Geodesic
Existence of one-signed solutions of nonlinear four-point boundary value problems
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 593-612
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ -u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ) $$ and $$ u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ), $$ where $\varepsilon \in (0,{1}/{2})$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C([0,1],(0,+\infty ))$, $f\in C(\mathbb {R},\mathbb {R})$ with $sf(s)>0$ for $s\neq 0$. The proof of the main results is based upon bifurcation techniques.
In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ -u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ) $$ and $$ u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ), $$ where $\varepsilon \in (0,{1}/{2})$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C([0,1],(0,+\infty ))$, $f\in C(\mathbb {R},\mathbb {R})$ with $sf(s)>0$ for $s\neq 0$. The proof of the main results is based upon bifurcation techniques.
DOI : 10.1007/s10587-012-0052-3
Classification : 34B10, 34B15, 34B18, 34C23
Keywords: four-point boundary value problem; one-signed solution; bifurcation method 
@article{10_1007_s10587_012_0052_3,
     author = {Ma, Ruyun and Chen, Ruipeng},
     title = {Existence of one-signed solutions of nonlinear four-point boundary value problems},
     journal = {Czechoslovak Mathematical Journal},
     pages = {593--612},
     year = {2012},
     volume = {62},
     number = {3},
     doi = {10.1007/s10587-012-0052-3},
     mrnumber = {2984621},
     zbl = {1265.34053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0052-3/}
}
TY  - JOUR
AU  - Ma, Ruyun
AU  - Chen, Ruipeng
TI  - Existence of one-signed solutions of nonlinear four-point boundary value problems
JO  - Czechoslovak Mathematical Journal
PY  - 2012
SP  - 593
EP  - 612
VL  - 62
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0052-3/
DO  - 10.1007/s10587-012-0052-3
LA  - en
ID  - 10_1007_s10587_012_0052_3
ER  - 
%0 Journal Article
%A Ma, Ruyun
%A Chen, Ruipeng
%T Existence of one-signed solutions of nonlinear four-point boundary value problems
%J Czechoslovak Mathematical Journal
%D 2012
%P 593-612
%V 62
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0052-3/
%R 10.1007/s10587-012-0052-3
%G en
%F 10_1007_s10587_012_0052_3
Ma, Ruyun; Chen, Ruipeng. Existence of one-signed solutions of nonlinear four-point boundary value problems. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 593-612. doi: 10.1007/s10587-012-0052-3

[1] Chu, J., Sun, Y., Chen, H.: Positive solutions of Neumann problems with singularities. J. Math. Anal. Appl. 337 (2008), 1267-1272 \MR 2386375. | DOI | MR | Zbl

[2] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) \MR 0787404. | MR | Zbl

[3] Jiang, D., Liu, H.: Existence of positive solutions to second order Neumann boundary value problems. J. Math. Res. Expo. 20 (2000), 360-364. | MR | Zbl

[4] Li, X., Jiang, D.: Optimal existence theory for single and multiple positive solutions to second order Neumann boundary value problems. Indian J. Pure Appl. Math. 35 (2004), 573-586. | MR | Zbl

[5] Li, Z.: Positive solutions of singular second-order Neumann boundary value problem. Ann. Differ. Equations 21 (2005), 321-326. | MR | Zbl

[6] Ma, R., Thompson, B.: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Anal., Theory Methods Appl. 59 (2004), 707-718. | DOI | MR | Zbl

[7] Miciano, A. R., Shivaji, R.: Multiple positive solutions for a class of semipositone Neumann two-point boundary value problems. J. Math. Anal. Appl. 178 (1993), 102-115. | DOI | MR | Zbl

[8] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487-513. | DOI | MR | Zbl

[9] Rachůnková, I., Staněk, S., Tvrdý, M.: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi Publishing Corporation, New York (2008). | MR

[10] Sun, J., Li, W.: Multiple positive solutions to second-order Neumann boundary value problems. Appl. Math. Comput. 146 (2003), 187-194. | DOI | MR | Zbl

[11] Sun, J., Li, W., Cheng, S.: Three positive solutions for second-order Neumann boundary value problems. Appl. Math. Lett. 17 (2004), 1079-1084. | DOI | MR | Zbl

[12] Sun, Y., Cho, Y. J., O'Regan, D.: Positive solution for singular second order Neumann boundary value problems via a cone fixed point theorem. Appl. Math. Comput. 210 (2009), 80-86 \MR 2504122. | DOI | MR

Cité par Sources :