GENERALIZED QUASILINEARIZATION METHOD FOR NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

We develop a generalized quasilinearization method for nonlinear initial value problems involving functional differential equations and obtain a sequence of approximate solutions converging monotonically and quadratically to the solution of the problem. In addition, we obtain a monotone sequence of approximate solutions converging uniformly to the solution of the problem, possessing the rate of convergence higher than quadratic.


Introduction
The method of quasilinearization pioneered by Bellman and Kalaba [1] provides a descent approach for obtaining approximate solutions to a nonlinear differential equation provided the nonlinearity involved is convex. Recently, this method has been generalized by relaxing the convexity assumption. This development was so significant that it received much attention and the generalized quasilinearization method was applied to a variety of problems [2,3,4,5,6,7,9,10,11,12]. For a complete survey of the generalized quasilinearization technique, see [8].
The future state of a physical system depends not only on the present state but also on its past history. Functional differential equations provide a mathematical model for such physical systems in which the rate of change of the system may not depend on the influence of its hereditary effects. The impetus has mainly been due to developments in control theory, mathematical biology, mathematical economics, and the theory of systems which communicate through less channels. The simplest type of such a system is a differential-difference equation of the form x (t) = f (t, x(t), x(t − τ )), where τ > 0 is a positive constant. More general systems may be described by x (t) = f (t, x t ) where f is a suitable functional.
The aim of this paper is to consider a nonlinear initial value problem (IVP)involving functional differential equation and develop a method of quasilinearization for this problem without requiring the function involved to be convex/concave. A monotone sequence of approximate solutions converging monotonically to a solution of the problem, with convergence higher than quadratic (k ≥ 2) is obtained.

Preliminaries
Given any τ > 0, let Γ = C[[−τ, 0], R] and for any φ ∈ Γ, define the norm For any t ≥ 0, let x t denotes a translation of the restriction of x ∈ C[[−τ, T ], R] to the interval [−τ, 0] and it is defined by Now, consider the IVP for the functional differential equation Now, we state the following results, which play an important role in the sequel (for the proof, see p: 34-35, [4]).

Theorem 2.2: Let α(t), β(t) be lower and upper solutions of
Theorem 2.4: Let y(u), z(u) be lower and upper solutions of (2.1) and Then,

Main Result
Theorem 3.1: Assume that are lower and upper solutions of (1) satisfying The derivatives f φ (t, φ) and f φφ (t, φ) exist and are continuous and satisfying, Then there exists a monotone sequence {u n (t)}, which converges uniformly to the unique solution of (2.1) on J and that the convergence is quadratic. Proof: In view of the assumption (A 3 ), we can write Define the functional F (t, φ, ψ) as We observe that Setting α = u o , and consider the IVP for functional differential equation: is quasimonotone nondecreasing and satisfies one-sided Lipschitz condition, it follows that (3.5) has a unique solution u 1 (t), with u 1,0 = φ o on J. Now, in view of (A 2 ) and (3.4), we have It follows that u o (t), β(t) are lower and upper solutions of (3.5). Also, Thus, by Theorem 2.2, we conclude that Now, consider the IVP for the functional differential equation Repeating the procedure used earlier, (3.7) has a unique solution u 2 (t), with In view of (3.4), the quasinondecreasing nature of F (t, φ, ψ), and the fact that u 1 (t) is a solution of (3.5), we obtain It follows that u 1 (t), β(t) are lower and upper solutions of (3.7), and since by Theorem 2.2, we have that Continuing in the same way, we obtain a monotone sequence{u n,t } satisfying where the element u n,t of the sequence is a solution of the IVP Since the sequence {u n,t } is monotone, it follows that it has a pointwise limit x t . To show that x t is in fact a solution of (2.1) we notice that u n,t is a solution of the following linear IVP for functional differential equation: Thus, from (3.11), we have u n,t = t 0 σ n,s ds Taking limit n → ∞,we obtain which is a solution of (2.1). Finally, we have to show that the convergence is quadratic.
For that, we define e n (t) = x(t) − u n (t), t ∈J. (3.14) Observe that e n (t) ≥ 0 and e n (s) = Clearly, w (t) ≥ 0. Since e n (t) ≤ w(t) and w(t) is nondecreasing in t, we get Note that w(0) = 0. By Gronwall's inequality [8], (3.18) can be written as Consequently, e n (t) ≤ L 2 e Lτ T Lτ max e 2 n−1,t where t ∈ J. This completes the proof. Theorem 3.2: Assume that are lower and upper solutions of (1) satisfying 1, 2, 3, ..., k − 1) exist and are continuous functions Then there exists a monotone sequence {u n (t)} , which converges uniformly to the unique solution of (2.1) on J and that the convergence is of order k ≥ 2.
Proof: In view of assumption (A 3 ) and generalized mean value theorem, we have Define the functional F (t, φ, ψ) as Observe that and F (t, φ, ψ) is nondecreasing in ψ for each (t, φ). Furthermore, Clearly, F (t, φ, ψ) satisfies one-sided Lipschitz condition with respect to φ for each (t, ψ). Now, consider the IVP for the functional differential equation Since F (t, u(t), u 0,t ) is quasimonotone nondecreasing and satisfies one-sided Lipschitz condition, it follows that (19) has a unique solution u 1 (t), with imply that u 0 (t) and v 0 (t) are lower and upper solutions of (3.21), respectively. Also, Thus, it follows from Theorem 2.4 that u 0,t ≤ u 1,t ≤ v 0,t for every t ∈ J. (3.24) Now, consider the following IVP for functional differential equation: Employing the earlier arguments, we find that (3.25) has a unique solution u 2 (t), with In view of the nondecreasing nature of F (t, φ, ψ), it follows that which implies that u 1 (t) is a lower solution of (3.25). Similarly, it can be shown that v 0 (t) is an upper solution of (3.25), and Hence, by Theorem 2.4, there exists a solution u 2,t such that Continuing in this way, we obtain a monotone sequence{u n,t } satisfying where the element u n,t of the sequence is a solution of the IVP Since the sequence {u n,t } is monotone, it follows that it has a pointwise limit x t . To show that x t is in fact a solution of (2.1), we notice that u n,t is a solution of the following linear IVP for functional differential equation: Thus, from (3.26), we have This proves that {u n,t } is uniformly bounded on J. Passing on to the limit n → ∞, we obtain which is a solution of (2.1). Now, we show that the convergence is of order k ≥ 2. For that, we define e n (t) = x(t) − u n (t), a n (t) = u n+1 (t) − u n (t), t ∈J, so that, e n (t) ≥ 0, a n (t) ≥ 0 and e n (s) = x(s) − u n (s) = x 0 − u n,0 = φ o − φ o = 0, a n (s) = 0, s ∈ [−τ, 0]. In view of assumption (A 3 ) and the generalized mean value theorem, we have where C = M + m k k! k! , e n+1,t = e n,t − a n,t , ∂ k ∂φ k f (t, ξ t ) ≤ M, e n,t ≥ a n,t . Taking the expression (3.29) becomes e n+1 (t) ≤ Q n,t e n+1,t + Ce k n,t .
Notice that lim n→∞ Q n,t = f φ (t, x t ).
Clearly, w (t) ≥ 0. Since e n (t) ≤ w(t) and w(t) is nondecreasing in t, we get