Abstract
New proofs of existence and uniqueness results for the solution of the Cauchy problem with delay are obtained by use of Mann-Ishikawa iteration.
Authors
Ş.M. Şoltuz
Universidad de los Andes, Colombia and Tiberiu Popoviciu Institute of Numerical Analysis, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
delay differential equationș Mann iteration; Ishikawa iteration
Paper coordinates
Ş.M. Şoltuz, D. Otrocol, Classical results via Mann-Ishikawa iteration, Rev. Anal. Numer. Theor. Approx., 36 (2007) no. 2, 193-197, https://doi.org/10.33993/jnaat362-868
About this paper
Journal
Rev. Anal. Numer. Theor. Approx.
Publisher Name
Romanian Academy
Print ISSN
1972-1974
Online ISSN
google scholar link
[1] Coman, Gh., Pavel, G., Rus, I.andRus, I. A.,Introduction in the theory of operatorialequation, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).
[2] Hartman, P.,Ordinary differential equations, John Wiley & Sons, Inc., New York,London, Sydney, 1964.
[3] Ishikawa, S.,Fixed points by a new iteration method, Proc. Amer. Math. Soc.,44,pp. 147–150, 1974.
[4] Mann, W. R.,Mean value in iteration, Proc. Amer. Math. Soc.,4, pp. 506–510, 1953.
[2] Hartman, P.,Ordinary differential equations, John Wiley & Sons, Inc., New York,London, Sydney, 1964.
[3] Ishikawa, S.,Fixed points by a new iteration method, Proc. Amer. Math. Soc.,44,pp. 147–150, 1974.
[4] Mann, W. R.,Mean value in iteration, Proc. Amer. Math. Soc.,4, pp. 506–510, 1953.
[5] Otrocol, D.,Data dependence for the solution of a Lotka-Volterra system with twodelays,Mathematica, Tome48(71), 1, pp. 61–68, 2006.
[6] Rus, I. A.,Principles and applications of the fixed point theory,Ed. Dacia, Cluj Napoca,1979 (in Romanian).
[7] Soltuz, S. M.,The equivalence of Picard, Mann and Ishikawa iteration dealing withquasi-contractive operators, Math. Comm.10, pp. 81–88, 2005.
[8] Soltuz, S. M.,An equivalence between the convergence of Ishikawa, Mann and Picarditerations, Math. Comm.8, pp. 15–22, 2003.Received by the editors: November 20, 2006.
[6] Rus, I. A.,Principles and applications of the fixed point theory,Ed. Dacia, Cluj Napoca,1979 (in Romanian).
[7] Soltuz, S. M.,The equivalence of Picard, Mann and Ishikawa iteration dealing withquasi-contractive operators, Math. Comm.10, pp. 81–88, 2005.
[8] Soltuz, S. M.,An equivalence between the convergence of Ishikawa, Mann and Picarditerations, Math. Comm.8, pp. 15–22, 2003.Received by the editors: November 20, 2006.
Paper (preprint) in HTML form
CLASSICAL RESULTS VIA MANN-ISHIKAWA ITERATION ^(‡){ }^{\ddagger}
Abstract
New proofs of existence and uniqueness results for the solution of the Cauchy problem with delay are obtained by use of Mann-Ishikawa iteration.
MSC 2000. 47H10.
Keywords. Delay differential equation, Mann iteration, Ishikawa iteration.
Keywords. Delay differential equation, Mann iteration, Ishikawa iteration.
1. INTRODUCTION
Consider the following delay differential equation
{:(1)x^(')(t)=f(t","x(t)","x(t-tau))","t in[t_(0),b]:}\begin{equation*}
x^{\prime}(t)=f(t, x(t), x(t-\tau)), t \in\left[t_{0}, b\right] \tag{1}
\end{equation*}
with t_(0),b,tau inR,tau > 0,f in C([t_(0),b]xxR^(2),R)t_{0}, b, \tau \in \mathbb{R}, \tau>0, f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}^{2}, \mathbb{R}\right).
