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Positive solutions to a class of elastic beam equations with semipositone nonlinearity

Qingliu Yao (2010)

Annales Polonici Mathematici

Let h ∈ L¹[0,1] ∩ C(0,1) be nonnegative and f(t,u,v) + h(t) ≥ 0. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem u ( 4 ) ( t ) = f ( t , u ( t ) , u ' ( t ) ) , 0 < t < 1, u(0) = u’(0) = u’(1) =u”’(1) =0, where the nonlinear term f(t,u,v) may be singular at t=0 and t=1. By constructing a suitable cone and integrating certain height functions of f(t,u,v) on some bounded sets, several new results are obtained. In mechanics, the problem models the deflection of...

Post-buckling range of plates in axial compression with uncertain initial geometric imperfections

Ivan Hlaváček (2002)

Applications of Mathematics

The method of reliable solutions alias the worst scenario method is applied to the problem of von Kármán equations with uncertain initial deflection. Assuming two-mode initial and total deflections and using Galerkin approximations, the analysis leads to a system of two nonlinear algebraic equations with one or two uncertain parameters-amplitudes of initial deflections. Numerical examples involve (i) minimization of lower buckling loads and (ii) maximization of the maximal mean reduced stress.

Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping

Alexander Y. Khapalov (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We show that the set of nonnegative equilibrium-like states, namely, like ( y d , 0 ) of the semilinear vibrating string that can be reached from any non-zero initial state ( y 0 , y 1 ) H 0 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) , by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace L 2 ( 0 , 1 ) × { 0 } of L 2 ( 0 , 1 ) × H - 1 ( 0 , 1 ) . Our main results deal with nonlinear terms which admit at most the linear growth at infinity in y and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

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