Properties of the solutions of a system of differential equations with maxima, via weakly picard operator theory

Diana Otrocol
(original), paper

Abstract

In this paper we present some properties of the solutions of a system of differential equation with maxima. Existence, uniqueness, inequalities of ˇCaplygin type and data  dependence (monotony, continuity) results for the solution of the Cauchy problem of this system are obtained using weakly Picard operator technique.

Authors

D. Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy,

Keywords

Differential equations with maxima, data dependence, Picard operator technique

Cite this paper as:

D. Otrocol, Properties of the solutions of a system of differential equations with maxima, via weakly picard operator theory, Commun. Appl. Anal. Vol. 17 (2013), no. 1, pp. 99–108

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https://ictp.acad.ro/otrocol/papers/2013-Otrocol-CAA-Properties%20of%20the%20solutions%20ddem.pdf

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Journal

Communications in Applied Analysis

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DOI
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1083-2564

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MR

MR3075771

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https://scholar.google.ro/scholar?oi=bibs&hl=en&cites=1330602022970784257

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[2] L. Georgiev, V.G. Angelov, On the existence and uniqueness of solutions for maximum equations, Glasnik Matematicki, 37 (2002), no. 2, 275–281.

[3] P. Gonzales, M. Pinto, Component-wise conditions for the asymptotic equivalence for nonlinear differential equations with maxima, Dynamic Systems and Applications, 20 (2011), 439–454.

[4] D. Otrocol, I.A. Rus, Functional-differential equations with “maxima” via weakly Picard operators theory, Bull. Math. Soc. Sci. Math. Roumanie, 51(99) (2008), No. 3, 253–261.

[5] D. Otrocol, I.A. Rus, Functional-differential equations with maxima of mixed type argument, Fixed Point Theory, 9 (2008), no. 1, pp. 207–220.

[6] I.A. Rus, Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003), No.1, 191–219.

[7] I.A. Rus, Generalized contractions, Cluj University Press, 2001.

[8] I.A. Rus, Functional-differential equations of mixed type, via weakly Picard operators, Seminar on fixed point theory, Cluj-Napoca, 3(2002), 335-345.

[9] E. Stepanov, On solvability of same boundary value problems for differential equations with “maxima”, Topological Methods in Nonlinear Analysis, 8 (1996), 315–326.

Communications in Applied Analysis xx (200x), no. N, xxx–xxx


Properties of the solutions of a system of differential equations with maxima,
via weakly Picard operator theory

Diana Otrocol
Abstract

“T. Popoviciu” Institute of Numerical Analysis of Romanian Academy

Cluj-Napoca 400110, ROMANIA

E-mail: dotrocol@ictp.acad.ro

††thanks: Received September 10, 2007 1083-2564 $15.00 ©Dynamic Publishers, Inc.

ABSTRACT. In this paper we present some properties of the solutions of a system of differential equation with maxima. Existence, uniqueness, inequalities of Čaplygin type and data dependence (monotony, continuity) results for the solution of the Cauchy problem of this system are obtained using weakly Picard operator technique.

AMS (MOS) Subject Classification. 45N05, 47H10.

1. INTRODUCTION

In this work, we study the solutions of the nonlinear differential system with maxima of the type

y′​(t)=A​(t)​y​(t)+f​(t,y​(t),maxa≤ξ≤t​y​(ξ))y^{\prime}(t)=A(t)y(t)+f(t,y(t),\underset{a\leq\xi\leq t}{\max}y(\xi)) (1.1)

as a perturbed equations of

x′​(t)=A​(t)​x​(t).x^{\prime}(t)=A(t)x(t). (1.2)

Existence and uniqueness, inequalities of Čaplygin type and data dependence (monotony, continuity) results for the solution of the Cauchy problem of the system (1.1) shall be obtain using weakly Picard operator technique.

Differential equations with maxima arise naturally when solving practical phenomenon problems, in particular, in those which appear in the study of systems with automatic regulation and automatic control of various technical systems. In connections with many possible applications it is absolutely necessary to be developed qualitative theory of differential equations with maxima (see the monograph [1] and papers [2], [3], [4], [5], [9]).

