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Positive solutions for a fourth order boundary value problem. (English) Zbl 1081.34025

The following boundary value problem is considered: \[ u^{(4)}(t)=g(t)f(u(t)),\quad t\in [0,1],\qquad u(0)=u'(0)=u''(1)=u'''(1)=0. \tag{1} \] The author establishes several results on the existence of at least one positive solution to (1) applying the Krasnoselskii-Guo fixed-point theorem on cone expansion and compression in the space \(C[0,1]\) with the cones \[ P=\{v\in C[0,1]:v(1)\geq 0,a(t)v(1)\leq v(t)\leq tv(1), t\in[0,1]\} \] and \[ P_1=\{v\in C[0,1]:v(1)\geq 0, v \text{ nondecreasing on } [0,1],\;a(t)v(1)\leq v(t)\leq b_1(t)v(1), t\in[0,1]\}, \] with \(a(t)=\frac 32 t^2-\frac12 t^3\) and \(b_1(t)=2t^2-\frac43 t^3+\frac13 t^4\), \(t\in [0,1]\).

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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