Iterative functional-differential system with retarded argument

Diana Otrocol
(original), paper

Abstract

Existence, uniqueness and data dependence results of solution to theCauchy problem for iterative functional-differential system with delays are ob-tained using weakly Picard operator theory

Authors

Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Iterative functional-differential equation; weakly Picard operator; delay; data dependence.

Paper coordinates

D. Otrocol, Iterative functional-differential system with retarded argument, Rev. Anal. Numér. Théor. Approx., 35 (2006), no. 2, 147-160.

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https://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no2-art3/Otrocol-pdf

About this paper

Journal

Rev. Anal. Numér. Théor. Approx.

Publisher Name

Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no2-art3

Print ISSN

2457-6794

Online ISSN

 2501-059X

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[3]Mures ̧an, V.,Functional-Integral Equations, Editura Mediamira, Cluj-Napoca, 2003.
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[5]Otrocol, D.,Data dependence for the solution of a Lotka-Volterra system with twodelays, Mathematica, Tome48(71), no. 1, pp. 61–68, 2006.
[6]Otrocol, D.,Smooth dependence on parameters for some Lotka-Volterra system withdelays(to appear).
[7]Rus, I. A.,Principii ̧si aplicat ̧ii ale teoriei punctului fix, Editura Dacia, Cluj-Napoca,1979.
[8]Rus, I. A.,Weakly Picard mappings, Comment. Math. Univ. Caroline,34, pp. 769–773,1993.
[9]Rus, I. A.,Functional-differential equation of mixed type, via weakly Picard operators,Seminar of Fixed Point Theory, Cluj-Napoca,3, pp. 335–346, 2002.
[10]Rus, I. A. andEgri, E.,Boundary value problems for iterative functional-differentialequations, Studia Univ. “Babe ̧s-Bolyai”, Matematica,51(2) pp. 109–126, 2006.[11]Si, J. G.,Li, W. R. andCheng, S. S.,Analytic solution of on iterative functional-differential equation, Comput. Math. Appl.,33(6), pp. 47–51, 1997.
[12]Stanek, S.,Global properties of decreasing solutions of equationx′(t) =x(x(t)) +x(t),Funct. Diff. Eq.,4(1–2), pp. 191–213, 1997.
[13]S ̧erban, M. A.,Fiberφ-contractions, Studia Univ. “Babe ̧s-Bolyai”, Mathematica,44(3), pp. 99–108, 1999.Received by the editors: February 14, 2006.

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 obtained using weakly Picard operator theory.

Keywords:
Iterative functional-differential equation, weakly Picard operator, delay, data dependence.
1991 Mathematics Subject Classification:
34L05, 47H10.
∗This work has been supported by CEEX 2532/2006
†“Tiberiu Popoviciu” Institute of Numerical Analysis, P.O.Box. 68-1, Cluj-Napoca, Romania

1. INTRODUCTION

The aim of this paper is to study the following iterative system with delays

(1) xi′​(t)=fi​(t,x1​(t),x2​(t),x1​(x1​(t−τ1)),x2​(x2​(t−τ2))),t∈[t0,b],i=1,2,x_{i}^{\prime}(t)=f_{i}(t,x_{1}(t),x_{2}(t),x_{1}(x_{1}(t-\tau_{1})),x_{2}(x_{2}(t-\tau_{2}))),\ t\in[t_{0},b],\ i=1,2,

with the initial conditions

(2) xi​(t)=φi​(t),t∈[t0−τi,t0],i=1,2,x_{i}(t)=\varphi_{i}(t),\ t\in[t_{0}-\tau_{i},t_{0}],\ i=1,2,

where

  1. (H1)

    t0<b,τ1,τ2>0,τ1<τ2;t_{0}<b,\ \tau_{1},\tau_{2}>0,\ \tau_{1}<\tau_{2};

  2. (H2)

    fi∈C​([t0,b]×([t0−τ1,b]×[t0−τ2,b])2,ℝ),i=1,2;f_{i}\in C([t_{0},b]\times([t_{0}-\tau_{1},b]\times[t_{0}-\tau_{2},b])^{2},\mathbb{R}),\ i=1,2;

  3. (H3)

    φ1∈C​([t0−τ1,t0],[t0−τ1,b]),φ2∈C​([t0−τ2,t0],[t0−τ2,b]);\varphi_{1}\in C([t_{0}-\tau_{1},t_{0}],[t_{0}-\tau_{1},b]),\ \varphi_{2}\in C([t_{0}-\tau_{2},t_{0}],[t_{0}-\tau_{2},b]);

  4. (H4)

    there exists Lfi>0L_{f_{i}}>0 such that:

    |fi​(t,u1,u2,u3,u4)−fi​(t,v1,v2,v3,v4)|≤Lfi​(∑k=14|uk−vk|),\left|f_{i}(t,u_{1},u_{2},u_{3},u_{4})-f_{i}(t,v_{1},v_{2},v_{3},v_{4})\right|\leq L_{f_{i}}(\sum_{k=1}^{4}\left|u_{k}-v_{k}\right|),

    for all t∈[t0,b],(u1,u2,u3,u4),(v1,v2,v3,v4)∈([t0−τ1,b]×[t0−τ2,b])2,i=1,2.t\!\in\![t_{0},b],\!(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2},\\ i=1,2.

By a solution of (1)–(2) we understand a function (x1,x2)(x_{1},x_{2}) with

x1∈C​([t0−τ1,b],[t0−τ1,b])∩C1​([t0,b],[t0−τ1,b])\displaystyle x_{1}\in C([t_{0}-\tau_{1},b],[t_{0}-\tau_{1},b])\cap C^{1}([t_{0},b],[t_{0}-\tau_{1},b])
x2∈C​([t0−τ2,b],[t0−τ2,b])∩C1​([t0,b],[t0−τ2,b])\displaystyle x_{2}\in C([t_{0}-\tau_{2},b],[t_{0}-\tau_{2},b])\cap C^{1}([t_{0},b],[t_{0}-\tau_{2},b])

which satisfies (1)–(2).

The problem (1)–(2) is equivalent with the following fixed point equations:

(3a) x1​(t)={φ1​(t),t∈[t0−τ1,t0],φ1​(t0)+∫t0tf1​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s,t∈[t0,b],x_{1}\!(t)\!=\!\left\{\!\begin{array}[]{l}\!\varphi_{1}(t),\ t\in[t_{0}-\tau_{1},t_{0}],\\ \!\varphi_{1}(t_{0})\!+\!\int_{t_{0}}^{t}\!f_{1}(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))\!ds,\!t\!\in\![t_{0},b],\end{array}\right.
(3b) x2​(t)={φ2​(t),t∈[t0−τ2,t0],φ2​(t0)+∫t0tf2​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s,t∈[t0,b],x_{2}\!(t)\!=\!\left\{\!\begin{array}[]{l}\!\varphi_{2}(t),\ t\in[t_{0}-\tau_{2},t_{0}],\\ \!\varphi_{2}(t_{0})\!+\!\int_{t_{0}}^{t}\!f_{2}(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))\!ds,\!t\!\in\![t_{0},b],\end{array}\right.

where x1∈C​([t0−τ1,b],[t0−τ1,b]),x2∈C​([t0−τ2,b],[t0−τ2,b])x_{1}\in C([t_{0}-\tau_{1},b],[t_{0}-\tau_{1},b]),x_{2}\in C([t_{0}-\tau_{2},b],[t_{0}-\tau_{2},b]).

On the other hand, the system (1) is equivalent with

(4a) x1​(t)={x1​(t),t∈[t0−τ1,t0],x1​(t0)+∫t0tf1​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s,t∈[t0,b],x_{1}\!(t)\!=\!\left\{\!\begin{array}[]{l}\!x_{1}(t),\ t\in[t_{0}-\tau_{1},t_{0}],\\ \!x_{1}(t_{0})\!+\!\int_{t_{0}}^{t}\!f_{1}(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))\!ds,\!t\!\in\![t_{0},b],\end{array}\right.
(4b) x2​(t)={x2​(t),t∈[t0−τ2,t0],x2​(t0)+∫t0tf2​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s,t∈[t0,b],x_{2}\!(t)\!=\!\left\{\!\begin{array}[]{l}\!x_{2}(t),\ t\in[t_{0}-\tau_{2},t_{0}],\\ \!x_{2}(t_{0})\!+\!\int_{t_{0}}^{t}\!f_{2}(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))\!ds,\!t\!\in\![t_{0},b],\end{array}\right.

and x1∈C​([t0−τ1,b],[t0−τ1,b]),x2∈C​([t0−τ2,b],[t0−τ2,b])x_{1}\in C([t_{0}-\tau_{1},b],[t_{0}-\tau_{1},b]),\ x_{2}\in C([t_{0}-\tau_{2},b],[t_{0}-\tau_{2},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 [3], Y. Kuang [5], V. Mureşan [4], I.A. Rus [8] and to our papers [6], [7]. The case of iterative system with retarded arguments has been studied by many authors: I.A. Rus and E. Egri [11], J. G. Si, W. R. Li and S. S. Cheng [12], S. Stanek [13]. So our paper complement in this respect the existing literature.

Let us mention that the results from this paper are obtained as a concequence of those from [11] 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 [10], [9], M. Serban [14]).