We consider the following Cauchy problem

y′(t)=A(t)y(t)+f(t,y(t),maxa≤ξ≤ty(ξ)),t∈[a,∞[,y^{\prime}(t)=A(t)y(t)+f(t,y(t),\underset{a\leq\xi\leq t}{\max}y(\xi)),\ t\in[a,\infty[, (1.3)
y​(a)=y0.y(a)=y_{0}. (1.4)

Let ℝn\mathbb{R}^{n} be the Euclidian nn-space. For u=(u1,…,un)T∈ℝnu=(u_{1},\ldots,u_{n})^{T}\in\mathbb{R}^{n}, let ‖u‖:=max⁡{|u1|,…,|un|}\left\|u\right\|:=\max\{\left|u_{1}\right|,\ldots,\left|u_{n}\right|\} be the norm of uu. For a matrix A∈Mn×n​(ℝ),A=(ai​j),A\in M_{n\times n}(\mathbb{R}),\ A=(a_{ij}), we define the norm |A|\left|A\right| of AA by |A|:=sup‖u‖≤1​‖A​u‖\left|A\right|:=\underset{\left\|u\right\|\leq 1}{\sup}\left\|Au\right\|. Then |A|=max1≤i≤n​∑j=1n|ai​j|.\left|A\right|=\underset{1\leq i\leq n}{\max}\sum\limits_{j=1}^{n}\left|a_{ij}\right|.

Throughout this paper we consider that y0∈ℝn,y_{0}\in\mathbb{R}^{n}, f∈C([a,∞[×ℝn×ℝn,ℝn)f\in C([a,\infty[\times\mathbb{R}^{n}\times\mathbb{R}^{n},\mathbb{R}^{n}) and X​(t)X(t) is the fundamental matrix of the system (1.2).

We remark that if y∈C1([a,∞[,ℝn)y\in C^{1}([a,\infty[,\mathbb{R}^{n}) is a solution of the problem (1.3)–(1.4), then yy is a solution of

y(t)=X(t)X−1(a)y0+∫atX(t)X−1(s)f(s,y(s),maxa≤ξ≤sy(ξ))ds,t∈[a,∞[y(t)=X(t)X^{-1}(a)y_{0}+\int_{a}^{t}X(t)X^{-1}(s)f(s,y(s),\underset{a\leq\xi\leq s}{\max}y(\xi))ds,t\in[a,\infty[ (1.5)

and if y∈C([a,∞[,ℝn)y\in C([a,\infty[,\mathbb{R}^{n}) is a solution of (1.5), then y∈C1([a,∞[,ℝn)y\in C^{1}([a,\infty[,\mathbb{R}^{n}) and is a solution of (1.3)–(1.4).

Also, if y∈C1([a,∞[,ℝn)y\in C^{1}([a,\infty[,\mathbb{R}^{n}) is a solution of the problem (1.3), then yy is a solution of

y(t)=X(t)X−1(a)y(a)+∫atX(t)X−1(s)f(s,y(s),maxa≤ξ≤sy(ξ))ds,t∈[a,∞[y(t)=X(t)X^{-1}(a)y(a)+\int_{a}^{t}X(t)X^{-1}(s)f(s,y(s),\underset{a\leq\xi\leq s}{\max}y(\xi))ds,t\in[a,\infty[ (1.6)

and if y∈C([a,∞[,ℝn)y\in C([a,\infty[,\mathbb{R}^{n}) is a solution of (1.6) then y∈C1([a,∞[,ℝn)y\in C^{1}([a,\infty[,\mathbb{R}^{n}) and is a solution of (1.3).

Let us consider the following operators:

Bf,Ef:C([a,∞[,ℝn)→C([a,∞[,ℝn),B_{f},E_{f}:C([a,\infty[,\mathbb{R}^{n})\rightarrow C([a,\infty[,\mathbb{R}^{n}),

defined by

Bf​(y)​(t):=X​(t)​X−1​(a)​y0+∫atX​(t)​X−1​(s)​f​(s,y​(s),maxa≤ξ≤s​y​(ξ))​𝑑s,B_{f}(y)(t):=X(t)X^{-1}(a)y_{0}+\int_{a}^{t}X(t)X^{-1}(s)f(s,y(s),\underset{a\leq\xi\leq s}{\max}y(\xi))ds,

and

Ef​(y)​(t):=X​(t)​X−1​(a)​y​(a)+∫atX​(t)​X−1​(s)​f​(s,y​(s),maxa≤ξ≤s​y​(ξ))​𝑑s.E_{f}(y)(t):=X(t)X^{-1}(a)y(a)+\int_{a}^{t}X(t)X^{-1}(s)f(s,y(s),\underset{a\leq\xi\leq s}{\max}y(\xi))ds.

For y0∈ℝn,y_{0}\in\mathbb{R}^{n}, we consider Xy0:={y∈C([a,∞[,ℝn)|y(a)=y0}.X_{y_{0}}:=\{y\in C([a,\infty[,\mathbb{R}^{n})|\ y(a)=y_{0}\}.

