Previous |  Up |  Next

# Article

Full entry | PDF   (0.3 MB)
Keywords:
nontrivial solutions; nonhomogeneous boundary conditions; cone; Krein-Rutman theorem; Leray-Schauder degree
Summary:
The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \gather u''+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber \\ u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \endgather Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term $f(t,x)$ may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of $f(t, x)/x$ for $x$ near $0$ and $\pm \infty$, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.
References:
[1] Deimling, K.: Nonlinear Functional Analysis. Springer New York (1985). MR 0787404 | Zbl 0559.47040
[2] Graef, J. R., Kong, L.: Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems. Nonlinear Anal. 68 (2008), 1529-1552. DOI 10.1016/j.na.2006.12.037 | MR 2388830 | Zbl 1139.34017
[3] Graef, J. R., Kong, L.: Existence results for nonlinear periodic boundary value problems. Proc. Edinb. Math. Soc., II. Ser. 52 (2009), 79-95. DOI 10.1017/S0013091507000788 | MR 2475882 | Zbl 1178.34024
[4] Graef, J. R., Kong, L.: Periodic solutions for functional differential equations with sign-changing nonlinearities. Proc. R. Soc. Edinb., Sect. A, Math. 140 (2010), 597-616. DOI 10.1017/S0308210509000523 | MR 2651375 | Zbl 1200.34077
[5] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press Orlando (1988). MR 0959889 | Zbl 0661.47045
[6] Guo, Y., Shan, W., Ge, W.: Positive solutions for second-order $m$-point boundary value problems. J. Comput. Appl. Math. 151 (2003), 415-424. DOI 10.1016/S0377-0427(02)00739-2 | MR 1956792 | Zbl 1026.34016
[7] Han, G., Wu, Y.: Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms. J. Math. Anal. Appl. 325 (2007), 1327-1338. DOI 10.1016/j.jmaa.2006.02.076 | MR 2270087 | Zbl 1111.34019
[8] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (I). Math. Nachr. 278 (2005), 173-193. DOI 10.1002/mana.200410234 | MR 2111808 | Zbl 1060.34005
[9] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (II). J. Math. Anal. Appl. 330 (2007), 1393-1411. DOI 10.1016/j.jmaa.2006.08.064 | MR 2308449 | Zbl 1119.34009
[10] Kong, L., Kong, Q.: Uniqueness and parameter-dependence of solutions of second order boundary value problems. Appl. Math. Lett. 22 (2009), 1633-1638. DOI 10.1016/j.aml.2009.05.009 | MR 2569054 | Zbl 1181.34021
[11] Krasnosel'skii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press New York (1964). MR 0159197
[12] Liu, L., Liu, B., Wu, Y.: Nontrivial solutions of $m$-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear terms. J. Comput. Appl. Math. 224 (2009), 373-382. DOI 10.1016/j.cam.2008.05.007 | MR 2474239
[13] Ma, R.: Positive solutions for second-order three-point boundary value problems. Appl. Math. Lett. 14 (2001), 1-5. DOI 10.1016/S0893-9659(00)00102-6 | MR 1758592 | Zbl 0989.34009
[14] Sun, W., Chen, S., Zhang, Q., Wang, C.: Existence of positive solutions to $n$-point nonhomogeneous boundary value problems. J. Math. Anal. Appl. 330 (2007), 612-621. DOI 10.1016/j.jmaa.2006.08.022 | MR 2302947

Partner of