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Solvability of a fourth-order beam equation with all-order derivatives. (English) Zbl 1174.34365
Summary: By using the Leray-Schauder fixed point theorem and choosing a suitable equivalent norm, two existence theorems are established for a nonlinear fourth-order elastic beam equation with all-order derivatives. This equation describes the equilibrium state of an elastic beam simply supported at both ends. The main results show that the equation has at least one solution provided the “height” of the nonlinear term is appropriate on a bounded set.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 47N20 Applications of operator theory to differential and integral equations