We remark that

C([a,∞[,ℝn)=∪y0∈ℝnXy0C([a,\infty[,\mathbb{R}^{n})=\underset{y_{0}\in\mathbb{R}^{n}}{\cup}X_{y_{0}}

is a partition of C([a,∞[,ℝn).C([a,\infty[,\mathbb{R}^{n}).

The following lemma is important for our further considerations.

Lemma 1.1.

(I.A. Rus, [8]) For y0∈ℝny_{0}\in\mathbb{R}^{n} and f∈C([a,∞[×ℝn×ℝn,ℝn),f\in C([a,\infty[\times\mathbb{R}^{n}\times\mathbb{R}^{n},\mathbb{R}^{n}), the following conditions hold:

  • (a)

    Bf(C([a,∞[,ℝn))⊂Xy0B_{f}(C([a,\infty[,\mathbb{R}^{n}))\subset X_{y_{0}} and Ef​(Xy0)⊂Xy0,∀y0∈ℝn;E_{f}(X_{y_{0}})\subset X_{y_{0}},\ \forall y_{0}\in\mathbb{R}^{n};

  • (b)

    Bf|Xy0=Ef|Xy0,∀y0∈ℝn.B_{f}|_{X_{y_{0}}}=E_{f}|_{X_{y_{0}}},\ \forall y_{0}\in\mathbb{R}^{n}.

In this paper we prove that the operator EfE_{f} is weakly Picard operator (see [7]), and we study the equation (1.3) in the terms of the weakly Picard operators theory.

2. WEAKLY PICARD OPERATORS

We start this section by presenting some notions and results from the weakly Picard operators theory.

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 subsets of AA.

We will denote by HH the Pompeiu-Housdorff functional, H:P​(X)×P​(X)→ℝ+∪{+∞}H:P(X)\times P(X)\rightarrow\mathbb{R}_{+}\cup\{+\infty\} defined as

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)\}.
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.

Definition 2.2.

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.

Definition 2.3.

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).
Remark 2.4.

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

Definition 2.5.

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.

For the theory of weakly Picard operator, see [6], [7], [8].

3. CAUCHY PROBLEM

Let us consider the following Banach space (BC([a,∞[,ℝn),∥⋅∥)(BC([a,\infty[,\mathbb{R}^{n}),\left\|\cdot\right\|) where BC([a,∞[,ℝn):={y∈C([a,∞[,ℝn)|yBC([a,\infty[,\mathbb{R}^{n}):=\{y\in C([a,\infty[,\mathbb{R}^{n})|\ y is bounded}\} with ‖y‖:=maxt∈[a,∞]​{|y1​(t)|,…,|yn​(t)|}\left\|y\right\|:=\underset{t\in[a,\infty]}{\max}\{\left|y_{1}(t)\right|,\ldots,\linebreak\left|y_{n}(t)\right|\}. We have the following existence and uniqueness theorem.

Theorem 3.1.

We suppose that:

  • (i)

    there exists Lf:[a,∞[→ℝ+L_{f}:[a,\infty[\rightarrow\mathbb{R}_{+} with ∫a∞Lf​(s)​𝑑s<∞\int_{a}^{\infty}L_{f}(s)ds<\infty such that

    ‖f​(t,u1,u2)−f​(t,v1,v2)‖≤Lf​(t)​max⁡{|u1−v1|,|u2−v2|},\displaystyle\left\|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right\|\leq L_{f}(t)\max\{\left|u_{1}-v_{1}\right|,\left|u_{2}-v_{2}\right|\},
    ∀t∈[a,∞[ and ui,vi∈ℝn,i=1,2.\displaystyle\quad\forall t\in[a,\infty[\text{ and }u_{i},v_{i}\in\mathbb{R}^{n},i=1,2.
  • (ii)

    ‖X​(t)​X−1​(s)‖≤K\left\|X(t)X^{-1}(s)\right\|\leq K, for a≤s≤t<∞;a\leq s\leq t<\infty;

  • (iii)

    ∫at‖f​(s,0,0)‖​𝑑s<∞;\int_{a}^{t}\left\|f(s,0,0)\right\|ds<\infty;

  • (iv)

    K​∫atLf​(s)​𝑑s<1.K\int_{a}^{t}L_{f}(s)ds<1.

Then the problem (1.3)–(1.4) has, in BC([a,∞[,ℝn)BC([a,\infty[,\mathbb{R}^{n}), a unique solution and this solution is the uniform limit of the successive approximations.

