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In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε. [ABSTRACT FROM AUTHOR]

We correct assumption (4.6) in Theorem 4.1 in [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 2981-3012], and we consequently change the proof of Theorem 4.1. We correct the proof of Lemma 5.4 in [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 2981-3012]. Moreover, throughout the paper, starting from formulas (3.13) and (3.14), we notice that the current cost of the control problem cannot depend on the state for the results of the paper to work. This change in the current cost must be made in the paper [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 3013-3038], too. [ABSTRACT FROM AUTHOR]

We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math., 133 (2016), pp. 525-555], [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2771-2793], and [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794-2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable to such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical since it is directly linked to the course of time. Finally, we prove that the finite element method is convergent, while we limit ourselves to commenting on the infeasibility of this approach as far as the spectral element method is concerned. [ABSTRACT FROM AUTHOR]

We give a sufficient condition for a chemical reaction network with time delays to be persistent, that is, any positive solution to the delay differential equation describing the time evolution of concentrations of species in the system does not approach the boundary of the positive orthant. We also prove that persistence implies consistence for reaction network with delay. These results are natural analogues of the theorems proven for chemical reaction network without time delay [D. Angeli, P. D. Leenheer, and E. D. Sontag, Math. Biosci., 210 (2007), pp. 598--618; D. Angeli, P. D. Leenheer, and E. D. Sontag, SIAM J. Appl. Math., 71 (2011), pp. 128--146]. Our results are applied to proving an example network to be persistence, which network is derived from a biochemical reaction network composed of cardiac hypertrophy factors. [ABSTRACT FROM AUTHOR]