Integro-differential equation with two times modifications

Diana Otrocol
(original), Fixed point theory, Nonlinear analysis, paper

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.

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https://ictp.acad.ro/otrocol/papers/2011-IleaOtrocol-CJM-Integro%20diff%20eq.pdf

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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2011-IleaOtrocol-CJM-Integro diff eq

INTEGRO-DIFFERENTIAL EQUATION WITH TWO TIME MODIFICATIONS

VERONICA-ANA ILEA, DIANA OTROCOL

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 − τ ) ) + ∫ t − h t K ( s , x ( s ) ) d s , t ∈ I (1.1) x ′ ( t ) = g ( t , x ( t ) , x ( t − τ ) ) + ∫ t − h t   K ( s , x ( s ) ) d s , t ∈ I {:(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*}(1.1)x′(t)=g(t,x(t),x(t−τ))+∫t−htK(s,x(s))ds,t∈I
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 , ∞ ) I = [ 0 , ∞ ) I=[0,oo)I=[0, \infty)I=[0,∞).
Regarding the two delays we have the following cases: h > 0 , τ > 0 , τ > h h > 0 , τ > 0 , τ > h h > 0,tau > 0,tau > hh>0, \tau>0, \tau>hh>0,τ>0,τ>h, discussed in [8], and here we take the case: τ < 0 , h > 0 , h = | τ | τ < 0 , h > 0 , h = | τ | tau < 0,h > 0,h=|tau|\tau<0, h>0, h=|\tau|τ<0,h>0,h=|τ|.
The equation becomes
(1.2) x ′ ( t ) = g ( t , x ( t ) , x ( t + h ) ) + ∫ t − h t K ( s , x ( s ) ) d s , t ∈ [ 0 , ∞ [ (1.2) x ′ ( t ) = g ( t , x ( t ) , x ( t + h ) ) + ∫ t − h t   K ( s , x ( s ) ) d s , t ∈ [ 0 , ∞ [ {:(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*}(1.2)x′(t)=g(t,x(t),x(t+h))+∫t−htK(s,x(s))ds,t∈[0,∞[
with the condition
(1.3) x ( t ) = φ ( t ) , t ∈ [ − h , h ] . (1.3) x ( t ) = φ ( t ) , t ∈ [ − h , h ] . {:(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*}(1.3)x(t)=φ(t),t∈[−h,h].
Relative to (1.2)-(1.3) we consider the following conditions:
( C 1 ) ( B , | ⋅ | ) C 1 ( B , | ⋅ | ) (C_(1))(B,|*|)\left(C_{1}\right)(\mathbb{B},|\cdot|)(C1)(B,|⋅|) is a Banach space, g ∈ C ( [ 0 , ∞ [ × B 2 , B ) , K ∈ C ( [ 0 , ∞ [ × B , B ) g ∈ C 0 , ∞ × B 2 , B , K ∈ C ( [ 0 , ∞ [ × B , B ) 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.g∈C([0,∞[×B2,B),K∈C([0,∞[×B,B), φ ∈ C ( [ − h , h ] , B ) ; φ ∈ C ( [ − h , h ] , B ) ; varphi in C([-h,h],B);\varphi \in C([-h, h], \mathbb{B}) ;φ∈C([−h,h],B);
( C 1 ′ ) ( B , | ⋅ | ) C 1 ′ ( B , | ⋅ | ) (C_(1)^('))(B,|*|)\left(C_{1}^{\prime}\right)(\mathbb{B},|\cdot|)(C1′)(B,|⋅|) is a Banach space, g ∈ C ∞ ( [ 0 , ∞ [ × B 2 , B ) , K ∈ C ∞ ( [ 0 , ∞ [ × B , B ) g ∈ C ∞ 0 , ∞ × B 2 , B , K ∈ C ∞ ( [ 0 , ∞ [ × B , B ) 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.g∈C∞([0,∞[×B2,B),K∈C∞([0,∞[×B,B), φ ∈ C ∞ ( [ − h , h ] , B ) ; φ ∈ C ∞ ( [ − h , h ] , B ) ; varphi inC^(oo)([-h,h],B);\varphi \in C^{\infty}([-h, h], \mathbb{B}) ;φ∈C∞([−h,h],B);
( C 2 ) C 2 (C_(2))\left(C_{2}\right)(C2) there exists L 1 , L 2 > 0 L 1 , L 2 > 0 L_(1),L_(2) > 0L_{1}, L_{2}>0L1,L2>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 ∈ B , t ∈ [ 0 , ∞ [ ; 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 ∈ B , t ∈ [ 0 , ∞ [ ; |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[;|g(t,u1,v1)−g(t,u2,v2)|≤L1|u1−u2|+L2|v1−v2|,ui,vi∈B,t∈[0,∞[;
( C 3 ) C 3 (C_(3))\left(C_{3}\right)(C3) there exists L 3 > 0 L 3 > 0 L_(3) > 0L_{3}>0L3>0 such that
| K ( s , u ) − K ( s , v ) | ≤ L 3 | u − v | , u , v ∈ B , t ∈ [ 0 , ∞ [ ; | K ( s , u ) − K ( s , v ) | ≤ L 3 | u − v | , u , v ∈ B , t ∈ [ 0 , ∞ [ ; |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[;|K(s,u)−K(s,v)|≤L3|u−v|,u,v∈B,t∈[0,∞[;
( C 4 ) ( L 1 + L 2 + 2 L 3 h ) h < 1 ; C 4 L 1 + L 2 + 2 L 3 h h < 1 ; (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 ;(C4)(L1+L2+2L3h)h<1;
( C 5 ) φ ′ ( 0 ) = g ( h , φ ( 0 ) , φ ( h ) ) + ∫ − h 0 K ( s , φ ( s ) ) d s C 5 φ ′ ( 0 ) = g ( h , φ ( 0 ) , φ ( h ) ) + ∫ − h 0   K ( s , φ ( 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(C5)φ′(0)=g(h,φ(0),φ(h))+∫−h0K(s,φ(s))ds.