Proof.

The problem (1.3)–(1.4) is equivalent with the fixed point equation

Bf(y)=y,y∈BC([a,∞[,ℝn).B_{f}(y)=y,\ y\in BC([a,\infty[,\mathbb{R}^{n}).

We show that the space BC([a,∞[,ℝn)BC([a,\infty[,\mathbb{R}^{n}) is invariant for the operator BfB_{f}.

If x∈BC([a,∞[,ℝn)x\in BC([a,\infty[,\mathbb{R}^{n}), then

|Bf​(x)​(t)|\displaystyle\left|B_{f}(x)(t)\right|
≤|X​(t)​X−1​(a)​y0|+∫at|X​(t)​X−1​(a)|​|f​(s,y​(s),maxa≤ξ≤s​y​(ξ))−f​(s,0,0)|​𝑑s\displaystyle\leq\left|X(t)X^{-1}(a)y_{0}\right|+\int_{a}^{t}\left|X(t)X^{-1}(a)\right|\left|f(s,y(s),\underset{a\leq\xi\leq s}{\max}y(\xi))-f(s,0,0)\right|ds
+∫at|X​(t)​X−1​(a)|​|f​(s,0,0)|​𝑑s\displaystyle\quad+\int_{a}^{t}\left|X(t)X^{-1}(a)\right|\left|f(s,0,0)\right|ds
≤K​y0+∫atK​Lf​(s)​max⁡{|y​(s)|,|maxa≤ξ≤s​y​(ξ)|}​𝑑s+∫atK​|f​(s,0,0)|​𝑑s<∞.\displaystyle\leq Ky_{0}+\int_{a}^{t}KL_{f}(s)\max\left\{\left|y(s)\right|,\left|\underset{a\leq\xi\leq s}{\max}y(\xi)\right|\right\}ds+\int_{a}^{t}K\left|f(s,0,0)\right|ds<\infty.

So Bf(BC([a,∞[,ℝn))⊆BC([a,∞[,ℝn).B_{f}(BC([a,\infty[,\mathbb{R}^{n}))\subseteq BC([a,\infty[,\mathbb{R}^{n}).

On the other hand we have that (see [4])

|Bf​(y1)​(t)−Bf​(y2)​(t)|\displaystyle\left|B_{f}(y_{1})(t)\!-\!B_{f}(y_{2})(t)\right|\!
=|X(t)X−1(a)y0+∫atX(t)X−1(s)f(s,y1(s),maxa≤ξ≤sy1(ξ))ds\displaystyle=\left|X(t)X^{-1}(a)y_{0}+\int_{a}^{t}X(t)X^{-1}(s)f(s,y_{1}(s),\underset{a\leq\xi\leq s}{\max}y_{1}(\xi))ds\right.
−X(t)X−1(a)y0−∫atX(t)X−1(s)f(s,y2(s),maxa≤ξ≤sy2(ξ))ds|\displaystyle\quad\left.-X(t)X^{-1}(a)y_{0}-\int_{a}^{t}X(t)X^{-1}(s)f(s,y_{2}(s),\underset{a\leq\xi\leq s}{\max}y_{2}(\xi))ds\right|
≤∫atLf​(s)​|X​(t)​X−1​(s)|​max⁡{|y1​(s)−y2​(s)|,|maxa≤ξ≤s​y1​(ξ)−max⁡y2​(ξ)a≤ξ≤s|}​𝑑s\displaystyle\leq\int_{a}^{t}L_{f}(s)\left|X(t)X^{-1}(s)\right|\max\left\{\left|y_{1}(s)\!-\!y_{2}(s)\right|,\left|\underset{a\leq\xi\leq s}{\max}y_{1}(\xi)\!-\!\underset{a\leq\xi\leq s}{\max y_{2}(\xi)}\right|\!\right\}\!ds
≤K​∫atLf​(s)​max⁡{|y1​(s)−y2​(s)|,maxa≤ξ≤s​|y1​(ξ)−y2​(ξ)|}​𝑑s\displaystyle\leq K\int_{a}^{t}L_{f}(s)\max\left\{\left|y_{1}(s)\!-\!y_{2}(s)\right|,\underset{a\leq\xi\leq s}{\max}\left|y_{1}(\xi)\!-\!y_{2}(\xi)\right|\!\right\}ds
≤(K∫atLf(s)ds)∥y1−y2∥,∀t∈[a,∞[,∀y1,y2∈BC([a,∞[,ℝn).\displaystyle\leq\left(K\int_{a}^{t}L_{f}(s)ds\right)\left\|y_{1}-y_{2}\right\|,\forall t\in[a,\infty[,\ \forall y_{1},y_{2}\in BC([a,\infty[,\mathbb{R}^{n}).

