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Schauder linearization technique; Schauder differential equation; functional boundary conditions; boundary value problem
Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle $ with the sup norm and $\alpha ,\beta \in X \rightarrow {R}$ be continuous increasing functionals, $\alpha (0)= \beta (0)=0$. This paper deals with the functional differential equation (1) $x^{\prime \prime \prime } (t) = Q [ x, x^\prime , x^{\prime \prime }(t)] (t)$, where $Q:{X}^2 \times {R} \rightarrow {X}$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha (x)=0=\beta (x^\prime )$, $x^{\prime \prime }(1)-x^{\prime \prime }(0)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional differential equations.
[1] Castro A., and Shivaji R.: Nonnegative solutions for a class of radially symmetric non- positone problems. Proc. Amer. Math. Soc., in press.
[2] Chiappinelli R., Mawhin J. and Nugari R.: Generalized Ambrosetti - Prodi conditions for nonlinear two-point boundary value problems. J. Differential Equations 69 (1987), 422-434. MR 0903395
[3] Dancer E.N. and Schmitt K.: On positive solutions of semilinear elliptic equations. Proc. Amer. Soc. 101 (1987), 445-452. MR 0908646
[4] Ding S.H. and Mawhim J.: A multiplicity result for periodic solutions of higher order ordinary differential equations. Differential and Integral Equations 1, 1 (1988), 31-39. MR 0920487
[5] Fabry C., Mawhin J. and Nkashama M.N.: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 18 (1986), 173-180. MR 0818822
[6] Gaete S. and Manasevich R.F.: Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear elasticity, via variational method. J. Math. Anal. Appl. 134 (1988), 257-271. MR 0961337
[7] Islamov G. and Shneiberg I.: Existence of nonnegative solutions for linear differential equations. J. Differential Equations 16 (1980), 237-242. MR 0569766
[8] Kolesov J.: Positive periodic solutions of a class of differential equations of the second order. Soviet Math. Dokl. 8 (1967), 68-79. MR 0206400 | Zbl 0189.38902
[9] Kiguradze I.T. and Půža B.: Some boundary-value problems for a system of ordinary differential equation. Differentsial’nye Uravneniya 12, 12 (1976), 2139-2148. (Russian) MR 0473302
[10] Kiguradze I.T.: Boundary Problems for Systems of Ordinary Differential Equations. Itogi nauki i tech. Sovr. problemy mat. 30 (1987), Moscow. (Russian)
[11] Nkashama M.N.: A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations. J. Math. Anal. Appl. 140 (1989), 381-395. MR 1001864 | Zbl 0674.34009
[12] Nkashama M.N. and Santanilla J.: Existence of multiple solutions for some nonlinear boundary value problems. J. Differential Equations 84 (1990), 148-164. MR 1042663
[13] Mawhin J.: First order ordinary differential equations with several solutions. Z. Angew. Math. Phys. 38 (1987), 257-265. MR 0885688
[14] Rachůnková I.: Multiplicity results for four-point boundary value problems. Nonlinear Analysis, TMA 18 5 (1992), 497-505. MR 1152724
[15] Ruf B. and Srikanth P.N.: Multiplicity results for ODE’s with nonlinearities crossing all but a finite number of eigenvalues. Nonlinear Analysis, TMA 10 2 (1986), 157-163. MR 0825214
[16] Santanilla J.: Nonnegative solutions to boundary value problems for nonlinear first and second order differential equations.  J. Math. Anal. Appl. 126 (1987), 397-408. MR 0900756
[17] Schaaf R. and Schmitt K.: A class of nonlinear Sturm-Liouville problems with infinitely many solutions. Trans. Amer. Math. Soc. 306 (1988), 853-859. MR 0933322
[18] Schmitt K.: Boundary value problems with jumping nonlinearities. Rocky Mountain J. Math. 16 (1986), 481-496. MR 0862276 | Zbl 0631.35032
[19] Smoller J. and Wasserman A.: Existence of positive solutions for semimilear elliptic equations in general domains. Arch. Rational Mech. Anal. 98 (1987), 229-249. MR 0867725
[20] Šenkyřík M.: Existence of multiple solutions for a third-order three-point regular boundary value problem. preprint. MR 1293243
[21] Vidossich G.: Multiple periodic solutions for first-order ordinary differential equations. J. Math. Anal. Appl. 127 (1987), 459-469. MR 0915071 | Zbl 0652.34050
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