Rev. Anal. Num´er. Th´eor. Approx., vol. 35 (2006) no. 2, pp. 147–160 ictp.acad.ro/jnaat
ITERATIVE FUNCTIONAL-DIFFERENTIAL SYSTEM WITH RETARDED ARGUMENT∗
DIANA OTROCOL†
Abstract. Existence, uniqueness and data dependence results of solution to the Cauchy problem for iterative functional-differential system with delays are ob- tained using weakly Picard operator theory.
MSC 2000. 34L05, 47H10.
Keywords. Iterative functional-differential equation, weakly Picard operator, delay, data dependence.
1. INTRODUCTION
The aim of this paper is to study the following iterative system with delays (1.1)
x0i(t) =fi(t, x1(t), x2(t), x1(x1(t−τ1)), x2(x2(t−τ2))), t∈[t0, b], i= 1,2, with the initial conditions
(1.2) xi(t) =ϕi(t), t∈[t0−τi, t0], i= 1,2, where
(H1) t0 < b, τ1, τ2>0, τ1< τ2;
(H2) fi∈C([t0, b]×([t0−τ1, b]×[t0−τ2, b])2,R), i= 1,2;
(H3) ϕ1 ∈C([t0−τ1, t0],[t0−τ1, b]), ϕ2 ∈C([t0−τ2, t0],[t0−τ2, b]);
(H4) there exists Lfi >0 such that:
|fi(t, u1, u2, u3, u4)−fi(t, v1, v2, v3, v4)| ≤Lfi(
4
X
k=1
|uk−vk|), for allt∈[t0, b],(u1,u2, u3, u4),(v1,v2,v3,v4)∈([t0−τ1, b]×[t0−τ2, b])2, i= 1,2.
By a solution of (1.1)–(1.2) we understand a function (x1, x2) with x1∈C([t0−τ1, b],[t0−τ1, b])∩C1([t0, b],[t0−τ1, b]) x2∈C([t0−τ2, b],[t0−τ2, b])∩C1([t0, b],[t0−τ2, b])
∗This work has been supported by MEdC-ANCS under grant 2CEEX-06-11-96.
†“Tiberiu Popoviciu” Institute of Numerical Analysis, P.O. Box. 68-1, Cluj-Napoca, Romania, e-mail: [email protected].
which satisfies (1.1)–(1.2).
The problem (1.1)–(1.2) is equivalent with the following fixed point equa- tions:
(3a) x1(t) =
(ϕ1(t), t∈[t0−τ1, t0], ϕ1(t0)+Rtt
0f1(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds,t∈[t0, b], (3b)
x2(t) =
(ϕ2(t), t∈[t0−τ2, t0], ϕ2(t0)+Rtt
0f2(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds,t∈[t0, b], wherex1∈C([t0−τ1, b],[t0−τ1, b]), x2∈C([t0−τ2, b],[t0−τ2, b]).
On the other hand, the system (1.1) is equivalent with (4a)
x1(t) =
(x1(t), t∈[t0−τ1, t0], x1(t0)+Rtt
0f1(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds,t∈[t0, b], (4b)
x2(t) =
(x2(t), t∈[t0−τ2, t0], x2(t0)+Rtt
0f2(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds,t∈[t0, b], and x1 ∈C([t0−τ1, b],[t0−τ1, b]), x2∈C([t0−τ2, b],[t0−τ2, b]).
We shall use the weakly Picard operators technique to study the systems (3a)–(3b) and (4a)–(4b).
The literature in differential equations with modified arguments, especially of retarded type, is now very extensive. We refer the reader to the following monographs: J. Hale [2], Y. Kuang [4], V. Mure¸san [3], I. A. Rus [7] and to our papers [5], [6]. The case of iterative system with retarded arguments has been studied by many authors: I. A. Rus and E. Egri [10], J. G. Si, W. R. Li and S. S. Cheng [11], S. Stanek [12]. So our paper complement in this respect the existing literature.
Let us mention that the results from this paper are obtained as a con- cequence of those from [10] where is considered the case of boundary value problems.
2. WEAKLY PICARD OPERATORS
In this paper we need some notions and results from the weakly Picard operator theory (for more details see I. A. Rus [9], [8], M. Serban [13]).
