572 NGUYEN HOANG LOC AND KLAUS SCHMITT I T zr tic 12 v K Le and K. Schmitt Sub-supersolution theorems ,or m)ear; A variational approdch, Elect. J. Diff. Equations 2004 {2004 , 1- · . . 13. - --, Some general concepts of sub-supersolutions for nonlmear elhpttc problems, Topol. Methods Nonlin. Anal. 28 (2006), 87-103. . . 14 G M Lieberman The natural generalization of the . . . , . C Partta Jueren ta Ladyzhenskaya and Ural'tseva for elliptic equatwns, omm. Equations 16 (1991), 311- 361. · · (I..., IP- 2\7 ) + ..\ l uiP-2u = 0 Proc. Amer. 15. P. Lindqvist, On the equatwn d'v v u u , Math. Soc. 109 {1990), 157-164. . . 16. s. Sakaguchi, Concavity properties of elliptic Dirichlet problems, Ann. Scuola Norm. up. tsa · .!. • • • 17. S. Sun and S. Wu, Iterative solution for a singular nonlmear elltpttc problem, Appl. Math. Comput. 118 {2001), 53-62. . 18 y Sun S. Wu andY. Long, Combined effects of superhr:ear nonline;rities' in some singular boundary value problems, J. Dtfferential EquatiOns 176 (2001), 511-531. · · l I some quasilinear elliptic 19. J.L. Vazquez, A strong maximum prmctp e ,or equations, Appl. Math. Optim. 12 (1984), 191- 202. . 20. M. Yao and J. Zhao, Positive solution of a singular non-lmear elliptic boundary value problem, Appl. Math. Comput. 148 (2004), 773-782. . 21 z Zhang and J. Cheng, Existence and optimal estimates of solutwns for nonlinear Dirichlet problems, Nonlinear Anal. 57 {2004), 473-484. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY, UT 84112 Email address : loc@math. utah.edu 155 SOUTH 1400 EAST, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, 155 SOUTH 1400 EAST, SALT LAKE C ITY, UT 84112 Email address: schmitt@math. utah .edu ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 41, Number 2, 2011 NAGUMO CONDITIONS AND SECOND-ORDER QUASILINEAR EQUATIONS WITH COMPATIBLE NONLINEAR FUNCTIONAL BOUNDARY CONDITIONS JEAN MAWHIN AND H.B. THOMPSON Dedicated to the memory of Lloyd Jackson ABSTRACT. We establish existence results for solutions to nonlinear functional boundary value problems for nonlinear second-order ordinary differential equations assuming there are lower and upper solutions and the right side satisfies a Nagumo growth bound. Our results contain as special cases many results for the p- and <f>-Laplacians as well as many results where the boundary conditions depend on n-points or even functionals. 1. Introduction. (1) - = f(t,x,x(t),x'(t)), for a.e. t E [0,1], subject to general functional boundary conditions of the form (2) G(x(O), x(1) , x, x'(O), x' (1)) = (0, 0), where cp E C([O, 1] x C[O, 1] x R 2 ), f : [0, 1J x C[O, 1] x R2 -+ R satisfies the Carathedory conditions and G E C(R 2 x C[O, 1] x R 2; R2). Our assumptions on cp and f are due to Cabada and Pouso [7]. By a solution x we mean a function x E C 1 [0, 1 J satisfying (2) such that cp(t,x,x(t),x'(t)) is absolutely continuous and satisfies (1) almost everywhere on [0, 1]. We assume that there are ordered lower and upper solutions, a and (3, respectively, for (1) and that the functional boundary conditions are compatible, in a sense defined below. The assumptions on cp are sufficiently general to apply to (r(t)x' + q(t)x)' = f(t, x, x(t), x'(t)) 2010 AMS Mathematics 'subject classification. Primary 34B10, 34B15. Received by the editors on June 20, 2010. DOI: l0. 1216/RMJ-2011-4 1-2-573 Copyright @2011 Rocky Mountain Mathematics Consortium 573