So,

‖Bf​(y1)−Bf​(y2)‖≤(K​∫atLf​(s)​𝑑s)​‖y1−y2‖,\left\|B_{f}(y_{1})-B_{f}(y_{2})\right\|\leq\left(K\int_{a}^{t}L_{f}(s)ds\right)\left\|y_{1}-y_{2}\right\|,

∀t∈[a,∞[,∀y1,y2∈BC([a,∞[,ℝn)\forall t\in[a,\infty[,\forall y_{1},y_{2}\in BC([a,\infty[,\mathbb{R}^{n}), i.e., BfB_{f} is a contraction w.r.t. the norm ∥⋅∥\left\|\cdot\right\| on BC([a,∞[,ℝn)BC([a,\infty[,\mathbb{R}^{n}). The proof follows from the Banach fixed point theorem. ∎

Remark 3.2.

In the conditions of Theorem 3.1, the operator BfB_{f} is Picard operator. But

Bf|Xy0=Ef|Xy0,∀y0∈ℝn.B_{f}|_{X_{y_{0}}}=E_{f}|_{X_{y_{0}}},\ \forall y_{0}\in\mathbb{R}^{n}.

Hence, the operator EfE_{f} is weakly Picard operator and FEf∩Xy0={yy0∗},∀y0∈ℝn,F_{E_{f}}\cap X_{y_{0}}=\{y_{y_{0}}^{\ast}\},\forall y_{0}\in\mathbb{R}^{n}, where FEf={yy0∗∈Xy0|Ef​(yy0∗)=yy0∗}F_{E_{f}}=\{y_{y_{0}}^{\ast}\in X_{y_{0}}|E_{f}(y_{y_{0}}^{\ast})=y_{y_{0}}^{\ast}\} and yy0∗y_{y_{0}}^{\ast} is the unique solution of the problem (1.3)–(1.4).

From the WPO theory, we present in the following sections inequalities of Čaplygin type and data dependence results for the solution of the system of differential equations.

4. INEQUALITIES OF ČAPLYGIN TYPE

From the weakly Picard operator theory we have

Theorem 4.1.

(Theorem of Čaplygin type) We suppose that:

  • (a)

    the hypothesis of Theorem 3.1 are satisfied;

  • (b)

    f​(t,⋅,⋅):ℝ2​n→ℝnf(t,\cdot,\cdot):\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n} is increasing, i.e., ui≤vi⇒f(t,u1,u2)≤f(t,v1,v2),∀t∈[a,∞[u_{i}\leq v_{i}\Rightarrow f(t,u_{1},u_{2})\leq f(t,v_{1},v_{2}),\ \forall t\in[a,\infty[ and ui,vi∈ℝn,i=1,2.u_{i},v_{i}\in\mathbb{R}^{n},i=1,2.

Let yy be a solution of equation (1.3) and xx a solution of the inequality

x′(t)≤A(t)x(t)+f(t,x(t),maxa≤ξ≤tx(ξ)),t∈[a,∞[.x^{\prime}(t)\leq A(t)x(t)+f(t,x(t),\underset{a\leq\xi\leq t}{\max}x(\xi)),\ t\in[a,\infty[.

Then

x​(a)≤y​(a)​ implies that ​x≤y.x(a)\leq y(a)\text{ implies that }x\leq y.
Proof.

In the terms of the operator Ef,E_{f}, we have

y=Ef​(y)​ and ​x≤Ef​(x),y=E_{f}(y)\text{ and }x\leq E_{f}(x),

and y​(a)≤x​(a).y(a)\leq x(a).

From Remark 3.2, EfE_{f} is weakly Picard operator. From the condition (b), Ef∞E_{f}^{\infty} is increasing ([7]). If y0∈ℝn,y_{0}\in\mathbb{R}^{n}, then we denote by y~0\widetilde{y}_{0} the following function

y~0:[a,∞[→ℝn,y~0(t)=y0,∀t∈[a,∞[.\widetilde{y}_{0}:[a,\infty[\rightarrow\mathbb{R}^{n},\ \widetilde{y}_{0}(t)=y_{0},\ \forall t\in[a,\infty[.