Let (X,d)(X,d) be a metric space and A:X→XA:X\rightarrow X an operator. We shall use the following notations:

FA:={x∈X∣A​(x)=x}F_{A}:=\{x\in X\mid A(x)=x\} - the fixed point set of AA;

I​(A):={Y⊂X∣A​(Y)⊂Y,Y≠∅}I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\} - the family of the nonempty invariant subset of AA;

An+1:=A∘An,A0=1X,A1=A,n∈ℕA^{n+1}:=A\circ A^{n},\;A^{0}=1_{X},\;A^{1}=A,\;n\in\mathbb{N};

P​(X):={Y⊂X∣Y≠∅}P(X):=\{Y\subset X\mid Y\neq\emptyset\} - the set of the parts of X;X;

H​(Y,Z):=max⁡{supy∈Y​infz∈Z​d​(y,z),supz∈Z​infy∈Y​d​(y,z)}H(Y,Z):=\max\{\underset{y\in Y}{\sup}\underset{z\in Z}{\inf}d(y,z),\underset{z\in Z}{\sup}\underset{y\in Y}{\inf}d(y,z)\} -the Pompeiu–Housdorff functional on P​(X)×P​(X)P(X)\times P(X).

Definition 2.1.

Let (X,d)(X,d) be a metric space. An operator A:X→XA:X\rightarrow X is a Picard operator (PO) if there exists x∗∈Xx^{\ast}\in X such that:

  1. (i)

    FA={x∗},F_{A}=\{x^{\ast}\},

  2. (ii)

    the sequence (An​(x0))n∈ℕ(A^{n}(x_{0}))_{n\in\mathbb{N}} converges to x∗x^{\ast} for all x0∈Xx_{0}\in X.

Remark 2.2.

Accordingly to the definition, the contraction principle insures that, if A:X→XA:X\rightarrow X is a α\alpha -contraction on the complet metric space XX, then it is a Picard operator.

Theorem 2.3.

(Data dependence theorem). Let (X,d)(X,d) be a complete metric space and A,B:X→XA,B:X\rightarrow X two operators. We suppose that

  1. (i)

    the operator AA is a α\alpha -contraction;

  2. (ii)

    FB≠∅;F_{B}\neq\emptyset;

  3. (iii)

    there exists η>0\eta>0 such that

    d​(A​(x),B​(x))≤η,∀x∈X.d(A(x),B(x))\leq\eta,\ \forall x\in X.

Then if FA={xA∗}F_{A}=\{x_{A}^{\ast}\} and xB∗∈FB,x_{B}^{\ast}\in F_{B}, we have

d​(xA∗,xB∗)≤η1−α.d(x_{A}^{\ast},x_{B}^{\ast})\leq\frac{\eta}{1-\alpha}.
Definition 2.4.

Let (X,d)(X,d) be a metric space. An operator A:X→XA:X\rightarrow X is a weakly Picard operator (WPO) if the sequence (An​(x))n∈ℕ(A^{n}(x))_{n\in\mathbb{N}} converges for all x∈Xx\in X, and its limit ( which may depend on xx ) is a fixed point of AA.

Theorem 2.5.

Let (X,d)(X,d) be a metric space and A:X→XA:X\rightarrow X an operator. The operator AA is weakly Picard operator if and only if there exists a partition of XX,

X=∪λ∈Λ​XλX=\underset{\lambda\in\Lambda}{\cup}X_{\lambda}

where Λ\Lambda is the indices set of partition, such that:

  1. (a)

    Xλ∈I​(A),λ∈ΛX_{\lambda}\in I(A),\ \lambda\in\Lambda;

  2. (b)

    A|Xλ:Xλ→XλA|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda} is a Picard operator for all λ∈Λ\lambda\in\Lambda.

Definition 2.6.

If AA is weakly Picard operator then we consider the operator A∞A^{\infty} defined by

A∞:X→X,A∞​(x):=limn→∞​An​(x).A^{\infty}:X\rightarrow X,\ A^{\infty}(x):=\underset{n\rightarrow\infty}{\lim}A^{n}(x).

It is clear that A∞​(X)=FA.A^{\infty}(X)=F_{A}.

Definition 2.7.

Let AA be a weakly Picard operator and c>0.c>0. The operator AA\ is cc -weakly Picard operator if

d​(x,A∞​(x))≤c​d​(x,A​(x)),∀x∈X.d(x,A^{\infty}(x))\leq cd(x,A(x)),\ \forall x\in X.
Example 2.8.

Let (X,d)(X,d) be a complete metric space and A:X→XA:X\rightarrow X a continuous operator. We suppose that there exists α∈[0,1)\alpha\in[0,1) such that

d​(A2​(x),A​(x))≤α​(x,A​(x)),∀x∈X.d(A^{2}(x),A(x))\leq\alpha(x,A(x)),\ \forall x\in X.

Then AA is cc -weakly Picard operator with c=11−α.c=\dfrac{1}{1-\alpha}.

Theorem 2.9.

Let (X,d)(X,d) be a metric space and Ai:X→X,i=1,2.A_{i}:X\rightarrow X,\ i=1,2. Suppose that

  1. (i)

    the operator AiA_{i} is cic_{i} -weakly Picard operator, i=1,2;i=1,2;

  2. (ii)

    there exists η>0\eta>0 such that

    d​(A1​(x),A2​(x))≤η,∀x∈X.d(A_{1}(x),A_{2}(x))\leq\eta,\ \forall x\in X.

Then

H​(FA1,FA2)≤η​max⁡(c1,c2).H(F_{A_{1}},F_{A_{2}})\leq\eta\max(c_{1},c_{2}).
Theorem 2.10.

(Fibre contraction principle). Let (X,d)(X,d) and (Y,ρ)(Y,\rho) be two metric spaces and A:X×Y→X×Y,A=(B,C),(B:X→X,C:X×Y→Y)A:X\times Y\rightarrow X\times Y,\ A=(B,C),\ (\ B:X\rightarrow X,\ C:X\times Y\rightarrow Y\ ) a triangular operator. We suppose that

  1. (i)

    (Y,ρ)(Y,\rho) is a complete metric space;

  2. (ii)

    the operator BB is Picard operator;

  3. (iii)

    there exists l∈[0,1)l\in[0,1) such that C​(x,⋅):Y→YC(x,\cdot):Y\rightarrow Y is a ll-contraction, for all x∈Xx\in X;

  4. (iv)

    if (x∗,y∗)∈FA(x^{\ast},y^{\ast})\in F_{A}, then C​(⋅,y∗)C(\cdot,y^{\ast}) is continuous in x∗x^{\ast}.

Then the operator AA 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),A_{f}\!:\!C(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\!\rightarrow\!C(\![t_{0}\!-\!\tau_{1},b]\!,\!\mathbb{R}\!)\!\times\!C(\![t_{0}\!-\!\tau_{2},b]\!,\!\mathbb{R}\!),

given by the relation

Af​(x1,x2)=(Af1​(x1,x2),Af2​(x1,x2)),A_{f}(x_{1},x_{2})=(A_{f_{1}}(x_{1},x_{2}),A_{f_{2}}(x_{1},x_{2})),

where Af1​(x1,x2)​(t):=A_{f_{1}}(x_{1},x_{2})(t):= the right hand side of (3a) and Af2​(x1,x2)​(t):=A_{f_{2}}(x_{1},x_{2})(t):= the right hand side of (3b).

Let L1,L2>0L_{1},L_{2}>0, L=max⁡{L1,L2}L=\max\{L_{1},L_{2}\} and

CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]):=\displaystyle C_{L}([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\times C_{L}([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b]):=
={(x1,x2)∈C([t0−τ1,b],[t0−τ1,b])×C([t0−τ2,b],[t0−τ2,b]):\displaystyle\ =\{\!(x_{1},x_{2})\!\in\!C([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\!\times\!C([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b])\!:
|xi(t1)−xi(t2)|≤Li|t1−t2|,∀(t1,t2)∈[t0−τ2,b],i=1,2}.\displaystyle\left|x_{i}(t_{1})-x_{i}(t_{2})\right|\leq L_{i}\left|t_{1}-t_{2}\right|,\ \forall(t_{1},t_{2})\in[t_{0}-\tau_{2},b],\ i=1,2\}.

It is clear that CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])C_{L}([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\times C_{L}([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b]) is a complete metric space with respect to the metric

d​(x,x¯):=maxt0≤t≤b​|x​(t)−x¯​(t)|.d(x,\overline{x}):=\underset{t_{0}\leq t\leq b}{\max}\left|x(t)-\overline{x}(t)\right|.

We remark that CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])C_{L}([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\times C_{L}([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b]) is a closed subset in C​([t0−τ1,b],[t0−τ1,b])×C​([t0−τ2,b],[t0−τ2,b])C([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\times C([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b]).

We have

Theorem 3.1.

We suppose that

  1. (i)

    the conditions (H1{}_{\text{1}})–(H4{}_{\text{4}}) are satisfied;

  2. (ii)

    φ1∈CL​([t0−τ1,t0],[t0−τ1,b]),φ2∈CL​([t0−τ2,t0],[t0−τ2,b]);\varphi_{1}\in C_{L}([t_{0}-\tau_{1},t_{0}],[t_{0}-\tau_{1},b]),\ \varphi_{2}\in C_{L}([t_{0}-\tau_{2},t_{0}],[t_{0}-\tau_{2},b]);

  3. (iii)

    mfim_{f_{i}} and Mfi∈ℝ,i=1,2M_{f_{i}}\in\mathbb{R},\ i=1,2 are such that

    1. (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,m_{f_{i}}\!\leq f_{i}(\!t,u_{1},u_{2},u_{3},u_{4}\!)\!\leq M_{f_{i}},\forall t\in[t_{0},b],(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\\ \in\!(\![t_{0}\!-\!\tau_{1},b]\!\times\![t_{0}\!-\!\tau_{2},b]\!)^{2},

    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,\begin{array}[]{ll}t_{0}-\tau_{i}\leq\varphi_{i}(t_{0})+m_{f_{i}}(b-t_{0})&\text{for }m_{f_{i}}<0,\\ t_{0}-\tau_{i}\leq\varphi_{i}(t_{0})&\text{for }m_{f_{i}}\geq 0,\\ b\geq\varphi_{i}(t_{0})&\text{for }M_{f_{i}}\leq 0,\\ b\geq\varphi_{i}(t_{0})+M_{f_{i}}(b-t_{0})&\text{for }M_{f_{i}}>0,\end{array}
    3. (iiic)

      L+Mfi<1;L+M_{f_{i}}<1;

  4. (iv)

    (b−t0)​(Lf1+Lf2)​(L+2)<1.(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2)<1.

