MSC 2000. 47H10.
Keywords. Delay differential equation, Mann iteration, Ishikawa iteration.

Consider the following delay differential equation

with $t_0,b,\tau \in \mathbb{R},\tau >0,f\in C([t_0,b]\times {\mathbb{R}}^2,\mathbb{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)$t_(0),b,tau inR,tau > 0,f in C([t_(0),b]xxR^(2),R)t_{0}, b, \tau \in \mathbb{R}, \tau>0, f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}^{2}, \mathbb{R}\right)$t_0,b,\tau \in \mathbb{R},\tau >0,f\in C([t_0,b]\times {\mathbb{R}}^2,\mathbb{R})$.
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 $X$$X$XX$X$ be a real Banach space and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a given operator, let $u_0,x_0\in X$$u_0,x_0\in X$u_(0),x_(0)in Xu_{0}, x_{0} \in X$u_0,x_0\in X$.

The Mann iteration is defined by

The Ishikawa iteration is defined by

where $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ and both sequences satisfy
$$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=\underset{n\to \mathrm{\infty }}{lim}{\beta }_n=0,\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$$$$\underset{n\to \mathrm{\infty }}{lim} {\alpha }_n=\underset{n\to \mathrm{\infty }}{lim} {\beta }_n=0,\sum _{n=0}^{\mathrm{\infty }} {\alpha }_n=\mathrm{\infty }$$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$$\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n=\underset{n\to \mathrm{\infty }}{lim}{\beta }_n=0,\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$$

We consider the delay differential equation

with initial condition

We suppose that the following conditions are fulfilled
$\left({\mathrm{H}}_1\right)t_0,b\in \mathbb{R},\tau >0;$$\left({\mathrm{H}}_1\right)t_0,b\in \mathbb{R},\tau >0;$(H_(1))t_(0),b inR,tau > 0;\left(\mathrm{H}_{1}\right) t_{0}, b \in \mathbb{R}, \tau>0 ;$\left({\mathrm{H}}_1\right)t_0,b\in \mathbb{R},\tau >0;$
$\left({\mathrm{H}}_2\right)f\in C([t_0,b]\times {\mathbb{R}}^2,\mathbb{R})$$\left({\mathrm{H}}_2\right)f\in C\left(\left[t_0,b\right]\times {\mathbb{R}}^2,\mathbb{R}\right)$(H_(2))f in C([t_(0),b]xxR^(2),R)\left(\mathrm{H}_{2}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}^{2}, \mathbb{R}\right)$\left({\mathrm{H}}_2\right)f\in C([t_0,b]\times {\mathbb{R}}^2,\mathbb{R})$;
$\left({\mathrm{H}}_3\right)\phi \in C([t_0-\tau ,b],\mathbb{R});$$\left({\mathrm{H}}_3\right)\phi \in C\left(\left[t_0-\tau ,b\right],\mathbb{R}\right);$(H_(3))varphi in C([t_(0)-tau,b],R);\left(\mathrm{H}_{3}\right) \varphi \in C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) ;$\left({\mathrm{H}}_3\right)\phi \in C([t_0-\tau ,b],\mathbb{R});$
$\left({\mathrm{H}}_4\right)$$\left({\mathrm{H}}_4\right)$(H_(4))\left(\mathrm{H}_{4}\right)$\left({\mathrm{H}}_4\right)$ there exist $L_f>0$$L_f>0$L_(f) > 0L_{f}>0$L_f>0$ such that

$\left({\mathrm{H}}_5\right)2L_f(b-t_0)<1$$\left({\mathrm{H}}_5\right)2L_f\left(b-t_0\right)<1$(H_(5))2L_(f)(b-t_(0)) < 1\left(\mathrm{H}_{5}\right) 2 L_{f}\left(b-t_{0}\right)<1$\left({\mathrm{H}}_5\right)2L_f(b-t_0)<1$.
By a solution of the problem (4)-(5) we mean the function $x\in C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$$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)$x in C([t_(0)-:}tau,b],R)nnC^(1)([t_(0),b],R)x \in C\left(\left[t_{0}-\right.\right. \tau, b], \mathbb{R}) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right)$x\in C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$.

