Abstract
We consider an integro-differential equation with two times modifications. Existence, uniqueness and monotony results of solution for the Cauchy problem are obtained using weakly Picard operator theory.
In the last section we present a step method for this type of equation.
Authors
Veronica-Ana Ilea
Babes-Bolyai University Department of Applied Mathematics Cluj-Napoca, Romania
Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
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Paper coordinates
V.A. Ilea, D. Otrocol, Integro-differential equation with two times modifications, Carpathian J. Math., 27 (2011) no. 2, pp. 209-216.
About this paper
Journal
Carpathian Journal Mathematics
Publisher Name
North University Centre at Baia Mare (Technical University of Cluj-Napoca), Romania
DOI
Print ISSN
1584-2851
Online ISSN
1843-4401
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Paper (preprint) in HTML form
INTEGRO-DIFFERENTIAL EQUATION WITH TWO TIME MODIFICATIONS
Abstract
We consider an integro-differential equation with two time modifications. Existence, uniqueness and monotony results of solution for the Cauchy problem are obtained using weakly Picard operator theory. In the last section we present a step method for this type of equation.
1. Introduction
This paper is concerned with the following integro-differential equation
{:(1.1)x^(')(t)=g(t","x(t)","x(t-tau))+int_(t-h)^(t)K(s","x(s))ds","t in I:}\begin{equation*}
x^{\prime}(t)=g(t, x(t), x(t-\tau))+\int_{t-h}^{t} K(s, x(s)) d s, t \in I \tag{1.1}
\end{equation*}
The aim of this paper is to obtain existence and uniqueness theorems using contraction principle, step method and monotony results for the Cauchy problem, see [7] and [11]. Such kind of results have been proved for an integro delay equation in [17]. The approach proposed in the present paper is different to the ones in [4], [17] and [18] and it is based on the different time modifications.
In our paper we consider I=[0,oo)I=[0, \infty).
Regarding the two delays we have the following cases: h > 0,tau > 0,tau > hh>0, \tau>0, \tau>h, discussed in [8], and here we take the case: tau < 0,h > 0,h=|tau|\tau<0, h>0, h=|\tau|.
Regarding the two delays we have the following cases: h > 0,tau > 0,tau > hh>0, \tau>0, \tau>h, discussed in [8], and here we take the case: tau < 0,h > 0,h=|tau|\tau<0, h>0, h=|\tau|.
The equation becomes
{:(1.2)x^(')(t)=g(t","x(t)","x(t+h))+int_(t-h)^(t)K(s","x(s))ds","t in[0","oo[:}\begin{equation*}
x^{\prime}(t)=g(t, x(t), x(t+h))+\int_{t-h}^{t} K(s, x(s)) d s, t \in[0, \infty[ \tag{1.2}
\end{equation*}
with the condition
{:(1.3)x(t)=varphi(t)","t in[-h","h].:}\begin{equation*}
x(t)=\varphi(t), t \in[-h, h] . \tag{1.3}
\end{equation*}
Relative to (1.2)-(1.3) we consider the following conditions:
(C_(1))(B,|*|)\left(C_{1}\right)(\mathbb{B},|\cdot|) is a Banach space, g in C([0,oo[xxB^(2),B),K in C([0,oo[xxB,B):}g \in C\left(\left[0, \infty\left[\times \mathbb{B}^{2}, \mathbb{B}\right), K \in C([0, \infty[\times \mathbb{B}, \mathbb{B})\right.\right., varphi in C([-h,h],B);\varphi \in C([-h, h], \mathbb{B}) ;
(C_(1)^('))(B,|*|)\left(C_{1}^{\prime}\right)(\mathbb{B},|\cdot|) is a Banach space, g inC^(oo)([0,oo[xxB^(2),B),K inC^(oo)([0,oo[xxB,B):}g \in C^{\infty}\left(\left[0, \infty\left[\times \mathbb{B}^{2}, \mathbb{B}\right), K \in C^{\infty}([0, \infty[\times \mathbb{B}, \mathbb{B})\right.\right., varphi inC^(oo)([-h,h],B);\varphi \in C^{\infty}([-h, h], \mathbb{B}) ;
(C_(2))\left(C_{2}\right) there exists L_(1),L_(2) > 0L_{1}, L_{2}>0 such that
(C_(1))(B,|*|)\left(C_{1}\right)(\mathbb{B},|\cdot|) is a Banach space, g in C([0,oo[xxB^(2),B),K in C([0,oo[xxB,B):}g \in C\left(\left[0, \infty\left[\times \mathbb{B}^{2}, \mathbb{B}\right), K \in C([0, \infty[\times \mathbb{B}, \mathbb{B})\right.\right., varphi in C([-h,h],B);\varphi \in C([-h, h], \mathbb{B}) ;
(C_(1)^('))(B,|*|)\left(C_{1}^{\prime}\right)(\mathbb{B},|\cdot|) is a Banach space, g inC^(oo)([0,oo[xxB^(2),B),K inC^(oo)([0,oo[xxB,B):}g \in C^{\infty}\left(\left[0, \infty\left[\times \mathbb{B}^{2}, \mathbb{B}\right), K \in C^{\infty}([0, \infty[\times \mathbb{B}, \mathbb{B})\right.\right., varphi inC^(oo)([-h,h],B);\varphi \in C^{\infty}([-h, h], \mathbb{B}) ;
(C_(2))\left(C_{2}\right) there exists L_(1),L_(2) > 0L_{1}, L_{2}>0 such that
|g(t,u_(1),v_(1))-g(t,u_(2),v_(2))| <= L_(1)|u_(1)-u_(2)|+L_(2)|v_(1)-v_(2)|,u_(i),v_(i)inB,t in[0,oo[;\left|g\left(t, u_{1}, v_{1}\right)-g\left(t, u_{2}, v_{2}\right)\right| \leq L_{1}\left|u_{1}-u_{2}\right|+L_{2}\left|v_{1}-v_{2}\right|, u_{i}, v_{i} \in \mathbb{B}, t \in[0, \infty[;
(C_(3))\left(C_{3}\right) there exists L_(3) > 0L_{3}>0 such that
|K(s,u)-K(s,v)| <= L_(3)|u-v|,u,v inB,t in[0,oo[;|K(s, u)-K(s, v)| \leq L_{3}|u-v|, u, v \in \mathbb{B}, t \in[0, \infty[;
(C_(4))(L_(1)+L_(2)+2L_(3)h)h < 1;\left(C_{4}\right)\left(L_{1}+L_{2}+2 L_{3} h\right) h<1 ;
(C_(5))varphi^(')(0)=g(h,varphi(0),varphi(h))+int_(-h)^(0)K(s,varphi(s))ds\left(C_{5}\right) \varphi^{\prime}(0)=g(h, \varphi(0), \varphi(h))+\int_{-h}^{0} K(s, \varphi(s)) d s.