Let (X, d) be a metric space andA:X →X an operator. We shall use the following notations:
FA:={x∈X|A(x) =x}- the fixed point set of A;
I(A) :={Y ⊂X|A(Y)⊂Y, Y 6=∅}- the family of the nonempty invariant subset of A;
An+1 :=A◦An, A0 = 1X, A1 =A, n∈N;
P(X) :={Y ⊂X|Y 6=∅} - the set of the parts ofX;
H(Y, Z) := max{sup
y∈Y
z∈Zinfd(y, z),sup
z∈Z
y∈Yinfd(y, z)} -the Pompeiu–Housdorff functional on P(X)×P(X).
Definition 2.1. Let (X, d) be a metric space. An operator A :X → X is a Picard operator (PO) if there exists x∗ ∈X such that:
(i) FA={x∗},
(ii) the sequence(An(x0))n∈N converges to x∗ for all x0∈X.
Remark 2.2. Accordingly to the definition, the contraction principle in- sures that, if A:X →X is a α -contraction on the complet metric spaceX, then it is a Picard operator.
Theorem2.3. (Data dependence theorem). Let(X, d)be a complete metric space and A, B:X→X two operators. We suppose that
(i) the operatorA is a α -contraction;
(ii) FB 6=∅;
(iii) there existsη >0 such that
d(A(x), B(x))≤η, ∀x∈X.
Then if FA={x∗A} and x∗B ∈FB, we have d(x∗A, x∗B)≤ 1−αη .
Definition 2.4. Let (X, d) be a metric space. An operator A :X → X is a weakly Picard operator (WPO) if the sequence (An(x))n∈N converges for all x∈X, and its limit ( which may depend on x ) is a fixed point of A.
Theorem 2.5. Let (X, d) be a metric space and A: X → X an operator.
The operator Ais weakly Picard operator if and only if there exists a partition of X,
X= ∪
λ∈ΛXλ where Λ is the indices set of partition, such that:
(a) Xλ ∈I(A), λ∈Λ;
(b) A|Xλ:Xλ →Xλ is a Picard operator for all λ∈Λ.
Definition 2.6. If A is weakly Picard operator then we consider the oper- ator A∞ defined by
A∞:X →X, A∞(x) := lim
n→∞An(x).
It is clear that A∞(X) =FA.
Definition 2.7. Let Abe a weakly Picard operator andc >0.The operator A is c-weakly Picard operator if
d(x, A∞(x))≤cd(x, A(x)), ∀x∈X.
Example 2.8. Let (X, d) be a complete metric space and A : X → X a continuous operator. We suppose that there exists α∈[0,1) such that
d(A2(x), A(x))≤α(x, A(x)), ∀x∈X.
Then A isc-weakly Picard operator with c= 1−α1 .
Theorem 2.9. Let (X, d) be a metric space and Ai : X → X, i = 1,2.
Suppose that
(i) the operatorAi is ci-weakly Picard operator, i= 1,2;
(ii) there existsη >0 such that
d(A1(x), A2(x))≤η, ∀x∈X.
Then
H(FA1, FA2)≤ηmax(c1, c2).
Theorem 2.10. (Fibre contraction principle). Let (X, d) and (Y, ρ) be two metric spaces and A : X×Y → X ×Y, A = (B, C), ( B : X → X, C : X×Y →Y ) a triangular operator. We suppose that
(i) (Y, ρ) is a complete metric space;
(ii) the operatorB is Picard operator;
(iii) there exists l∈[0,1)such that C(x,·) :Y →Y is a l-contraction, for allx∈X;
(iv) if(x∗, y∗)∈FA, then C(·, y∗) is continuous in x∗. Then the operator A is Picard operator.
3. CAUCHY PROBLEM
In what follows we consider the fixed point equations (3a) and (3b).
Let
Af:C([t0−τ1, b],[t0−τ1, b])×C([t0−τ2, b],[t0−τ2, b])→C([t0−τ1, b],R)×C([t0−τ2, b],R), given by the relation
Af(x1, x2) = (Af1(x1, x2), Af2(x1, x2)),
where Af1(x1, x2)(t) := the right hand side of (3a) andAf2(x1, x2)(t) := the right hand side of (3b).
LetL1, L2 >0,L= max{L1, L2}and
CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]) :=
={(x1, x2)∈C([t0−τ1, b],[t0−τ1, b])×C([t0−τ2, b],[t0−τ2, b]) :
|xi(t1)−xi(t2)| ≤Li|t1−t2|, ∀(t1, t2)∈[t0−τ2, b], i= 1,2}.
