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Positive solutions of the nonlinear fourth-order beam equation with three parameters. (English) Zbl 1077.34027

The authors study the existence of positive solutions to the beam equation \[ u^{(4)}(t)+\eta u''(t)-\zeta u(t)=\lambda f(t,u(t)) \tag{1} \] with the boundary conditions \[ u(0)=u(1)=u''(0)=u''(1)=0, \tag{2} \] where \(t\in (0,1)\) and \(\lambda\), \(\zeta\) and \(\eta\) are parameters. Using properties of the fixed-point index for completely continuous operators in cones, the authors show that the number of positive solutions of (1)–(2) is determined by the parameter \(\lambda\). Moreover, the monotone iterative technique is applied to obtain a uniqueness criterion for a positive solution of (1)–(2).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

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