Some new bounds for the blow-up time of solutions for certain nonlinear Volterra integral equations
The author studies the nonlinear Volterra integral equation \N\[\Nu(t)= \int_0^t k(t-s)r(s)g\bigl (u(s)+h(s) \bigr)\, ds , \qquad t\geq 0, \N\]\Nand derives some new new upper and lower bounds for the blow-up time of solutions to this equation. Here it is assumed (as in [\textit{C. A. Roberts} et al., J. Integral Equations Appl. 5, No. 4, 531--546 (1993; Zbl 0804.45002)]) that \(g(u)>0\), \(g'(u)>0\), \(g''(u)>0\) for \(u>0\) and \(r(t)>0\), \(r'(t)\geq 0\), \(0< h_0\leq h(t) \leq h_\infty <\infty\), \(h'(t)\geq 0\), \(k(t)>0\) and \(k'(t)< 0\) for \(t>0\).\N\NSome of the bounds are given in closed form and others as the unique solution to a certain nonlinear equation. A crucial part of the argument is the use of the fact that if \(f_1\) and \(f_2\) are nondecreasing functions and \(X\) is a random variable, then the covariance of \(f_1(X)\) and \(f_2(X)\) (provided it exists) is non-negative.
scientific article; zbMATH DE number 7866536
scientific article