A Theorem on Rearrangements and Its Application to Certain Delay Differential Equations

Uniform Stability In Nonlinear Infinite Delay Volterra Integro-differential Equations Using Lyapunov Functionals

“…The function g(x) : R → R and is continuous in x. Equation (1) has its roots in the study of nuclear reactors and the stability of its zero solution was studied by Brownwell and Ergen [1] in 1954. Later on, the same study was revisited by Nohel [9], in 1960 and then by Levin and Nohel [8], in 1964.…”

“…Recently, in [2] and [3], Burton used the notion of xed point theory to alleviate some of the di culties that arise from the use of Lyapunov functionals and obtained results concerning the stability and asymptotic stability of the zero solution of (1) when it is scalar. As it is mentioned above, in [10] the author obtained results concerning the exponential stability of (1), which generalized the papers of [1], [8] and [9]. In [10], the author proposed the open problem of extending the results of [10] to the in nite delay nonlinear Volterra integro-di erential equation…”

Fixed points, stability, and harmless perturbations

“…All of these are important classical problems and are not merely contrived to make our point here. Equation (2.1) was studied by Volterra [25] in connection with a biological application, by Ergen [11] and Brownell and Ergen [4] in the study of a circulating-fuel nuclear reactor, by Levin and Nohel [22] in numerous contexts, and by Hale [16, page 458] in stability theory, all with convex kernels. We ask much less on the kernels here, but more on g. Equation (2.1) was studied by MacCamy and Wong [23, page 16] concerning positive kernel theory and they note that their methods fail to establish stability for that equation.…”

Fixed points and stability of a nonconvolution equation

“…For (2) it is supposed that xg(x) > 0 for small x = 0, plus other conditions including (3) and (4), to ensure stability and asymptotic stability.…”

“…In 1954 Brownell and Ergen [2] studied a form of (2) in connection with reactor dynamics. Nohel [10] picked up that work in 1960 and four years later was joined by Levin [9].…”