We consider the nonlinear Volterra equation \[ ({V}) u(t) + (b * Au)(t) f(t),0 t < ∞ \] in the general setting b:[0,∞ ) → R a given kernel, A a nonlinear m-accretive operator on a real Banach space X,f:[0,∞ ) → X a given function and $ * $ the convolution. We study the existence of positive solutions of (V) and their asymptotic behavior as t → ∞, together with estimates of their rates of decay, under physically reasonable assumptions on b, A, f motivated by the problem of heat flow in materials with memory. The concept of complete positivity of the kernel b and its characterization play a crucial role in the analysis.
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Clément et al. (1981) studied this question.