Nonconvolution nonlinear integral Volterra equations with monotone operators. (English) Zbl 1088.45004

The authors study nonlinear Volterra integral equations of the form \[ u(x) = \int_0^x k(x,s)g(u(s))\, ds, \quad t \geq 0, \] where \(k\) is locally bounded, the function \(x\mapsto \int_0^x k(x,s)\, ds\) is strictly increasing and \(g\) is continuous, and strictly increasing. They prove results about existence, uniqueness and attractive behaviour of positive solutions to this equation by showing that under certain additional assumptions the solutions behave in the same way as solutions to convolution equations. Finally some examples are given showing the wide range of situations that appear if the nonconvolution kernel is not assumed to be locally bounded.


45G10 Other nonlinear integral equations
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