We have

x≤Ef​(x)≤…≤Ef∞​(x)=Ef∞​(x~​(a))≤Ef∞​(y~​(a))=y.x\leq E_{f}(x)\leq\ldots\leq E_{f}^{\infty}(x)=E_{f}^{\infty}(\widetilde{x}(a))\leq E_{f}^{\infty}(\widetilde{y}(a))=y.

∎

5. DATA DEPENDENCE: MONOTONY

The following concept is important for our further considerations.

Lemma 5.1.

(Comparison principle, [6]) Let (X,d,≤)(X,d,\leq) an ordered metric space and A,B,C:X→XA,B,C:X\rightarrow X be such that

  • (a)

    A≤B≤C;A\leq B\leq C;

  • (b)

    the operator A,B,CA,B,C, are WPOs;

  • (c)

    the operator BB is increasing.

Then x≤y≤zx\leq y\leq z imply that A∞​(x)≤B∞​(y)≤C∞​(z).A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(z).

From this abstract result we have

Theorem 5.2.

Let fi∈C([a,∞[×ℝ2​n,ℝn),i=1,2,f_{i}\in C([a,\infty[\times\mathbb{R}^{2n},\mathbb{R}^{n}),i=1,2, be as in Theorem 3.1. We suppose that:

  • (i)

    f1≤f2≤f3;f_{1}\leq f_{2}\leq f_{3};

  • (ii)

    f2​(t,⋅,⋅):ℝ2​n→ℝnf_{2}(t,\cdot,\cdot):\mathbb{R}^{2n}\rightarrow\mathbb{R}^{n} is increasing;

Let yi∈BC1([a,∞[,ℝn)y_{i}\in BC^{1}([a,\infty[,\mathbb{R}^{n}) be a solution of the equation

yi′(t)=A(t)yi(t)+fi(t,y(t),maxa≤ξ≤ty(ξ)),t∈[a,∞[ and i=1,2,3.y_{i}^{\prime}(t)=A(t)y_{i}(t)+f_{i}(t,y(t),\underset{a\leq\xi\leq t}{\max}y(\xi)),\ t\in[a,\infty[\text{ and }i=1,2,3.

If y1​(a)≤y2​(a)≤y3​(a)y_{1}(a)\leq y_{2}(a)\leq y_{3}(a), then y1≤y2≤y3.y_{1}\leq y_{2}\leq y_{3}.

Proof.

From Theorem 3.1 we have that the operator Efi,i=1,2,3,E_{f_{i}},i=1,2,3,\ are WPOs. From the condition (ii) the operator Ef2E_{f_{2}} is monotone increasing. From the condition (i) it follows that

Ef1≤Ef2≤Ef3.E_{f_{1}}\leq E_{f_{2}}\leq E_{f_{3}}.

Let y~i(a)∈BC([a,∞[,ℝn)\widetilde{y}_{i}(a)\in BC([a,\infty[,\mathbb{R}^{n}) be defined by y~i(a)(t)=yi(a),∀t∈[a,∞[\widetilde{y}_{i}(a)(t)=y_{i}(a),\ \forall t\in[a,\infty[. It is clear that

y~1(a)(t)≤y~2(a)(t)≤y~3(a)(t),∀t∈[a,∞[.\widetilde{y}_{1}(a)(t)\leq\widetilde{y}_{2}(a)(t)\leq\widetilde{y}_{3}(a)(t),\ \forall t\in[a,\infty[.

From Lemma 5.1 we have that

Ef1∞​(y~1​(a))≤Ef2∞​(y~2​(a))≤Ef3∞​(y~3​(a)).E_{f_{1}}^{\infty}(\widetilde{y}_{1}(a))\leq E_{f_{2}}^{\infty}(\widetilde{y}_{2}(a))\leq E_{f_{3}}^{\infty}(\widetilde{y}_{3}(a)).

But yi=Ef​i∞​(y~i​(a)),y_{i}=E_{fi}^{\infty}(\widetilde{y}_{i}(a)), and y1≤y2≤y3.y_{1}\leq y_{2}\leq y_{3}. ∎

6. DATA DEPENDENCE: CONTINUITY

Consider the Cauchy problem (1.3)–(1.4) and suppose the conditions of Theorem 3.1 are satisfied. Denote by y∗​(⋅;y0,f)y^{\ast}(\cdot;y_{0},f)\ the solution of this problem.

In order to study the continuous dependence of the fixed points we will use the following result:

Theorem 6.1.

