EXISTENCE THEORY FOR FUNCTIONAL INITIAL VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper the existence of a solution of general nonlinear functional dierential equations is proved under mixed generalized Lipschitz and Carath eodory condition. An existence theorem for the extremal solutions is also proved under certain monotonicity and weaker continuity conditions. Examples are provided to illustrate the abstract theory developed in this paper.

for x = (x 1 , ..., x n ) ∈ R n .Let a, r ∈ R be such that a > 0, r > 0 and let I 0 = [−r, 0] and I = [0, a] be two closed and bounded intervals in R. Let C = C(I 0 , R n ) denote a Banach space of all continuous R n -valued functions on I 0 with the usual supremum norm • C .For every t ∈ I we define a continuous function x t : I 0 → R by x t (θ) = x(t + θ) for each θ ∈ I 0 .Let J = [−r, a] and let BM (J, R n ) denote the space of bounded and measurable R n -valued functions on J. Define a maximum norm • in BM (J, R n ) by x = max t∈J |x(t)|.Given a bounded operator G : X ⊂ BM (J, R n ) → Y ⊂ BM (J, R n ), consider the perturbed functional differential equation (in short FDE) x (t) = f (t, x(t), Sx) a.e.t ∈ I x(t) = Gx(t), t ∈ I 0 (1.1)where f : EJQTDE, 2004 No. 15, p. 1 By a solution of FDE (1.1) we mean a function x ∈ C(J, R n )∩B(I 0 , R n )∩AC(I, R n ) that satisfies the equations in (1.1), where AC(I, R n ) is the space of all absolutely continuous functions on I with J = I 0 I.
The FDE (1.1) seems to be new and special cases of it have been discussed in the literature since long time.These special cases to FDE (1.1) can be obtained by defining the operators G and S appropriately.The operators G and S are called functional operators of the functional differential equation (1.1).As far as the author is aware there is no previous work on the existence theory for the FDE (1.1) in the framework of Carathéodory as well as monotonicity conditions.Now take X = BM (I 0 , R) ∩ AC(I, R) ∩ BM (J, R) ⊂ BM (J, R).Let G : X → BM (I 0 , R) and define the operator S : X → X by Sx(t) = x(t), t ∈ J. Then the FDE (1.1) takes the form which is the functional differential equation discussed in Liz and Pouso [8] for the existence of solution in the framework of upper and lower solutions.Further as mentioned in Liz and Pouso [8], the FDE (1.2) includes several important classes of functional differential equations as special cases.Again when S, G : X → C(I 0 , R) are two operators defined by Sx(t) = x t , t ∈ I and Gx(t) = φ(t), t ∈ I 0 , the FDE (1.1) reduces to the following FDE where f : We note that the FDE (1.3) again covers several important classes of functional differential equations discussed earlier as special cases.See Haddock and Nkashama [4], Lee and O'Regan [6], Leela and Oguztoreli [7], Stepanov [9], Xu and Liz [11] and references therein.
We shall apply fixed point theorems for proving the existence theorems for FDE (1.1) under the generalized Lipschitz and monotonicity conditions.