(C_(5))varphi^(')(0)=g(h,varphi(0),varphi(h))+int_(-h)^(0)K(s,varphi(s))ds\left(C_{5}\right) \varphi^{\prime}(0)=g(h, \varphi(0), \varphi(h))+\int_{-h}^{0} K(s, \varphi(s)) d s.
In what follow we shall present some notions that will help us obtaining the results bellow.
Let XX be a nonempty set,
s(X):={(x_(n))_(n inN^(**))∣x_(n)in X,n inN^(**)}s(X):=\left\{\left(x_{n}\right)_{n \in \mathbb{N}^{*}} \mid x_{n} \in X, n \in \mathbb{N}^{*}\right\}
and
M(X):={(x_(ij))_(1)^(oo)∣x_(ij)in X,i,j inN^(**)}M(X):=\left\{\left(x_{i j}\right)_{1}^{\infty} \mid x_{i j} \in X, i, j \in \mathbb{N}^{*}\right\}
where
(x_(ij))_(1)^(oo):=([x_(11),x_(12),x_(13),cdots],[x_(21),x_(22),x_(23),cdots],[x_(31),x_(32),x_(33),cdots],[cdots,cdots,cdots,cdots])\left(x_{i j}\right)_{1}^{\infty}:=\left(\begin{array}{cccc}
x_{11} & x_{12} & x_{13} & \cdots \\
x_{21} & x_{22} & x_{23} & \cdots \\
x_{31} & x_{32} & x_{33} & \cdots \\
\cdots & \cdots & \cdots & \cdots
\end{array}\right)
is a infinite matrix.
For A in M(B)A \in M(\mathbb{B}) we denote
For A in M(B)A \in M(\mathbb{B}) we denote
|A|:=s u p_(1 <= i <= oo)sum_(j inN^(**))|a_(ij)|.|A|:=\sup _{1 \leq i \leq \infty} \sum_{j \in \mathbb{N}^{*}}\left|a_{i j}\right| .
Let d:X xx X rarr s(B)d: X \times X \rightarrow s(\mathbb{B}) be the generalized metric.
Remark 1.1. [13] A functional d:X xx X rarr s(B),(x,y)|->(d_(k)(x,y))_(k inN^(**))d: X \times X \rightarrow s(\mathbb{B}),(x, y) \mapsto\left(d_{k}(x, y)\right)_{k \in \mathbb{N}^{*}} is a generalized metric of XX iff
(a) d_(k)d_{k} is a pseudometric, AA k inN^(**)\forall k \in \mathbb{N}^{*};
(b) AA x,y in X,x!=y\forall x, y \in X, x \neq y, there exist k inN^(**)k \in \mathbb{N}^{*} such as d_(k)(x,y)!=0d_{k}(x, y) \neq 0.
Remark 1.1. [13] A functional d:X xx X rarr s(B),(x,y)|->(d_(k)(x,y))_(k inN^(**))d: X \times X \rightarrow s(\mathbb{B}),(x, y) \mapsto\left(d_{k}(x, y)\right)_{k \in \mathbb{N}^{*}} is a generalized metric of XX iff
(a) d_(k)d_{k} is a pseudometric, AA k inN^(**)\forall k \in \mathbb{N}^{*};
(b) AA x,y in X,x!=y\forall x, y \in X, x \neq y, there exist k inN^(**)k \in \mathbb{N}^{*} such as d_(k)(x,y)!=0d_{k}(x, y) \neq 0.
Definition 1.2. [13] Let (X,d)(X, d) be a complete generalized metric space, A:X rarr XA: X \rightarrow X and S in M(B)S \in M(\mathbb{B}). The operator AA is a SS-contraction iff:
(i) SS is row and column finite (meaning that there are only a finite number of nonzero elements in each row and each column);
(ii) SS is a Neumann matrix (meaning that if S^(n)S^{n} is definite for all n inNn \in \mathbb{N} and
(..) sum_(n inN)S^(n)\sum_{n \in \mathbb{N}} S^{n} converges for all x,y in Xx, y \in X );
(iii) sum_(n inN)S^(n)d(x,y)\sum_{n \in \mathbb{N}} S^{n} d(x, y) converges AA x,y in X\forall x, y \in X;
(iv) d(A(x),A(y)) <= Sd(x,y)AA x,y in Xd(A(x), A(y)) \leq S d(x, y) \forall x, y \in X.
(i) SS is row and column finite (meaning that there are only a finite number of nonzero elements in each row and each column);
(ii) SS is a Neumann matrix (meaning that if S^(n)S^{n} is definite for all n inNn \in \mathbb{N} and
(..) sum_(n inN)S^(n)\sum_{n \in \mathbb{N}} S^{n} converges for all x,y in Xx, y \in X );
(iii) sum_(n inN)S^(n)d(x,y)\sum_{n \in \mathbb{N}} S^{n} d(x, y) converges AA x,y in X\forall x, y \in X;
(iv) d(A(x),A(y)) <= Sd(x,y)AA x,y in Xd(A(x), A(y)) \leq S d(x, y) \forall x, y \in X.
