Reducing increasing monotonicity of kernels (Q1061341)

Consider the equation \(f(t)=1-\int^{t}_{0}K(t-s)f(s)ds,\) \(t\geq 0\) where K is given and f is the unknown. Let \(K(t)>0\), \(K'(t)>0\), \(K''(t)<0\), ln K'(t) convex, \(t\geq 0\), and assume \(K(0)+K''(0)[K'(0)]^{-1}\geq 2[K'(0)]^{1/2}.\) The author shows how this equation can be transformed to the equation \(f(t)=e^{-\gamma t}- \int^{t}_{0}L(t-s)f(s)ds\) where \(L(t)>0\), \(L'(t)<0\) and ln L(t) is convex, i.e. to an equation with a kernel of positive type (f is the same in both case). Further results and the general case are also presented. The article continues an earlier work by the author [J. Math. Anal. Appl. 64, 381-397 (1978; Zbl 0403.45001)].