It is clear thatCL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]) is a complete metric space with respect to the metric
d(x, x) := max
t0≤t≤b|x(t)−x(t)|.
We remark thatCL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]) is a closed subset inC([t0−τ1, b],[t0−τ1, b])×C([t0−τ2, b],[t0−τ2, b]).
We have
Theorem 3.1. We suppose that
(i) the conditions (H1)–(H4) are satisfied;
(ii) ϕ1 ∈CL([t0−τ1, t0],[t0−τ1, b]), ϕ2∈CL([t0−τ2, t0],[t0−τ2, b]);
(iii) mfi and Mfi ∈R, i= 1,2 are such that
(iiia) mfi≤fi(t, u1, u2, u3, u4)≤Mfi,∀t∈[t0, b],(u1,u2, u3, u4),(v1,v2,v3,v4)
∈([t0−τ1, b]×[t0−τ2, b])2, (iiib)
t0−τi ≤ϕi(t0) +mfi(b−t0) for mfi <0, t0−τi ≤ϕi(t0) for mfi ≥0,
b≥ϕi(t0) for Mfi ≤0,
b≥ϕi(t0) +Mfi(b−t0) for Mfi >0, (iiic) L+Mfi <1;
(iv) (b−t0)(Lf1 +Lf2)(L+ 2)<1.
Then the Cauchy problem (1.1)–(1.2) has, in CL([t0−τ1, b],[t0−τ1, b])× CL([t0−τ2, b],[t0−τ2, b]) a unique solution. Moreover the operator
Af:CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b])→
CL([t0−τ1, b],CL([t0−τ1, b],[t0−τ1, b]))×CL([t0−τ2, b],CL([t0−τ2, b],[t0−τ2, b])) is a c-Picard operator with c= (b−t 1
0)(Lf1+Lf2)(L+2).
Proof. (a) CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]) is an invariant subset forAf.
Indeed,
t0−τi≤Afi(x1, x2)(t)≤b, (x1, x2)(t)∈[t0−τ1, b]×[t0−τ2, b], t∈[t0, b], i= 1,2.
From (iiia) we have mfi and Mfi ∈Rsuch that mfi ≤fi(t, u1, u2, u3, u4)≤Mfi,
∀t∈[t0, b], (u1,u2, u3, u4),(v1,v2,v3,v4)∈([t0−τ1, b]×[t0−τ2, b])2, i= 1,2.
This implies that Rt
t0mfids≤Rtt
0fi(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds≤Rtt
0Mfids,
∀t∈[t0, b],that is
ϕi(t0) +mfi(b−t0)≤Afi(x1, x2)(t)≤ϕi(t0) +Mfi(b−t0), t∈[t0, b].
Therefor if condition (iii) holds, we have satisfied the invariance property for the operatorAf inC([t0−τ1, b],[t0−τ1, b])×C([t0−τ2, b],[t0−τ2, b]).
Now, consider t1, t2∈[t0−τ1, t0] :
|Af1(x1, x2)(t1)−Af1(x1, x2)(t2)|=|ϕ1(t1)−ϕ1(t2)| ≤L1|t1−t2|, because ϕ1 ∈CL([t0−τ1, t0],[t0−τ1, b]).
Similarly, for t1, t2 ∈[t0−τ2, t0] :
|Af2(x1, x2)(t1)−Af2(x1, x2)(t2)|=|ϕ2(t1)−ϕ2(t2)| ≤L2|t1−t2|, that follows from (ii), too.
On the other hand, if t1, t2 ∈[t0, b], we have
|Afi(x1, x2)(t1)−Afi(x1, x2)(t2)|=
=
ϕi(t1)−ϕi(t2)+
Z t1
t0
fi(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds−
− Z t2
t0
fi(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))ds
≤
≤Li|t1−t2|+Mfi|t1−t2| ≤(L+Mfi)|t1−t2|, i= 1,2.
So we can affirm that∀t1, t2∈[t0, b], t1≤t2,and doe to (iii),Af isL-Lipshitz.
Thus, according to the above, we haveCL([t0−τ1, b],[t0−τ1, b])×CL([t0− τ2, b],[t0−τ2, b])∈I(Af).
(b) Af is a LAf -contraction withLAf = (b−t0)(Lf1 +Lf2)(L+ 2).