We consider the space X=C([-h,oo[,B)X=C([-h, \infty[, \mathbb{B}) endowed with the norm
||*||:X rarr s(R_(+)),||x||:=([||x||_(0)],[vdots],[||x||_(m)],[vdots])\|\cdot\|: X \rightarrow s\left(\mathbb{R}_{+}\right),\|x\|:=\left(\begin{array}{c}
\|x\|_{0} \\
\vdots \\
\|x\|_{m} \\
\vdots
\end{array}\right)
where ||x||_(0)=max_(-h <= t <= h)|x(t)|\|x\|_{0}=\max _{-h \leq t \leq h}|x(t)| and ||x||_(m)=max_(mh <= t <= (m+1)h)|x(t)|,m >= 1\|x\|_{m}=\max _{m h \leq t \leq(m+1) h}|x(t)|, m \geq 1.
This generalized norm induces a generalized metric, d(x,y):=||x-y||d(x, y):=\|x-y\|.
This generalized norm induces a generalized metric, d(x,y):=||x-y||d(x, y):=\|x-y\|.
2. Preliminaries
Let ( X,dX, d ) be a generalized metric space and A:X rarr XA: X \rightarrow X an operator. In this paper we shall use the terminologies and notations from [13]-[15]. For the convenience of the reader we shall recall some of them.
We denote by A_(0):=1_(X),A^(1):=A,A^(n+1):=A@A^(n),n inNA_{0}:=1_{X}, A^{1}:=A, A^{n+1}:=A \circ A^{n}, n \in \mathbb{N}, the iterate operators of the operator AA. Also we shall use the following notations:
F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\} - the fixed point set of AA;
I(A):={Y sub X∣A(Y)sub Y,Y!=O/}I(A):=\{Y \subset X \mid A(Y) \subset Y, Y \neq \emptyset\} - the family of the nonempty invariant subset of AA;
F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\} - the fixed point set of AA;
I(A):={Y sub X∣A(Y)sub Y,Y!=O/}I(A):=\{Y \subset X \mid A(Y) \subset Y, Y \neq \emptyset\} - the family of the nonempty invariant subset of AA;
Definition 2.1. A:X rarr XA: X \rightarrow X is called a Picard operator (briefly PO) if:
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) A^(n)(x)rarrx^(**)A^{n}(x) \rightarrow x^{*} as n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in X.
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) A^(n)(x)rarrx^(**)A^{n}(x) \rightarrow x^{*} as n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in X.
Definition 2.2. A:X rarr XA: X \rightarrow X is said to be a weakly Picard operator (briefly WPO) if the sequence (A^(n)(x))_(n inN)\left(A^{n}(x)\right)_{n \in \mathbb{N}} converges for all x in Xx \in X and the limit (which may depend on xx ) is a fixed point of AA.
If A:X rarr XA: X \rightarrow X is a WPO, then we may define the operator
A^(oo):X rarr XA^{\infty}: X \rightarrow X by
A^(oo):X rarr XA^{\infty}: X \rightarrow X by
A^(oo)(x):=lim_(n rarr oo)A^(n)(x).A^{\infty}(x):=\lim _{n \rightarrow \infty} A^{n}(x) .
Obviously A^(oo)(X)=F_(A)A^{\infty}(X)=F_{A}. Moreover, if AA is a PO and we denote by x^(**)x^{*} its unique fixed point, then A^(oo)(x)=x^(**)A^{\infty}(x)=x^{*}, for each x in Xx \in X.
Lemma 2.3. Let (X,d, <= )(X, d, \leq) be an ordered metric space and A:X rarr XA: X \rightarrow X an operator. We suppose that:
(i) AA is WPO;
(ii) AA is increasing.
(i) AA is WPO;
(ii) AA is increasing.
Then, the operator A^(oo)A^{\infty} is increasing.
Lemma 2.4. Let ( X,d, <=X, d, \leq ) an ordered metric space and A,B,C:X rarr XA, B, C: X \rightarrow X be such that:
(i) the operator A,B,CA, B, C are WPOs;
(ii) A <= B <= CA \leq B \leq C;
(iii) the operator BB is increasing.
Lemma 2.4. Let ( X,d, <=X, d, \leq ) an ordered metric space and A,B,C:X rarr XA, B, C: X \rightarrow X be such that:
(i) the operator A,B,CA, B, C are WPOs;
(ii) A <= B <= CA \leq B \leq C;
(iii) the operator BB is increasing.
Then x <= y <= zx \leq y \leq z implies that A^(oo)(x) <= B^(oo)(y) <= C^(oo)(z)A^{\infty}(x) \leq B^{\infty}(y) \leq C^{\infty}(z).
Theorem 2.5. [13] Let (X,d)(X, d) be a complete metric space and A:X rarr XA: X \rightarrow X a SS-contraction. Then we have
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) A^(n)(x)rarr"d"x^(**)A^{n}(x) \xrightarrow{d} x^{*}, as n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in X;
(iii) d(A^(n)(x),x^(**)) <= (E-S)^(-1)S^(n)d(x,A(x))d\left(A^{n}(x), x^{*}\right) \leq(E-S)^{-1} S^{n} d(x, A(x));
(iv) d(x,x^(**)) <= (E-S)^(-1)d(x,A(x))d\left(x, x^{*}\right) \leq(E-S)^{-1} d(x, A(x)).