(I.A. Rus, [7]) Let (X,d)(X,d) be a complete metric space and A,B:X→XA,B:X\rightarrow X two operators. We suppose that

  • (i)

    the operator AA is a α\alpha-contraction;

  • (ii)

    FB≠∅;F_{B}\neq\emptyset;

  • (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}.

Then, accordingly to Theorem 6.1 we have the result as follows.

Theorem 6.2.

Let y0i,fi,i=1,2y_{0}^{i},f_{i},i=1,2 be as in Theorem 3.1. Furthermore, we suppose that there exists ηi∈ℝ+n,i=1,2\eta_{i}\in\mathbb{R}_{+}^{n},i=1,2 with ∫atη2​(s)​𝑑s<∞\int_{a}^{t}\eta_{2}(s)ds<\infty such that

  1. (i)

    ‖y01−y02‖≤η1;\left\|y_{0}^{1}-y_{0}^{2}\right\|\leq\eta_{1};

  2. (ii)

    ‖X​(t)​X−1​(s)‖≤K\left\|X(t)X^{-1}(s)\right\|\leq K for a≤s≤t<∞;a\leq s\leq t<\infty;

  3. (iii)

    ∥f1(t,u,v)−f2(t,u,v)∥≤η2(t),∀t∈[a,∞[,u∈ℝn.\left\|f_{1}(t,u,v)-f_{2}(t,u,v)\right\|\leq\eta_{2}(t),\forall t\in[a,\infty[,u\in\mathbb{R}^{n}.

Then

‖y1∗​(t;y01,f1)−y2∗​(t;y02,f2)‖≤K​η1+K​∫atη2​(s)​𝑑s1−K​∫atLf​(s)​𝑑s,\left\|y_{1}^{\ast}(t;y_{0}^{1},f_{1})-y_{2}^{\ast}(t;y_{0}^{2},f_{2})\right\|\leq\frac{K\eta_{1}+K\int_{a}^{t}\eta_{2}(s)ds}{1-K\int_{a}^{t}L_{f}(s)ds},

where yi∗​(t;x0i,fi),i=1,2y_{i}^{\ast}(t;x_{0}^{i},f_{i}),i=1,2 are the solution of the problem (1.3)–(1.4) with respect to y0i,fiy_{0}^{i},f_{i} and Lf=max⁡{Lf1,Lf2}.L_{f}=\max\left\{L_{f_{1}},L_{f_{2}}\right\}.

Proof.

Consider the operators By0i,fi,i=1,2.B_{y_{0}^{i},f_{i}},i=1,2. From Theorem 3.1 these operators are contractions.

Additionally

∥By01,f1(y)−By02,f2(y)∥≤Kη1+K∫atη2(s)ds,∀y∈BC([a,∞[,ℝn).\left\|B_{y_{0}^{1},f_{1}}(y)-B_{y_{0}^{2},f_{2}}(y)\!\right\|\leq K\eta_{1}+K\int_{a}^{t}\eta_{2}(s)ds,\ \forall y\in BC([a,\infty[,\mathbb{R}^{n}).

Now the proof follows from the Theorem 6.1, with A:=By01,f1,B=By02,f2,η=K​η1+K​∫atη2​(s)​𝑑sA\!:=\!B_{y_{0}^{1},f_{1}},\ B\!=\!\!B_{y_{0}^{2},f_{2}},\ \eta\!=\!K\eta_{1}+K\int_{a}^{t}\eta_{2}(s)ds and α:=K​∫atLf​(s)​𝑑s\alpha:=K\int_{a}^{t}L_{f}(s)ds, where Lf=max⁡{Lf1,Lf2}.L_{f}=\max\left\{L_{f_{1}},L_{f_{2}}\right\}. ∎

We shall use the cc-WPOs techniques to give some data dependence results.

Theorem 6.3.

(I.A. Rus, [7]) 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

  • (i)

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

  • (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\left\{c_{1},c_{2}\right\}.

Based upon Theorem 6.3 we have the next result.

Theorem 6.4.

