EXACT AND APPROXIMATE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH NONLOCAL HISTORY CONDITIONS

We study the exact and approximate solutions of a delay differential equation with various types of nonlocal history conditions. We establish the existence and uniqueness of mild, strong, and classical solutions for a class of such problems using the method of semidiscretization in time. We also establish a result concerning the global existence of solutions. Finally, we consider some examples and discuss their exact and approximate solutions.

Nonlocal abstract differential and functional differential equations have been extensively studied in the literature.We refer to the works of Byszewski [6], Byszewski and Lakshmikantham [8], Balachandran and Chandrasekaran [5], and Lin and Liu [11].Most of them used semigroup theory and fixed point theorem to establish the unique existence and regularity of solution.In [7], Byszewski and Akca applied Schauder's fixed point principle to prove the theorems for existence of mild and classical solutions of nonlocal Cauchy problem of the form where −A is the infinitesimal generator of a compact C 0 semigroup in a Banach space.
In our recent work [1,2], we studied the functional differential equation (1.2) with the nonlocal history condition h(u [−τ,0] ) = φ, where h is a Volterra-type operator from Ꮿ 0 into itself and φ ∈ Ꮿ 0 .We made use of method of semidiscretization in time to derive the existence and uniqueness of a strong solution.Many authors have used and developed the method of semidiscretization for nonlinear evolution and nonlinear functional evolution equations, see, for instance, the papers of Kartsatos and Parrott [9], Kartsatos and Zigler [10], Bahuguna and Raghavendra [4], and the references listed therein.
Our purpose here is to study the exact and approximate solutions of the delay differential equation (1.1) with a nonlocal condition.In doing so, we first use the method of semidiscretization to derive the existence of a unique strong solution, then we prove that strong solution is a classical solution if additional conditions are assumed on the operator.The global existence of a solution for (1.1), a nonconsidered problem in [1,2], is also established with an additional assumption (see Theorem 4.1).The result of the paper consists, among other things, in that we obtain a solution of problem of much stronger regularity than in [1,2].

Existence and uniqueness of solutions
The existence and uniqueness results have been established for the more general case of (2.2) in Bahuguna [3].For the sake of completeness, we briefly mention the ideas and the main result of the existence and uniqueness.
If we take H := L 2 (a,b), the real Hilbert space of all real-valued square-integrable functions on the interval (a,b), and the linear operator A defined by then it is well known that −A generates an analytic semigroup e tA , t ≥ 0, in H.If we define u : [−τ,T] → H given by u(t)(x) = w(x,t), then (1.1) may be rewritten as the following evolution equation: Suppose that there exists a χ ∈ Ꮿ 0 such that h(χ) = Φ.Let T be any number such that 0 for all t ∈ (0, T] and u ∈ C 1 ((0, T);H), and We have the following existence and uniqueness result for (2.2).
Theorem 2.1.Suppose that there exists a Lipschitz continuous χ ∈ Ꮿ 0 such that h(χ) = Φ and F satisfies the condition ) , where B r (Z,z 0 ) denotes the closed ball of radius r > 0 centered at z 0 in the Banach space Z.Then there exists a strong solution u of (2.2) either on the whole interval [−τ,T] or on a maximal interval [−τ,t max ), 0 < t max ≤ T, such that u is a strong solution of (2.2) on [−τ, T] for every 0 < T < t max , and in the latter case, If, in addition, S(t) is an analytic semigroup in H, then u is a classical Lipschitz continuous solution on every compact subinterval of the interval of existence.Furthermore, u is unique in {ψ ∈ Ꮿ T : ψ = χ on [−τ,0]} for every compact subinterval [−τ, T] of the interval of existence.

