Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. (English) Zbl 1177.34030

Consider the boundary value problems \[ u^{(4)}(t)=f(t,u,u'),\;0<t<1,\tag{1} \]
\[ u(0)=u'(0)=0,\;u'''(1)=g(u(1)),\tag{2} \]
\[ u''(1)=0, \tag{3} \]
\[ u'(1)=0\tag{4} \] describing bending equilibra of elastic beams. Using the method of lower and upper solutions coupled with monotone iteration, the authors derive conditions on the functions \(f\) and \(g\) such that the boundary value problems \((1),(2),(3)\) and \((1),(2),(4)\) have monotone positive solutions. They also present numerical simulations of the problem \((1),(2),(3)\).


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI


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