Then the Cauchy problem (1)–(2) has, in CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])C_{L}([t_{0}\!-\!\tau_{1},b],[t_{0}\!-\!\tau_{1},b])\times C_{L}([t_{0}\!-\!\tau_{2},b],[t_{0}\!-\!\tau_{2},b]) a unique solution. Moreover the operator

Af:CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])→\displaystyle A_{f}\!:\!C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\!\rightarrow
CL​([t0−τ1,b],CL​([t0−τ1,b],[t0−τ1,b]))×CL​([t0−τ2,b],CL​([t0−τ2,b],[t0−τ2,b]))\displaystyle\!C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\!C_{L}\!(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!)\!\times\!C_{L}\!(\![t_{0}\!-\!\tau_{2},b]\!,\!C_{L}\!(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\!)

is a cc -Picard operator with c=1(b−t0)​(Lf1+Lf2)​(L+2).c=\dfrac{1}{(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2)}.

(a) CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!) is an invariant subset for Af.A_{f}.

Indeed,

t0−τi≤Afi​(x1,x2)​(t)≤b,t_{0}-\tau_{i}\leq A_{f_{i}}(x_{1},x_{2})(t)\leq b,

(x1,x2)​(t)∈[t0−τ1,b]×[t0−τ2,b],t∈[t0,b],i=1,2.(x_{1},x_{2})(t)\in[t_{0}-\tau_{1},b]\times[t_{0}-\tau_{2},b],\ t\in[t_{0},b],\ i=1,2.

From (iiia) we have mfim_{f_{i}} and Mfi∈ℝM_{f_{i}}\in\mathbb{R} such that

mfi≤fi​(t,u1,u2,u3,u4)≤Mfi,m_{f_{i}}\leq f_{i}(t,u_{1},u_{2},u_{3},u_{4})\leq M_{f_{i}},

∀t∈[t0,b],(u1,u2,u3,u4),(v1,v2,v3,v4)∈([t0−τ1,b]×[t0−τ2,b])2,i=1,2.\forall t\in[t_{0},b],\ (\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2},\ i=1,2.

This implies that

∫t0tmfi​𝑑s≤∫t0tfi​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s≤∫t0tMfi​𝑑s\int_{t_{0}}^{t}m_{f_{i}}ds\leq\int_{t_{0}}^{t}f_{i}(s,x_{1}(s),x_{2}(s),x_{1}(x_{1}(s\!-\!\tau_{1})),x_{2}(x_{2}(s\!-\!\tau_{2})))ds\leq\int_{t_{0}}^{t}M_{f_{i}}ds, ∀t∈[t0,b],\forall t\in[t_{0},b], that is

φi​(t0)+mfi​(b−t0)≤Afi​(x1,x2)​(t)≤φi​(t0)+Mfi​(b−t0),t∈[t0,b].\varphi_{i}(t_{0})+m_{f_{i}}(b-t_{0})\leq A_{f_{i}}(x_{1},x_{2})(t)\leq\varphi_{i}(t_{0})+M_{f_{i}}(b-t_{0}),t\in[t_{0},b].

Therefor if condition (iii) holds, we have satisfied the invariance property for the operator AfA_{f} in C​([t0−τ1,b],[t0−τ1,b])×C​([t0−τ2,b],[t0−τ2,b]).C\!(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C\!(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!).

Now, consider t1,t2∈[t0−τ1,t0]:t_{1},t_{2}\in[t_{0}-\tau_{1},t_{0}]:

|Af1​(x1,x2)​(t1)−Af1​(x1,x2)​(t2)|=|φ1​(t1)−φ1​(t2)|≤L1​|t1−t2|,\left|A_{f_{1}}(x_{1},x_{2})(t_{1})-A_{f_{1}}(x_{1},x_{2})(t_{2})\right|=\left|\varphi_{1}(t_{1})-\varphi_{1}(t_{2})\right|\leq L_{1}\left|t_{1}-t_{2}\right|,

because φ1∈CL​([t0−τ1,t0],[t0−τ1,b])\varphi_{1}\in C_{L}(\![t_{0}\!-\!\tau_{1},t_{0}],[t_{0}\!-\!\tau_{1},b]\!).

Similarly, for t1,t2∈[t0−τ2,t0]:t_{1},t_{2}\in[t_{0}-\tau_{2},t_{0}]:

|Af2​(x1,x2)​(t1)−Af2​(x1,x2)​(t2)|=|φ2​(t1)−φ2​(t2)|≤L2​|t1−t2|,\left|A_{f_{2}}(x_{1},x_{2})(t_{1})-A_{f_{2}}(x_{1},x_{2})(t_{2})\right|=\left|\varphi_{2}(t_{1})-\varphi_{2}(t_{2})\right|\leq L_{2}\left|t_{1}-t_{2}\right|,

that follows from (ii), too.

On the other hand, if t1,t2∈[t0,b]t_{1},t_{2}\in[t_{0},b], we have

|Afi​(x1,x2)​(t1)−Afi​(x1,x2)​(t2)|=\displaystyle\left|A_{f_{i}}(x_{1},x_{2})(t_{1})-A_{f_{i}}(x_{1},x_{2})(t_{2})\right|=
=\displaystyle= |φi(t1)−φi(t2)+∫t0t1fi(s,x1(s),x2(s),x1(x1(s−τ1)),x2(x2(s−τ2)))ds−\displaystyle\left|\varphi_{i}(t_{1})\!-\!\varphi_{i}(t_{2})\!+\!\int_{t_{0}}^{t_{1}}\!f_{i}\!(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))ds\right.\!-
−∫t0t2fi​(s,x1​(s),x2​(s),x1​(x1​(s−τ1)),x2​(x2​(s−τ2)))​𝑑s|≤\displaystyle-\left.\int_{t_{0}}^{t_{2}}f_{i}(s,x_{1}(s),x_{2}(s),x_{1}\!(x_{1}(s\!-\!\tau_{1})),x_{2}\!(x_{2}(s\!-\!\tau_{2})))ds\right|\leq
≤\displaystyle\leq Li​|t1−t2|+Mfi​|t1−t2|≤(L+Mfi)​|t1−t2|,i=1,2.\displaystyle L_{i}\left|t_{1}-t_{2}\right|+M_{f_{i}}\left|t_{1}-t_{2}\right|\leq(L+M_{f_{i}})\left|t_{1}-t_{2}\right|,\ i=1,2.

So we can affirm that ∀t1,t2∈[t0,b],t1≤t2,\forall t_{1},t_{2}\in[t_{0},b],\ t_{1}\leq t_{2}, and doe to (iii), AfA_{f} is LL -Lipshitz.

Thus, according to the above, we have CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])∈I​(Af).C_{L}\!(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}\!(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\in I(A_{f}).

(b) AfA_{f} is a LAfL_{A_{f}} -contraction with LAf=(b−t0)​(Lf1+Lf2)​(L+2)L_{A_{f}}=(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2).

For t∈[t0−τ1,t0],t\in[t_{0}-\tau_{1},t_{0}], we have |Af1​(x1,x2)​(t)−Af1​(x¯1,x¯2)​(t)|=0.\left|A_{f_{1}}(x_{1},x_{2})(t)-A_{f_{1}}(\overline{x}_{1},\overline{x}_{2})(t)\right|=0.

For t∈[t0−τ2,t0],t\in[t_{0}-\tau_{2},t_{0}], we have |Af2​(x1,x2)​(t)−Af2​(x¯1,x¯2)​(t)|=0.\left|A_{f_{2}}(x_{1},x_{2})(t)-A_{f_{2}}(\overline{x}_{1},\overline{x}_{2})(t)\right|=0.

For t∈[t0,b]:t\in[t_{0},b]:

