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Existence, uniqueness and asymptotic behavior for solutions of the nonlinear beam equation. (English) Zbl 0704.45013
Of concern is the existence, uniqueness and asymptotic behavior of solutions to an initial value problem for the second-order equation \[ Ku''(t)+A^ 2u(t)+M(| A^{1/2}u(t)|^ 2)\quad Au(t)+u'(t)=0,\quad t\geq 0, \] in a Hilbert space H. Here \(| \cdot |\) denotes the norm in H, K: \(H\to H\) is linear monotone, A is a given linear unbounded operator of H, and M is a real function on \([0,\infty)\). The author extends earlier results by P. Biler [ibid. 10, 839-842 (1986; Zbl 0611.35057)], and E. H. Brito [ibid. 8, 1489-1496 (1984; Zbl 0524.35026); ibid. 11, 125-137 (1987; Zbl 0613.34013)].
Reviewer: S.Aizicovici

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45K05 Integro-partial differential equations
34G20 Nonlinear differential equations in abstract spaces
35L70 Second-order nonlinear hyperbolic equations
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