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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 \(x\geq 0.\)

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)
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References:

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