|Af1​(x1,x2)​(t)−Af1​(x¯1,x¯2)​(t)|=\displaystyle\left|A_{f_{1}}(x_{1},x_{2})(t)-A_{f_{1}}(\overline{x}_{1},\overline{x}_{2})(t)\right|=
=\displaystyle= |∫t0t[f1(s,x1(s),x2(s),x1(x1(s−τ1)),x2(x2(s−τ2)))−\displaystyle\!\left|\int_{t_{0}}^{t}\![\!f_{1}\!(\!s,x_{1}(s),x_{2}(s),x_{1}(x_{1}(s\!-\!\tau_{1})),x_{2}(x_{2}(s\!-\!\tau_{2}))\!)\!\right.-
−f1(s,x¯1(s),x¯2(s),x¯1(x¯1(s−τ1)),x¯2(x¯2(s−τ2)))]ds|≤\displaystyle-\left.\!f_{1}(\!s,\overline{x}_{1}(s),\overline{x}_{2}(s),\overline{x}_{1}(\overline{x}_{1}(s\!-\!\tau_{1})),\overline{x}_{2}(\overline{x}_{2}(s\!-\!\tau_{2}))\!)\!]ds\!\right|\leq
≤\displaystyle\leq Lf1(|x1(s)−x¯1(s)|+|x2(s)−x¯2(s)|+|x1(x1(s−τ1))−x¯1(x¯1(s−τ1))|+\displaystyle L_{f_{1}}\!(\left|x_{1}(s)\!-\!\overline{x}_{1}(s)\right|+\left|x_{2}(s)\!-\!\overline{x}_{2}(s)\right|+\left|x_{1}(x_{1}(s\!-\!\tau_{1}))\!-\!\overline{x}_{1}(\overline{x}_{1}(s\!-\!\tau_{1}))\right|\!+
+|x2(x2(s−τ2))−x¯2(x¯2(s−τ2))|)(b−t0)≤\displaystyle+\left|x_{2}(x_{2}(s\!-\!\tau_{2}))-\overline{x}_{2}(\overline{x}_{2}(s\!-\!\tau_{2}))\right|\!)(b-t_{0})\leq
≤\displaystyle\leq (b−t0)Lf1[∥x1−x¯1∥C+∥x2−x¯2∥C+|x1(x1(s−τ1))−x1(x¯1(s−τ1))|+\displaystyle(b\!-\!t_{0})L_{f_{1}}[\left\|x_{1}\!-\!\overline{x}_{1}\right\|_{C}+\left\|x_{2}\!-\!\overline{x}_{2}\right\|_{C}+\left|x_{1}(x_{1}(s\!-\!\tau_{1}))\!-\!x_{1}(\overline{x}_{1}(s\!-\!\tau_{1}))\right|\!+
+|x1​(x¯1​(s−τ1))−x¯1​(x¯1​(s−τ1))|+|x2​(x2​(s−τ2))−x2​(x¯2​(s−τ2))|+\displaystyle+\left|x_{1}(\overline{x}_{1}(s\!-\!\tau_{1}))\!-\!\overline{x}_{1}(\overline{x}_{1}(s\!-\!\tau_{1}))\right|+\left|x_{2}(x_{2}(s\!-\!\tau_{2}))\!-\!x_{2}(\overline{x}_{2}(s\!-\!\tau_{2}))\right|+
+|x2(x¯2(s−τ2))−x¯2(x¯2(s−τ2))|]≤(b−t0)Lf1[∥x1−x¯1∥C+∥x2−x¯2∥C+\displaystyle+\!\left|x_{2}\!(\overline{x}_{2}(s\!-\!\tau_{2}))\!-\!\overline{x}_{2}(\overline{x}_{2}\!(s\!-\!\tau_{2}))\right|]\!\leq\!(b\!-\!t_{0})L_{f_{1}}[\left\|x_{1}\!-\!\overline{x}_{1}\right\|_{C}\!+\!\left\|x_{2}\!-\!\overline{x}_{2}\right\|_{C}\!+
+L1∥x1−x¯1∥C+∥x1−x¯1∥C+L2∥x2−x¯2∥C+∥x2−x¯2∥C]≤\displaystyle+L_{1}\left\|x_{1}-\overline{x}_{1}\right\|_{C}+\left\|x_{1}-\overline{x}_{1}\right\|_{C}+L_{2}\left\|x_{2}-\overline{x}_{2}\right\|_{C}+\left\|x_{2}-\overline{x}_{2}\right\|_{C}]\leq
≤\displaystyle\leq (b−t0)​Lf1​(L+2)​(‖x1−x¯1‖C+‖x2−x¯2‖C).\displaystyle(b-t_{0})L_{f_{1}}(L+2)(\left\|x_{1}-\overline{x}_{1}\right\|_{C}+\left\|x_{2}-\overline{x}_{2}\right\|_{C}).

In the same way

|Af2​(x1,x2)​(t)−Af2​(x¯1,x¯2)​(t)|≤(b−t0)​Lf2​(L+2)​(‖x1−x¯1‖+‖x2−x¯2‖).\left|A_{f_{2}}(x_{1},x_{2})(t)-A_{f_{2}}(\overline{x}_{1},\overline{x}_{2})(t)\right|\leq(b-t_{0})L_{f_{2}}(L+2)(\left\|x_{1}-\overline{x}_{1}\right\|+\left\|x_{2}-\overline{x}_{2}\right\|).

Then we have the following relation

‖Af​(x1,x2)−Af​(x¯1,x¯2)‖C≤(b−t0)​(Lf1+Lf2)​(L+2)​‖(x1,x2)−(x¯1,x¯2)‖C\left\|A_{f}(x_{1},x_{2})-A_{f}(\overline{x}_{1},\overline{x}_{2})\right\|_{C}\leq(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2)\left\|(x_{1},x_{2})-(\overline{x}_{1},\overline{x}_{2})\right\|_{C}

So AfA_{f} is a cc -Picard operator with c=11−LAf.c=\dfrac{1}{1-L_{A_{f}}}. ∎

In what follows, consider the following operator

Bf:\displaystyle B_{f}\!:\!\! CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b])→\displaystyle C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\!\rightarrow
CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]),\displaystyle C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!),

given by the relation

Bf​(x1,x2)=(Bf1​(x1,x2),Bf2​(x1,x2)),B_{f}(x_{1},x_{2})=(B_{f_{1}}(x_{1},x_{2}),B_{f_{2}}(x_{1},x_{2})),

where Bf1​(x1,x2):=B_{f_{1}}(x_{1},x_{2}):= the right hand side of (4a) and Bf2​(x1,x2):=B_{f_{2}}(x_{1},x_{2}):= the right hand side of (4b).

Theorem 3.2.

In the conditions of Theorem 3.1, the 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])B_{f}:C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\rightarrow C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!) is WPO.

The operator BfB_{f} 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},\displaystyle X_{\varphi_{1}}:=\{x_{1}\in C(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)|\ x_{1}|_{[t_{0}-\tau_{1},t_{0}]}=\varphi_{1}\},
Xφ2:={x2∈C​([t0−τ2,b],[t0−τ2,b])|x2|[t0−τ2,t0]=φ2}.\displaystyle X_{\varphi_{2}}:=\{x_{2}\in C(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)|\ x_{2}|_{[t_{0}-\tau_{2},t_{0}]}=\varphi_{2}\}.

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.C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)\!=\!\underset{\varphi_{i}\in\!C_{L}\!(\![t_{0}\!-\!\tau_{i},t_{0}]\!,\![t_{0}\!-\!\tau_{i},b]\!)}{\bigcup\!}X_{\varphi_{1}}\!\times\!X_{\varphi_{2}}.

We have that Xφ1×Xφ2∈I​(Bf)X_{\varphi_{1}}\!\times\!X_{\varphi_{2}}\in I(B_{f}) and Bf|Xφ1×Xφ2B_{f}|_{X_{\varphi_{1}}\!\times\!X_{\varphi_{2}}} is a Picard operator because is the operator which appears in the proof of Theorem 3.1. By applying Theorem 2.5, we obtain that BfB_{f} is WPO. ∎

4. INCREASING SOLUTION OF (1)

4.1. Inequalities of Ĉapligin type

Theorem 4.1.

We suppose that

  1. (a)

    the conditions of the Theorem 3.1 are satisfied;

  2. (b)

    (u1,u2,u3,u4),(v1,v2,v3,v4)∈([t0−τ1,b]×[t0−τ2,b])2,uj≤vj,j=1,4¯,(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2},\ u_{j}\leq v_{j},\ j=\overline{1,4}, imply that

    fi​(t,u1,u2,u3,u4)≤fi​(t,v1,v2,v3,v4),f_{i}(\!t,\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!)\leq f_{i}(\!t,\!v_{1}\!,\!v_{2}\!,v_{3}\!,v_{4}\!),

    i=1,2,i=1,2, for all t∈[t0,b].t\in[t_{0},b].

Let (x1,x2)(x_{1},x_{2}) be an increasing solution of the system (1) and (y1,y2)(y_{1},y_{2}) an increasing solution for the system of inequalities

yi′​(t)≤fi​(t,y1​(t),y2​(t),y1​(y1​(t−τ1)),y2​(y2​(t−τ2))),t∈[t0,b],y_{i}^{\prime}(t)\leq f_{i}(t,y_{1}(t),y_{2}(t),y_{1}(y_{1}(t-\tau_{1})),y_{2}(y_{2}(t-\tau_{2}))),\ t\in[t_{0},b],

Then

yi​(t)≤xi​(t),t∈[t0−τi,t0],i=1,2⇒(y1,y2)≤(x1,x2).y_{i}(t)\leq x_{i}(t),\ t\in[t_{0}-\tau_{i},t_{0}],\ i=1,2\Rightarrow(y_{1},y_{2})\leq(x_{1},x_{2}).

In the terms of the operator BfB_{f}, we have

(x1,x2)=Bf​(x1,x2)​ and ​(y1,y2)≤Bf​(y1,y2)​.(x_{1},x_{2})=B_{f}(x_{1},x_{2})\text{ and }(y_{1},y_{2})\leq B_{f}(y_{1},y_{2})\text{.}

However, from the condition (b), we have that the operator Bf∞B_{f}^{\infty} is increasing,

(y1,y2)\displaystyle(y_{1},y_{2}) ≤\displaystyle\leq Bf∞​(y1,y2)=Bf∞​(y~1|[t0−τ1,t0],y~2|[t0−τ2,t0])≤\displaystyle B_{f}^{\infty}(y_{1},y_{2})=B_{f}^{\infty}(\widetilde{y}_{1}|_{[t_{0}-\tau_{1},t_{0}]},\widetilde{y}_{2}|_{[t_{0}-\tau_{2},t_{0}]})\leq
≤\displaystyle\leq Bf∞​(x~1|[t0−τ1,t0],x~2|[t0−τ2,t0])=(x1,x2).\displaystyle B_{f}^{\infty}(\widetilde{x}_{1}|_{[t_{0}-\tau_{1},t_{0}]},\widetilde{x}_{2}|_{[t_{0}-\tau_{2},t_{0}]})=(x_{1},x_{2}).

Thus (y1,y2)≤(x1,x2)(y_{1},y_{2})\leq(x_{1},x_{2}).