Fort∈[t0−τ1, t0],we have |Af1(x1, x2)(t)−Af1(x1, x2)(t)|= 0.
Fort∈[t0−τ2, t0],we have |Af2(x1, x2)(t)−Af2(x1, x2)(t)|= 0.
Fort∈[t0, b] :
|Af1(x1, x2)(t)−Af1(x1, x2)(t)|=
=
Z t t0
[f1(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))
−f1(s, x1(s), x2(s), x1(x1(s−τ1)), x2(x2(s−τ2)))]ds
≤Lf1(|x1(s)−x1(s)|+|x2(s)−x2(s)|+|x1(x1(s−τ1))−x1(x1(s−τ1))|
+|x2(x2(s−τ2))−x2(x2(s−τ2))|)(b−t0)
≤(b−t0)Lf1[kx1−x1kC+kx2−x2kC +|x1(x1(s−τ1))−x1(x1(s−τ1))|
+|x1(x1(s−τ1))−x1(x1(s−τ1))|+|x2(x2(s−τ2))−x2(x2(s−τ2))|
+|x2(x2(s−τ2))−x2(x2(s−τ2))|]≤(b−t0)Lf1[kx1−x1kC+kx2−x2kC +L1kx1−x1kC+kx1−x1kC+L2kx2−x2kC+kx2−x2kC]
≤(b−t0)Lf1(L+ 2)(kx1−x1kC+kx2−x2kC).
In the same way
|Af2(x1, x2)(t)−Af2(x1, x2)(t)| ≤(b−t0)Lf2(L+ 2)(kx1−x1k+kx2−x2k).
Then we have the following relation
kAf(x1, x2)−Af(x1, x2)kC ≤(b−t0)(Lf1+Lf2)(L+ 2)k(x1, x2)−(x1, x2)kC So Af is ac-Picard operator with c= 1−L1
Af.
In what follows, consider the following operator
Bf : CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b])→
CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]), given by the relation
Bf(x1, x2) = (Bf1(x1, x2), Bf2(x1, x2)),
where Bf1(x1, x2) := the right hand side of (4a) andBf2(x1, x2) := the right hand side of (4b).
Theorem3.2. In the conditions of Theorem 3.1, the operatorBf :CL([t0− τ1, b],[t0−τ1, b])×CL([t0 −τ2, b],[t0 −τ2, b]) → CL([t0−τ1, b],[t0−τ1, b])× CL([t0−τ2, b],[t0−τ2, b])is WPO.
Proof. The operatorBf is a continuous operator but it is not a contraction operator. Let take the following notation:
Xϕ1 :={x1∈C([t0−τ1, b],[t0−τ1, b])|x1|[t0−τ1,t0]=ϕ1}, Xϕ2 :={x2 ∈C([t0−τ2, b],[t0−τ2, b])|x2|[t0−τ2,t0]=ϕ2}.
Then we can write (5)
CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]) = [
ϕi∈CL([t0−τi,t0],[t0−τi,b])
Xϕ1×Xϕ2. We have thatXϕ1×Xϕ2 ∈I(Bf) andBf|Xϕ
1×Xϕ2 is a Picard operator be- cause is the operator which appears in the proof of Theorem 3.1. By applying
Theorem 2.5, we obtain that Bf is WPO.
4. INCREASING SOLUTION OF (??) 4.1. Inequalities of Chapligin type.
Theorem 4.1. We suppose that
(a) the conditions of the Theorem 3.1 are satisfied;
(b) (u1,u2, u3, u4),(v1,v2,v3,v4)∈([t0−τ1, b]×[t0−τ2, b])2, uj ≤ vj, j = 1,4, imply that
fi(t, u1, u2, u3, u4)≤fi(t, v1, v2, v3, v4), i= 1,2,for all t∈[t0, b].
Let (x1, x2) be an increasing solution of the system (1.1) and (y1, y2) an increasing solution for the system of inequalities
y0i(t)≤fi(t, y1(t), y2(t), y1(y1(t−τ1)), y2(y2(t−τ2))), t∈[t0, b], Then
yi(t)≤xi(t), t∈[t0−τi, t0], i= 1,2⇒(y1, y2)≤(x1, x2).
Proof. In the terms of the operatorBf, we have
(x1, x2) =Bf(x1, x2) and (y1, y2)≤Bf(y1, y2).