Theorem 2.5. [13] Let (X,d)(X, d) be a complete metric space and A:X rarr XA: X \rightarrow X a SS-contraction. Then we have
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) A^(n)(x)rarr"d"x^(**)A^{n}(x) \xrightarrow{d} x^{*}, as n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in X;
(iii) d(A^(n)(x),x^(**)) <= (E-S)^(-1)S^(n)d(x,A(x))d\left(A^{n}(x), x^{*}\right) \leq(E-S)^{-1} S^{n} d(x, A(x));
(iv) d(x,x^(**)) <= (E-S)^(-1)d(x,A(x))d\left(x, x^{*}\right) \leq(E-S)^{-1} d(x, A(x)).
In what follow we shall apply the above results to the problem (1.2)-(1.3). For other applications of these abstract results, see [2], [3], [8], [9], [12], [16].
3. Existence and uniqueness
From Theorem 2.5 we have
Theorem 3.1. In the condition (C_(1)),(C_(2))(C_(3))\left(C_{1}\right),\left(C_{2}\right)\left(C_{3}\right), and (C_(4))\left(C_{4}\right) the problem (1.2)-(1.3) has in C([-h,oo[,B)C([-h, \infty[, \mathbb{B}) a unique solution x^(**)\stackrel{*}{x} which is the limit of the sequence of successive approximation.
Theorem 3.1. In the condition (C_(1)),(C_(2))(C_(3))\left(C_{1}\right),\left(C_{2}\right)\left(C_{3}\right), and (C_(4))\left(C_{4}\right) the problem (1.2)-(1.3) has in C([-h,oo[,B)C([-h, \infty[, \mathbb{B}) a unique solution x^(**)\stackrel{*}{x} which is the limit of the sequence of successive approximation.
Proof. We consider the operator A:X rarr XA: X \rightarrow X defined by
{:(3.1)A(x)(t)={[varphi(t)","t in[-h","h]],[varphi(h)+int_(h)^(t)g(xi","x(xi)","x(xi+h))d xi+],[quad+int_(h)^(t)int_(xi-h)^(xi)K(s","x(s))dsd xi","t in[h","oo[]:}:}A(x)(t)=\left\{\begin{array}{l}
\varphi(t), t \in[-h, h] \tag{3.1}\\
\varphi(h)+\int_{h}^{t} g(\xi, x(\xi), x(\xi+h)) d \xi+ \\
\quad+\int_{h}^{t} \int_{\xi-h}^{\xi} K(s, x(s)) d s d \xi, t \in[h, \infty[
\end{array}\right.
(X,d)(X, d) is a complete metric space with d=(||*||_(m))_(m in{-1,0,1,dots})d=\left(\|\cdot\|_{m}\right)_{m \in\{-1,0,1, \ldots\}} where
d(x,y)=([d_(0)(x","y)],[vdots],[d_(m)(x","y)],[vdots]).d(x, y)=\left(\begin{array}{c}
d_{0}(x, y) \\
\vdots \\
d_{m}(x, y) \\
\vdots
\end{array}\right) .
For t in[-h,h]t \in[-h, h] we have
||A(x)(t)-A(y)(t)||_(0)=0,AA x,y in X\|A(x)(t)-A(y)(t)\|_{0}=0, \forall x, y \in X
For t in[h,2h]t \in[h, 2 h] we have
{:[|A(x)(t)-A(y)(t)|_(1) <= ],[ <= L_(1)int_(h)^(t)|x(xi)-y(xi)|d xi+L_(2)int_(h)^(t)|x(xi+h)-y(xi+h)|d xi+],[+L_(3)int_(h)^(t)int_(xi-h)^(xi)|x(s)-y(s)|dsd xi],[ <= L_(1)h||x-y||_(1)+L_(2)h||x-y||_(2)+L_(3)int_(h)^(t)(h||x-y||_(0)+h||x-y||_(1))d xi],[ <= L_(3)h^(2)||x-y||_(0)+(L_(1)h+L_(3)h^(2))||x-y||_(1)+L_(2)h||x-y||_(2)]:}\begin{aligned}
&|A(x)(t)-A(y)(t)|_{1} \leq \\
& \leq L_{1} \int_{h}^{t}|x(\xi)-y(\xi)| d \xi+L_{2} \int_{h}^{t}|x(\xi+h)-y(\xi+h)| d \xi+ \\
&+L_{3} \int_{h}^{t} \int_{\xi-h}^{\xi}|x(s)-y(s)| d s d \xi \\
& \leq L_{1} h\|x-y\|_{1}+L_{2} h\|x-y\|_{2}+L_{3} \int_{h}^{t}\left(h\|x-y\|_{0}+h\|x-y\|_{1}\right) d \xi \\
& \leq L_{3} h^{2}\|x-y\|_{0}+\left(L_{1} h+L_{3} h^{2}\right)\|x-y\|_{1}+L_{2} h\|x-y\|_{2}
\end{aligned}
So, ||A(x)(t)-A(y)(t)||_(1) <= L_(3)h^(2)||x-y||_(0)+(L_(1)h+L_(3)h^(2))||x-y||_(1)+L_(2)h||x-y||_(2)\|A(x)(t)-A(y)(t)\|_{1} \leq L_{3} h^{2}\|x-y\|_{0}+\left(L_{1} h+L_{3} h^{2}\right)\|x-y\|_{1}+L_{2} h\|x-y\|_{2}. For t in[2h,3h]t \in[2 h, 3 h] we have ||A(x)(t)-A(y)(t)||_(2) <= L_(3)h^(2)||x-y||_(1)+(L_(1)h+L_(3)h^(2))||x-y||_(2)+L_(2)h||x-y||_(3)\|A(x)(t)-A(y)(t)\|_{2} \leq L_{3} h^{2}\|x-y\|_{1}+\left(L_{1} h+L_{3} h^{2}\right)\|x-y\|_{2}+L_{2} h\|x-y\|_{3}. By induction, for t in[mh,(m+1)h]t \in[m h,(m+1) h] we have that
{:[||A(x)(t)-A(y)(t)||_(m) <= ],[ <= L_(3)h^(2)||x-y||_(m-1)+(L_(1)h+L_(3)h^(2))||x-y||_(m)+L_(2)h||x-y||_(m+1)]:}\begin{aligned}
& \|A(x)(t)-A(y)(t)\|_{m} \leq \\