Here, for (x~1,x~2)(\widetilde{x}_{1},\widetilde{x}_{2}) we used the notation x~1∈Xx1|[t0−τ1,t0],x~2∈Xx1|[t0−τ2,t0]\widetilde{x}_{1}\in X_{x_{1}|_{[t_{0}-\tau_{1},t_{0}]}},\widetilde{x}_{2}\in X_{x_{1}|_{[t_{0}-\tau_{2},t_{0}]}}. ∎

4.2. Comparison theorem

In the next result we want to study the monotony of the solution of the problem (1)–(2) with respect to φi\varphi_{i} and fi,i=1,2.f_{i},\ i=1,2. We shall use the result below:

Lemma 4.2.

(Abstract comparison lemma). Let (X,d,≤)(X,d,\leq) be an ordered metric space and A,B,C:X→XA,B,C:X\rightarrow X such that:

  1. (i)

    A≤B≤C;A\leq B\leq C;

  2. (ii)

    the operators A,B,CA,B,C are WPO;

  3. (iii)

    the operator BB is increasing.

Then

x≤y≤z⇒A∞​(x)≤B∞​(y)≤C∞​(z).x\leq y\leq z\Rightarrow A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(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.f_{i}^{j}\in C([t_{0},b]\times([t_{0}\!-\!\tau_{1},b]\times[t_{0}\!-\!\tau_{2},b])^{2}),\ i=1,2,j=1,2,3.

We suppose that

  1. (a)

    fi2​(t,⋅,⋅,⋅,⋅):([t0−τ1,b]×[t0−τ2,b])2→([t0−τ1,b]×[t0−τ2,b])2f_{i}^{2}(t,\cdot,\cdot,\cdot,\cdot)\!:\!([t_{0}\!-\!\tau_{1},b]\times[t_{0}\!-\!\tau_{2},b])^{2}\!\rightarrow\!([t_{0}\!-\!\tau_{1},b]\times[t_{0}\!-\!\tau_{2},b])^{2} are increasing;

  2. (b)

    fi1≤fi2≤fi3.f_{i}^{1}\leq f_{i}^{2}\leq f_{i}^{3}.

Let (x1j,x2j)(x_{1}^{j},x_{2}^{j}) be an increasing solution of the systems

xi′​(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.x_{i}^{\prime}(t)\!=\!f_{i}^{j}\!(\!t,x_{1}(t),x_{2}(t),x_{1}(x_{1}\!(t\!-\!\tau_{1})),x_{2}\!(x_{2}(t\!-\!\tau_{2}))\!),\!t\in[t_{0}\!,\!b],i=1,2,\!j=1,2,3.

If xi1​(t)≤xi2​(t)≤xi3​(t),t∈[t0−τi,t0]x_{i}^{1}(t)\leq x_{i}^{2}(t)\leq x_{i}^{3}(t),\ t\in[t_{0}-\tau_{i},t_{0}] then xi1≤xi2≤xi3,i=1,2.x_{i}^{1}\leq x_{i}^{2}\leq x_{i}^{3},\ i=1,2.

The operators Bfj,j=1,2,3B_{f}^{j},\ j=1,2,3 are WPO. Taking into consideration the condition (a) the operator Bf2B_{f}^{2} is increasing. From (b) we have that Bf1≤Bf2≤Bf3B_{f}^{1}\leq B_{f}^{2}\leq B_{f}^{3}. We note that (x1j,x2j)=Bfj​∞​(x~1j,x~2j),j=1,2,3(x_{1}^{j},x_{2}^{j})=B_{f}^{j\infty}(\widetilde{x}_{1}^{j},\widetilde{x}_{2}^{j}),\ j=1,2,3. Now, using the Abstract comparison lemma, the proof is complete. ∎

5. DATA DEPENDENCE: CONTINUITY

Consider the Cauchy problem (1)–(2) and suppose the conditions of Theorem 3.1 are satisfied. Denote by (x1,x2)​(⋅;φ1,φ2,f1,f2),i=1,2(x_{1},x_{2})(\cdot;\varphi_{1},\varphi_{2},f_{1},f_{2}),i=1,2 the solution of this problem. We can state the following result:

Theorem 5.1.

Let φ1j,φ2j,f1j,f2j,j=1,2\varphi_{1}^{j},\varphi_{2}^{j},f_{1}^{j},f_{2}^{j},\ j=1,2 be as in Theorem 3.1. We suppose that there exists η1,η2,ηi3,i=1,2\eta^{1},\eta^{2},\eta_{i}^{3},i=1,2 such that

  1. (i)

    |φ11​(t)−φ12​(t)|≤η1,∀t∈[t0−τ1,t0]\left|\varphi_{1}^{1}(t)-\varphi_{1}^{2}(t)\right|\leq\eta^{1},\ \forall t\in[t_{0}-\tau_{1},t_{0}] and |φ21​(t)−φ22​(t)|≤η2,∀t∈[t0−τ2,t0];\left|\varphi_{2}^{1}(t)-\varphi_{2}^{2}(t)\right|\leq\eta^{2},\ \forall t\in[t_{0}-\tau_{2},t_{0}];

  2. (ii)

    |fi1​(t,u1,u2,u3,u4)−fi2​(t,v1,v2,v3,v4)|≤ηi3,i=1,2,(u1,u2,u3,u4),(v1,v2,v3,v4)∈([t0−τ1,b]×[t0−τ2,b])2.\left|f_{i}^{1}(t,u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4})-f_{i}^{2}(t,v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4})\right|\leq\eta_{i}^{3},i=1,2,(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\\ \!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2}.

Then

|(x1,x2)​(t;φ11,φ21,f11,f21)−(x1,x2)​(t;φ12,φ22​f12,f22)|≤η1+η2+(η13+η23)​(b−t0)(b−t0)​(Lf1+Lf2)​(L+2),\left|(\!x_{1},x_{2}\!)(\!t;\varphi_{1}^{1}\!,\!\varphi_{2}^{1}\!,\!f_{1}^{1}\!,\!f_{2}^{1}\!)\!-\!(\!x_{1},x_{2}\!)(\!t;\varphi_{1}^{2}\!,\!\varphi_{2}^{2}\!\,\!f_{1}^{2}\!,\!f_{2}^{2}\!)\right|\!\leq\!\frac{\eta^{1}\!+\!\eta^{2}\!+\!(\eta_{1}^{3}\!+\!\eta_{2}^{3})(b\!-\!t_{0})}{(b\!-\!t_{0})(L_{f_{1}}\!+\!L_{f_{2}})(L\!+\!2)},

where Lfi=max⁡(Lfi1,Lfi2),i=1,2.L_{f_{i}}=\max(L_{f_{i}^{1}},L_{f_{i}^{2}}),i=1,2.

Consider the operators Aφ1j,φ2j,f1j,f2j,j=1,2.A_{\varphi_{1}^{j},\varphi_{2}^{j},f_{1}^{j},f_{2}^{j}},j=1,2. From Theorem 3.1 these operators are contractions.

Then

‖Aφ11,φ21,f11,f21​(x1,x2)−Aφ12,φ22,f12,f22​(x1,x2)‖C≤η1+η2+(η13+η23)​(b−t0),\left\|A_{\varphi_{1}^{1},\varphi_{2}^{1},f_{1}^{1},f_{2}^{1}}(x_{1},x_{2})-A_{\varphi_{1}^{2},\varphi_{2}^{2},f_{1}^{2},f_{2}^{2}}(x_{1},x_{2})\right\|_{C}\leq\eta^{1}+\eta^{2}+(\eta_{1}^{3}+\eta_{2}^{3})(b-t_{0}),

∀(x1,x2)∈CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]).\forall(x_{1},x_{2})\in C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!).

Now the proof follows from Theorem 2.3, with A:=Aφ11,φ21,f11,f21,B=Aφ12,φ22,f12,f22,η=η1+η2+(η13+η23)​(b−t0)A:=A_{\varphi_{1}^{1},\varphi_{2}^{1},f_{1}^{1},f_{2}^{1}},\ B=A_{\varphi_{1}^{2},\varphi_{2}^{2},f_{1}^{2},f_{2}^{2}},\ \eta=\eta^{1}+\eta^{2}+(\eta_{1}^{3}+\eta_{2}^{3})(b-t_{0}) and α:=LAf=(b−t0)​(Lf1+Lf2)​(L+2)\alpha:=L_{A_{f}}=(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2) where Lfi=max⁡(Lfi1,Lfi2),i=1,2.L_{f_{i}}=\max(L_{f_{i}^{1}},L_{f_{i}^{2}}),i=1,2. ∎

From the Theorem above we have:

Theorem 5.2.

Let fi1f_{i}^{1} and fi2f_{i}^{2} be as in Theorem 3.1, i=1,2i=1,2. Let SBfi1,SBfi2S_{B_{f_{i}^{1}}},S_{B_{f_{i}^{2}}} be the solution set of the system (1) corresponding to fi1f_{i}^{1} and fi2,i=1,2f_{i}^{2},i=1,2. Suppose that there exists ηi>0,i=1,2\eta_{i}>0,i=1,2 such that

(6) |fi1​(t,u1,u2,u3,u4)−fi2​(t,v1,v2,v3,v4)|≤ηi\left|f_{i}^{1}(\!t\!,\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4})\!-\!f_{i}^{2}(\!t\!,\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4})\!\right|\leq\eta_{i}

for all t∈[t0,b],(u1,u2,u3,u4),(v1,v2,v3,v4)∈([t0−τ1,b]×[t0−τ2,b])2,i=1,2.t\in[t_{0},b],(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\!(\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2},i=1,2.

