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Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń, Ireneusz Kubiaczyk (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we prove an existence theorem for the Hammerstein integral equation x ( t ) = p ( t ) + λ I K ( t , s ) f ( s , x ( s ) ) d s , where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

Exponential stability of a flexible structure with history and thermal effect

Roberto Díaz, Jaime Muñoz, Carlos Martínez, Octavio Vera (2020)

Applications of Mathematics

In this paper we study the asymptotic behavior of a system composed of an integro-partial differential equation that models the longitudinal oscillation of a beam with a memory effect to which a thermal effect has been given by the Green-Naghdi model type III, being physically more accurate than the Fourier and Cattaneo models. To achieve this goal, we will use arguments from spectral theory, considering a suitable hypothesis of smoothness on the integro-partial differential equation.

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