The problem (4)-(5) is equivalent with the integral equation

The operator $T:C([t_0-\tau ,b],\mathbb{R})\to C([t_0-\tau ,b],\mathbb{R})$$T:C\left(\left[t_0-\tau ,b\right],\mathbb{R}\right)\to C\left(\left[t_0-\tau ,b\right],\mathbb{R}\right)$T:C([t_(0)-tau,b],R)rarr C([t_(0)-tau,b],R)T: C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \rightarrow C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right)$T:C([t_0-\tau ,b],\mathbb{R})\to C([t_0-\tau ,b],\mathbb{R})$, is defined by

and the Banach space $C[t_0,b]$$C\left[t_0,b\right]$C[t_(0),b]C\left[t_{0}, b\right]$C[t_0,b]$ is embedded with Tchebyshev norm

Applying contraction principle we have
Theorem 1. 1 We suppose conditions $\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$(H_(1))-(H_(5))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{5}\right)$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$ are satisfied. Then the problem (4)-(5) has a unique solution in $C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$$C\left(\left[t_0-\tau ,b\right],\mathbb{R}\right)\cap C^1\left(\left[t_0,b\right],\mathbb{R}\right)$C([t_(0)-tau,b],R)nnC^(1)([t_(0),b],R)C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right)$C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$. Moreover, if $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ is the unique solution of the problem (4)-(5), then

The following lemma is well-known. For sake of completeness, we shall give a proof here.

Lemma 2. Let $\left\{a_n\right\}$$\left\{a_n\right\}${a_(n)}\left\{a_{n}\right\}$\left\{a_n\right\}$ be a nonnegative sequence satisfying

where $\left\{{\alpha }_n\right\}\subset (0,1),\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$$\left\{{\alpha }_n\right\}\subset (0,1),\sum _{n=0}^{\mathrm{\infty }} {\alpha }_n=\mathrm{\infty }${alpha_(n)}sub(0,1),sum_(n=0)^(oo)alpha_(n)=oo\left\{\alpha_{n}\right\} \subset(0,1), \sum_{n=0}^{\infty} \alpha_{n}=\infty$\left\{{\alpha }_n\right\}\subset (0,1),\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$. Then $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$$\underset{n\to \mathrm{\infty }}{lim} a_n=0$lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0$\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$.
Proof. Use $1-x\le {\mathrm{e}}^{-x},\mathrm{\forall }x\in (0,1)$$1-x\le {\mathrm{e}}^{-x},\mathrm{\forall }x\in (0,1)$1-x <= e^(-x),AA x in(0,1)1-x \leq \mathrm{e}^{-x}, \forall x \in(0,1)$1-x\le {\mathrm{e}}^{-x},\mathrm{\forall }x\in (0,1)$ to obtain $\prod _{k=1}^n(1-{\alpha }_k)\le {\mathrm{e}}^{-\sum _{k=1}^n{\alpha }_k}$$\prod _{k=1}^n \left(1-{\alpha }_k\right)\le {\mathrm{e}}^{-\sum _{k=1}^n {\alpha }_k}$prod_(k=1)^(n)(1-alpha_(k)) <= e^(-sum_(k=1)^(n)alpha_(k))\prod_{k=1}^{n}\left(1-\alpha_{k}\right) \leq \mathrm{e}^{-\sum_{k=1}^{n} \alpha_{k}}$\prod _{k=1}^n(1-{\alpha }_k)\le {\mathrm{e}}^{-\sum _{k=1}^n{\alpha }_k}$. Actually, one has

Theorem 3. We suppose that conditions $\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$(H_(1))-(H_(5))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{5}\right)$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_5\right)$ are satisfied. Then the problem (4)-(5) has a unique solution in $C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$$C\left(\left[t_0-\tau ,b\right],\mathbb{R}\right)\cap C^1\left(\left[t_0,b\right],\mathbb{R}\right)$C([t_(0)-tau,b],R)nnC^(1)([t_(0),b],R)C\left(\left[t_{0}-\tau, b\right], \mathbb{R}\right) \cap C^{1}\left(\left[t_{0}, b\right], \mathbb{R}\right)$C([t_0-\tau ,b],\mathbb{R})\cap C^1([t_0,b],\mathbb{R})$.

Proof. Consider Mann iteration

for the operator

Denote by $x^{\ast }:=T{x}^{\ast }$$x^{\ast }:=T{x}^{\ast }$x^(**):=Tx^(**)x^{*}:=T x^{*}$x^{\ast }:=T{x}^{\ast }$ the fixed point of $T$$T$TT$T$.
For $t\in [t_0-\tau ,t_0]$$t\in \left[t_0-\tau ,t_0\right]$t in[t_(0)-tau,t_(0)]t \in\left[t_{0}-\tau, t_{0}\right]$t\in [t_0-\tau ,t_0]$ we get

Consider Lemma 2 to obtain

For $t\in [t_0,b]$$t\in \left[t_0,b\right]$t in[t_(0),b]t \in\left[t_{0}, b\right]$t\in [t_0,b]$ we have