The existence of an approximative solution for equation (1) is given by theorem 1 from [1] (see also [2], [6], [5]). The proof of this theorem is based on the contraction principle. We shall prove it here by applying Mann iteration.
The existence of an approximative solution for equation (1) is given by theorem 1 from [1] (see also [2], [6], [5]). The proof of this theorem is based on the contraction principle. We shall prove it here by applying Mann iteration.
In the last decades, numerous papers were published on the iterative approximation of fixed points of contractive type operators in metric spaces, see for example [7], [8]. The Mann iteration [4] and the Ishikawa iteration [3] are certainly the most studied of these fixed point iteration procedures.
Let XX be a real Banach space and T:X rarr XT: X \rightarrow X a given operator, let u_(0),x_(0)in Xu_{0}, x_{0} \in X.
The Mann iteration is defined by
{:(2)u_(n+1)=(1-alpha_(n))u_(n)+alpha_(n)Tu_(n).:}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n}+\alpha_{n} T u_{n} . \tag{2}
\end{equation*}
The Ishikawa iteration is defined by
{:(3){[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)],[y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)]:}:}\left\{\begin{array}{l}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T y_{n} \tag{3}\\
y_{n}=\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}
\end{array}\right.
where {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) and both sequences satisfy
lim_(n rarr oo)alpha_(n)=lim_(n rarr oo)beta_(n)=0,sum_(n=0)^(oo)alpha_(n)=oo\lim _{n \rightarrow \infty} \alpha_{n}=\lim _{n \rightarrow \infty} \beta_{n}=0, \sum_{n=0}^{\infty} \alpha_{n}=\infty
lim_(n rarr oo)alpha_(n)=lim_(n rarr oo)beta_(n)=0,sum_(n=0)^(oo)alpha_(n)=oo\lim _{n \rightarrow \infty} \alpha_{n}=\lim _{n \rightarrow \infty} \beta_{n}=0, \sum_{n=0}^{\infty} \alpha_{n}=\infty
2. FIXED POINT THEOREMS
We consider the delay differential equation
{:(4)x^(')(t)=f(t","x(t)","x(t-tau))","t in[t_(0),b]:}\begin{equation*}
x^{\prime}(t)=f(t, x(t), x(t-\tau)), t \in\left[t_{0}, b\right] \tag{4}
\end{equation*}
with initial condition
{:(5)x(t)=varphi(t)","t in[t_(0)-tau,t_(0)].:}\begin{equation*}
x(t)=\varphi(t), t \in\left[t_{0}-\tau, t_{0}\right] . \tag{5}
\end{equation*}
We suppose that the following conditions are fulfilled
(H_(1))t_(0),b inR,tau > 0;\left(\mathrm{H}_{1}\right) t_{0}, b \in \mathbb{R}, \tau>0 ;
(H_(2))f in C([t_(0),b]xxR^(2),R)\left(\mathrm{H}_{2}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}^{2}, \mathbb{R}\right);
(H_(3))varphi in C([t_(0)-tau,b],R);\left(\mathrm{H}_{3}\right) \varphi \in C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) ;
(H_(4))\left(\mathrm{H}_{4}\right) there exist L_(f) > 0L_{f}>0 such that
(H_(1))t_(0),b inR,tau > 0;\left(\mathrm{H}_{1}\right) t_{0}, b \in \mathbb{R}, \tau>0 ;
(H_(2))f in C([t_(0),b]xxR^(2),R)\left(\mathrm{H}_{2}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}^{2}, \mathbb{R}\right);
(H_(3))varphi in C([t_(0)-tau,b],R);\left(\mathrm{H}_{3}\right) \varphi \in C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) ;
(H_(4))\left(\mathrm{H}_{4}\right) there exist L_(f) > 0L_{f}>0 such that
{:[|f(t,u_(1),u_(2))-f(t,v_(1),v_(2))| <= L_(f)(|u_(1)-v_(1)|+|u_(2)-v_(2)|)],[AAu_(i)","v_(i)inR","i=1","2","t in[t_(0),b]]:}\begin{aligned}
& \left|f\left(t, u_{1}, u_{2}\right)-f\left(t, v_{1}, v_{2}\right)\right| \leq L_{f}\left(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|\right) \\
& \forall u_{i}, v_{i} \in \mathbb{R}, i=1,2, t \in\left[t_{0}, b\right]
\end{aligned}
(H_(5))2L_(f)(b-t_(0)) < 1\left(\mathrm{H}_{5}\right) 2 L_{f}\left(b-t_{0}\right)<1.