In what follow we shall present some notions that will help us obtaining the results bellow.
Let X X XXX be a nonempty set,
s ( X ) := { ( x n ) n ∈ N ∗ ∣ x n ∈ X , n ∈ N ∗ } s ( X ) := x n n ∈ N ∗ ∣ x n ∈ X , n ∈ N ∗ 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\}s(X):={(xn)n∈N∗∣xn∈X,n∈N∗}
and
M ( X ) := { ( x i j ) 1 ∞ ∣ x i j ∈ X , i , j ∈ N ∗ } M ( X ) := x i j 1 ∞ ∣ x i j ∈ X , i , j ∈ N ∗ 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\}M(X):={(xij)1∞∣xij∈X,i,j∈N∗}
where
( x i j ) 1 ∞ := ( x 11 x 12 x 13 ⋯ x 21 x 22 x 23 ⋯ x 31 x 32 x 33 ⋯ ⋯ ⋯ ⋯ ⋯ ) x i j 1 ∞ := x 11 x 12 x 13 ⋯ x 21 x 22 x 23 ⋯ x 31 x 32 x 33 ⋯ ⋯ ⋯ ⋯ ⋯ (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)(xij)1∞:=(x11x12x13⋯x21x22x23⋯x31x32x33⋯⋯⋯⋯⋯)
is a infinite matrix.
For A ∈ M ( B ) A ∈ M ( B ) A in M(B)A \in M(\mathbb{B})A∈M(B) we denote
| A | := sup 1 ≤ i ≤ ∞ ∑ j ∈ N ∗ | a i j | . | A | := sup 1 ≤ i ≤ ∞   ∑ j ∈ N ∗   a i j . |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| .|A|:=sup1≤i≤∞∑j∈N∗|aij|.
Let d : X × X → s ( B ) d : X × X → s ( B ) d:X xx X rarr s(B)d: X \times X \rightarrow s(\mathbb{B})d:X×X→s(B) be the generalized metric.
Remark 1.1. [13] A functional d : X × X → s ( B ) , ( x , y ) ↦ ( d k ( x , y ) ) k ∈ N ∗ d : X × X → s ( B ) , ( x , y ) ↦ d k ( x , y ) k ∈ N ∗ 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}^{*}}d:X×X→s(B),(x,y)↦(dk(x,y))k∈N∗ is a generalized metric of X X XXX iff
(a) d k d k d_(k)d_{k}dk is a pseudometric, ∀ k ∈ N ∗ ∀ k ∈ N ∗ AA k inN^(**)\forall k \in \mathbb{N}^{*}∀k∈N∗;
(b) ∀ x , y ∈ X , x ≠ y ∀ x , y ∈ X , x ≠ y AA x,y in X,x!=y\forall x, y \in X, x \neq y∀x,y∈X,x≠y, there exist k ∈ N ∗ k ∈ N ∗ k inN^(**)k \in \mathbb{N}^{*}k∈N∗ such as d k ( x , y ) ≠ 0 d k ( x , y ) ≠ 0 d_(k)(x,y)!=0d_{k}(x, y) \neq 0dk(x,y)≠0.
Definition 1.2. [13] Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a complete generalized metric space, A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X and S ∈ M ( B ) S ∈ M ( B ) S in M(B)S \in M(\mathbb{B})S∈M(B). The operator A A AAA is a S S SSS-contraction iff:
(i) S S SSS is row and column finite (meaning that there are only a finite number of nonzero elements in each row and each column);
(ii) S S SSS is a Neumann matrix (meaning that if S n S n S^(n)S^{n}Sn is definite for all n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N and
(..) ∑ n ∈ N S n ∑ n ∈ N   S n sum_(n inN)S^(n)\sum_{n \in \mathbb{N}} S^{n}∑n∈NSn converges for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X );
(iii) ∑ n ∈ N S n d ( x , y ) ∑ n ∈ N   S n d ( x , y ) sum_(n inN)S^(n)d(x,y)\sum_{n \in \mathbb{N}} S^{n} d(x, y)∑n∈NSnd(x,y) converges ∀ x , y ∈ X ∀ x , y ∈ X AA x,y in X\forall x, y \in X∀x,y∈X;
(iv) d ( A ( x ) , A ( y ) ) ≤ S d ( x , y ) ∀ x , y ∈ X d ( A ( x ) , A ( y ) ) ≤ S d ( x , y ) ∀ x , y ∈ X 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 Xd(A(x),A(y))≤Sd(x,y)∀x,y∈X.
We consider the space X = C ( [ − h , ∞ [ , B ) X = C ( [ − h , ∞ [ , B ) X=C([-h,oo[,B)X=C([-h, \infty[, \mathbb{B})X=C([−h,∞[,B) endowed with the norm
‖ ⋅ ‖ : X → s ( R + ) , ‖ x ‖ := ( ‖ x ‖ 0 ⋮ ‖ x ‖ m ⋮ ) ‖ ⋅ ‖ : X → s R + , ‖ x ‖ := ‖ x ‖ 0 ⋮ ‖ x ‖ m ⋮ ||*||: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)‖⋅‖:X→s(R+),‖x‖:=(‖x‖0⋮‖x‖m⋮)
where ‖ x ‖ 0 = max − h ≤ t ≤ h | x ( t ) | ‖ x ‖ 0 = max − h ≤ t ≤ h   | x ( t ) | ||x||_(0)=max_(-h <= t <= h)|x(t)|\|x\|_{0}=\max _{-h \leq t \leq h}|x(t)|‖x‖0=max−h≤t≤h|x(t)| and ‖ x ‖ m = max m h ≤ t ≤ ( m + 1 ) h | x ( t ) | , m ≥ 1 ‖ x ‖ m = max m h ≤ t ≤ ( m + 1 ) h   | x ( t ) | , m ≥ 1 ||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‖x‖m=maxmh≤t≤(m+1)h|x(t)|,m≥1.
This generalized norm induces a generalized metric, d ( x , y ) := ‖ x − y ‖ d ( x , y ) := ‖ x − y ‖ d(x,y):=||x-y||d(x, y):=\|x-y\|d(x,y):=‖x−y‖.