Then

H∥⋅∥C​(SBfi1,SBfi2)≤(η1+η2)​(b−t0)1−(Lf1+Lf2)​(L+2)​(b−t0),H_{\left\|\cdot\right\|_{C}}(S_{B_{f_{i}^{1}}},S_{B_{f_{i}^{2}}})\leq\frac{(\eta_{1}+\eta_{2})(b\!-\!t_{0})}{1-(L_{f_{1}}+L_{f_{2}})(L+2)(b\!-\!t_{0})},

where Lfi:=max⁡(Lfi1,Lfi2)L_{f_{i}}:=\max(L_{f_{i}^{1}},L_{f_{i}^{2}}) and H∥⋅∥CH_{\left\|\cdot\right\|_{C}} denotes the Pompeiu-Housdorff functional with respect to ∥⋅∥C\left\|\cdot\right\|_{C} on CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]).C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!).

We will look for those c1c_{1} and c2c_{2} for which in condition of Theorem 3.1 the operators Bfi1B_{f_{i}^{1}} and Bfi2,i=1,2B_{f_{i}^{2}},i=1,2 are c1c_{1}-WPO and c2c_{2}-WPO.

Let

Xφ1:={x1∈C​([t0−τ1,b],[t0−τ1,b])|x1|[t0−τ1,t0]=φ1},\displaystyle X_{\varphi_{1}}:=\{x_{1}\in C(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)|\ x_{1}|_{[t_{0}-\tau_{1},t_{0}]}=\varphi_{1}\},
Xφ2:={x2∈C​([t0−τ2,b],[t0−τ2,b])|x2|[t0−τ2,t0]=φ2}.\displaystyle X_{\varphi_{2}}:=\{x_{2}\in C(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!)|\ x_{2}|_{[t_{0}-\tau_{2},t_{0}]}=\varphi_{2}\}.

It is clear that Bfi1|X​φ1×Xφ2=Afi1,Bfi2|X​φ1×Xφ2=Afi2.B_{f_{i}^{1}}|_{X\varphi_{1}\times X_{\varphi_{2}}}=A_{f_{i}^{1}},\ B_{f_{i}^{2}}|_{X\varphi_{1}\times X_{\varphi_{2}}}=A_{f_{i}^{2}}. So from Theorem 2.5 and Theorem 3.1 we have

‖Bfi12​(x1,x2)−Bfi1​(x1,x2)‖C\displaystyle\left\|\!B_{f_{i}^{1}}^{2}\!(\!x_{1}\!,\!x_{2}\!)\!-\!B_{f_{i}^{1}}\!(\!x_{1}\!,\!x_{2}\!)\!\right\|_{C}\! ≤\displaystyle\leq (b−t0)​(Lf11+Lf21)​(L+2)​‖Bfi1​(x1,x2)−(x1,x2)‖C,\displaystyle\!(b\!-\!t_{0})\!(L_{f_{1}^{1}}\!+\!L_{f_{2}^{1}})\!(L\!+\!2)\!\left\|\!B_{f_{i}^{1}}\!(\!x_{1}\!,\!x_{2}\!)\!-\!(\!x_{1}\!,\!x_{2}\!)\!\right\|_{C},
‖Bfi22​(x1,x2)−Bfi2​(x1,x2)‖C\displaystyle\left\|\!B_{f_{i}^{2}}^{2}\!(\!x_{1}\!,\!x_{2}\!)\!-\!B_{f_{i}^{2}}\!(\!x_{1}\!,\!x_{2}\!)\!\right\|_{C}\! ≤\displaystyle\leq (b−t0)​(Lf12+Lf22)​(L+2)​‖Bfi2​(x1,x2)−(x1,x2)‖C,\displaystyle\!(b\!-\!t_{0})\!(L_{f_{1}^{2}}\!+\!L_{f_{2}^{2}})\!(L\!+\!2)\!\left\|\!B_{f_{i}^{2}}\!(\!x_{1}\!,\!x_{2}\!)\!-\!(\!x_{1}\!,\!x_{2}\!)\!\right\|_{C},

for all (x1,x2)∈CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]),i=1,2.(x_{1},x_{2})\in C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!),i=1,2.

Now choosing

α1\displaystyle\alpha_{1} =\displaystyle= (b−t0)​(Lf11+Lf21)​(L+2),\displaystyle(b-t_{0})(L_{f_{1}^{1}}+L_{f_{2}^{1}})(L+2),
α2\displaystyle\alpha_{2} =\displaystyle= (b−t0)​(Lf12+Lf22)​(L+2),\displaystyle(b-t_{0})(L_{f_{1}^{2}}+L_{f_{2}^{2}})(L+2),

we get that Bfi1B_{f_{i}^{1}} and Bfi2B_{f_{i}^{2}} are c1c_{1}-WPO and c2c_{2}-WPO with c1=(1−α1)−1,c2=(1−α2)−1c_{1}=(1-\alpha_{1})^{-1},\ c_{2}=(1-\alpha_{2})^{-1}. From (6) we obtain that

‖Bfi1​(x1,x2)−Bfi2​(x1,x2)‖C≤(η1+η2)​(b−t0),\left\|B_{f_{i}^{1}}(x_{1},x_{2})-B_{f_{i}^{2}}(x_{1},x_{2})\right\|_{C}\leq(\eta_{1}+\eta_{2})(b-t_{0}),

for all (x1,x2)∈CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]),i=1,2.(x_{1},x_{2})\in C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!),i=1,2. Applying Theorem 2.9 we have that

H∥⋅∥C​(SBfi1,SBfi2)≤(η1+η2)​(b−t0)1−(b−t0)​(Lf1+Lf2)​(L+2),H_{\left\|\cdot\right\|_{C}}(S_{B_{f_{i}^{1}}},S_{B_{f_{i}^{2}}})\leq\frac{(\eta_{1}+\eta_{2})(b-t_{0})}{1-(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2)},

where Lfi:=max⁡(Lfi1,Lfi2)L_{f_{i}}:=\max(L_{f_{i}^{1}},L_{f_{i}^{2}}) and H∥⋅∥CH_{\left\|\cdot\right\|_{C}} denotes the Pompeiu-Housdorff functional with respect to ∥⋅∥C\left\|\cdot\right\|_{C} on CL​([t0−τ1,b],[t0−τ1,b])×CL​([t0−τ2,b],[t0−τ2,b]),i=1,2.C_{L}(\![t_{0}\!-\!\tau_{1},b]\!,\![t_{0}\!-\!\tau_{1},b]\!)\!\times\!C_{L}(\![t_{0}\!-\!\tau_{2},b]\!,\![t_{0}\!-\!\tau_{2},b]\!),\ i=1,2. ∎

6. DATA DEPENDENCE: DIFFERENTIABILITY

Consider the following Cauchy problem with parameter

(7) xi′​(t)=fi​(t,x1​(t),x2​(t),x1​(x1​(t−τ1)),x2​(x2​(t−τ2));λ),t∈[t0,b],i=1,2,x_{i}^{\prime}(t)=f_{i}(\!t,x_{1}(t),x_{2}(t),x_{1}(x_{1}(t\!-\!\tau_{1})),x_{2}(x_{2}(t\!-\!\tau_{2}));\lambda\!),\ t\in[t_{0},b],i=1,2,
(8) xi​(t)=φi​(t),t∈[t0−τi,t0],i=1,2.x_{i}(t)=\varphi_{i}(t),\ t\in[t_{0}-\tau_{i},t_{0}],i=1,2.

Suppose that we have satisfied the following conditions:

  1. (C1)

    t0<b,τ1,τ2>0,τ1<τ2,J⊂ℝt_{0}<b,\tau_{1},\tau_{2}>0,\tau_{1}<\tau_{2},J\subset\mathbb{R} a compact interval;

  2. (C2)

    φi∈CL​([t0−τi,t0],[t0−τi,b]),i=1,2;\varphi_{i}\in C_{L}([t_{0}-\tau_{i},t_{0}],[t_{0}-\tau_{i},b]),\ i=1,2;

  3. (C3)

    fi∈C1​([t0,b]×([t0−τ1,b]×[t0−τ2,b])2×J,ℝ)​i=1,2;f_{i}\in C^{1}([t_{0},b]\times([t_{0}-\tau_{1},b]\times[t_{0}-\tau_{2},b])^{2}\times J,\mathbb{R})\ i=1,2;

  4. (C4)

    there exists Lfi>0L_{f_{i}}>0 such that

    |∂fi​(t,u1,u2,u3,u4;λ)∂ui|≤Lfi\left|\frac{\partial f_{i}(t,u_{1},u_{2},u_{3},u_{4};\lambda)}{\partial u_{i}}\right|\leq L_{f_{i}}

    for all t∈[t0,b],(u1,u2,u3,u4)∈([t0−τ1,b]×[t0−τ2,b])2,i=1,2,λ∈J;t\in[t_{0},b],(\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!)\!\in\!(\![t_{0}-\tau_{1},b]\!\times\![t_{0}-\tau_{2},b]\!)^{2},\ i=1,2,\lambda\in J;

  5. (C5)

    mfim_{f_{i}} and Mfi∈ℝ,i=1,2M_{f_{i}}\in\mathbb{R},\ i=1,2 are such that

    1. (a)

      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,m_{f_{i}}\leq f_{i}(t,u_{1},u_{2},u_{3},u_{4})\leq M_{f_{i}},\forall t\in[t_{0},b],\ (\!u_{1}\!,\!u_{2}\!,u_{3}\!,u_{4}\!),\\ (\!v_{1}\!,\!v_{2}\!,\!v_{3}\!,\!v_{4}\!)\!\in\!(\![t_{0}\!-\!\tau_{1},b]\!\times\![t_{0}\!-\!\tau_{2},b]\!)^{2},

    2. (b)
      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,\begin{array}[]{ll}t_{0}-\tau_{i}\leq\varphi_{i}(t_{0})+m_{f_{i}}(b-t_{0})&\text{for }m_{f_{i}}<0,\\ t_{0}-\tau_{i}\leq\varphi_{i}(t_{0})&\text{for }m_{f_{i}}\geq 0,\\ b\geq\varphi_{i}(t_{0})&\text{for }M_{f_{i}}\leq 0,\\ b\geq\varphi_{i}(t_{0})+M_{f_{i}}(b-t_{0})&\text{for }M_{f_{i}}>0,\end{array}
    3. (c)

      L+Mfi<1;L+M_{f_{i}}<1;

  6. (C6)

    (b−t0)​(Lf1+Lf2)​(L+2)<1.(b-t_{0})(L_{f_{1}}+L_{f_{2}})(L+2)<1.

