##
**Three positive fixed points of nonlinear operators on ordered Banach spaces.**
*(English)*
Zbl 1005.47051

The authors generalize the triple fixed-point theorem of Leggett and Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. As an application of the abstract result, the authors prove the existence of three positive symmetric solutions of the discrete second-order nonlinear conjugate boundary value problem
\[
\Delta^2 x(t-1)+f(x(t))=0, \text{for all} t\in [a+1,b+1],
\]

\[ x(a)=0=x(b+2), \] where \(f: \mathbb R\to \mathbb R\) is continuous and \(f\) is nonnegative for \(x\geq 0.\)

\[ x(a)=0=x(b+2), \] where \(f: \mathbb R\to \mathbb R\) is continuous and \(f\) is nonnegative for \(x\geq 0.\)

Reviewer: S.K.Ntouyas (Ioannina)

### MSC:

47H10 | Fixed-point theorems |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

39A05 | General theory of difference equations |

47N20 | Applications of operator theory to differential and integral equations |

65J15 | Numerical solutions to equations with nonlinear operators |

65Q05 | Numerical methods for functional equations (MSC2000) |

### Keywords:

fixed point theorems; difference equations; positive solutions; boundary; positive symmetric solutions; discrete second-order nonlinear conjugate boundary value problem
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\textit{R. I. Avery} and \textit{A. C. Peterson}, Comput. Math. Appl. 42, No. 3--5, 313--322 (2001; Zbl 1005.47051)

Full Text:
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### References:

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