2. Preliminaries

Let ( X , d X , d X,dX, dX,d ) be a generalized metric space and A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→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 ∈ N A 0 := 1 X , A 1 := A , A n + 1 := A ∘ A n , n ∈ N 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}A0:=1X,A1:=A,An+1:=A∘An,n∈N, the iterate operators of the operator A A AAA. Also we shall use the following notations:
F A := { x ∈ X ∣ A ( x ) = x } F A := { x ∈ X ∣ A ( x ) = x } F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\}FA:={x∈X∣A(x)=x} - the fixed point set of A A AAA;
I ( A ) := { Y ⊂ X ∣ A ( Y ) ⊂ Y , Y ≠ ∅ } I ( A ) := { Y ⊂ X ∣ A ( Y ) ⊂ Y , Y ≠ ∅ } I(A):={Y sub X∣A(Y)sub Y,Y!=O/}I(A):=\{Y \subset X \mid A(Y) \subset Y, Y \neq \emptyset\}I(A):={Y⊂X∣A(Y)⊂Y,Y≠∅} - the family of the nonempty invariant subset of A A AAA;
Definition 2.1. A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X is called a Picard operator (briefly PO) if:
(i) F A = { x ∗ } F A = x ∗ F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\}FA={x∗};
(ii) A n ( x ) → x ∗ A n ( x ) → x ∗ A^(n)(x)rarrx^(**)A^{n}(x) \rightarrow x^{*}An(x)→x∗ as n → ∞ , ∀ x ∈ X n → ∞ , ∀ x ∈ X n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in Xn→∞,∀x∈X.
Definition 2.2. A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X is said to be a weakly Picard operator (briefly WPO) if the sequence ( A n ( x ) ) n ∈ N A n ( x ) n ∈ N (A^(n)(x))_(n inN)\left(A^{n}(x)\right)_{n \in \mathbb{N}}(An(x))n∈N converges for all x ∈ X x ∈ X x in Xx \in Xx∈X and the limit (which may depend on x x xxx ) is a fixed point of A A AAA.
If A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X is a WPO, then we may define the operator
A ∞ : X → X A ∞ : X → X A^(oo):X rarr XA^{\infty}: X \rightarrow XA∞:X→X by
A ∞ ( x ) := lim n → ∞ A n ( x ) . A ∞ ( x ) := lim n → ∞   A n ( x ) . A^(oo)(x):=lim_(n rarr oo)A^(n)(x).A^{\infty}(x):=\lim _{n \rightarrow \infty} A^{n}(x) .A∞(x):=limn→∞An(x).
Obviously A ∞ ( X ) = F A A ∞ ( X ) = F A A^(oo)(X)=F_(A)A^{\infty}(X)=F_{A}A∞(X)=FA. Moreover, if A A AAA is a PO and we denote by x ∗ x ∗ x^(**)x^{*}x∗ its unique fixed point, then A ∞ ( x ) = x ∗ A ∞ ( x ) = x ∗ A^(oo)(x)=x^(**)A^{\infty}(x)=x^{*}A∞(x)=x∗, for each x ∈ X x ∈ X x in Xx \in Xx∈X.
Lemma 2.3. Let ( X , d , ≤ ) ( X , d , ≤ ) (X,d, <= )(X, d, \leq)(X,d,≤) be an ordered metric space and A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X an operator. We suppose that:
(i) A A AAA is WPO;
(ii) A A AAA is increasing.
Then, the operator A ∞ A ∞ A^(oo)A^{\infty}A∞ is increasing.
Lemma 2.4. Let ( X , d , ≤ X , d , ≤ X,d, <=X, d, \leqX,d,≤ ) an ordered metric space and A , B , C : X → X A , B , C : X → X A,B,C:X rarr XA, B, C: X \rightarrow XA,B,C:X→X be such that:
(i) the operator A , B , C A , B , C A,B,CA, B, CA,B,C are WPOs;
(ii) A ≤ B ≤ C A ≤ B ≤ C A <= B <= CA \leq B \leq CA≤B≤C;
(iii) the operator B B BBB is increasing.
Then x ≤ y ≤ z x ≤ y ≤ z x <= y <= zx \leq y \leq zx≤y≤z implies that A ∞ ( x ) ≤ B ∞ ( y ) ≤ C ∞ ( z ) A ∞ ( x ) ≤ B ∞ ( y ) ≤ C ∞ ( z ) A^(oo)(x) <= B^(oo)(y) <= C^(oo)(z)A^{\infty}(x) \leq B^{\infty}(y) \leq C^{\infty}(z)A∞(x)≤B∞(y)≤C∞(z).
Theorem 2.5. [13] Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a complete metric space and A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X a S S SSS-contraction. Then we have
(i) F A = { x ∗ } F A = x ∗ F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\}FA={x∗};
(ii) A n ( x ) → d x ∗ A n ( x ) → d x ∗ A^(n)(x)rarr"d"x^(**)A^{n}(x) \xrightarrow{d} x^{*}An(x)→dx∗, as n → ∞ , ∀ x ∈ X n → ∞ , ∀ x ∈ X n rarr oo,AA x in Xn \rightarrow \infty, \forall x \in Xn→∞,∀x∈X;
(iii) d ( A n ( x ) , x ∗ ) ≤ ( E − S ) − 1 S n d ( x , A ( x ) ) d A n ( x ) , x ∗ ≤ ( E − S ) − 1 S n d ( x , A ( x ) ) 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))d(An(x),x∗)≤(E−S)−1Snd(x,A(x));
(iv) d ( x , x ∗ ) ≤ ( E − S ) − 1 d ( x , A ( x ) ) d x , x ∗ ≤ ( E − S ) − 1 d ( x , A ( x ) ) d(x,x^(**)) <= (E-S)^(-1)d(x,A(x))d\left(x, x^{*}\right) \leq(E-S)^{-1} d(x, A(x))d(x,x∗)≤(E−S)−1d(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 ) C 1 , C 2 C 3 (C_(1)),(C_(2))(C_(3))\left(C_{1}\right),\left(C_{2}\right)\left(C_{3}\right)(C1),(C2)(C3), and ( C 4 ) C 4 (C_(4))\left(C_{4}\right)(C4) the problem (1.2)-(1.3) has in C ( [ − h , ∞ [ , B ) C ( [ − h , ∞ [ , B ) C([-h,oo[,B)C([-h, \infty[, \mathbb{B})C([−h,∞[,B) a unique solution x ∗ x ∗ x^(**)\stackrel{*}{x}x∗ which is the limit of the sequence of successive approximation.