& \leq L_{3} h^{2}\|x-y\|_{m-1}+\left(L_{1} h+L_{3} h^{2}\right)\|x-y\|_{m}+L_{2} h\|x-y\|_{m+1}
\end{aligned}
Then
{:[([|A(x)(t)-A(y)(t)|_(0)],[|A(x)(t)-A(y)(t)|_(1)],[|A(x)(t)-A(y)(t)|_(2)],[vdots],[|A(x)(t)-A(y)(t)|_(m)],[vdots]) <= ],[([0,0,0,cdots,0,0,cdots],[L_(3)h^(2),L_(1)h+L_(3)h^(2),L_(2)h,cdots,0,0,cdots],[0,L_(3)h^(2),L_(1)h+L_(3)h^(2),cdots,0,0,cdots],[vdots,vdots,vdots,vdots,vdots,vdots,vdots],[0,0,0,cdots,L_(1)h+L_(3)h^(2),L_(2)h,cdots],[vdots,vdots,vdots,vdots,vdots,vdots,vdots])([||x-y||_(0)],[||x-y||_(1)],[||x-y||_(2)],[vdots],[||x-y||_(m)],[vdots]).]:}\begin{aligned}
& \left(\begin{array}{c}
|A(x)(t)-A(y)(t)|_{0} \\
|A(x)(t)-A(y)(t)|_{1} \\
|A(x)(t)-A(y)(t)|_{2} \\
\vdots \\
|A(x)(t)-A(y)(t)|_{m} \\
\vdots
\end{array}\right) \leq \\
& \left(\begin{array}{ccccccc}
0 & 0 & 0 & \cdots & 0 & 0 & \cdots \\
L_{3} h^{2} & L_{1} h+L_{3} h^{2} & L_{2} h & \cdots & 0 & 0 & \cdots \\
0 & L_{3} h^{2} & L_{1} h+L_{3} h^{2} & \cdots & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & L_{1} h+L_{3} h^{2} & L_{2} h & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots
\end{array}\right)\left(\begin{array}{c}
\|x-y\|_{0} \\
\|x-y\|_{1} \\
\|x-y\|_{2} \\
\vdots \\
\|x-y\|_{m} \\
\vdots
\end{array}\right) .
\end{aligned}
So d(A(x),A(y)) <= Sd(x,y)d(A(x), A(y)) \leq S d(x, y), where S:s(R)rarr s(R),||S||:=s u p_(i inN)sum_(j=0)^(oo)|L_(ij)|=(L_(1)+L_(2)+2L_(3)h)hS: s(\mathbb{R}) \rightarrow s(\mathbb{R}),\|S\|:=\sup _{i \in \mathbb{N}} \sum_{j=0}^{\infty}\left|L_{i j}\right|= \left(L_{1}+L_{2}+2 L_{3} h\right) h, which proves that AA is Lipschitz with
S=([0,0,0,cdots,0,0,cdots],[L_(3)h^(2),L_(1)h+L_(3)h^(2),L_(2)h,cdots,0,0,cdots],[0,L_(3)h^(2),L_(1)h+L_(3)h^(2),cdots,0,0,cdots],[vdots,vdots,vdots,vdots,vdots,vdots,vdots],[0,0,0,cdots,L_(1)h+L_(3)h^(2),L_(2)h,cdots],[vdots,vdots,vdots,vdots,vdots,vdots,vdots]).S=\left(\begin{array}{ccccccc}
0 & 0 & 0 & \cdots & 0 & 0 & \cdots \\
L_{3} h^{2} & L_{1} h+L_{3} h^{2} & L_{2} h & \cdots & 0 & 0 & \cdots \\
0 & L_{3} h^{2} & L_{1} h+L_{3} h^{2} & \cdots & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & L_{1} h+L_{3} h^{2} & L_{2} h & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots
\end{array}\right) .
From condition (C_(4))\left(C_{4}\right) we have that AA is SS-contraction. Applying Theorem 2.5 we have the conclusion.
Remark 3.2. From the proof of Theorem 3.1, it follows that the operator AA is POP O in ((C[-h,oo],B),d)((C[-h, \infty], \mathbb{B}), d).
4. Inequalities of Čaplygin type
In this section we shall study the relation between the solution of the problem (1.2)-(1.3) and the subsolution of the same problem.
Let x^(**)\stackrel{*}{x} be the unique solution of the problem (1.2)-(1.3) and yy a subsolution of the same problem, i.e.
{:(4.1)y^(')(t) <= g(t","y(t)","y(t+h))+int_(t-h)^(t)K(s","y(s))ds","t in[0","oo[:}\begin{equation*}
y^{\prime}(t) \leq g(t, y(t), y(t+h))+\int_{t-h}^{t} K(s, y(s)) d s, t \in[0, \infty[ \tag{4.1}
\end{equation*}
where gg and KK satisfy the conditions (C_(1))-(C_(3))\left(C_{1}\right)-\left(C_{3}\right) and
{:(4.2)y(t)=varphi(t)","t in[-h","h].:}\begin{equation*}
y(t)=\varphi(t), t \in[-h, h] . \tag{4.2}
\end{equation*}
In this section we consider an ordered Banach space ( B,|*|, <=\mathbb{B},|\cdot|, \leq ) and the operator AA defined by (3.1) on the ordered Banach space X=((C[a,b],B),||*||, <= )X=((C[a, b], \mathbb{B}),\|\cdot\|, \leq). We have the following theorem
Theorem 4.1. We suppose that:
(a) the conditions (C_(1))-(C_(4))\left(C_{1}\right)-\left(C_{4}\right) are satisfied;
(b) g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B} and K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B} are increasing, AA t in[0,oo[\forall t \in[0, \infty[.