Existence Theorem
An operator T : X → X is called compact if T (X) is a compact subset of X. Similarly T : X → X is called totally bounded if T maps a bounded subset of X into the relatively compact subset of X. Finally T : X → X is called completely continuous operator if it is continuous and totally bounded operator on X. EJQTDE, 2004 No. 15, p. 2 In this paper we shall prove existence theory for the FDE (1.1) via the following nonlinear alternative of Leray-Schauder [2].Theorem 2.1 Let X be a Banach space and let T : X → X be a completely continuous operator.Then either (i) the equation λT x = x has a solution for λ = 1, or Let M (J, R n ) and B(J, R n ) denote respectively the spaces of measurable and bounded R n -valued functions on J.We shall seek the solution of FDE (1.1) in the space C(J, R n ), of all continuous real-valued functions on J. Define a norm Clearly C(J, R n ) becomes a Banach space with this norm.We need the following definition in the sequel.Definition 2.1 A mapping β : J × R n × C → R n is said to satisfy Carathéodory's conditions or simply is called L 1 -Carathéodory if (i) t → β(t, x, y) is measurable for each x ∈ R n and y ∈ BM (J, R n ), (ii) (x, y) → β(t, x, y) is continuous almost everywhere for t ∈ J, and (iii) for each real number k > 0, there exists a function We will need the following hypotheses: Theorem 2.2 Assume that the hypotheses (A 1 )-(A 4 ) hold.Suppose that Then the FIE (1.1) has a solution on J.
Proof.Now the FDE (1.1) is equivalent to the functional integral equation (in short FIE) Let X = AC(J, R n ).Define a mappings T on X by (2.3) Obviously T defines the operator T : X → X.We show that T is completely continuous on X.Using the standard arguments as in Granas et al. [3], it is shown that T is a continuous operator on X, with respect to the norm • .Let Y be a bounded set in X.Then there is real number r > 0 such that x ≤ r for all x ∈ Y .We shall show that T (Y ) is a uniformly bounded and equi-continuous set in X.Since Y is bounded, then there exists a constant r > 0 such that x ≤ r for all x ∈ Y. Now by (A 1 ), Similarly if τ, t ∈ I 0 , then we obtain Consequently it is a equi-continuous set in C(I 0 , R n ) and hence we have for all x ∈ Y .If τ ∈ I 0 and t ∈ I, then we obtain Hence T (Y ) is an equi-continuous set and consequently T (Y ) is relatively compact by Arzelá-Ascoli theorem.Consequently T is a completely continuous operator on X.Thus all the conditions of Theorem 3.1 are satisfied and a direct application of it yields that either conclusion (i) or conclusion (ii) holds.We show that the conclusion (ii) is not possible.Let u ∈ X be any solution to FDE (1.1).Then we have, for any λ ∈ (0, 1), that is, By the change of variables in the above integral gives that Now an application of mean value theorem yields that there is a constant M > 0 such that w(t) ≤ M for all t ∈ J.This further implies that for all t ∈ I. Again if t ∈ I 0 , then we have Hence we have |u(t)| ≤ M for all t ∈ J. Thus the conclusion (ii) of Theorem 2.1 does not hold.Therefore the operator equation T x = x and consequently the FDE (1.1) has a solution on J.This completes the proof.
where p ∈ L 1 (I, R + ) and Define functional operator S and operator G on BM (J, R) by Sx = x t ∈ C(I 0 , R) for t ∈ I and Gx(t) = cos t for all t ∈ I 0 .Obviously S is continuous and G is completely continuous with N = max{ Gx : It is very easy to prove that the function f (t, x, y) is L 1 -Carathéodory.Again we have and so hypothesis (A 5 ) is satisfied with φ(r) = 1 + r.Now by the definition of φ we obtain Now we apply Theorem 3.1 to yields that the FDE (1.1) has a solution on J = I 0 I.

Uniqueness Theorem
Let X be a Banach space with norm • .A mapping T : X → X is called D-Lipschitzian if there exists a continuous nondecreasing function ψ : R for all x, y ∈ X with ψ(0) = 0. Sometimes we call the function ψ a D-function of T .
In the special case when ψ(r) = αr, α > 0, T is called a Lipchitzian with a Lipschitz constant α.In particular if α < 1, T is called a contraction with a contraction constant α.Further if ψ(r) < r for r > 0 , then T is called a nonlinear contraction on X.
Finally if ψ(r) = r, then T is called a nonexpansive operator on X.

EJQTDE, 2004
No. 15, p. 7 The following fixed point theorem for the nonlinear contraction is well-known and useful for proving the existence and the uniqueness theorems for the nonlinear differential and integral equations.Theorem 3.1 (Browder [1]) Let X be a Banach space and let T : X → X be a nonlinear contraction.Then T has a unique fixed point.
We will need the following hypotheses: , a.e.t ∈ I 0 for all x, y ∈ BM (J, R).
Theorem 3.2 Assume that the hypotheses (B 1 ) − (B 3 ) hold.Then the FDE (1.1) has a unique solution on J.
Proof : Let X = C(J, R n ) and define an operator T on X by (2.2).We show that T is a nonlinear contraction on X.Let x, y ∈ X.By hypothesis (B 1 ), No. 15, p. 8 for all t ∈ J. Taking supremum over t we obtain Ax − Ay ≤ ψ( x − y ), for all x, y ∈ X where ψ(r) = ar a + r < r, which shows that T is a nonlinear contraction on X.We now apply Theorem 3.1 to yield that the operator T has a unique fixed point.This further implies that the FDE (1.1) has a unique solution on J.This completes the proof. where Define the functional operator S and the operator G on BM (J, R) by Sx = x t ∈ C(I 0 , R) for t ∈ I and Gx(t) = cos t for all t ∈ I 0 .Obviously S is continuous and G is bounded with C = max{ Gx : x ∈ BM (J, R)} = 1.Clearly S is nonexpansive on BM (J, R).