Proof. We consider the operator A : X → X A : X → X A:X rarr XA: X \rightarrow XA:X→X defined by
(3.1) A ( x ) ( t ) = { φ ( t ) , t ∈ [ − h , h ] φ ( h ) + ∫ h t g ( ξ , x ( ξ ) , x ( ξ + h ) ) d ξ + + ∫ h t ∫ ξ − h ξ K ( s , x ( s ) ) d s d ξ , t ∈ [ h , ∞ [ (3.1) A ( x ) ( t ) = φ ( t ) , t ∈ [ − h , h ] φ ( h ) + ∫ h t   g ( ξ , x ( ξ ) , x ( ξ + h ) ) d ξ + + ∫ h t   ∫ ξ − h ξ   K ( s , x ( s ) ) d s d ξ , t ∈ [ h , ∞ [ {:(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.(3.1)A(x)(t)={φ(t),t∈[−h,h]φ(h)+∫htg(ξ,x(ξ),x(ξ+h))dξ++∫ht∫ξ−hξK(s,x(s))dsdξ,t∈[h,∞[
( X , d ) ( X , d ) (X,d)(X, d)(X,d) is a complete metric space with d = ( ‖ ⋅ ‖ m ) m ∈ { − 1 , 0 , 1 , … } d = ‖ ⋅ ‖ m m ∈ { − 1 , 0 , 1 , … } d=(||*||_(m))_(m in{-1,0,1,dots})d=\left(\|\cdot\|_{m}\right)_{m \in\{-1,0,1, \ldots\}}d=(‖⋅‖m)m∈{−1,0,1,…} where
d ( x , y ) = ( d 0 ( x , y ) ⋮ d m ( x , y ) ⋮ ) . d ( x , y ) = d 0 ( x , y ) ⋮ d m ( x , y ) ⋮ . 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) .d(x,y)=(d0(x,y)⋮dm(x,y)⋮).
For t ∈ [ − h , h ] t ∈ [ − h , h ] t in[-h,h]t \in[-h, h]t∈[−h,h] we have
‖ A ( x ) ( t ) − A ( y ) ( t ) ‖ 0 = 0 , ∀ x , y ∈ X ‖ A ( x ) ( t ) − A ( y ) ( t ) ‖ 0 = 0 , ∀ x , y ∈ X ||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‖A(x)(t)−A(y)(t)‖0=0,∀x,y∈X
For t ∈ [ h , 2 h ] t ∈ [ h , 2 h ] t in[h,2h]t \in[h, 2 h]t∈[h,2h] we have
| A ( x ) ( t ) − A ( y ) ( t ) | 1 ≤ ≤ L 1 ∫ h t | x ( ξ ) − y ( ξ ) | d ξ + L 2 ∫ h t | x ( ξ + h ) − y ( ξ + h ) | d ξ + + L 3 ∫ h t ∫ ξ − h ξ | x ( s ) − y ( s ) | d s d ξ ≤ L 1 h ‖ x − y ‖ 1 + L 2 h ‖ x − y ‖ 2 + L 3 ∫ h t ( h ‖ x − y ‖ 0 + h ‖ x − y ‖ 1 ) d ξ ≤ 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 ≤ ≤ L 1 ∫ h t   | x ( ξ ) − y ( ξ ) | d ξ + L 2 ∫ h t   | x ( ξ + h ) − y ( ξ + h ) | d ξ + + L 3 ∫ h t   ∫ ξ − h ξ   | x ( s ) − y ( s ) | d s d ξ ≤ L 1 h ‖ x − y ‖ 1 + L 2 h ‖ x − y ‖ 2 + L 3 ∫ h t   h ‖ x − y ‖ 0 + h ‖ x − y ‖ 1 d ξ ≤ 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) <= ],[ <= 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}|A(x)(t)−A(y)(t)|1≤≤L1∫ht|x(ξ)−y(ξ)|dξ+L2∫ht|x(ξ+h)−y(ξ+h)|dξ++L3∫ht∫ξ−hξ|x(s)−y(s)|dsdξ≤L1h‖x−y‖1+L2h‖x−y‖2+L3∫ht(h‖x−y‖0+h‖x−y‖1)dξ≤L3h2‖x−y‖0+(L1h+L3h2)‖x−y‖1+L2h‖x−y‖2
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 ≤ 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) <= 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}‖A(x)(t)−A(y)(t)‖1≤L3h2‖x−y‖0+(L1h+L3h2)‖x−y‖1+L2h‖x−y‖2. For t ∈ [ 2 h , 3 h ] t ∈ [ 2 h , 3 h ] t in[2h,3h]t \in[2 h, 3 h]t∈[2h,3h] 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 ≤ 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) <= 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}‖A(x)(t)−A(y)(t)‖2≤L3h2‖x−y‖1+(L1h+L3h2)‖x−y‖2+L2h‖x−y‖3. By induction, for t ∈ [ m h , ( m + 1 ) h ] t ∈ [ m h , ( m + 1 ) h ] t in[mh,(m+1)h]t \in[m h,(m+1) h]t∈[mh,(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 ‖ 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 {:[||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}‖A(x)(t)−A(y)(t)‖m≤≤L3h2‖x−y‖m−1+(L1h+L3h2)‖x−y‖m+L2h‖x−y‖m+1
Then
