Constrained Mesh Methods for Functional Differential Equations

Consider the following Volterra Delay Integro Differential Equation (VDIDE): $$ y'(t) = f(t,y(t),\int\limits_{{t_0}}^t {k(t,s,y(s))ds,y(t - \tau (t)),\int\limits_{{t_0}}^{t - \sigma (t)} {k'} (t,s,y(s))ds} ) $$ (1) with initial conditions: $$ \begin{gathered} y({t_0}) = {y_0} \hfill \\ and\quad y(t): = g(t)\quad for\,\;t < {t_0} \hfill \\ \end{gathered} $$ where y: [to,b]→ℝn, f∈[to,b]×ℝ4n→ℝn and both K,K′ map: {(t,s): to≦s≦t≦b} × ℝn→ℝn. Moreover assume the delays τ and σ are continuous and strictly positive. For K=K′=0 (1) reduces to a Delay Differential Equation (DDE), and for τ=σ=0 to a Volterra Integro Differential Equation (VIDE).

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