Assumption ( $H_5$$H_5$H_(5)H_{5}$H_5$ ) leads us to

We take ${\alpha }_n:={\alpha }_n(1-2L_f(b-t_0)),a_n:=\Vert u_n-x^{\ast }\Vert$${\alpha }_n:={\alpha }_n\left(1-2L_f\left(b-t_0\right)\right),a_n:=‖u_n-x^{\ast }‖$alpha_(n):=alpha_(n)(1-2L_(f)(b-t_(0))),a_(n):=||u_(n)-x^(**)||\alpha_{n}:=\alpha_{n}\left(1-2 L_{f}\left(b-t_{0}\right)\right), a_{n}:=\left\|u_{n}-x^{*}\right\|${\alpha }_n:={\alpha }_n(1-2L_f(b-t_0)),a_n:=\Vert u_n-x^{\ast }\Vert$ and use Lemma 2 to obtain $\underset{n\to \mathrm{\infty }}{lim}\Vert u_n-x^{\ast }\Vert =0$$\underset{n\to \mathrm{\infty }}{lim} ‖u_n-x^{\ast }‖=0$lim_(n rarr oo)||u_(n)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|u_{n}-x^{*}\right\|=0$\underset{n\to \mathrm{\infty }}{lim}\Vert u_n-x^{\ast }\Vert =0$.

Theorem 4. [7] Let $X$$X$XX$X$ be a normed space, $D$$D$DD$D$ a nonempty, convex, closed subset of $X$$X$XX$X$ and $T:D\to D$$T:D\to D$T:D rarr DT: D \rightarrow D$T:D\to D$ a contraction. If $u_0,x_0\in D$$u_0,x_0\in D$u_(0),x_(0)in Du_{0}, x_{0} \in D$u_0,x_0\in D$, then the following are equivalent:
(i) the Mann iteration (2) converges to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$;
(ii) the Ishikawa iteration (3) converges to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$.

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$tau=0\tau=0$\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:

with initial condition

We suppose that the following conditions are fulfilled
$\left({\mathrm{H}}_1^{\prime }\right)t_0,{\phi }_0,b\in \mathbb{R}$$\left({\mathrm{H}}_1^{\prime }\right)t_0,{\phi }_0,b\in \mathbb{R}$(H_(1)^('))t_(0),varphi_(0),b inR\left(\mathrm{H}_{1}^{\prime}\right) t_{0}, \varphi_{0}, b \in \mathbb{R}$\left({\mathrm{H}}_1^{\prime }\right)t_0,{\phi }_0,b\in \mathbb{R}$;
$\left({\mathrm{H}}_2^{\prime }\right)f\in C([t_0,b]\times \mathbb{R},\mathbb{R})$$\left({\mathrm{H}}_2^{\prime }\right)f\in C\left(\left[t_0,b\right]\times \mathbb{R},\mathbb{R}\right)$(H_(2)^('))f in C([t_(0),b]xxR,R)\left(\mathrm{H}_{2}^{\prime}\right) f \in C\left(\left[t_{0}, b\right] \times \mathbb{R}, \mathbb{R}\right)$\left({\mathrm{H}}_2^{\prime }\right)f\in C([t_0,b]\times \mathbb{R},\mathbb{R})$;
$\left({\mathrm{H}}_3^{\prime }\right)$$\left({\mathrm{H}}_3^{\prime }\right)$(H_(3)^('))\left(\mathrm{H}_{3}^{\prime}\right)$\left({\mathrm{H}}_3^{\prime }\right)$ there exist $L_f>0$$L_f>0$L_(f) > 0L_{f}>0$L_f>0$ such that

$\left({\mathrm{H}}_4^{\prime }\right)L_f(b-t_0)<1$$\left({\mathrm{H}}_4^{\prime }\right)L_f\left(b-t_0\right)<1$(H_(4)^('))L_(f)(b-t_(0)) < 1\left(\mathrm{H}_{4}^{\prime}\right) L_{f}\left(b-t_{0}\right)<1$\left({\mathrm{H}}_4^{\prime }\right)L_f(b-t_0)<1$.
Note that we have supplied here a new proof for the following result using Mann-Ishikawa iteration.

Theorem 6. We suppose conditions $\left({\mathrm{H}}_1^{\prime }\right)-\left({\mathrm{H}}_4^{\prime }\right)$$\left({\mathrm{H}}_1^{\prime }\right)-\left({\mathrm{H}}_4^{\prime }\right)$(H_(1)^('))-(H_(4)^('))\left(\mathrm{H}_{1}^{\prime}\right)-\left(\mathrm{H}_{4}^{\prime}\right)$\left({\mathrm{H}}_1^{\prime }\right)-\left({\mathrm{H}}_4^{\prime }\right)$ are satisfied. Then the problem (7)-(8) has a unique solution in $C([t_0,b],\mathbb{R})$$C\left(\left[t_0,b\right],\mathbb{R}\right)$C([t_(0),b],R)C\left(\left[t_{0}, b\right], \mathbb{R}\right)$C([t_0,b],\mathbb{R})$. Moreover, if $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ is the unique solution of the problem (7)-(8), then

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[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.