MSC 2000. 65J22, 47N40, 47H10.
Keywords. contractive like operators.

Let $X$$X$XX$X$ be a real Banach space, $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ be an operator. The following result of Barnsley, see 1, becomes "a classic".

Theorem 1. (Collage Theorem) Let $x\in X$$x\in X$x in Xx \in X$x\in X$ be given and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a contraction with contraction factor $L\in (0,1)$$L\in (0,1)$L in(0,1)L \in(0,1)$L\in (0,1)$, (i.e. $\Vert Tx-Ty\Vert \le L\Vert x-y\Vert$$\Vert Tx-Ty\Vert \le L\Vert x-y\Vert$||Tx-Ty|| <= L||x-y||\|T x-T y\| \leq L\|x-y\|$\Vert Tx-Ty\Vert \le L\Vert x-y\Vert$, $\mathrm{\forall }x,y\in X$$\mathrm{\forall }x,y\in X$AA x,y in X\forall x, y \in X$\mathrm{\forall }x,y\in X$ ), and fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$. Then

In fractal-based applications, $T$$T$TT$T$ produces a union of shrunken copies of $x$$x$xx$x$, i.e. "a collage" of itself. The term $\Vert x-Tx\Vert$$\Vert x-Tx\Vert$||x-Tx||\|x-T x\|$\Vert x-Tx\Vert$ is referred as "collage distance". Such "collages" are sufficient to be taken in finite number, in order to have a good approximation of an denoised image; fact which is very useful in Image Compression (both Analysis and Synthesis of an image). Kunze et all., see [5], 6], 7), were able to apply the Collage Theorem to inverse problems in ODE, that is to reconstruct the field of an ODE, from a given "target" (trajectory). Our aim is to generalize the above Collage result for a larger operatorial class than contractions. Recently, similar results were introduced for other operatorial classes, see [11] and [12.

The following operatorial class, satisfying (1), was introduced in [4]. Since they failed to named it, we shall do so here. It is just a convention.

Definition 2. The operator $T$$T$TT$T$ is contractive-like, or $CL$$CL$CLC L$CL$ for short, if there exist a constant $q\in (0,1)$$q\in (0,1)$q in(0,1)q \in(0,1)$q\in (0,1)$ and a monotone increasing and continuous function

$\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$$\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$psi:[0,oo)rarr[0,oo)\psi:[0, \infty) \rightarrow[0, \infty)$\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\psi (0)=0$$\psi (0)=0$psi(0)=0\psi(0)=0$\psi (0)=0$ such that for each $x,y\in X$$x,y\in X$x,y in Xx, y \in X$x,y\in X$,
$$\begin{array}{}\text{(1)}& \Vert Tx-Ty\Vert \le q\Vert x-y\Vert +\psi (\Vert x-Tx\Vert ).\end{array}$$$$\begin{array}{}\text{(1)}& \Vert Tx-Ty\Vert \le q\Vert x-y\Vert +\psi (\Vert x-Tx\Vert ).\end{array}$${:(1)||Tx-Ty|| <= q||x-y||+psi(||x-Tx||).:}\begin{equation*} \|T x-T y\| \leq q\|x-y\|+\psi(\|x-T x\|) . \tag{1} \end{equation*}$$\begin{array}{}\text{(1)}& \Vert Tx-Ty\Vert \le q\Vert x-y\Vert +\psi (\Vert x-Tx\Vert ).\end{array}$$

Let $F(T)$$F(T)$F(T)F(T)$F(T)$ denote the fixed point set with respect to $X$$X$XX$X$ for the map $T$$T$TT$T$. Suppose that $x^{\ast }\in F(T)$$x^{\ast }\in F(T)$x^(**)in F(T)x^{*} \in F(T)$x^{\ast }\in F(T)$. The following operators are called Zamfirescu operators, see [13] or [10].

