### A Liapunov functional for a linear integral equation.

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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.

Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further...

2000 Mathematics Subject Classification: 45F15, 45G10, 46B38.An existence theorem is proved for a class of quasi-linear singular integro-differential equations with Cauchy kernel.

A class of non-linear singular integral equations with Hilbert kernel and a related class of quasi-linear singular integro-differential equations are investigated by applying Schauder's fixed point theorem in Banach spaces.

The paper studies a construction of nontrivial solution for a class of Hammerstein–Nemytskii type nonlinear integral equations on half-line with noncompact Hammerstein integral operator, which belongs to space ${L}_{1}(0,+\infty )\cap {L}_{\infty}(0,+\infty )$. This class of equations is the natural generalization of Wiener-Hopf type conservative integral equations. Examples are given to illustrate the results. For one type of considering equations continuity and uniqueness of the solution is established.