Stability Theorems for a Class of Functional Differential Equations

The equations studied have the form \[ u = - mg(u(t)) - ∫_0^t {a(t - τ )} g(u(τ ))dτ + f(t), m 0, \] on a Hilbert space H. g can be nonlinear and unbounded. The case $m = 0$ was studied earlier by the author [9], [11]. a is to be a strongly positive kernel in the terminology of the earlier reports. It is shown that if g is a nonlinear elliptic operator of a special type and f = f₀ t + f₁ + ρ (t),ρ (t) → 0 as t → ∞, the equation has a generalized solution and that this solution has a finite limit as t → ∞. Implications in the theory of approach to steady state are discussed.