon. Math. 8 (1960), 139–152.
M. Nagumo, Uber die differentialgleichung y“ = ƒ (t, y, y’), Proc. PhysMath. Soc. Japan 19 (1937), 861–866.
M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (1954), 207–229.
G. Peano, Sull’integrabilità delle equazioni differenziali di primo or-dine, Atti Acad. Torino 21 (1885), 677–685.
O. Perron, Ein neuer existenzbeweis für die integrale der differentialgleichung y’ = f (x, y), Math. Ann. 76 (1915), 471–484.
E. Picard, Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires, J. de Math. 9 (1893), 217–271.
G. Prodi, Teoremi di esistenza per equazioni alle derivate parziali non lineari di tipo parabolico, Nota I e II, Rend. Ist. Lombardo 86 (1953), 1–47.
A. Rosenblatt, Sur les théorèmes de M. Picard dans la théorie des problèmes aux limites des équations différentielles ordinaires non linéaires, Bull. Sc. Math. 57 (1933), 100–106.
K. Schmitt, Boundary value problems for quasilinear second order elliptic equations, Nonlinear Anal. T.M.A. 2 (1978), 263–309.
G. Scorza Dragoni, Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine, Giornale di Mat. (Battaglini) 69 (1931), 77–112.
G. Scorza Dragoni, Intorno a un criterio di esistenza per un problema di valori ai limiti, Rend. Semin. R. Accad. Naz. Lincei 28 (1938), 317–325.
G.M. Troianiello, On solutions to quasilinear parabolic unilateral problems, Boll. U.M.I. 1-B (1982), 535–552.
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De Coster, C., Habets, P. (2001). An Overview of the Method of Lower and Upper Solutions for ODEs. 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_1
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