Nonlinear integrodifferential equations of the form \[ \tau \dot \varphi (t) = 3D - \varphi (t) - \int^t_0 m(t - t') \dot \varphi (t') dt', \quad m(t) = 3DF \bigl( \varphi (t) \bigr),\;\varphi (0) = 3D1, \tag{1} \] are studied, where \(\tau\) is a real positive number, \(\varphi\) is a real function defined on \(\mathbb{R}^+\), of class \(C^1\) and \(F\) is an absolutely monotone function on \([0,1 + d^*)\), \(d^* > 0\). Equations related to (1) arise in probability theory and in statistical mechanics theory of many particle systems, in connection with Newtonian friction.
The authors prove some general properties of (1), such as existence and uniqueness of solutions and their possible representations as a power series and as a Fourier transform of some measure, the long time limit of them, etc. A special attention is dedicated to show bifurcation scenarios and to reveal the main differences with respect to those known, e.g., for ordinary differential equations. Also, some generalizations of (1) are studied, such as systems of equations and the case where the function \(F\) may depend on some control parameter vector \(v \in K \subset\mathbb{R}^n\).