Then, from Theorem 3.1, we have that the problem (1)–(2) has a unique solution (x1∗​(⋅,λ),x2∗​(⋅,λ)).(x_{1}^{\ast}(\cdot,\lambda),x_{2}^{\ast}(\cdot,\lambda)).

We will prove that

xi∗​(⋅,λ)∈C1​(J), for all ​t∈[t0−τi,t0],i=1,2.x_{i}^{\ast}(\cdot,\lambda)\in C^{1}(J),\text{ for all }t\in[t_{0}-\tau_{i},t_{0}],i=1,2.

For this we consider the system

(9) xi′​(t,λ)=fi​(t,x1​(t;λ),x2​(t;λ),x1​(x1​(t−τ1;λ);λ),x2​(x2​(t−τ2;λ);λ);λ),x_{i}^{\prime}(t,\lambda)=f_{i}(t,x_{1}(t;\lambda),x_{2}(t;\lambda),x_{1}(x_{1}(t-\tau_{1};\lambda);\lambda),x_{2}(x_{2}(t-\tau_{2};\lambda);\lambda);\lambda),

t∈[t0,b],λ∈J,xi∈C​([t0−τi,b]×J,[t0−τi,b]×J)∩C1​([t0,b]×J,[t0−τi,b]×J),i=1,2.t\in[t_{0},b],\ \lambda\in J,\ x_{i}\in C([t_{0}-\tau_{i},b]\times J,[t_{0}-\tau_{i},b]\times J)\cap C^{1}([t_{0},b]\times J,[t_{0}-\tau_{i},b]\times J),\ i=1,2.

Theorem 6.1.

Consider the problem (9)–(8), and suppose the conditions (C1{}_{\text{1}})–(C6{}_{\text{6}}) holds. Then,

  1. (i)

    (9)–(8) has a unique solution (x1∗,x2∗)(x_{1}^{\ast},x_{2}^{\ast}), in C​([t0−τ1,b]×J,[t0−τ1,b])×C​([t0−τ2,b]×J,[t0−τ2,b]);C([t_{0}-\tau_{1},b]\times J,[t_{0}-\tau_{1},b])\times C([t_{0}-\tau_{2},b]\times J,[t_{0}-\tau_{2},b]);

  2. (ii)

    xi∗​(⋅,λ)∈C1​(J),x_{i}^{\ast}(\cdot,\lambda)\in C^{1}(J), for all t∈[t0−τi,t0],i=1,2.t\in[t_{0}-\tau_{i},t_{0}],i=1,2.

The problem (9)–(8) is equivalent with the following functional integral equations

(10a) x1​(t;λ)={φ1​(t),t∈[t0−τ1,t0]φ1​(t)+∫t0tf1​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ),t∈[t0,b]x_{1}\!(\!t\!;\!\lambda\!)\!=\!\left\{\!\begin{array}[]{l}\!\varphi_{1}(t),\ t\in[t_{0}-\tau_{1},t_{0}]\\ \!\varphi_{1}\!(\!t\!)\!+\!\int_{t_{0}}^{t}\!f_{1}\!(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}\!(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}\!(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)\!,t\!\in\![\!t_{0}\!,\!b\!]\end{array}\right.
(10b) x2​(t;λ)={φ2​(t),t∈[t0−τ2,t0]φ2​(t)+∫t0tf2​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ),t∈[t0,b]x_{2}\!(\!t\!;\!\lambda\!)\!=\!\left\{\!\begin{array}[]{l}\varphi_{2}(t),\ t\in[t_{0}-\tau_{2},t_{0}]\\ \!\varphi_{2}\!(\!t\!)\!+\!\int_{t_{0}}^{t}\!f_{2}\!(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)\!,t\!\in\![\!t_{0}\!,\!b\!]\end{array}\right.

Now, let take the operator

A\displaystyle A\! :\displaystyle: CL​([t0−τ1,b]×J,[t0−τ1,b]×J)×CL​([t0−τ2,b]×J,[t0−τ2,b]×J)→\displaystyle\!C_{L}\!([t_{0}-\tau_{1},b]\times J,[t_{0}-\tau_{1},b]\times J)\!\times\!C_{L}\!([t_{0}-\tau_{2},b]\times J,[t_{0}-\tau_{2},b]\times J)\!\rightarrow
CL​([t0−τ1,b]×J,[t0−τ1,b]×J)×CL​([t0−τ2,b]×J,[t0−τ2,b]×J),\displaystyle C_{L}\!([t_{0}-\tau_{1},b]\times J,[t_{0}-\tau_{1},b]\times J)\!\times\!C_{L\!}([t_{0}-\tau_{2},b]\times J,[t_{0}-\tau_{2},b]\times J),

given by the relation

A​(x1,x2)=(A1​(x1,x2),A2​(x1,x2)),A(x_{1},x_{2})=(A_{1}(x_{1},x_{2}),A_{2}(x_{1},x_{2})),

where A1​(x1,x2)​(t;λ):=A_{1}(x_{1},x_{2})(t;\lambda):= the right hand side of (10a) and A2​(x1,x2)​(t;λ):=A_{2}(x_{1},x_{2})(t;\lambda):= the right hand side of (10b).

Let X=CL​([t0−τ1,b]×J,[t0−τ1,b])×CL​([t0−τ2,b]×J,[t0−τ2,b]).X=C_{L}([t_{0}-\tau_{1},b]\times J,[t_{0}-\tau_{1},b])\times C_{L}([t_{0}-\tau_{2},b]\times J,[t_{0}-\tau_{2},b]).

It is clear from the proof of Theorem 3.1 that in the conditions (C1{}_{\text{1}})–(C6{}_{\text{6}}) the operator

A:(X,∥⋅∥C)→(X,∥⋅∥C)A:(X,\left\|\cdot\right\|_{C})\rightarrow(X,\left\|\cdot\right\|_{C})

is a PO.

Let (x1∗,x2∗)(x_{1}^{\ast},x_{2}^{\ast}) be the unique fixed point of A.A.

We consider the subset X1⊂X,X_{1}\subset X,

X1:={(x1,x2)∈X|∂x1∂t∈[t0−τ1,t0],∂x2∂t∈[t0−τ2,t0]}.X_{1}:=\{(x_{1},x_{2})\in X|\ \frac{\partial x_{1}}{\partial t}\in[t_{0}-\tau_{1},t_{0}],\ \frac{\partial x_{2}}{\partial t}\in[t_{0}-\tau_{2},t_{0}]\}.

We remark that (x1∗,x2∗)∈X1,A​(X1)⊂X1\ (x_{1}^{\ast},x_{2}^{\ast})\in X_{1},A(X_{1})\subset X_{1} and A:(X1,∥⋅∥C)→(X1,∥⋅∥C)A:(X_{1},\left\|\cdot\right\|_{C})\rightarrow(X_{1},\left\|\cdot\right\|_{C}) is PO.

Let Y:=C​([t0−τ1,b]×J)×C​([t0−τ2,b]×J).Y:=C([t_{0}-\tau_{1},b]\times J)\times C([t_{0}-\tau_{2},b]\times J).

Supposing that there exists ∂x1∗∂λ\dfrac{\partial x_{1}^{\ast}}{\partial\lambda} and ∂x2∗∂λ,\dfrac{\partial x_{2}^{\ast}}{\partial\lambda}, from (10a)–(10b) we have that