( | A ( x ) ( t ) − A ( y ) ( t ) | 0 | A ( x ) ( t ) − A ( y ) ( t ) | 1 | A ( x ) ( t ) − A ( y ) ( t ) | 2 ⋮ | A ( x ) ( t ) − A ( y ) ( t ) | m ⋮ ) ≤ ( 0 0 0 ⋯ 0 0 ⋯ L 3 h 2 L 1 h + L 3 h 2 L 2 h ⋯ 0 0 ⋯ 0 L 3 h 2 L 1 h + L 3 h 2 ⋯ 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ L 1 h + L 3 h 2 L 2 h ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ) ( ‖ x − y ‖ 0 ‖ x − y ‖ 1 ‖ x − y ‖ 2 ⋮ ‖ x − y ‖ m ⋮ ) . | A ( x ) ( t ) − A ( y ) ( t ) | 0 | A ( x ) ( t ) − A ( y ) ( t ) | 1 | A ( x ) ( t ) − A ( y ) ( t ) | 2 ⋮ | A ( x ) ( t ) − A ( y ) ( t ) | m ⋮ ≤ 0 0 0 ⋯ 0 0 ⋯ L 3 h 2 L 1 h + L 3 h 2 L 2 h ⋯ 0 0 ⋯ 0 L 3 h 2 L 1 h + L 3 h 2 ⋯ 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ L 1 h + L 3 h 2 L 2 h ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ‖ x − y ‖ 0 ‖ x − y ‖ 1 ‖ x − y ‖ 2 ⋮ ‖ x − y ‖ m ⋮ . {:[([|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}(|A(x)(t)−A(y)(t)|0|A(x)(t)−A(y)(t)|1|A(x)(t)−A(y)(t)|2⋮|A(x)(t)−A(y)(t)|m⋮)≤(000⋯00⋯L3h2L1h+L3h2L2h⋯00⋯0L3h2L1h+L3h2⋯00⋯⋮⋮⋮⋮⋮⋮⋮000⋯L1h+L3h2L2h⋯⋮⋮⋮⋮⋮⋮⋮)(‖x−y‖0‖x−y‖1‖x−y‖2⋮‖x−y‖m⋮).
So d ( A ( x ) , A ( y ) ) ≤ S d ( x , y ) d ( A ( x ) , A ( y ) ) ≤ S d ( x , y ) d(A(x),A(y)) <= Sd(x,y)d(A(x), A(y)) \leq S d(x, y)d(A(x),A(y))≤Sd(x,y), where S : s ( R ) → s ( R ) , ‖ S ‖ := sup i ∈ N ∑ j = 0 ∞ | L i j | = ( L 1 + L 2 + 2 L 3 h ) h S : s ( R ) → s ( R ) , ‖ S ‖ := sup i ∈ N   ∑ j = 0 ∞   L i j = L 1 + L 2 + 2 L 3 h h 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) hS:s(R)→s(R),‖S‖:=supi∈N∑j=0∞|Lij|=(L1+L2+2L3h)h, which proves that A A AAA is Lipschitz with
S = ( 0 0 0 ⋯ 0 0 ⋯ L 3 h 2 L 1 h + L 3 h 2 L 2 h ⋯ 0 0 ⋯ 0 L 3 h 2 L 1 h + L 3 h 2 ⋯ 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ L 1 h + L 3 h 2 L 2 h ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ) . S = 0 0 0 ⋯ 0 0 ⋯ L 3 h 2 L 1 h + L 3 h 2 L 2 h ⋯ 0 0 ⋯ 0 L 3 h 2 L 1 h + L 3 h 2 ⋯ 0 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 ⋯ L 1 h + L 3 h 2 L 2 h ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ . 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) .S=(000⋯00⋯L3h2L1h+L3h2L2h⋯00⋯0L3h2L1h+L3h2⋯00⋯⋮⋮⋮⋮⋮⋮⋮000⋯L1h+L3h2L2h⋯⋮⋮⋮⋮⋮⋮⋮).
From condition ( C 4 ) C 4 (C_(4))\left(C_{4}\right)(C4) we have that A A AAA is S S SSS-contraction. Applying Theorem 2.5 we have the conclusion.
Remark 3.2. From the proof of Theorem 3.1, it follows that the operator A A AAA is P O P O POP OPO in ( ( C [ − h , ∞ ] , B ) , d ) ( ( C [ − h , ∞ ] , B ) , d ) ((C[-h,oo],B),d)((C[-h, \infty], \mathbb{B}), d)((C[−h,∞],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 ∗ x ∗ x^(**)\stackrel{*}{x}x∗ be the unique solution of the problem (1.2)-(1.3) and y y yyy a subsolution of the same problem, i.e.
(4.1) y ′ ( t ) ≤ g ( t , y ( t ) , y ( t + h ) ) + ∫ t − h t K ( s , y ( s ) ) d s , t ∈ [ 0 , ∞ [ (4.1) y ′ ( t ) ≤ g ( t , y ( t ) , y ( t + h ) ) + ∫ t − h t   K ( s , y ( s ) ) d s , t ∈ [ 0 , ∞ [ {:(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*}(4.1)y′(t)≤g(t,y(t),y(t+h))+∫t−htK(s,y(s))ds,t∈[0,∞[
where g g ggg and K K KKK satisfy the conditions ( C 1 ) − ( C 3 ) C 1 − C 3 (C_(1))-(C_(3))\left(C_{1}\right)-\left(C_{3}\right)(C1)−(C3) and
(4.2) y ( t ) = φ ( t ) , t ∈ [ − h , h ] . (4.2) y ( t ) = φ ( t ) , t ∈ [ − h , h ] . {:(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*}(4.2)y(t)=φ(t),t∈[−h,h].