By a solution of the problem (4)-(5) we mean the function x in C([t_(0)-:}tau,b],R)nnC^(1)([t_(0),b],R)x \in C\left(\left[t_{0}-\right.\right. \tau, b], \mathbb{R}) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right).
By a solution of the problem (4)-(5) we mean the function x in C([t_(0)-:}tau,b],R)nnC^(1)([t_(0),b],R)x \in C\left(\left[t_{0}-\right.\right. \tau, b], \mathbb{R}) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right).
The problem (4)-(5) is equivalent with the integral equation
{:(6)x(t)={[varphi(t)",",t in[t_(0)-tau,t_(0)]],[varphi(t_(0))+int_(t_(0))^(t)f(s","x(s)","x(s-tau))ds",",t in[t_(0),b]]:}:}x(t)= \begin{cases}\varphi(t), & t \in\left[t_{0}-\tau, t_{0}\right] \tag{6}\\ \varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f(s, x(s), x(s-\tau)) \mathrm{d} s, & t \in\left[t_{0}, b\right]\end{cases}
The operator T:C([t_(0)-tau,b],R)rarr C([t_(0)-tau,b],R)T: C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \rightarrow C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right), is defined by
T(x)(t)={[varphi(t)",",t in[t_(0)-tau,t_(0)]],[varphi(t_(0))+int_(t_(0))^(t)f(s","x(s)","x(s-tau))ds",",t in[t_(0),b]]:}T(x)(t)= \begin{cases}\varphi(t), & t \in\left[t_{0}-\tau, t_{0}\right] \\ \varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f(s, x(s), x(s-\tau)) \mathrm{d} s, & t \in\left[t_{0}, b\right]\end{cases}
and the Banach space C[t_(0),b]C\left[t_{0}, b\right] is embedded with Tchebyshev norm
d(y,z)=max_(t_(0) <= t <= b)|y(t)-z(t)|d(y, z)=\max _{t_{0} \leq t \leq b}|y(t)-z(t)|
Applying contraction principle we have
Theorem 1. 1 We suppose conditions (H_(1))-(H_(5))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{5}\right) are satisfied. Then the problem (4)-(5) has a unique solution in C([t_(0)-tau,b],R)nnC^(1)([t_(0),b],R)C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right). Moreover, if x^(**)x^{*} is the unique solution of the problem (4)-(5), then
Theorem 1. 1 We suppose conditions (H_(1))-(H_(5))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{5}\right) are satisfied. Then the problem (4)-(5) has a unique solution in C([t_(0)-tau,b],R)nnC^(1)([t_(0),b],R)C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right). Moreover, if x^(**)x^{*} is the unique solution of the problem (4)-(5), then
x^(**)=lim_(n rarr oo)T^(n)(x)" for any "x in C([t_(0)-tau,b],R)x^{*}=\lim _{n \rightarrow \infty} T^{n}(x) \text { for any } x \in C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right)
The following lemma is well-known. For sake of completeness, we shall give a proof here.
