CONVEXITY AND QUADRATIC MONOTONE APPROXIMATION IN DELAY DIFFERENTIAL EQUATIONS

. In this paper the method of quasiliniarization, an application of Newton’s method, recently generalized in [1], is used for the quadratic, monotonic, bilateral approximation of the solution of the delay problem (5). The result is applied to an integral equation from biomathematics.


INTRODUCTION
In the papers [3]- [6] we investigated the following delay integral equation (1) x (t) = t t−τ f (s, x (s)) ds arising naturally from the study of the spread of virus diseases or, more generally, of the growth of single species populations.For example, we dealt with the initial values problem for (1) and looked for a continuous solution x (t) of (1), for −τ ≤ t ≤ T , when it is known that (2) x (t) = ϕ (t) for − τ ≤ t ≤ 0.
We obtained several existence and approximation results for the solutions of ( 4) by means of the monotone iterative method, assuming that f (t, x) is monotone (increasing or decreasing) with respect to x.Such a result is the following one: and there is L ≥ 0 such that Then (4) has a unique solution x ∈ C 1 [0, T ] such that u 0 ≤ x ≤ v 0 , and x n , y n → x uniformly on [0, T ], where x 0 = u 0 , y 0 = v 0 and Moreover, the sequences (x 2n ) and (y 2n+1 ) are increasing, while (x 2n+1 ) and (y 2n ) are decreasing.
Unfortunately, the convergence of the sequences (x n ) and (y n ) is only linear, more exactly In this paper we prove that, if f is also convex, then there exist two monotone sequences (u n ) and (v n ) whose members are solutions of some linear equations, that converge quadratically to the unique solution of (4) from both directions.We say that the convergence of (u n ) and (v n ) is quadratic provided that We succeed this by adapting to (4) the recent quasilinearization method used in [1] for equations without delay.A second ingredient is the step method which is well known in the theory of delay equations.

RESULTS
We shall discuss a more general problem of type (4), namely (5) x and (H2) f ∈ C(Ω), g ∈ C( Ω), the derivatives f x , f xx , g x and g xx exist and are continuous on Ω, and satisfy (6) f xx ≥ 0, g x ≥ 0 and g xx ≤ 0 on Ω.
Then there exist the sequences (u n ) increasing and (v n ) decreasing which converge uniformly on [0, T ] to the unique solution x ∈ C 1 [0, T ] of (5) satisfying u 0 ≤ x ≤ v 0 , and the convergence is quadratic.
Proof.We use the convexity of f and concavity of g by means of the following two inequalities: which are true for all (t, x) , (t, y) ∈ Ω. Suppose we have already constructed the functions Then we u n+1 = α and v n+1 = β, α and β being the unique solutions of the following linear initial value problems with delay ( 8) where and From ( 6) and ( 7) we easily see that the following inequalities hold: Next we need the following lemma: The proof of the Lemma can be given by using Corollary 3.1.2in [2].Now, we successively apply Lemma to the intervals [0, τ ] , [τ, 2τ ] , ..., [kτ, T ], where kτ < T ≤ (k + 1) τ.Thus we prove the existence of the solution α = u n+1 of (8) satisfying u n ≤ α ≤ v n on [0, T ] .Similarly, we find a solution β of (9) such that u , by a comparison result (see Theorem 2.3 in [1]), and making use again of the step method, we can derive the inequality α ≤ β, that is Finally, by similar arguments, we can prove that the sequences (u n ) and (v n ) converge to the unique solution x, uniformly and quadratically.
and let H (t, x) be continuous for t ∈ [a, b] and u (t) ≤ x ≤ v (t) .Suppose that