Geodesic
Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions
Mathematica Bohemica, Tome 136 (2011) no. 4, pp. 337-356

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2011.141693
Classification : 34B08, 34B10, 34B15
Keywords: nontrivial solutions; nonhomogeneous boundary conditions; cone; Krein-Rutman theorem; Leray-Schauder degree 
Graef, John R.; Kong, Lingju; Kong, Qingkai; Yang, Bo. Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions. Mathematica Bohemica, Tome 136 (2011) no. 4, pp. 337-356. doi: 10.21136/MB.2011.141693
@article{10_21136_MB_2011_141693,
     author = {Graef, John R. and Kong, Lingju and Kong, Qingkai and Yang, Bo},
     title = {Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions},
     journal = {Mathematica Bohemica},
     pages = {337--356},
     year = {2011},
     volume = {136},
     number = {4},
     doi = {10.21136/MB.2011.141693},
     mrnumber = {2985544},
     zbl = {1249.34055},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141693/}
}
TY  - JOUR
AU  - Graef, John R.
AU  - Kong, Lingju
AU  - Kong, Qingkai
AU  - Yang, Bo
TI  - Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions
JO  - Mathematica Bohemica
PY  - 2011
SP  - 337
EP  - 356
VL  - 136
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141693/
DO  - 10.21136/MB.2011.141693
LA  - en
ID  - 10_21136_MB_2011_141693
ER  - 
%0 Journal Article
%A Graef, John R.
%A Kong, Lingju
%A Kong, Qingkai
%A Yang, Bo
%T Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions
%J Mathematica Bohemica
%D 2011
%P 337-356
%V 136
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2011.141693/
%R 10.21136/MB.2011.141693
%G en
%F 10_21136_MB_2011_141693

[1] Deimling, K.: Nonlinear Functional Analysis. Springer New York (1985). | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[5] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press Orlando (1988). | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[8] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (I). Math. Nachr. 278 (2005), 173-193. | DOI | MR | Zbl

[9] Kong, L., Kong, Q.: Second-order boundary value problems with nonhomogeneous boundary conditions (II). J. Math. Anal. Appl. 330 (2007), 1393-1411. | DOI | MR | Zbl

[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 | MR | Zbl

[11] Krasnosel'skii, M. A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press New York (1964). | MR

[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 | MR

[13] Ma, R.: Positive solutions for second-order three-point boundary value problems. Appl. Math. Lett. 14 (2001), 1-5. | DOI | MR | Zbl

[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 | MR

Cité par Sources :