Definition 3. [13] The operator $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ satisfies condition $Z$$Z$ZZ$Z$ (or is a quasi-contraction) if and only if there exist the real numbers $a,b,c$$a,b,c$a,b,ca, b, c$a,b,c$ satisfying $0<a<1,0<b<1/2,0<c<1/2$$0<a<1,0<b<1/2,0<c<1/2$0 < a < 1,0 < b < 1//2,0 < c < 1//20<a<1,0<b<1 / 2,0<c<1 / 2$0<a<1,0<b<1/2,0<c<1/2$ such that for each pair $x,y$$x,y$x,yx, y$x,y$ in $X$$X$XX$X$, at least one condition is true
$\left(z_1\right)\Vert Tx-Ty\Vert \le a\Vert x-y\Vert$$\left(z_1\right)\Vert Tx-Ty\Vert \le a\Vert x-y\Vert$(z_(1))||Tx-Ty|| <= a||x-y||\left(z_{1}\right)\|T x-T y\| \leq a\|x-y\|$\left(z_1\right)\Vert Tx-Ty\Vert \le a\Vert x-y\Vert$,
$\left(z_2\right)\Vert Tx-Ty\Vert \le b(\Vert x-Tx\Vert +\Vert y-Ty\Vert )$$\left(z_2\right)\Vert Tx-Ty\Vert \le b(\Vert x-Tx\Vert +\Vert y-Ty\Vert )$(z_(2))||Tx-Ty|| <= b(||x-Tx||+||y-Ty||)\left(z_{2}\right)\|T x-T y\| \leq b(\|x-T x\|+\|y-T y\|)$\left(z_2\right)\Vert Tx-Ty\Vert \le b(\Vert x-Tx\Vert +\Vert y-Ty\Vert )$,
$\left(z_3\right)\Vert Tx-Ty\Vert \le c(\Vert x-Ty\Vert +\Vert y-Tx\Vert )$$\left(z_3\right)\Vert Tx-Ty\Vert \le c(\Vert x-Ty\Vert +\Vert y-Tx\Vert )$(z_(3))||Tx-Ty|| <= c(||x-Ty||+||y-Tx||)\left(z_{3}\right)\|T x-T y\| \leq c(\|x-T y\|+\|y-T x\|)$\left(z_3\right)\Vert Tx-Ty\Vert \le c(\Vert x-Ty\Vert +\Vert y-Tx\Vert )$.
It has been shown in 2 (see also [3]) that conditions $\left(z_1\right)-\left(z_3\right)$$\left(z_1\right)-\left(z_3\right)$(z_(1))-(z_(3))\left(z_{1}\right)-\left(z_{3}\right)$\left(z_1\right)-\left(z_3\right)$ lead to

where

Set

to obtain (1). Thus, relation (1) generalizes (2). In 4 was introduced this more general class of operators satisfying (1).

Remark 4. Set $\psi (t)=2t$$\psi (t)=2t$psi(t)=2t\psi(t)=2 t$\psi (t)=2t$ to see that a CL operators need not have a fixed point, as pointed in [8] or [9]. Therefore, we shall suppose implicitly throughout this paper that all CL operators involved have a fixed point.

A typical inverse problem is the following:
Problem 5. For given $\epsilon >0$$\epsilon >0$epsi > 0\varepsilon>0$\epsilon >0$ and a "target" $\overline{x}$$\overline{x}$bar(x)\bar{x}$\overline{x}$, find $T_{\epsilon }\in CL$$T_{\epsilon }\in CL$T_(epsi)in CLT_{\varepsilon} \in C L$T_{\epsilon }\in CL$ such that $\Vert \overline{x}-x_{T_{\epsilon }}^{\ast }\Vert <\epsilon$$‖\overline{x}-x_{T_{\epsilon }}^{\ast }‖<\epsilon$||( bar(x))-x_(T_(epsi))^(**)|| < epsi\left\|\bar{x}-x_{T_{\varepsilon}}^{*}\right\|<\varepsilon$\Vert \overline{x}-x_{T_{\epsilon }}^{\ast }\Vert <\epsilon$, where $x_{T_{\epsilon }}^{\ast }=T_{\epsilon }\left(x_{T_{\epsilon }}^{\ast }\right)$$x_{T_{\epsilon }}^{\ast }=T_{\epsilon }\left(x_{T_{\epsilon }}^{\ast }\right)$x_(T_(epsi))^(**)=T_(epsi)(x_(T_(epsi))^(**))x_{T_{\varepsilon}}^{*}=T_{\varepsilon}\left(x_{T_{\varepsilon}}^{*}\right)$x_{T_{\epsilon }}^{\ast }=T_{\epsilon }\left(x_{T_{\epsilon }}^{\ast }\right)$ is the unique fixed point of the $CL$$CL$CLC L$CL$ mapping $T_{\epsilon }$$T_{\epsilon }$T_(epsi)T_{\varepsilon}$T_{\epsilon }$.

Randomly selecting various maps in $CL$$CL$CLC L$CL$, finding their fixed points and computing the distance from our target is an extremely tedious procedure. Consider now the following problem which we shall fit in our framework and which is very useful for practitioners, see 5 .

