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Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity. (English) Zbl 1249.34064
Summary: We study the existence of a solution to the nonlinear fourth-order elastic beam equation $u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)) \text{ a. e. } t\in [0,1]$ with nonhomogeneous boundary conditions $u(0)=a, \;u'(0)=b, \;u(1)=c, \;u''(1)=d,$ 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.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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