##
**Symmetric solutions of a multi-point boundary value problem.**
*(English)*
Zbl 1085.34011

The author considers the second-order nonlinear multipoint boundary value problem
\[
-u''(t)=a(t)f(t,u(t),| u'(t)| ), \quad t\in (0,1),\tag{1}
\]

\[ u(0)=\sum_{i=1}^{n} \mu_{i}u(\xi_i),\tag{2} \]

\[ u(t)=u(1-t), \quad t\in [0,1],\tag{3} \]

with \(0<\xi_1<\xi_2<\cdots<\xi_n \leq \frac{1}{2},\) \(\mu_i>0, i=1,2,\cdots,n\) and \(\sum_{i=1}^{n} \mu_{i}<1, n\geq 2.\) It is assumed that the inhomogeneous term of (1) satisfies the following conditions: \(f\) is continuous and \(a\) belongs to \(L^p\) for some \(1\leq p \leq \infty.\) Moreover, \(a(t)\geq 0\) and, in fact, \(a(t)\geq m\) a.e. on \([0,1]\) for some positive constant \(m\). Adding other suitable growth conditions on \(f\), the author proves the existence of at least three symmetric positive solutions of the boundary value problem. In order to reach that, he transforms this question into an equivalent problem of showing the existence of fixed-points for the completely continuous operator \[ Su=\int_{0}^{1}G(t,s)a(s)f(s,u(s),| u'(t)| )ds \] whose kernel \(G(t,s)\) is precisely the Green function of the equation \(u''(t)=-g(t),\) \(g\in C[0,1],\) subject to conditions \((2)-(3),\) and then he applies a fixed-point result due to R.I. Avery and A.C. Peterson [Comput. Math. Appl. 42, 313-322 (2001; Zbl 1005.47051)].

\[ u(0)=\sum_{i=1}^{n} \mu_{i}u(\xi_i),\tag{2} \]

\[ u(t)=u(1-t), \quad t\in [0,1],\tag{3} \]

with \(0<\xi_1<\xi_2<\cdots<\xi_n \leq \frac{1}{2},\) \(\mu_i>0, i=1,2,\cdots,n\) and \(\sum_{i=1}^{n} \mu_{i}<1, n\geq 2.\) It is assumed that the inhomogeneous term of (1) satisfies the following conditions: \(f\) is continuous and \(a\) belongs to \(L^p\) for some \(1\leq p \leq \infty.\) Moreover, \(a(t)\geq 0\) and, in fact, \(a(t)\geq m\) a.e. on \([0,1]\) for some positive constant \(m\). Adding other suitable growth conditions on \(f\), the author proves the existence of at least three symmetric positive solutions of the boundary value problem. In order to reach that, he transforms this question into an equivalent problem of showing the existence of fixed-points for the completely continuous operator \[ Su=\int_{0}^{1}G(t,s)a(s)f(s,u(s),| u'(t)| )ds \] whose kernel \(G(t,s)\) is precisely the Green function of the equation \(u''(t)=-g(t),\) \(g\in C[0,1],\) subject to conditions \((2)-(3),\) and then he applies a fixed-point result due to R.I. Avery and A.C. Peterson [Comput. Math. Appl. 42, 313-322 (2001; Zbl 1005.47051)].

Reviewer: Antonio Linero Bas (Murcia)

### MSC:

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34B27 | Green’s functions for ordinary differential equations |

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

47H10 | Fixed-point theorems |

### Keywords:

multipoint boundary value problems; symmetric positive solutions; Green functions; fixed-point theorems; cone; real Banach space; nonnegative continuous convex (concave) functional### Citations:

Zbl 1005.47051
PDF
BibTeX
XML
Cite

\textit{N. Kosmatov}, J. Math. Anal. Appl. 309, No. 1, 25--36 (2005; Zbl 1085.34011)

Full Text:
DOI

### References:

[1] | Avery, R.I., A generalization of leggett – williams fixed point theorem, Math. sci. res. hot-line, 3, 9-14, (1999) · Zbl 0965.47038 |

[2] | Avery, R.I.; Davis, J.M.; Henderson, J., Three symmetric positive solutions for lidstone problems by a generalization of the leggett – williams theorem, Electron. J. differential equations, 2000, 1-15, (2000) · Zbl 0958.34020 |

[3] | Avery, R.I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. math. lett., 13, 1-7, (2000) · Zbl 0961.34014 |

[4] | Avery, R.I.; Henderson, J., Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. appl. nonlinear anal., 8, 27-36, (2001) · Zbl 1014.47025 |

[5] | Avery, R.I.; Peterson, A.C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. math. appl., 42, 313-322, (2001) · Zbl 1005.47051 |

[6] | Z. Bai, W. Ge, Existence of three positive solutions for some second-order boundary-value problems, Comput. Math. Appl., in press · Zbl 1066.34019 |

[7] | Bai, Z.; Wang, W.; Ge, W., Triple positive solutions for a class of two-point boundary-value problems, Electron. J. differential equations, 2004, 1-8, (2004) |

[8] | Gupta, C.P., A new a priori estimate for multi-point boundary-value problem, Electron. J. differential equations, conf., 07, 47-59, (2001) · Zbl 0980.34009 |

[9] | He, X.; Ge, W., Triple solutions for second-order three-point boundary value problems, J. math. anal. appl., 268, 256-265, (2002) · Zbl 1043.34015 |

[10] | Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the first kind for a sturm – liouville operator in its differential and finite difference aspects, Differential equations, 23, 803-810, (1987) · Zbl 0668.34025 |

[11] | Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liouville operator, Differential equations, 23, 979-987, (1987) · Zbl 0668.34024 |

[12] | E.R. Kaufmann, N. Kosmatov, A singular three-point boundary value problem, Comm. Appl. Anal., in press · Zbl 1102.34010 |

[13] | L. Kong, Q. Kong, Multi-point boundary value problems of second order differential equations (I), Nonlinear Anal., in press · Zbl 1066.34012 |

[14] | N. Kosmatov, Symmetric solutions of a multi-point boundary value problem at resonance, preprint · Zbl 1085.34011 |

[15] | Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 |

[16] | Ma, R., Positive solutions for second order three-point boundary-value problems, Appl. math. lett., 14, 1-5, (2001) · Zbl 0989.34009 |

[17] | Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.