However, from the condition (b), we have that the operatorBf∞is increasing, (y1, y2) ≤ Bf∞(y1, y2) =Bf∞(ye1|[t0−τ1,t0],ye2|[t0−τ2,t0])
≤ Bf∞(xe1|[t0−τ1,t0],xe2|[t0−τ2,t0]) = (x1, x2).
Thus (y1, y2)≤(x1, x2).
Here, for (xe1,xe2) we used the notation xe1 ∈Xx1|[t
0−τ1,t0],xe2 ∈Xx1|[t
0−τ2,t0].
4.2. Comparison theorem. In the next result we want to study the monotony of the solution of the problem (1.1)–(1.2) with respect to ϕi and fi, i= 1,2.We shall use the result below:
Lemma 4.2. (Abstract comparison lemma). Let (X, d,≤) be an ordered metric space and A, B, C :X→X such that:
(i) A≤B≤C;
(ii) the operators A, B, C are WPO;
(iii) the operatorB is increasing.
Then
x≤y≤z⇒A∞(x)≤B∞(y)≤C∞(z).
In this case we can establish the theorem.
Theorem 4.3. Let fij ∈C([t0, b]×([t0−τ1, b]×[t0−τ2, b])2), i= 1,2, j = 1,2,3.
We suppose that
(a) fi2(t,·,·,·,·) : ([t0−τ1, b]×[t0−τ2, b])2→([t0−τ1, b]×[t0−τ2, b])2 are increasing;
(b) fi1 ≤fi2 ≤fi3.
Let (xj1, xj2) be an increasing solution of the systems
x0i(t) =fij(t, x1(t), x2(t), x1(x1(t−τ1)), x2(x2(t−τ2))),t∈[t0,b], i= 1,2,j = 1,2,3.
If x1i(t)≤x2i(t)≤x3i(t), t∈[t0−τi, t0] thenx1i ≤x2i ≤x3i, i= 1,2.
Proof. The operators Bfj, j = 1,2,3 are WPO. Taking into consideration the condition (a) the operator Bf2 is increasing. From (b) we have that Bf1 ≤ Bf2 ≤Bf3. We note that (xj1, xj2) =Bfj∞(xej1,xej2), j = 1,2,3. Now, using the
Abstract comparison lemma, the proof is complete.
5. DATA DEPENDENCE: CONTINUITY
Consider the Cauchy problem (1.1)–(1.2) and suppose the conditions of Theorem 3.1 are satisfied. Denote by (x1, x2)(·;ϕ1, ϕ2, f1, f2), i = 1,2 the solution of this problem. We can state the following result:
Theorem5.1. Letϕj1, ϕj2, f1j, f2j, j = 1,2be as in Theorem 3.1. We suppose that there exists η1, η2, ηi3, i= 1,2 such that
(i) ϕ11(t)−ϕ21(t) ≤η1, ∀t ∈[t0−τ1, t0] and ϕ12(t)−ϕ22(t)≤ η2, ∀t ∈ [t0−τ2, t0];
(ii) fi1(t, u1, u2, u3, u4)−fi2(t, v1, v2, v3, v4)≤η3i, i= 1,2,(u1, u2, u3, u4), (v1, v2, v3, v4)∈([t0−τ1, b]×[t0−τ2, b])2.
Then
(x1, x2)(t;ϕ11, ϕ12, f11, f21)−(x1, x2)(t;ϕ21, ϕ22, f12, f22)≤(b−tη1+η2+(η13+η23)(b−t0)
0)(Lf1+Lf2)(L+2), where Lfi = max(Lf1
i, Lf2
i), i= 1,2.
Proof. Consider the operatorsAϕj
1,ϕj2,f1j,f2j, j= 1,2.From Theorem 3.1 these operators are contractions.
Then Aϕ1
1,ϕ12,f11,f21(x1, x2)−Aϕ2
1,ϕ22,f12,f22(x1, x2)
C ≤η1+η2+ (η13+η23)(b−t0),
∀(x1, x2)∈CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]).
Now the proof follows from Theorem 2.3, with A := Aϕ1
1,ϕ12,f11,f21, B = Aϕ2
1,ϕ22,f12,f22, η = η1 +η2 + (η13+η23)(b−t0) and α := LAf = (b−t0)(Lf1 + Lf2)(L+ 2) whereLfi = max(Lf1
i, Lf2
i), i= 1,2.
