Geodesic
Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity
Applications of Mathematics, Tome 56 (2011) no. 6, pp. 543-555.

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We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions \[ \begin {cases} u^{(4)}(t)=f\bigl (t,u(t),u'(t),u''(t),u'''(t)\bigr ),\quad \text {a.e.} \ t\in [0,1],\\ u(0)=a, \ u'(0)=b, \ u(1)=c, \ u''(1)=d, \end {cases} \] where the nonlinear term $f(t,u_{0},u_{1},u_{2},u_{3})$ is a strong Carathéodory function. By constructing suitable height functions of the nonlinear term $f(t,u_{0},u_{1},u_{2},u_{3})$ on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.
DOI : 10.1007/s10492-011-0032-1
Classification : 34B15, 34B16, 47N20, 74K10
Keywords: nonlinear ordinary differential equation; boundary value problem; existence; fixed point theorem 
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Yao, Qingliu. Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity. Applications of Mathematics, Tome 56 (2011) no. 6, pp. 543-555. doi : 10.1007/s10492-011-0032-1. http://geodesic.mathdoc.fr/articles/10.1007/s10492-011-0032-1/

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