Approximations
In this section, we consider the application of the method of semidiscretization in time and the convergence of the approximate solutions.We first establish the existence and uniqueness of a strong solution of (2.2) for any given χ ∈ Ꮿ 0 and χ(0 where, For n ∈ N, let h n = t 0 /n.We set u n 0 = χ(0) for all n ∈ N and define each of {u n j } n j=1 as the unique solution of the equation where , and for 2 ≤ j ≤ n, ( The existence of a unique u n j ∈ D(A) satisfying (3.2) is a consequence of the m-monotonicity of A. We define the sequence {U n } ⊂ Ꮿ t0 of polygonal functions and prove the convergence of {U n } to a unique strong solution u of (2.2) as n → ∞.Before proving the convergence, we state and prove some lemmas which will be used to establish the main result.
Lemma 3.2.There exists a positive constant K independent of the discretization parameters n, j, and h n such that Proof.In this proof and subsequently, K will represent a generic constant independent of j, h n , and n.From (3.2), for j = 1 and monotonicity of A, we have Now, for 2 ≤ j ≤ n, using monotonicity of A and local Lipschitz-like condition (2.7) of F, we get where C is a positive constant independent of j, h n , and n.Repeating the above inequality, we obtain This completes the proof of the lemma.
We introduce another sequence {X n } of step functions from [−h n ,t 0 ] into H by (3.13) For notational convenience, let To prove the remaining part of Theorem 2.1, we assume the interval of existence [−τ, T].The proof may be modified for the interval [−τ,t max ).Also −A is the infinitesimal generator of C 0 semigroup.The function F : [0,T] → H := L 2 (a,b) given by is Lipschitz continuous and therefore continuous on [0, T] and F ∈ L 1 ((0,T);H).Now it is easy to see that if u is the strong solution of (2.2), then u is given by and therefore is a mild solution of (2.2).If S(t) is an analytic semigroup in H, then by the use of Corollary 3.3 in Pazy [12, page 113], we obtain that u is a classical solution of (2.2).Clearly, if χ ∈ Ꮿ 0 satisfying that h(χ) = Φ is unique on [−τ,0], u is unique since for two χ, χ in Ꮿ 0 satisfying h(χ) = h( χ) = Φ with χ = χ, the corresponding solutions u χ and u χ belonging to {ψ ∈ Ꮿ T : ψ = χ on [−τ,0]} and {ψ ∈ Ꮿ T : ψ = χ on [−τ,0]}, respectively, are different.

Global existence
We turn now to global existence.Here further assumptions are made, under the consideration of which, the existence of a global solution is established.Theorem 4.1.Let −A be the infinitesimal generator of a compact C 0 semigroup S(t), t ≥ 0, on H. Let F : [0,∞) × H → H be continuous and map bounded sets in [0,∞) × H into bounded sets in H. Also there exist two locally integrable functions k 1 (s) and k 2 (s) such that Then, for every Proof.We know that the corresponding solution u exists on the interval [−τ,T) and is given by We also know that S(t) ≤ Me ωt for some M ≥ 1 and ω ≥ 0. Let The function ξ thus defined is obviously continuous on [0,∞).
and for t ∈ [0,T), The above inequality implies that By the application of Gronwall's inequality, we have which implies the boundedness of u(t) by a continuous function.Consequently, there exists a global solution u of (2.2) (see Theorem 2.2 on page 193 in Pazy [12]).

Examples
In this section, to illustrate the applicability of our work, we discuss the exact and approximate solutions of some initial boundary value problems.
[2]of of Theorem 2.1.By proceeding as in Agarwal and Bahuguna[2]we can show the existence and uniqueness of the strong solution on [−τ,t 0 ] as well as the continuation of the solution on [−τ,T].Thus we have that there exists a strong solution of (2.2) either on the whole interval [−τ,T] or on the maximal interval of existence [−τ,t max ), 0 < t max ≤ T. In the latter case, if lim t→tmax− u(t) < ∞, we have that lim t→tmax− u(t) is in the closure of D(A) in H, and if it is in D(A), then, following the same steps as before, u(t) can be extended beyond t max , which contradicts the definition of the maximal interval of existence.