(a) the conditions (C_(1))-(C_(4))\left(C_{1}\right)-\left(C_{4}\right) are satisfied;
(b) g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B} and K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B} are increasing, AA t in[0,oo[\forall t \in[0, \infty[.
Then y <= x^(**)y \leq \stackrel{*}{x} for all t in[0,oo[t \in[0, \infty[.
Proof. In terms of the operator AA defined by the relation (3.1), we have x^(**)=A(x^(**))\stackrel{*}{x}= A(\stackrel{*}{x}) and y <= A(y)y \leq A(y). On the other hand from condition (b) and Lemma 2.3, we have that the operator A^(oo)A^{\infty} is increasing. Hence y <= A(y) <= A^(2)(y) <= cdots <= A^(oo)(y) <= A^(oo)(x^(**))=x^(**)y \leq A(y) \leq A^{2}(y) \leq \cdots \leq A^{\infty}(y) \leq A^{\infty}(\stackrel{*}{x})=\stackrel{*}{x}. So, y <= x^(**)y \leq \stackrel{*}{x}.
Proof. In terms of the operator AA defined by the relation (3.1), we have x^(**)=A(x^(**))\stackrel{*}{x}= A(\stackrel{*}{x}) and y <= A(y)y \leq A(y). On the other hand from condition (b) and Lemma 2.3, we have that the operator A^(oo)A^{\infty} is increasing. Hence y <= A(y) <= A^(2)(y) <= cdots <= A^(oo)(y) <= A^(oo)(x^(**))=x^(**)y \leq A(y) \leq A^{2}(y) \leq \cdots \leq A^{\infty}(y) \leq A^{\infty}(\stackrel{*}{x})=\stackrel{*}{x}. So, y <= x^(**)y \leq \stackrel{*}{x}.
5. Data dependence: monotony
In this section we study the monotony of the system (1.2)-(1.3) with respect to gg and KK. For this we use the abstract comparison Lemma from Section 2.
Consider the following equations
{:(5.1)x_(i)^(')(t)=g_(i)(t,x_(i)(t),x_(i)(t+h))+int_(t-h)^(t)K_(i)(s,x_(i)(s))ds","t in[0","oo[","i= bar(1,3):}\begin{equation*}
x_{i}^{\prime}(t)=g_{i}\left(t, x_{i}(t), x_{i}(t+h)\right)+\int_{t-h}^{t} K_{i}\left(s, x_{i}(s)\right) d s, t \in[0, \infty[, i=\overline{1,3} \tag{5.1}
\end{equation*}
with the conditions (1.3) for each problem and let x^(**)_(i),i= bar(1,3)\stackrel{*}{x}_{i}, i=\overline{1,3} the unique solutions of these problems. Then we need the operators A_(i):X rarr XA_{i}: X \rightarrow X defined by
A_(i)(x)(t)={[varphi(t)","t in[-h","h]],[varphi(h)+int_(h)^(t)g(xi,x_(i)(xi),x_(i)(xi+h))d xi+],[quadquad+int_(h)^(t)int_(xi-h)^(xi)K(s,x_(i)(s))dsd xi","t in[h","oo[]:}A_{i}(x)(t)=\left\{\begin{array}{l}
\varphi(t), t \in[-h, h] \\
\varphi(h)+\int_{h}^{t} g\left(\xi, x_{i}(\xi), x_{i}(\xi+h)\right) d \xi+ \\
\quad \quad+\int_{h}^{t} \int_{\xi-h}^{\xi} K\left(s, x_{i}(s)\right) d s d \xi, t \in[h, \infty[
\end{array}\right.
Theorem 5.1. Let g_(i),K_(i),i= bar(1,3)g_{i}, K_{i}, i=\overline{1,3}, that satisfy the conditions (C_(1))-(C_(4))\left(C_{1}\right)-\left(C_{4}\right).
We suppose that we have
(i) g_(1) <= g_(2) <= g_(3)g_{1} \leq g_{2} \leq g_{3};
(ii) g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B} and K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B} are increasing.
We suppose that we have
(i) g_(1) <= g_(2) <= g_(3)g_{1} \leq g_{2} \leq g_{3};
(ii) g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B} and K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B} are increasing.
Let x^(**)_(i)\stackrel{*}{x}_{i} the solutions of the equations (5.1), i= bar(1,3)i=\overline{1,3}.
Then x_(1)^(**)(t) <= x_(2)^(**)(t) <= x_(3)^(**)(t),AA t in[0,oo[\stackrel{*}{x_{1}}(t) \leq \stackrel{*}{x_{2}}(t) \leq \stackrel{*}{x_{3}}(t), \forall t \in[0, \infty[.
Proof. From Theorem 3.1 the operators A_(i)A_{i} are POs. From the condition (ii) it follows that the operator A_(2)A_{2} is monotone increasing and from condition (i) we have A_(1) <= A_(2) <= A_(3)A_{1} \leq A_{2} \leq A_{3}. But {:x^(**)_(1)=A_(1)oo((x_(1)^(**))),(x_(2)^(**))=A_(2)((x_(2):}^(oo)))\left.\stackrel{*}{x}_{1}=A_{1} \infty\left(\stackrel{*}{x_{1}}\right), \stackrel{*}{x_{2}}=A_{2} \stackrel{\infty}{\left(x_{2}\right.}\right) and {:(x_(3)^(**))=A_(3)((x_(3):}^(oo)))\left.\stackrel{*}{x_{3}}=A_{3} \stackrel{\infty}{\left(x_{3}\right.}\right).