Lemma 2. Let {a_(n)}\left\{a_{n}\right\} be a nonnegative sequence satisfying
a_(n+1) <= (1-alpha_(n))a_(n)a_{n+1} \leq\left(1-\alpha_{n}\right) a_{n}
where {alpha_(n)}sub(0,1),sum_(n=0)^(oo)alpha_(n)=oo\left\{\alpha_{n}\right\} \subset(0,1), \sum_{n=0}^{\infty} \alpha_{n}=\infty. Then lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
Proof. Use 1-x <= e^(-x),AA x in(0,1)1-x \leq \mathrm{e}^{-x}, \forall x \in(0,1) to obtain prod_(k=1)^(n)(1-alpha_(k)) <= e^(-sum_(k=1)^(n)alpha_(k))\prod_{k=1}^{n}\left(1-\alpha_{k}\right) \leq \mathrm{e}^{-\sum_{k=1}^{n} \alpha_{k}}. Actually, one has
Proof. Use 1-x <= e^(-x),AA x in(0,1)1-x \leq \mathrm{e}^{-x}, \forall x \in(0,1) to obtain prod_(k=1)^(n)(1-alpha_(k)) <= e^(-sum_(k=1)^(n)alpha_(k))\prod_{k=1}^{n}\left(1-\alpha_{k}\right) \leq \mathrm{e}^{-\sum_{k=1}^{n} \alpha_{k}}. Actually, one has
a_(n+1) <= prod_(k=1)^(n)(1-alpha_(k))a_(1) <= a_(1)e^(-sum_(k=1)^(n)alpha_(k))rarr0a_{n+1} \leq \prod_{k=1}^{n}\left(1-\alpha_{k}\right) a_{1} \leq a_{1} \mathrm{e}^{-\sum_{k=1}^{n} \alpha_{k}} \rightarrow 0
3. MAIN RESULT
Theorem 3. We suppose that conditions (H_(1))-(H_(5))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{5}\right) are satisfied. Then the problem (4)-(5) has a unique solution in C([t_(0)-tau,b],R)nnC^(1)([t_(0),b],R)C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right).
Proof. Consider Mann iteration
u_(n+1)=(1-alpha_(n))u_(n)+alpha_(n)Tu_(n)u_{n+1}=\left(1-\alpha_{n}\right) u_{n}+\alpha_{n} T u_{n}
for the operator
Tu_(n)={[varphi(t)",",t in[t_(0)-tau,t_(0)]],[varphi(t_(0))+int_(t_(0))^(t)f(s,u_(n)(s),u_(n)(s-tau))ds",",t in[t_(0),b]]:}T u_{n}= \begin{cases}\varphi(t), & t \in\left[t_{0}-\tau, t_{0}\right] \\ \varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f\left(s, u_{n}(s), u_{n}(s-\tau)\right) \mathrm{d} s, & t \in\left[t_{0}, b\right]\end{cases}
Denote by x^(**):=Tx^(**)x^{*}:=T x^{*} the fixed point of TT.
For t in[t_(0)-tau,t_(0)]t \in\left[t_{0}-\tau, t_{0}\right] we get
For t in[t_(0)-tau,t_(0)]t \in\left[t_{0}-\tau, t_{0}\right] we get
||u_(n+1)-x^(**)|| <= (1-alpha_(n))||u_(n)-x^(**)||\left\|u_{n+1}-x^{*}\right\| \leq\left(1-\alpha_{n}\right)\left\|u_{n}-x^{*}\right\|
Consider Lemma 2 to obtain
lim_(n rarr oo)||u_(n)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|u_{n}-x^{*}\right\|=0
For t in[t_(0),b]t \in\left[t_{0}, b\right] we have
{:[||u_(n+1)-x^(**)||=||(1-alpha_(n))u_(n)+alpha_(n)(varphi(t_(0))+int_(t_(0))^(t)f(s,u_(n)(s),u_(n)(s-tau))ds)-],[-(1-alpha_(n))x^(**)+alpha_(n)(varphi(t_(0))+int_(t_(0))^(t)f(s,x^(**)(s),x^(**)(s-tau))ds)||],[ <= (1-alpha_(n))||u_(n)-x^(**)||+alpha_(n)L_(f)(b-t_(0))(|u_(n)(s)-x^(**)(s)|+:}],[{:+|u_(n)(s-tau)-x^(**)(s-tau)|)],[ <= (1-alpha_(n)+2alpha_(n)L_(f)(b-t_(0)))||u_(n)-x^(**)||],[ <= (1-alpha_(n)(1-2L_(f)(b-t_(0)))||u_(n)-x^(**)||:}]:}\begin{aligned}
\left\|u_{n+1}-x^{*}\right\|= & \|\left(1-\alpha_{n}\right) u_{n}+\alpha_{n}\left(\varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f\left(s, u_{n}(s), u_{n}(s-\tau)\right) \mathrm{d} s\right)- \\
& -\left(1-\alpha_{n}\right) x^{*}+\alpha_{n}\left(\varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f\left(s, x^{*}(s), x^{*}(s-\tau)\right) \mathrm{d} s\right) \| \\
\leq & \left(1-\alpha_{n}\right)\left\|u_{n}-x^{*}\right\|+\alpha_{n} L_{f}\left(b-t_{0}\right)\left(\left|u_{n}(s)-x^{*}(s)\right|+\right. \\
& \left.+\left|u_{n}(s-\tau)-x^{*}(s-\tau)\right|\right) \\
\leq & \left(1-\alpha_{n}+2 \alpha_{n} L_{f}\left(b-t_{0}\right)\right)\left\|u_{n}-x^{*}\right\| \\
\leq & \left(1-\alpha_{n}\left(1-2 L_{f}\left(b-t_{0}\right)\right)\left\|u_{n}-x^{*}\right\|\right.