Let f1f_{1} and f2f_{2} be as in Theorem 3.1. Let SEf1,SEf2S_{E_{f_{1}}},S_{E_{f_{2}}} be the solution sets of system (1.3) corresponding to f1f_{1} and f2f_{2}. Suppose that

  • (i)

    ‖X​(t)​X−1​(s)‖≤K\left\|X(t)X^{-1}(s)\right\|\leq K for a≤s≤t<∞;a\leq s\leq t<\infty;

  • (ii)

    there exists η∈ℝ+n,\eta\in\mathbb{R}_{+}^{n}, ∫atη​(s)​𝑑s<∞\int_{a}^{t}\eta(s)ds<\infty such that

    ∥f1(t,u,v)−f2(t,u,v)∥≤η(t),∀t∈[a,∞[,u∈ℝn.\left\|f_{1}(t,u,v)-f_{2}(t,u,v)\right\|\leq\eta(t),\forall t\in[a,\infty[,u\in\mathbb{R}^{n}. (6.1)

Then

H∥⋅∥C​(SEf1,SEf2)≤K​∫atη​(s)​𝑑s1−K​∫atLf​(s)​𝑑s,H_{\left\|\cdot\right\|_{C}}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{K\int_{a}^{t}\eta(s)ds}{1-K\int_{a}^{t}L_{f}(s)ds},

where Lf=max⁡{Lf1,Lf2}L_{f}=\max\left\{L_{f_{1}},L_{f_{2}}\right\} and H∥⋅∥H_{\left\|\cdot\right\|} denotes the Pompeiu-Housdorff functional with respect to ∥⋅∥\left\|\cdot\right\| on BC([a,∞[,ℝn).BC([a,\infty[,\mathbb{R}^{n}).

Proof.

In the condition of Theorem 3.1, the operators Ef1E_{f_{1}} and Ef2E_{f_{2}} are cic_{i}-weakly Picard operators, i=1,2i=1,2. Let Xy0:={y∈BC([a,∞[,ℝn)|y(a)=y0}.X_{y_{0}}:=\{y\in BC([a,\infty[,\mathbb{R}^{n})|\ y(a)=y_{0}\}. It is clear that Ef1|Xy0=Bf1,Ef2|Xy0=Bf2.E_{f_{1}}|_{X_{y_{0}}}=B_{f_{1}},\ E_{f_{2}}|_{X_{y_{0}}}=B_{f_{2}}. Therefore,

∥Ef12(y)−Ef1(y)∥≤(K∫atLf1(s)ds)∥Ef1(y)−y∥,∀y∈BC([a,∞[,ℝn),\left\|E_{f_{1}}^{2}(y)-E_{f_{1}}(y)\right\|\leq\left(K\int_{a}^{t}L_{f_{1}}(s)ds\right)\left\|E_{f_{1}}(y)-y\right\|,\ \forall y\in BC([a,\infty[,\mathbb{R}^{n}),
∥Ef22(y)−Ef2(y)∥≤(K∫atLf2(s)ds)∥Ef2(y)−y∥,∀y∈BC([a,∞[,ℝn).\left\|E_{f_{2}}^{2}(y)-E_{f_{2}}(y)\right\|\leq\left(K\int_{a}^{t}L_{f_{2}}(s)ds\right)\left\|E_{f_{2}}(y)-y\right\|,\ \forall y\in BC([a,\infty[,\mathbb{R}^{n}).

Now, choosing y01=K​∫atLf1​(s)​𝑑sy_{0}^{1}=K\int_{a}^{t}L_{f_{1}}(s)ds\ and y02=K​∫atLf2​(s)​𝑑s,y_{0}^{2}=K\int_{a}^{t}L_{f_{2}}(s)ds, we get that Ef1E_{f_{1}} and Ef2E_{f_{2}} are cic_{i}-weakly Picard operators, i=1,2i=1,2 with c1=(1−y01)−1c_{1}=(1-y_{0}^{1})^{-1}\ and c2=(1−y02)−1\ c_{2}=(1-y_{0}^{2})^{-1}. From (6.1) we obtain that

∥Ef1(y)−Ef2(y)∥≤K∫atη(s)ds,∀y∈BC([a,∞[,ℝn).\left\|E_{f_{1}}(y)-E_{f_{2}}(y)\right\|\leq K\int_{a}^{t}\eta(s)ds,\ \forall y\in BC([a,\infty[,\mathbb{R}^{n}).

Applying Theorem 6.3 we have that

H∥⋅∥​(SEf1,SEf2)≤K​∫atη​(s)​𝑑s1−K​∫atLf​(s)​𝑑s,H_{\left\|\cdot\right\|}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{K\int_{a}^{t}\eta(s)ds}{1-K\int_{a}^{t}L_{f}(s)ds},

where Lf=max⁡{Lf1,Lf2}L_{f}=\max\left\{L_{f_{1}},L_{f_{2}}\right\} and H∥⋅∥H_{\left\|\cdot\right\|} is the Pompeiu-Housdorff functional with respect to ∥⋅∥\left\|\cdot\right\| on BC([a,∞[,ℝn).BC([a,\infty[,\mathbb{R}^{n}). ∎

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