Define a function f
It is very easy to prove that the function f is continuous on I × R × BM (J, R).Finally we show that the function f satisfies the inequality given in (B 1 ).Let x 1 , x 2 ∈ R and y 1 , y 2 ∈ BM (J, R) be arbitrary.Then we have for all t ∈ J. Hence the hypothesis (B 1 ) of Theorem 3.1 is satisfied.Therefore an application of Theorem 3.1 yields that the FDE (3.8 has a unique solution on [−π/2, 1].EJQTDE, 2004 No. 15, p. 9 Let x, y ∈ R n be such that x = (x 1 , . . ., x n ) and y = (y 1 , . . ., y n ).We define the co-ordinate wise order relation in R n , that is, x ≤ y ⇔ x i ≤ y i ∀ i = 1, . . ., n.We equip the Banach space C(J, R n ) with the order relation ≤ by ξ 1 ≤ ξ 2 i and only if We use the following fixed point theorem of Heikkila and Lakshmikantham [5] in the sequel.We need the following definitions in the sequel.
(ii) The function β(t, x, y) is nondecreasing in x and y almost everywhere for t ∈ J, and (iii) for each real number k > 0, there exists a function We consider the following set of assumptions: Finally let {x n } be a monotone sequence in [u, v].We shall show that the sequence {T x n } converges in T ([u, v]).Obviously the sequence {T x n } is monotone in T ([u, v]).Now it can be shown as in the proof of Theorem 2.2 that the sequence {T x n } is uniformly bounded and equi-continuous in T ([u, v]) with the function h playing the role of h k .Hence an application of Arzela-Ascoli theorem yields that the sequence {T x n } converges in T ([u, v]).Thus all the conditions of Theorem 4.1 are satisfied and hence the operator T has a least and a greatest fixed point in [u, v].This further implies that the the FDE (1.1) has maximal and minimal solutions on J.This completes the proof.

Remark 4.2
The main existence result proved in Liz and Pouso [8] seems to be not correct wherein the authors assume the function f (t, x, y) to be nondecreasing only in y, whereas we need both in x and y.In consequence the example 4.1 quoted in Liz and Pouso [8] also goes wrong.Therefore our existence theorem proved in this section is an improvement of the result of Liz and Pouso [8] with correct proof.

1
Statement of ProblemLet R n denote the n-dimensional Euclidean space with a norm | • | defined by |x| = |x 1 | + ... + |x n | is a uniformly bounded set in X.Now we show that T (Y ) is an equi-continuous set.Let EJQTDE, 2004 No. 15, p. 4 t, τ ∈ I. Then for any x ∈ Y we have by (2.3),

Theorem 4 . 1
Let [a, b] be an order interval in a subset Y of an ordered Banach space X and let Q : [a, b] → [a, b] be a nondecreasing mapping.If each sequence {Qx n } ⊆ Q([a, b]) converges, whenever {x n } is a monotone sequence in [a, b], then the sequence of Q-iteration of a converges to the least foxed point x * of Q and the sequence of Qiteration of b converges to the greatest fixed point x * of Q. Moreover x * = min{y ∈ [a, b] | y ≥ Qy} and x * = max{y ∈ [a, b] | y ≤ Qy}.
R) such that |β(t, x, y)| ≤ h k (t), a.e.t ∈ J for all x ∈ R n and y ∈ BM (J, R n ) with |x| ≤ k, y ≤ k.
Definition 4.3 A solution x M of the FDE (1.1) is said to be maximal if for any other solution x to FDE(1.1) one has x(t) ≤ x M (t), ∀t ∈ J. Again a solution x m of the FDE (1.1) is said to be minimal if x m (t) ≤ x(t), ∀t ∈ J, where x is any solution of the FDE (1.1) on J.
To see this, let x, y ∈ [u, v] be such that x ≤ y.