\end{aligned}
Assumption ( H_(5)H_{5} ) leads us to
1-alpha_(n)(1-2L_(f)(b-t_(0)) < 1:}1-\alpha_{n}\left(1-2 L_{f}\left(b-t_{0}\right)<1\right.
We take alpha_(n):=alpha_(n)(1-2L_(f)(b-t_(0))),a_(n):=||u_(n)-x^(**)||\alpha_{n}:=\alpha_{n}\left(1-2 L_{f}\left(b-t_{0}\right)\right), a_{n}:=\left\|u_{n}-x^{*}\right\| and use Lemma 2 to obtain lim_(n rarr oo)||u_(n)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|u_{n}-x^{*}\right\|=0.
Theorem 4. [7] Let XX be a normed space, DD a nonempty, convex, closed subset of XX and T:D rarr DT: D \rightarrow D a contraction. If u_(0),x_(0)in Du_{0}, x_{0} \in D, then the following are equivalent:
(i) the Mann iteration (2) converges to x^(**)x^{*};
(ii) the Ishikawa iteration (3) converges to x^(**)x^{*}.
(i) the Mann iteration (2) converges to x^(**)x^{*};
(ii) the Ishikawa iteration (3) converges to x^(**)x^{*}.
Remark 5. Because Mann iteration and Ishikawa iteration are equivalent, it is possible to consider Ishikawa iteration in order to prove Theorem 3.
Set tau=0\tau=0, in (4), to obtain the classical existence and uniqueness result, i.e. Theorem 6, for the Cauchy problem. This problem, see [1], 6], is proved by use of contraction principle.
Consider the following equation:
{:(7)x^(')(t)=f(t","x(t))","t in[t_(0),b]",":}\begin{equation*}
x^{\prime}(t)=f(t, x(t)), t \in\left[t_{0}, b\right], \tag{7}
\end{equation*}
with initial condition
{:(8)x(t_(0))=varphi_(0).:}\begin{equation*}
x\left(t_{0}\right)=\varphi_{0} . \tag{8}
\end{equation*}
We suppose that the following conditions are fulfilled
(H_(1)^('))t_(0),varphi_(0),b inR\left(\mathrm{H}_{1}^{\prime}\right) t_{0}, \varphi_{0}, b \in \mathbb{R};
(H_(2)^('))f in C([t_(0),b]xxR,R)\left(\mathrm{H}_{2}^{\prime}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}, \mathbb{R}\right);
(H_(3)^('))\left(\mathrm{H}_{3}^{\prime}\right) there exist L_(f) > 0L_{f}>0 such that
(H_(1)^('))t_(0),varphi_(0),b inR\left(\mathrm{H}_{1}^{\prime}\right) t_{0}, \varphi_{0}, b \in \mathbb{R};
(H_(2)^('))f in C([t_(0),b]xxR,R)\left(\mathrm{H}_{2}^{\prime}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}, \mathbb{R}\right);
(H_(3)^('))\left(\mathrm{H}_{3}^{\prime}\right) there exist L_(f) > 0L_{f}>0 such that
|f(t,u_(1))-f(t,v_(1))| <= L_(f)(|u_(1)-v_(1)|),AAu_(1),v_(1)inR,t in[t_(0),b];\left|f\left(t, u_{1}\right)-f\left(t, v_{1}\right)\right| \leq L_{f}\left(\left|u_{1}-v_{1}\right|\right), \forall u_{1}, v_{1} \in \mathbb{R}, t \in\left[t_{0}, b\right] ;
(H_(4)^('))L_(f)(b-t_(0)) < 1\left(\mathrm{H}_{4}^{\prime}\right) L_{f}\left(b-t_{0}\right)<1.
