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A singular initial value problem for the equation u ( n ) ( x ) = g ( u ( x ) )

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

We consider the problem of the existence of positive solutions u to the problem u ( n ) ( x ) = g ( u ( x ) ) , u ( 0 ) = u ' ( 0 ) = . . . = u ( n - 1 ) ( 0 ) = 0 (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition δ 1 / s [ s / g ( s ) ] 1 / n d s < is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

A stability result for a class of nonlinear integrodifferential equations with L¹ kernels

Piermarco Cannarsa, Daniela Sforza (2008)

Applicationes Mathematicae

We study second order nonlinear integro-differential equations in Hilbert spaces with weakly singular convolution kernels obtaining energy estimates for the solutions, uniform in t. Then we show that the solutions decay exponentially at ∞ in the energy norm. Finally, we apply these results to a problem in viscoelasticity.

An abstract nonlinear second order differential equation

Jan Bochenek (1991)

Annales Polonici Mathematici

By using the theory of strongly continuous cosine families of linear operators in Banach space the existence of solutions of a semilinear second order differential initial value problem (1) as well as the existence of solutions of the linear inhomogeneous problem corresponding to (1) are proved. The main result of the paper is contained in Theorem 5.

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