From the Theorem above we have:
Theorem5.2. Let fi1 andfi2 be as in Theorem 3.1,i= 1,2. LetSB
f1 i
, SB
f2 i
be the solution set of the system (1.1) corresponding to fi1 and fi2, i = 1,2.
Suppose that there exists ηi>0, i= 1,2 such that
(6) fi1(t,u1,u2, u3, u4)−fi2(t,v1,v2,v3,v4)≤ηi
for all t∈[t0, b],(u1,u2, u3, u4),(v1,v2,v3,v4)∈([t0−τ1, b]×[t0−τ2, b])2, i= 1,2.
Then
Hk·k
C(SB
f1 i
, SB
f2 i
)≤ 1−(L(η1+η2)(b−t0)
f1+Lf2)(L+2)(b−t0), where Lfi := max(Lf1
i, Lf2
i) and Hk·kC denotes the Pompeiu-Housdorff func- tional with respect to k·kC onCL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]).
Proof. We will look for thosec1 and c2 for which in condition of Theorem 3.1 the operatorsBf1
i and Bf2
i, i= 1,2 are c1-WPO and c2-WPO.
Let
Xϕ1 :={x1 ∈C([t0−τ1, b],[t0−τ1, b])|x1|[t0−τ1,t0]=ϕ1}, Xϕ2 :={x2 ∈C([t0−τ2, b],[t0−τ2, b])|x2|[t
0−τ2,t0]=ϕ2}.
It is clear that Bf1
i|Xϕ1×Xϕ2 = Af1
i, Bf2
i|Xϕ1×Xϕ2 = Af2
i. So from Theorem 2.5 and Theorem 3.1 we have
Bf21
i
(x1,x2)−Bf1 i(x1,x2)
C ≤ (b−t0)(Lf1 1+Lf1
2)(L+2)Bf1
i(x1,x2)−(x1,x2)
C,
Bf22
i
(x1,x2)−Bf2 i(x1,x2)
C ≤ (b−t0)(Lf2
1+Lf2
2)(L+2)Bf2
i(x1,x2)−(x1,x2)
C, for all (x1, x2)∈CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]), i= 1,2.
Now choosing
α1 = (b−t0)(Lf1
1 +Lf1
2)(L+ 2), α2 = (b−t0)(Lf2
1 +Lf2
2)(L+ 2), we get thatBf1
i andBf2
i arec1-WPO andc2-WPO withc1= (1−α1)−1, c2 = (1−α2)−1. From (6) we obtain that
Bf1
i(x1, x2)−Bf2
i(x1, x2)
C ≤(η1+η2)(b−t0),
for all (x1, x2) ∈ CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−τ2, b]), i = 1,2.
Applying Theorem 2.9 we have that Hk·k
C(SB
f1 i
, SB
f2 i
)≤ 1−(b−t(η1+η2)(b−t0)
0)(Lf1+Lf2)(L+2), where Lfi := max(Lf1
i, Lf2
i) and Hk·kC denotes the Pompeiu-Housdorff func- tional with respect tok·kC on CL([t0−τ1, b],[t0−τ1, b])×CL([t0−τ2, b],[t0−
τ2, b]), i= 1,2.
6. DATA DEPENDENCE: DIFFERENTIABILITY
Consider the following Cauchy problem with parameter
(7) x0i(t) =fi(t, x1(t), x2(t), x1(x1(t−τ1)), x2(x2(t−τ2));λ), t∈[t0, b], i= 1,2, (8) xi(t) =ϕi(t), t∈[t0−τi, t0], i= 1,2.
Suppose that we have satisfied the following conditions:
(C1) t0 < b, τ1, τ2 >0, τ1 < τ2, J ⊂Ra compact interval;
(C2) ϕi ∈CL([t0−τi, t0],[t0−τi, b]), i= 1,2;
(C3) fi∈C1([t0, b]×([t0−τ1, b]×[t0−τ2, b])2×J,R)i= 1,2;
(C4) there exists Lfi >0 such that
∂fi(t,u1,u2,u3,u4;λ)
∂ui
≤Lfi
for allt∈[t0, b],(u1,u2, u3, u4)∈([t0−τ1, b]×[t0−τ2, b])2, i= 1,2, λ∈J;
(C5) mfi and Mfi ∈R, i= 1,2 are such that