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The method of lower and upper solutions for a bending of an elastic beam equation. (English) Zbl 1016.34010
The author considers the fourth-order nonlinear boundary value problem $u^{(4)}(x)= f(x, u(x), u''(x)),\quad 0< x< 1,\quad u(0)= u(1)= u''(0)= u''(1)= 0.$ An appropriate maximum principle for the linear case is obtained and used to prove the existence of monotone sequences of functions that converge to solutions to the nonlinear problem. It is assumed that $$f(x,u,v)$$ is continuous and satisfies some inequalities, which can be considered as a relaxation of monotonicity in $$u$$, $$v$$. No growth restrictions are imposed on $$f$$. An example is given.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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