In this section we consider an ordered Banach space ( B , | ⋅ | , ≤ B , | ⋅ | , ≤ B,|*|, <=\mathbb{B},|\cdot|, \leqB,|⋅|,≤ ) and the operator A A AAA defined by (3.1) on the ordered Banach space X = ( ( C [ a , b ] , B ) , ‖ ⋅ ‖ , ≤ ) X = ( ( C [ a , b ] , B ) , ‖ ⋅ ‖ , ≤ ) X=((C[a,b],B),||*||, <= )X=((C[a, b], \mathbb{B}),\|\cdot\|, \leq)X=((C[a,b],B),‖⋅‖,≤). We have the following theorem
Theorem 4.1. We suppose that:
(a) the conditions ( C 1 ) − ( C 4 ) C 1 − C 4 (C_(1))-(C_(4))\left(C_{1}\right)-\left(C_{4}\right)(C1)−(C4) are satisfied;
(b) g ( t , ⋅ , ⋅ ) : B 2 → B g ( t , ⋅ , ⋅ ) : B 2 → B g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B}g(t,⋅,⋅):B2→B and K ( t , ⋅ ) : B → B K ( t , ⋅ ) : B → B K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B}K(t,⋅):B→B are increasing, ∀ t ∈ [ 0 , ∞ [ ∀ t ∈ [ 0 , ∞ [ AA t in[0,oo[\forall t \in[0, \infty[∀t∈[0,∞[.
Then y ≤ x ∗ y ≤ x ∗ y <= x^(**)y \leq \stackrel{*}{x}y≤x∗ for all t ∈ [ 0 , ∞ [ t ∈ [ 0 , ∞ [ t in[0,oo[t \in[0, \infty[t∈[0,∞[.
Proof. In terms of the operator A A AAA defined by the relation (3.1), we have x ∗ = A ( x ∗ ) x ∗ = A ( x ∗ ) x^(**)=A(x^(**))\stackrel{*}{x}= A(\stackrel{*}{x})x∗=A(x∗) and y ≤ A ( y ) y ≤ A ( y ) y <= A(y)y \leq A(y)y≤A(y). On the other hand from condition (b) and Lemma 2.3, we have that the operator A ∞ A ∞ A^(oo)A^{\infty}A∞ is increasing. Hence y ≤ A ( y ) ≤ A 2 ( y ) ≤ ⋯ ≤ A ∞ ( y ) ≤ A ∞ ( x ∗ ) = x ∗ y ≤ A ( y ) ≤ A 2 ( y ) ≤ ⋯ ≤ A ∞ ( y ) ≤ A ∞ ( x ∗ ) = x ∗ 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}y≤A(y)≤A2(y)≤⋯≤A∞(y)≤A∞(x∗)=x∗. So, y ≤ x ∗ y ≤ x ∗ y <= x^(**)y \leq \stackrel{*}{x}y≤x∗.

5. Data dependence: monotony

In this section we study the monotony of the system (1.2)-(1.3) with respect to g g ggg and K K KKK. 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 ) ) + ∫ t − h t K i ( s , x i ( s ) ) d s , t ∈ [ 0 , ∞ [ , i = 1 , 3 ― (5.1) x i ′ ( t ) = g i t , x i ( t ) , x i ( t + h ) + ∫ t − h t   K i s , x i ( s ) d s , t ∈ [ 0 , ∞ [ , i = 1 , 3 ¯ {:(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*}(5.1)xi′(t)=gi(t,xi(t),xi(t+h))+∫t−htKi(s,xi(s))ds,t∈[0,∞[,i=1,3―
with the conditions (1.3) for each problem and let x ∗ i , i = 1 , 3 ― x ∗ i , i = 1 , 3 ¯ x^(**)_(i),i= bar(1,3)\stackrel{*}{x}_{i}, i=\overline{1,3}x∗i,i=1,3― the unique solutions of these problems. Then we need the operators A i : X → X A i : X → X A_(i):X rarr XA_{i}: X \rightarrow XAi:X→X defined by
A i ( x ) ( t ) = { φ ( t ) , t ∈ [ − h , h ] φ ( h ) + ∫ h t g ( ξ , x i ( ξ ) , x i ( ξ + h ) ) d ξ + + ∫ h t ∫ ξ − h ξ K ( s , x i ( s ) ) d s d ξ , t ∈ [ h , ∞ [ A i ( x ) ( t ) = φ ( t ) , t ∈ [ − h , h ] φ ( h ) + ∫ h t   g ξ , x i ( ξ ) , x i ( ξ + h ) d ξ + + ∫ h t   ∫ ξ − h ξ   K s , x i ( s ) d s d ξ , t ∈ [ h , ∞ [ 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.Ai(x)(t)={φ(t),t∈[−h,h]φ(h)+∫htg(ξ,xi(ξ),xi(ξ+h))dξ++∫ht∫ξ−hξK(s,xi(s))dsdξ,t∈[h,∞[
Theorem 5.1. Let g i , K i , i = 1 , 3 ― g i , K i , i = 1 , 3 ¯ g_(i),K_(i),i= bar(1,3)g_{i}, K_{i}, i=\overline{1,3}gi,Ki,i=1,3―, that satisfy the conditions ( C 1 ) − ( C 4 ) C 1 − C 4 (C_(1))-(C_(4))\left(C_{1}\right)-\left(C_{4}\right)(C1)−(C4).
We suppose that we have
(i) g 1 ≤ g 2 ≤ g 3 g 1 ≤ g 2 ≤ g 3 g_(1) <= g_(2) <= g_(3)g_{1} \leq g_{2} \leq g_{3}g1≤g2≤g3;
(ii) g ( t , ⋅ , ⋅ ) : B 2 → B g ( t , ⋅ , ⋅ ) : B 2 → B g(t,*,*):B^(2)rarrBg(t, \cdot, \cdot): \mathbb{B}^{2} \rightarrow \mathbb{B}g(t,⋅,⋅):B2→B and K ( t , ⋅ ) : B → B K ( t , ⋅ ) : B → B K(t,*):BrarrBK(t, \cdot): \mathbb{B} \rightarrow \mathbb{B}K(t,⋅):B→B are increasing.
Let x ∗ i x ∗ i x^(**)_(i)\stackrel{*}{x}_{i}x∗i the solutions of the equations (5.1), i = 1 , 3 ― i = 1 , 3 ¯ i= bar(1,3)i=\overline{1,3}i=1,3―.