Problem 6. Let $\overline{x}\in X$$\overline{x}\in X$bar(x)in X\bar{x} \in X$\overline{x}\in X$ be a target and let $\delta >0$$\delta >0$delta > 0\delta>0$\delta >0$ be given. Find $T_{\delta }\in CL$$T_{\delta }\in CL$T_(delta)in CLT_{\delta} \in C L$T_{\delta }\in CL$, such that $\Vert \overline{x}-T_{\delta }\overline{x}\Vert <\delta$$‖\overline{x}-T_{\delta }\overline{x}‖<\delta$||( bar(x))-T_(delta)( bar(x))|| < delta\left\|\bar{x}-T_{\delta} \bar{x}\right\|<\delta$\Vert \overline{x}-T_{\delta }\overline{x}\Vert <\delta$.

In other words, instead of searching for $CL$$CL$CLC L$CL$ maps whose fixed points lie close to target $\overline{x}$$\overline{x}$bar(x)\bar{x}$\overline{x}$, we search for $CL$$CL$CLC L$CL$ maps that send $\overline{x}$$\overline{x}$bar(x)\bar{x}$\overline{x}$ close to itself.

We shall give a lower bound to approximation error in terms of the collage error.

Proposition 7. Let $X$$X$XX$X$ be a real Banach space and $T$$T$TT$T$ a $CL$$CL$CLC L$CL$ map with contraction factor $q\in (0,1)$$q\in (0,1)$q in(0,1)q \in(0,1)$q\in (0,1)$ and fixed point $x^{\ast }\in X$$x^{\ast }\in X$x^(**)in Xx^{*} \in X$x^{\ast }\in X$. Then for any $x\in X$$x\in X$x in Xx \in X$x\in X$,

Proof. For any $x\in X$$x\in X$x in Xx \in X$x\in X$ satisfying $x=x^{\ast }$$x=x^{\ast }$x=x^(**)x=x^{*}$x=x^{\ast }$, the above inequality holds. If $x\ne x^{\ast },\mathrm{\forall }x\in X$$x\ne x^{\ast },\mathrm{\forall }x\in X$x!=x^(**),AA x in Xx \neq x^{*}, \forall x \in X$x\ne x^{\ast },\mathrm{\forall }x\in X$, then one obtains

From which one gets the conclusion.
Theorem 8. (Collage theorem for contractive-like maps) Let $X$$X$XX$X$ be a real Banach space and $T$$T$TT$T$ a $CL$$CL$CLC L$CL$ map with contraction factor $q\in (0,1)$$q\in (0,1)$q in(0,1)q \in(0,1)$q\in (0,1)$ and fixed point $x^{\ast }\in X$$x^{\ast }\in X$x^(**)in Xx^{*} \in X$x^{\ast }\in X$. Then for any $x\in X$$x\in X$x in Xx \in X$x\in X$,

Proof. The $CL$$CL$CLC L$CL$ condition assures that the fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ is unique. If $x=x^{\ast }$$x=x^{\ast }$x=x^(**)x=x^{*}$x=x^{\ast }$, the above inequality holds. If $x\ne x^{\ast },\mathrm{\forall }x\in X$$x\ne x^{\ast },\mathrm{\forall }x\in X$x!=x^(**),AA x in Xx \neq x^{*}, \forall x \in X$x\ne x^{\ast },\mathrm{\forall }x\in X$, then one obtains

From which one gets the conclusion.
Remark 9. To summarize, we have the following bounds

The above "Collage Theorem" allows us to reformulate the inverse Problem 5 in the particular and more convenient Problem 6.

Theorem 10. If Problem 6 has a solution, then Problem 5 has a solution too.

Proof. Let $\epsilon >0$$\epsilon >0$epsi > 0\varepsilon>0$\epsilon >0$ and $\overline{x}\in X$$\overline{x}\in X$bar(x)in X\bar{x} \in X$\overline{x}\in X$ be given. For $\delta :=(1-q)\epsilon$$\delta :=(1-q)\epsilon$delta:=(1-q)epsi\delta:=(1-q) \varepsilon$\delta :=(1-q)\epsilon$, let $T_{\delta }\in CL$$T_{\delta }\in CL$T_(delta)in CLT_{\delta} \in C L$T_{\delta }\in CL$ be such that $\Vert \overline{x}-T_{\delta }\overline{x}\Vert <\delta$$‖\overline{x}-T_{\delta }\overline{x}‖<\delta$||( bar(x))-T_(delta)( bar(x))|| < delta\left\|\bar{x}-T_{\delta} \bar{x}\right\|<\delta$\Vert \overline{x}-T_{\delta }\overline{x}\Vert <\delta$. If $x_{T_{\delta }}^{\ast }$$x_{T_{\delta }}^{\ast }$x_(T_(delta))^(**)x_{T_{\delta}}^{*}$x_{T_{\delta }}^{\ast }$ is the unique fixed point of the $CL$$CL$CLC L$CL$ mapping $T_{\delta }$$T_{\delta }$T_(delta)T_{\delta}$T_{\delta }$, then, by Theorem 8,

Note that shrinking the distance between two operators, one of them from $CL$$CL$CLC L$CL$, reduces the distance between their fixed points.

