## 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

### Keywords:

beam equation; positive solution; existence; uniqueness
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### References:

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