Then x 1 ∗ ( t ) ≤ x 2 ∗ ( t ) ≤ x 3 ∗ ( t ) , ∀ t ∈ [ 0 , ∞ [ x 1 ∗ ( t ) ≤ x 2 ∗ ( t ) ≤ x 3 ∗ ( t ) , ∀ t ∈ [ 0 , ∞ [ 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[x1∗(t)≤x2∗(t)≤x3∗(t),∀t∈[0,∞[.
Proof. From Theorem 3.1 the operators A i A i A_(i)A_{i}Ai are POs. From the condition (ii) it follows that the operator A 2 A 2 A_(2)A_{2}A2 is monotone increasing and from condition (i) we have A 1 ≤ A 2 ≤ A 3 A 1 ≤ A 2 ≤ A 3 A_(1) <= A_(2) <= A_(3)A_{1} \leq A_{2} \leq A_{3}A1≤A2≤A3. But x ∗ 1 = A 1 ∞ ( x 1 ∗ ) , x 2 ∗ = A 2 ( x 2 ∞ ) x ∗ 1 = A 1 ∞ x 1 ∗ , x 2 ∗ = A 2 x 2 ∞ {: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)x∗1=A1∞(x1∗),x2∗=A2(x2∞) and x 3 ∗ = A 3 ( x 3 ∞ ) x 3 ∗ = A 3 x 3 ∞ {:(x_(3)^(**))=A_(3)((x_(3):}^(oo)))\left.\stackrel{*}{x_{3}}=A_{3} \stackrel{\infty}{\left(x_{3}\right.}\right)x3∗=A3(x3∞).
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 A A AAA.
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 ) C 1 ′ , C 2 , C 3 (C_(1)^(')),(C_(2)),(C_(3))\left(C_{1}^{\prime}\right),\left(C_{2}\right),\left(C_{3}\right)(C1′),(C2),(C3) and ( C 5 C 5 C_(5)C_{5}C5 ) and we suppose also the condition
( C 6 ) C 6 (C_(6))\left(C_{6}\right)(C6) For all t ∈ [ − h , ∞ ) , u 1 , u 2 , u 3 ∈ B t ∈ [ − h , ∞ ) , u 1 , u 2 , u 3 ∈ B t in[-h,oo),u_(1),u_(2),u_(3)inBt \in[-h, \infty), u_{1}, u_{2}, u_{3} \in \mathbb{B}t∈[−h,∞),u1,u2,u3∈B there exists a unique u 2 ∈ B , u 2 = f ( t , u 1 , u 3 ) u 2 ∈ B , u 2 = f t , u 1 , u 3 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)u2∈B,u2=f(t,u1,u3) such as u 3 = g ( t , u 1 , u 2 ) + ∫ t − h t K ( s , u 1 ) d s u 3 = g t , u 1 , u 2 + ∫ t − h t   K s , u 1 d s 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 su3=g(t,u1,u2)+∫t−htK(s,u1)ds.
Note that if x ∈ C 1 ( B ) x ∈ C 1 ( B ) x inC^(1)(B)x \in C^{1}(\mathbb{B})x∈C1(B) is a solution for (1.2)-(1.3) then, by mathematical induction, follows that x ∈ C ∞ ( B ) x ∈ C ∞ ( B ) x inC^(oo)(B)x \in C^{\infty}(\mathbb{B})x∈C∞(B).
Theorem 6.1. Suppose that we have ( C 1 ′ ) , ( C 2 ) , ( C 3 ) , ( C 5 ) C 1 ′ , C 2 , C 3 , C 5 (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)(C1′),(C2),(C3),(C5) and ( C 6 ) C 6 (C_(6))\left(C_{6}\right)(C6). Then the problem (1.2)-(1.3) has a solution if and only if
φ ( n + 1 ) ( 0 ) = g ( n ) ( 0 , φ ( 0 ) , φ ( h ) ) + [ ∫ t − h t K ( s , φ ( s ) ) d s ] ( n ) | t = 0 , n ∈ N φ ( n + 1 ) ( 0 ) = g ( n ) ( 0 , φ ( 0 ) , φ ( h ) ) + ∫ t − h t   K ( s , φ ( s ) ) d s ( n ) t = 0 , n ∈ N 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}φ(n+1)(0)=g(n)(0,φ(0),φ(h))+[∫t−htK(s,φ(s))ds](n)|t=0,n∈N
More, the solution is unique.
Proof. By the step method we have
( p 0 ) x 0 ( t ) = φ ( t ) , t ∈ [ 0 , h ] p 0 x 0 ( t ) = φ ( t ) , t ∈ [ 0 , h ] (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](p0)x0(t)=φ(t),t∈[0,h].
Also we have
x 0 ′ ( t ) = g ( t , x 0 ( t ) , x ( t + h ) ) + ∫ t − h t K ( s , x 0 ( s ) ) d s x 0 ′ ( t ) = g t , x 0 ( t ) , x ( t + h ) + ∫ t − h t   K s , x 0 ( s ) d s 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 sx0′(t)=g(t,x0(t),x(t+h))+∫t−htK(s,x0(s))ds
or
φ ′ ( t ) = g ( t , φ ( t ) , x ( t + h ) ) + ∫ t − h t K ( s , φ ( s ) ) d s φ ′ ( t ) = g ( t , φ ( t ) , x ( t + h ) ) + ∫ t − h t   K ( s , φ ( s ) ) d s 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 sφ′(t)=g(t,φ(t),x(t+h))+∫t−htK(s,φ(s))ds
From condition ( C 6 C 6 C_(6)C_{6}C6 ) we have that
x ( t ) := x 1 ( t ) = f ( t − h , φ ( t − h ) , φ ′ ( t − h ) ) , ∀ t ∈ [ h , 2 h ] . x ( t ) := x 1 ( t ) = f t − h , φ ( t − h ) , φ ′ ( t − h ) , ∀ t ∈ [ h , 2 h ] . 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] .x(t):=x1(t)=f(t−h,φ(t−h),φ′(t−h)),∀t∈[h,2h].