Then x_(1)^(**)(t) <= x_(2)^(**)(t) <= x_(3)^(**)(t),AA t in[0,oo[\stackrel{*}{x_{1}}(t) \leq \stackrel{*}{x_{2}}(t) \leq \stackrel{*}{x_{3}}(t), \forall t \in[0, \infty[.
Proof. From Theorem 3.1 the operators A_(i)A_{i} are POs. From the condition (ii) it follows that the operator A_(2)A_{2} is monotone increasing and from condition (i) we have A_(1) <= A_(2) <= A_(3)A_{1} \leq A_{2} \leq A_{3}. But {:x^(**)_(1)=A_(1)oo((x_(1)^(**))),(x_(2)^(**))=A_(2)((x_(2):}^(oo)))\left.\stackrel{*}{x}_{1}=A_{1} \infty\left(\stackrel{*}{x_{1}}\right), \stackrel{*}{x_{2}}=A_{2} \stackrel{\infty}{\left(x_{2}\right.}\right) and {:(x_(3)^(**))=A_(3)((x_(3):}^(oo)))\left.\stackrel{*}{x_{3}}=A_{3} \stackrel{\infty}{\left(x_{3}\right.}\right).
By applying the abstract comparison Lemma 2.4 follows that the unique solution of the problem (1.2)-(1.3) is increasing with respect to AA.
Remark 5.2. The conclusion of the Theorem 5.1. means that the unique solution of (1.2)-(1.3) is increasing with respect to the right hand.
6. Step method
Next we apply the step method for (1.2)-(1.3). Let the conditions (C_(1)^(')),(C_(2)),(C_(3))\left(C_{1}^{\prime}\right),\left(C_{2}\right),\left(C_{3}\right) and ( C_(5)C_{5} ) and we suppose also the condition
(C_(6))\left(C_{6}\right) For all t in[-h,oo),u_(1),u_(2),u_(3)inBt \in[-h, \infty), u_{1}, u_{2}, u_{3} \in \mathbb{B} there exists a unique u_(2)inB,u_(2)=f(t,u_(1),u_(3))u_{2} \in \mathbb{B}, u_{2}= f\left(t, u_{1}, u_{3}\right) such as u_(3)=g(t,u_(1),u_(2))+int_(t-h)^(t)K(s,u_(1))dsu_{3}=g\left(t, u_{1}, u_{2}\right)+\int_{t-h}^{t} K\left(s, u_{1}\right) d s.
Note that if x inC^(1)(B)x \in C^{1}(\mathbb{B}) is a solution for (1.2)-(1.3) then, by mathematical induction, follows that x inC^(oo)(B)x \in C^{\infty}(\mathbb{B}).
(C_(6))\left(C_{6}\right) For all t in[-h,oo),u_(1),u_(2),u_(3)inBt \in[-h, \infty), u_{1}, u_{2}, u_{3} \in \mathbb{B} there exists a unique u_(2)inB,u_(2)=f(t,u_(1),u_(3))u_{2} \in \mathbb{B}, u_{2}= f\left(t, u_{1}, u_{3}\right) such as u_(3)=g(t,u_(1),u_(2))+int_(t-h)^(t)K(s,u_(1))dsu_{3}=g\left(t, u_{1}, u_{2}\right)+\int_{t-h}^{t} K\left(s, u_{1}\right) d s.
Note that if x inC^(1)(B)x \in C^{1}(\mathbb{B}) is a solution for (1.2)-(1.3) then, by mathematical induction, follows that x inC^(oo)(B)x \in C^{\infty}(\mathbb{B}).
Theorem 6.1. Suppose that we have (C_(1)^(')),(C_(2)),(C_(3)),(C_(5))\left(C_{1}^{\prime}\right),\left(C_{2}\right),\left(C_{3}\right),\left(C_{5}\right) and (C_(6))\left(C_{6}\right). Then the problem (1.2)-(1.3) has a solution if and only if
varphi^((n+1))(0)=g^((n))(0,varphi(0),varphi(h))+[int_(t-h)^(t)K(s,varphi(s))ds]^((n))|_(t=0),n inN\varphi^{(n+1)}(0)=g^{(n)}(0, \varphi(0), \varphi(h))+\left.\left[\int_{t-h}^{t} K(s, \varphi(s)) d s\right]^{(n)}\right|_{t=0}, n \in \mathbb{N}
More, the solution is unique.
Proof. By the step method we have
(p_(0))x_(0)(t)=varphi(t),t in[0,h]\left(\mathrm{p}_{0}\right) x_{0}(t)=\varphi(t), t \in[0, h].
Also we have
Proof. By the step method we have
(p_(0))x_(0)(t)=varphi(t),t in[0,h]\left(\mathrm{p}_{0}\right) x_{0}(t)=\varphi(t), t \in[0, h].
Also we have
x_(0)^(')(t)=g(t,x_(0)(t),x(t+h))+int_(t-h)^(t)K(s,x_(0)(s))dsx_{0}^{\prime}(t)=g\left(t, x_{0}(t), x(t+h)\right)+\int_{t-h}^{t} K\left(s, x_{0}(s)\right) d s
or
varphi^(')(t)=g(t,varphi(t),x(t+h))+int_(t-h)^(t)K(s,varphi(s))ds\varphi^{\prime}(t)=g(t, \varphi(t), x(t+h))+\int_{t-h}^{t} K(s, \varphi(s)) d s
From condition ( C_(6)C_{6} ) we have that
x(t):=x_(1)(t)=f(t-h,varphi(t-h),varphi^(')(t-h)),AA t in[h,2h].x(t):=x_{1}(t)=f\left(t-h, \varphi(t-h), \varphi^{\prime}(t-h)\right), \forall t \in[h, 2 h] .