∂xi∗∂λ\displaystyle\dfrac{\partial x_{i}^{\ast}}{\partial\lambda}\!\! =∫t0t∂fi​(s,x1∗​(s;λ),x2∗​(s;λ),x1∗​(x1∗​(s−τ1;λ);λ),x2∗​(x2∗​(s−τ2;λ);λ);λ)∂u1⋅∂x1∗​(s,λ)∂λ​𝑑s+\displaystyle=\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}\!(\!s\!,\!x_{1}^{\ast}(\!s\!;\!\lambda\!)\!,\!x_{2}^{\ast}(\!s\!;\!\lambda\!)\!,\!x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{1}}\!\cdot\!\frac{\partial x_{1}^{\ast}\!(\!s,\lambda\!)}{\partial\lambda}\!ds\!+
+∫t0t∂fi​(s,x1∗​(s;λ),x2∗​(s;λ),x1∗​(x1∗​(s−τ1;λ);λ),x2∗​(x2∗​(s−τ2;λ);λ);λ)∂u2⋅∂x2∗​(s,λ)∂λ​𝑑s+\displaystyle\!+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}\!(\!s\!,\!x_{1}^{\ast}(\!s\!;\!\lambda\!)\!,\!x_{2}^{\ast}(\!s\!;\!\lambda\!)\!,\!x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{2}}\!\cdot\!\frac{\partial x_{2}^{\ast}\!(\!s\!,\!\lambda\!)}{\partial\lambda}\!ds\!+
+∫t0t∂fi​(s,x1∗​(s;λ),x2∗​(s;λ),x1∗​(x1∗​(s−τ1;λ);λ),x2∗​(x2∗​(s−τ2;λ);λ);λ)∂u3⋅\displaystyle\!+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}^{\ast}(\!s;\!\lambda\!)\!,\!x_{2}^{\ast}(\!s;\!\lambda\!)\!,\!x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{3}}\cdot
⋅[∂x1∗​(x1∗​(s−τ1;λ);λ)∂u1⋅∂x1∗​(s−τ1;λ)∂λ+∂x1∗​(x1∗​(s−τ1;λ);λ)∂λ]​d​s+\displaystyle\cdot\left[\frac{\partial x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)}{\partial u_{1}}\cdot\frac{\partial x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!)}{\partial\lambda}+\frac{\partial x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)}{\partial\lambda}\right]ds+
+∫t0t∂fi​(s,x1∗​(s;λ),x2∗​(s;λ),x1∗​(x1∗​(s−τ1;λ);λ),x2∗​(x2∗​(s−τ2;λ);λ);λ)∂u4⋅\displaystyle\!+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}^{\ast}(\!s;\!\lambda\!)\!,\!x_{2}^{\ast}(\!s;\!\lambda\!)\!,\!x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{4}}\cdot
⋅[∂x2∗​(x2∗​(s−τ1;λ);λ)∂u2⋅∂x2∗​(s−τ2;λ)∂λ+∂x2∗​(x2∗​(s−τ2;λ);λ)∂λ]​d​s+\displaystyle\cdot\left[\frac{\partial x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)}{\partial u_{2}}\cdot\frac{\partial x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!)}{\partial\lambda}+\frac{\partial x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!)}{\partial\lambda}\right]ds+
+∫t0t∂fi​(s,x1∗​(s;λ),x2∗​(s;λ),x1∗​(x1∗​(s−τ1;λ);λ),x2∗​(x2∗​(s−τ2;λ);λ);λ)∂λ​𝑑s,\displaystyle\!+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}^{\ast}(\!s;\!\lambda\!)\!,\!x_{2}^{\ast}(\!s;\!\lambda\!)\!,\!x_{1}^{\ast}\!(\!x_{1}^{\ast}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}^{\ast}\!(\!x_{2}^{\ast}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial\lambda}ds,

t∈[t0,b],λ∈J,i=1,2.t\in[t_{0},b],\lambda\in J,i=1,2.

The relation suggest us to consider the following operator

C:X1×Y→Y,(x1,x2,u,v)→C​(x1,x2,u,v),C:X_{1}\times Y\rightarrow Y,\ (x_{1},x_{2},u,v)\rightarrow C(x_{1},x_{2},u,v),

where

C​(x1,x2,u,v)​(t;λ)=0​ for ​t∈[t0−τi,t0],λ∈J,i=1,2C(x_{1},x_{2},u,v)(t;\lambda)=0\text{ for }t\in[t_{0}-\tau_{i},t_{0}],\lambda\in J,i=1,2

and

C​(x1,x2,u,v)​(t;λ):=\displaystyle C(x_{1},x_{2},u,v)(t;\lambda):=
=∫t0t∂fi​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ)∂u1​u​(s;λ)​𝑑s+\displaystyle=\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{1}}u(s;\lambda)ds\!+
+∫t0t∂fi​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ)∂u2​v​(s;λ)​𝑑s+\displaystyle+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{2}}v(s;\lambda)ds\!+
+∫t0t∂fi​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ)∂u3⋅\displaystyle\!+\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{3}}\cdot
⋅[∂x1​(x1​(s−τ1;λ);λ)∂u1⋅u​(s−τ1;λ)+∂x1​(x1​(s−τ1;λ);λ)∂λ]​d​s+\displaystyle\cdot\left[\frac{\partial x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)}{\partial u_{1}}\cdot u(\!s\!-\!\tau_{1};\!\lambda\!)+\frac{\partial x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)}{\partial\lambda}\right]ds\!+
+∫t0t∂fi​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ)∂u4⋅\displaystyle\!+\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial u_{4}}\cdot
⋅[∂x2​(x2​(s−τ2;λ);λ)∂u2⋅v​(s−τ2;λ)+∂x2​(x2​(s−τ2;λ);λ)∂λ]​d​s+\displaystyle\cdot\left[\frac{\partial x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!)}{\partial u_{2}}\cdot v(\!s\!-\!\tau_{2};\!\lambda\!)+\frac{\partial x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!)}{\partial\lambda}\right]ds\!+
+∫t0t∂fi​(s,x1​(s;λ),x2​(s;λ),x1​(x1​(s−τ1;λ);λ),x2​(x2​(s−τ2;λ);λ);λ)∂λ\displaystyle\!+\!\int_{t_{0}}^{t}\!\frac{\partial f_{i}(\!s\!,\!x_{1}(\!s;\!\lambda\!)\!,\!x_{2}(\!s;\!\lambda\!)\!,\!x_{1}\!(\!x_{1}(\!s\!-\!\tau_{1};\!\lambda\!);\!\lambda\!)\!,\!x_{2}\!(\!x_{2}(\!s\!-\!\tau_{2};\!\lambda\!);\!\lambda\!);\!\lambda\!)}{\partial\lambda}

for t∈[t0,b],λ∈J,i=1,2.t\in[t_{0},b],\lambda\in J,i=1,2.

In this way we have the triangular operator

D\displaystyle D :\displaystyle: X1×Y→X1×Y,\displaystyle X_{1}\times Y\rightarrow X_{1}\times Y,
(x1,x2,u,v)\displaystyle(x_{1},x_{2},u,v) →\displaystyle\rightarrow (A​(x1,x2),C​(x1,x2,u,v)),\displaystyle(A(x_{1},x_{2}),C(x_{1},x_{2},u,v)),

where AA is PO and C​(x1,x2,⋅,⋅):Y→YC(x_{1},x_{2},\cdot,\cdot):Y\rightarrow Y is an LCL_{C} -contraction with LC=(b−t0)​(L~f1+L~f2)​(L+2),L_{C}=(b-t_{0})(\widetilde{L}_{f_{1}}+\widetilde{L}_{f_{2}})(L+2), where L~fi=max⁡{Lfi,L⋅Lfi},i=1,2.\widetilde{L}_{f_{i}}=\max\{L_{f_{i}},L\cdot L_{f_{i}}\},i=1,2.

From the fibre contraction Theorem we have that the operator DD is PO, i.e. the sequences

(x1,n+1,x2,n+1):=A​(x1,n,x2,n),n∈ℕ,\displaystyle(x_{1,n+1},x_{2,n+1}):=A(x_{1,n},x_{2,n}),n\in\mathbb{N},
(un+1,vn+1):=C​(x1,n,x2,n,un,vn),n∈ℕ,\displaystyle(u_{n+1},v_{n+1}):=C(x_{1,n},x_{2,n},u_{n},v_{n}),n\in\mathbb{N},

converges uniformly, with respect to t∈X,λ∈J,t\in X,\ \lambda\in J, to (x1∗,x2∗,u∗,v∗)∈FD(x_{1}^{\ast},x_{2}^{\ast},u^{\ast},v^{\ast})\in F_{D}, for all (x1,0,x2,0)∈X1,(u0,v0)∈Y(x_{1,0},x_{2,0})\in X_{1},\ (u_{0},v_{0})\in Y.

If we take

x1,0=0,x2,0=0,u0=∂x1,0∂λ=0,v0=∂x2,0∂λ=0,x_{1,0}=0,\;x_{2,0}=0,u_{0}=\dfrac{\partial x_{1,0}}{\partial\lambda}=0,\;v_{0}=\dfrac{\partial x_{2,0}}{\partial\lambda}=0,

then

u1=∂x1,1∂λ,v1=∂x2,1∂λ.u_{1}=\frac{\partial x_{1,1}}{\partial\lambda},v_{1}=\frac{\partial x_{2,1}}{\partial\lambda}.

By induction we prove that

un\displaystyle u_{n} =\displaystyle= ∂x1,n∂λ,∀n∈ℕ,\displaystyle\frac{\partial x_{1,n}}{\partial\lambda},\;\forall n\in\mathbb{N},
vn\displaystyle v_{n} =\displaystyle= ∂x2,n∂λ,∀n∈ℕ.\displaystyle\frac{\partial x_{2,n}}{\partial\lambda},\;\forall n\in\mathbb{N}.

So

x1,n\displaystyle x_{1,n} →u​n​i​f​x1∗​ as ​n→∞,\displaystyle\overset{unif}{\rightarrow}x_{1}^{\ast}\text{ as }n\rightarrow\infty,
x2,n\displaystyle x_{2,n} →u​n​i​f​x2∗​ as ​n→∞,\displaystyle\overset{unif}{\rightarrow}x_{2}^{\ast}\text{ as }n\rightarrow\infty,
∂x1,n∂λ\displaystyle\frac{\partial x_{1,n}}{\partial\lambda} →u​n​i​f​u∗​ as ​n→∞,\displaystyle\overset{unif}{\rightarrow}u^{\ast}\text{ as }n\rightarrow\infty,
∂x2,n∂λ\displaystyle\frac{\partial x_{2,n}}{\partial\lambda} →u​n​i​f​v∗​ as ​n→∞.\displaystyle\overset{unif}{\rightarrow}v^{\ast}\text{ as }n\rightarrow\infty.

From a Weierstrass argument we have that there exists ∂xi∗∂λ,i=1,2\dfrac{\partial x_{i}^{\ast}}{\partial\lambda},\;i=1,2 and

∂x1∗∂λ=u∗​,​∂x2∗∂λ=v∗​.\frac{\partial x_{1}^{\ast}}{\partial\lambda}=u^{\ast}\text{,}\frac{\partial x_{2}^{\ast}}{\partial\lambda}=v^{\ast}\text{.}

∎

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