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Zbl 1005.47051
Avery, R.I.; Peterson, A.C.
Three positive fixed points of nonlinear operators on ordered Banach spaces.
(English)
[J] Comput. Math. Appl. 42, No.3-5, 313-322 (2001). ISSN 0898-1221

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 $f: \Bbb R\to \Bbb R$ is continuous and $f$ is nonnegative for $x\ge 0.$
[S.K.Ntouyas (Ioannina)]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
34B15 Nonlinear boundary value problems of ODE
39A05 General theory of difference equations
47N20 Appl. of operator theory to differential and integral equations
65J15 Equations with nonlinear operators (numerical methods)
65Q05 Numerical methods for functional equations

Keywords: fixed point theorems; difference equations; positive solutions; boundary; positive symmetric solutions; discrete second-order nonlinear conjugate boundary value problem

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