Note that we have supplied here a new proof for the following result using Mann-Ishikawa iteration.
Note that we have supplied here a new proof for the following result using Mann-Ishikawa iteration.
Theorem 6. We suppose conditions (H_(1)^('))-(H_(4)^('))\left(\mathrm{H}_{1}^{\prime}\right)-\left(\mathrm{H}_{4}^{\prime}\right) are satisfied. Then the problem (7)-(8) has a unique solution in C([t_(0),b],R)C\left(\left[t_{0}, b\right], \mathbb{R}\right). Moreover, if x^(**)x^{*} is the unique solution of the problem (7)-(8), then
x^(**)=lim_(n rarr oo)T^(n)(x)" for any "x in C([t_(0),b],R).x^{*}=\lim _{n \rightarrow \infty} T^{n}(x) \text { for any } x \in C\left(\left[t_{0}, b\right], \mathbb{R}\right) .
REFERENCES
[1] Coman, Gh., Pavel, G., Rus, I. and Rus, I. A., Introduction in the theory of operatorial equation, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).
[2] Hartman, P., Ordinary differential equations, John Wiley & Sons, Inc., New York, London, Sydney, 1964.
[3] Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, pp. 147-150, 1974.
[4] Mann, W. R., Mean value in iteration, Proc. Amer. Math. Soc., 4, pp. 506-510, 1953.
[5] Otrocol, D., Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, Tome 48 (71), 1, pp. 61-68, 2006.
[6] Rus, I. A., Principles and applications of the fixed point theory, Ed. Dacia, Cluj Napoca, 1979 (in Romanian).
[7] Şoltuz, Ş. M., The equivalence of Picard, Mann and Ishikawa iteration dealing with quasi-contractive operators, Math. Comm. 10, pp. 81-88, 2005.
[8] Şoltuz, Ş. M., An equivalence between the convergence of Ishikawa, Mann and Picard iterations, Math. Comm. 8, pp. 15-22, 2003.
[2] Hartman, P., Ordinary differential equations, John Wiley & Sons, Inc., New York, London, Sydney, 1964.
[3] Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, pp. 147-150, 1974.
[4] Mann, W. R., Mean value in iteration, Proc. Amer. Math. Soc., 4, pp. 506-510, 1953.
[5] Otrocol, D., Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, Tome 48 (71), 1, pp. 61-68, 2006.
[6] Rus, I. A., Principles and applications of the fixed point theory, Ed. Dacia, Cluj Napoca, 1979 (in Romanian).
[7] Şoltuz, Ş. M., The equivalence of Picard, Mann and Ishikawa iteration dealing with quasi-contractive operators, Math. Comm. 10, pp. 81-88, 2005.
[8] Şoltuz, Ş. M., An equivalence between the convergence of Ishikawa, Mann and Picard iterations, Math. Comm. 8, pp. 15-22, 2003.
Received by the editors: November 20, 2006.
- *"Tiberiu Popoviciu" Institute of Numerical Analysis, P.O. Box. 68-1, Cluj-Napoca, Romania, e-mail: {smsoltuz,dotrocol}@ictp.acad.ro.
^(†){ }^{\dagger} Departamento de Matematicas, Universidad de los Andes, Carrera 1, No. 18A-10, Bogota, Columbia, e-mail: smsoltuz@gmail.com.
^(‡){ }^{\ddagger} The work of the second author was supported by MEdC under Grant 2-CEx06-11-96/ 19.09.2006.