Proposition 11. Let $X$$X$XX$X$ be a real Banach space and $T_1\in CL$$T_1\in CL$T_(1)in CLT_{1} \in C L$T_1\in CL$ with contraction factor $q_1\in (0,1)$$q_1\in (0,1)$q_(1)in(0,1)q_{1} \in(0,1)$q_1\in (0,1)$ and $T_2:X\to X$$T_2:X\to X$T_(2):X rarr XT_{2}: X \rightarrow X$T_2:X\to X$ a map such that $x_1^{\ast },x_2^{\ast }\in X$$x_1^{\ast },x_2^{\ast }\in X$x_(1)^(**),x_(2)^(**)in Xx_{1}^{*}, x_{2}^{*} \in X$x_1^{\ast },x_2^{\ast }\in X$ are distinct fixed points for $T_1$$T_1$T_(1)T_{1}$T_1$ and $T_2$$T_2$T_(2)T_{2}$T_2$. Then,

Proof. Using (1) one obtains

from which we get the conclusion.
Theorem 12. Let $X$$X$XX$X$ be a real Banach space, $T:X\to X,\overline{x}=T\overline{x}$$T:X\to X,\overline{x}=T\overline{x}$T:X rarr X, bar(x)=T bar(x)T: X \rightarrow X, \bar{x}=T \bar{x}$T:X\to X,\overline{x}=T\overline{x}$ and suppose there exists $T_1\in CL$$T_1\in CL$T_(1)in CLT_{1} \in C L$T_1\in CL$ with contraction factor $q$$q$qq$q$, such that

Then

Proof. Let $x^{\ast }=T_1{x}^{\ast }$$x^{\ast }=T_1{x}^{\ast }$x^(**)=T_(1)x^(**)x^{*}=T_{1} x^{*}$x^{\ast }=T_1{x}^{\ast }$, and by use of Proposition 11 we obtain

Thus,

[1] Barnsley, M. F., Fractals everywhere, New York: Academic Press, 1988.
[2] Berinde, V., On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenianae, Vol. LXXIII, no. 1, pp. 119-126, 2004.
[3] Berinde, V., Iterative approximation of fixed points, Springer-Verlag Berlin Heidelberg, 2007.
[4] Imoru C.O. and Olatiwo, M.O., On the stability of Picard and Mann iteration processes, Carpathian J. Math., 19, pp. 155-160, 2003.
[5] Kunze H.E. and Vrscay, E.R. Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems, 15, pp. 745-770, 1999.
[6] Kunze, H.E. and Gomes, S., Solving an inverse problem for Urison-type integral equations using Banach's fixed point theorem, Inverse Problems, 19, pp. 411-418, 2003.
[7] Kunze, H.E., Hicken, J.E and Vrscay, E.R., Inverse problems for ODEs using contraction maps and suboptimality for the 'collage method', Inverse Problems, 20, pp. 977-991, 2004.
[8] Osilike, M.O., Stability results for fixed point iteration procedures, J. Nigerian Math. Soc., 14/15, pp. 17-29, 1995/96.
[9] Osilike, M.O., Stability of the Ishikawa iteration method for quasi-contractive maps, Indian J. Pure Appl. Math., 28, no. 9, pp. 1251-1265, 1997.
[10] Rhoades, B.E., Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc., 196, pp. 161-176, 1974.
[11] Şoltuz, Ş. M., Solving inverse problems via hemicontractive maps, Nonlinear Analysis, 71, pp. 2387-2390, 2009.
[12] Şoltuz, S. M., Solving inverse problems via weak-contractive maps, Rev. Anal. Numer. Theor. Approx., 37, No. 2, pp. 217-220, 2008. 뜸
[13] Zamfirescu, T., Fix Point Theorems in metric spaces, Arch. Math., 23, pp. 292-298, 1972.

Received by the editors: September 17, 2008.