From the regularity condition we have that x(t)inC^(oo)[-h,2h]x(t) \in C^{\infty}[-h, 2 h] where
{:(6.1)x(t)={[varphi(t)","t in[-h","h]],[f(t-h,varphi(t-h),varphi^(')(t-h))","t in[h","2h]]:}:}x(t)=\left\{\begin{array}{l}
\varphi(t), t \in[-h, h] \tag{6.1}\\
f\left(t-h, \varphi(t-h), \varphi^{\prime}(t-h)\right), t \in[h, 2 h]
\end{array}\right.
The next step is
( {:p_(1))x_(1)^(')(t)=g(t,x_(1)(t),x(t+h))+int_(t-h)^(t)K(s,x_(1)(s))ds\left.\mathrm{p}_{1}\right) x_{1}^{\prime}(t)=g\left(t, x_{1}(t), x(t+h)\right)+\int_{t-h}^{t} K\left(s, x_{1}(s)\right) d s.
From condition ( C5C 5 ) we have that
( {:p_(1))x_(1)^(')(t)=g(t,x_(1)(t),x(t+h))+int_(t-h)^(t)K(s,x_(1)(s))ds\left.\mathrm{p}_{1}\right) x_{1}^{\prime}(t)=g\left(t, x_{1}(t), x(t+h)\right)+\int_{t-h}^{t} K\left(s, x_{1}(s)\right) d s.
From condition ( C5C 5 ) we have that
x(t):=x_(2)(t)=f(t-h,x_(1)(t-h),x_(1)^(')(t-h)),AA t in[2h,3h].x(t):=x_{2}(t)=f\left(t-h, x_{1}(t-h), x_{1}^{\prime}(t-h)\right), \forall t \in[2 h, 3 h] .
From the regularity condition we have that x(t)inC^(oo)[-h,3h]x(t) \in C^{\infty}[-h, 3 h] where
{:(6.2)x(t)={[varphi(t)","t in[-h","h]],[f(t-h,varphi(t-h),varphi^(')(t-h))","t in[h","2h]],[f(t-h,x_(1)(t-h),x_(1)^(')(t-h))","t in[2h","3h]]:}:}x(t)=\left\{\begin{array}{l}
\varphi(t), t \in[-h, h] \tag{6.2}\\
f\left(t-h, \varphi(t-h), \varphi^{\prime}(t-h)\right), t \in[h, 2 h] \\
f\left(t-h, x_{1}(t-h), x_{1}^{\prime}(t-h)\right), t \in[2 h, 3 h]
\end{array}\right.
By induction we can obtain the solution on [-h,oo[[-h, \infty[ of the form
{:(6.3)x(t)={[varphi(t)","t in[-h","h]],[x_(1)","t in[h","2h]],[x_(2)","t in[2h","3h]],[cdots],[x_(n)","t in[nh","(n+1)h]]:}:}x(t)=\left\{\begin{array}{l}
\varphi(t), t \in[-h, h] \tag{6.3}\\
x_{1}, t \in[h, 2 h] \\
x_{2}, t \in[2 h, 3 h] \\
\cdots \\
x_{n}, t \in[n h,(n+1) h]
\end{array}\right.
In order to prove the necessity of the regularity condition we have x inC^(oo)[-h,oo[x \in C^{\infty}[-h, \infty[ a solution of the problem (1.2)-(1.3). By successive derivations we have
x^((n+1))(t)=g^((n))(t,varphi(t),varphi(t+h))+[int_(t-h)^(t)K(s,varphi(s))ds]^((n)),n inNx^{(n+1)}(t)=g^{(n)}(t, \varphi(t), \varphi(t+h))+\left[\int_{t-h}^{t} K(s, \varphi(s)) d s\right]^{(n)}, n \in \mathbb{N}
For t=0t=0 follows that
varphi^((n+1))(0)=g^((n))(0,varphi(0),varphi(h))+[int_(-h)^(0)K(s,varphi(s))ds]^((n))\varphi^{(n+1)}(0)=g^{(n)}(0, \varphi(0), \varphi(h))+\left[\int_{-h}^{0} K(s, \varphi(s)) d s\right]^{(n)}
Remark 6.2. If B=R^(n)\mathbb{B}=\mathbb{R}^{n}, then (1.2) is a finite system of equations, see [6], [10].
Remark 6.3. If B=l^(p)\mathbb{B}=l^{p}, then (1.2) is a infinite system of equations, see [1], [5], [19].
Remark 6.3. If B=l^(p)\mathbb{B}=l^{p}, then (1.2) is a infinite system of equations, see [1], [5], [19].
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[16] I.A. Rus, Some nonlinear functional differential and integral equations, via weakly Picard operator theory: a survey, Carpathian J. Math., 26 (2010), no. 2, 230-258
[17] I.A. Rus, M.A. Şerban, D. Trif, Step method for some integral equations from biomathematics, to appear
[18] S. Sakata, T. Hara, Stability regions for linear differential equations with two kinds of time lags, Funkcialaj Ekvacioj, 47 (2004), 129-144
[19] Sze-Bi Hsu, Ordinary differential equations with applications, Series on Applied Mathematics, vol. 16, Word Scientific, 2006
Babeş-Bolyai University
Department of Applied Mathematics
Kogălniceanu Str., No. 1, Cluj-Napoca, Romania
E-mail address: vdarzu@math.ubbcluj.ro
T. Popoviciu Institute of Numerical Analysis
Department of Applied Mathematics
Kogălniceanu Str., No. 1, Cluj-Napoca, Romania
E-mail address: vdarzu@math.ubbcluj.ro
T. Popoviciu Institute of Numerical Analysis
- 2010 Mathematics Subject Classification. 47H10, 47N20.
Key words and phrases. Integro-differential equation, two time modifications, step method, Picard operators.