From the regularity condition we have that x ( t ) ∈ C ∞ [ − h , 2 h ] x ( t ) ∈ C ∞ [ − h , 2 h ] x(t)inC^(oo)[-h,2h]x(t) \in C^{\infty}[-h, 2 h]x(t)∈C∞[−h,2h] where
(6.1) x ( t ) = { φ ( t ) , t ∈ [ − h , h ] f ( t − h , φ ( t − h ) , φ ′ ( t − h ) ) , t ∈ [ h , 2 h ] (6.1) x ( t ) = φ ( t ) , t ∈ [ − h , h ] f t − h , φ ( t − h ) , φ ′ ( t − h ) , t ∈ [ h , 2 h ] {:(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.(6.1)x(t)={φ(t),t∈[−h,h]f(t−h,φ(t−h),φ′(t−h)),t∈[h,2h]
The next step is
( p 1 ) x 1 ′ ( t ) = g ( t , x 1 ( t ) , x ( t + h ) ) + ∫ t − h t K ( s , x 1 ( s ) ) d s p 1 x 1 ′ ( t ) = g t , x 1 ( t ) , x ( t + h ) + ∫ t − h t   K s , x 1 ( s ) d s {: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 sp1)x1′(t)=g(t,x1(t),x(t+h))+∫t−htK(s,x1(s))ds.
From condition ( C 5 C 5 C5C 5C5 ) we have that
x ( t ) := x 2 ( t ) = f ( t − h , x 1 ( t − h ) , x 1 ′ ( t − h ) ) , ∀ t ∈ [ 2 h , 3 h ] . x ( t ) := x 2 ( t ) = f t − h , x 1 ( t − h ) , x 1 ′ ( t − h ) , ∀ t ∈ [ 2 h , 3 h ] . 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] .x(t):=x2(t)=f(t−h,x1(t−h),x1′(t−h)),∀t∈[2h,3h].
From the regularity condition we have that x ( t ) ∈ C ∞ [ − h , 3 h ] x ( t ) ∈ C ∞ [ − h , 3 h ] x(t)inC^(oo)[-h,3h]x(t) \in C^{\infty}[-h, 3 h]x(t)∈C∞[−h,3h] where
(6.2) x ( t ) = { φ ( t ) , t ∈ [ − h , h ] f ( t − h , φ ( t − h ) , φ ′ ( t − h ) ) , t ∈ [ h , 2 h ] f ( t − h , x 1 ( t − h ) , x 1 ′ ( t − h ) ) , t ∈ [ 2 h , 3 h ] (6.2) x ( t ) = φ ( t ) , t ∈ [ − h , h ] f t − h , φ ( t − h ) , φ ′ ( t − h ) , t ∈ [ h , 2 h ] f t − h , x 1 ( t − h ) , x 1 ′ ( t − h ) , t ∈ [ 2 h , 3 h ] {:(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.(6.2)x(t)={φ(t),t∈[−h,h]f(t−h,φ(t−h),φ′(t−h)),t∈[h,2h]f(t−h,x1(t−h),x1′(t−h)),t∈[2h,3h]
By induction we can obtain the solution on [ − h , ∞ [ [ − h , ∞ [ [-h,oo[[-h, \infty[[−h,∞[ of the form
(6.3) x ( t ) = { φ ( t ) , t ∈ [ − h , h ] x 1 , t ∈ [ h , 2 h ] x 2 , t ∈ [ 2 h , 3 h ] ⋯ x n , t ∈ [ n h , ( n + 1 ) h ] (6.3) x ( t ) = φ ( t ) , t ∈ [ − h , h ] x 1 , t ∈ [ h , 2 h ] x 2 , t ∈ [ 2 h , 3 h ] ⋯ x n , t ∈ [ n h , ( n + 1 ) h ] {:(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.(6.3)x(t)={φ(t),t∈[−h,h]x1,t∈[h,2h]x2,t∈[2h,3h]⋯xn,t∈[nh,(n+1)h]
In order to prove the necessity of the regularity condition we have x ∈ C ∞ [ − h , ∞ [ x ∈ C ∞ [ − h , ∞ [ x inC^(oo)[-h,oo[x \in C^{\infty}[-h, \infty[x∈C∞[−h,∞[ a solution of the problem (1.2)-(1.3). By successive derivations we have
x ( n + 1 ) ( t ) = g ( n ) ( t , φ ( t ) , φ ( t + h ) ) + [ ∫ t − h t K ( s , φ ( s ) ) d s ] ( n ) , n ∈ N x ( n + 1 ) ( t ) = g ( n ) ( t , φ ( t ) , φ ( t + h ) ) + ∫ t − h t   K ( s , φ ( s ) ) d s ( n ) , n ∈ N 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}x(n+1)(t)=g(n)(t,φ(t),φ(t+h))+[∫t−htK(s,φ(s))ds](n),n∈N
For t = 0 t = 0 t=0t=0t=0 follows that
φ ( n + 1 ) ( 0 ) = g ( n ) ( 0 , φ ( 0 ) , φ ( h ) ) + [ ∫ − h 0 K ( s , φ ( s ) ) d s ] ( n ) φ ( n + 1 ) ( 0 ) = g ( n ) ( 0 , φ ( 0 ) , φ ( h ) ) + ∫ − h 0   K ( s , φ ( s ) ) d s ( n ) 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)}φ(n+1)(0)=g(n)(0,φ(0),φ(h))+[∫−h0K(s,φ(s))ds](n)
Remark 6.2. If B = R n B = R n B=R^(n)\mathbb{B}=\mathbb{R}^{n}B=Rn, then (1.2) is a finite system of equations, see [6], [10].
Remark 6.3. If B = l p B = l p B=l^(p)\mathbb{B}=l^{p}B=lp, then (1.2) is a infinite system of equations, see [1], [5], [19].

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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
Romanian Academy
Cluj-Napoca, Romania
E-mail address: dotrocol@ictp.acad.ro
  1. 2010 Mathematics Subject Classification. 47H10, 47N20.
    Key words and phrases. Integro-differential equation, two time modifications, step method, Picard operators.
2011

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