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Weak solutions of differential equations in Banach spaces

Mieczysław Cichoń (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we prove a theorem for the existence of pseudo-solutions to the Cauchy problem, x' = f(t,x), x(0) = x₀ in Banach spaces. The function f will be assumed Pettis-integrable, but this assumption is not sufficient for the existence of solutions. We impose a weak compactness type condition expressed in terms of measures of weak noncompactness. We show that under some additionally assumptions our solutions are, in fact, weak solutions or even strong solutions. Thus, our theorem is an essential...

Well posedness and control of semilinear wave equations with iterated logarithms

Piermarco Cannarsa, Vilmos Komornik, Paola Loreti (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.

Wirtinger inequality and nonlinear differential systems

Jaroslav Jaroš (2013)

Archivum Mathematicum

Picone identity for a class of nonlinear differential equations is established and various qualitative results (such as Wirtinger-type inequality and the existence of zeros of first components of solutions) are obtained with the help of this new formula.

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