AMS (MOS) Subject Classification: 65J15, 65B05, 47H17, 49D15.

Key Words and Phrases: Steffensen-Aitken method, implicit function, projection operator.

Let $E,\mathrm{\Lambda }$$E,\mathrm{\Lambda }$E,LambdaE, \Lambda$E,\mathrm{\Lambda }$ be Banach spaces and denote by $U(x_0,R)$$U\left(x_0,R\right)$U(x_(0),R)U\left(x_{0}, R\right)$U(x_0,R)$ the closed ball with center $x_0\in E$$x_0\in E$x_(0)in Ex_{0} \in E$x_0\in E$ and of radius $R\ge 0$$R\ge 0$R >= 0R \geq 0$R\ge 0$. We will use the same symbol for the norm $\Vert \Vert$$\Vert \Vert$||||\|\|$\Vert \Vert$ in both spaces. Let $P$$P$PP$P$ be a projection operator ( $P=P^2$$P=P^2$P=P^(2)P=P^{2}$P=P^2$ ) which projects $E$$E$EE$E$ on its subspace $E_P$$E_P$E_(P)E_{P}$E_P$ and set $Q=I-P$$Q=I-P$Q=I-PQ=I-P$Q=I-P$. Suppose that the nonlinear operators $F(x,\lambda )$$F(x,\lambda )$F(x,lambda)F(x, \lambda)$F(x,\lambda )$ and $G(x,\lambda )$$G(x,\lambda )$G(x,lambda)G(x, \lambda)$G(x,\lambda )$ with values in $E$$E$EE$E$ are defined for $x\in D$$x\in D$x in Dx \in D$x\in D$, where $D$$D$DD$D$ is some open convex subset of $E$$E$EE$E$ containing $U(x_0,R)$$U\left(x_0,R\right)$U(x_(0),R)U\left(x_{0}, R\right)$U(x_0,R)$, and $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$ for some ${\lambda }_0\in \mathrm{\Lambda },S\ge 0$${\lambda }_0\in \mathrm{\Lambda },S\ge 0$lambda_(0)in Lambda,S >= 0\lambda_{0} \in \Lambda, S \geq 0${\lambda }_0\in \mathrm{\Lambda },S\ge 0$. For each fixed $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$ the operator $PF(w,\lambda )$$PF(w,\lambda )$PF(w,lambda)P F(w, \lambda)$PF(w,\lambda )$ will be assumed to be Fréchet-differentiable for all $w\in D$$w\in D$w in Dw \in D$w\in D$. Then $P{F}^{\prime }(x,\lambda )$$P{F}^{\prime }(x,\lambda )$PF^(')(x,lambda)P F^{\prime}(x, \lambda)$P{F}^{\prime }(x,\lambda )$ will denote the Fréchet-derivative of the operator $PF(w,\lambda )$$PF(w,\lambda )$PF(w,lambda)P F(w, \lambda)$PF(w,\lambda )$ with respect to the argument $w$$w$ww$w$ at $w=x$$w=x$w=xw=x$w=x$. Moreover for each fixed $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$ the operator $PG(w,\lambda )$$PG(w,\lambda )$PG(w,lambda)P G(w, \lambda)$PG(w,\lambda )$ will be assumed to be continuous for all $w\in D$$w\in D$w in Dw \in D$w\in D$.

In this study we are concerned with the problem of approximating a solution $x^{\ast }:=x^{\ast }(\lambda )$$x^{\ast }:=x^{\ast }(\lambda )$x^(**):=x^(**)(lambda)x^{*}:=x^{*}(\lambda)$x^{\ast }:=x^{\ast }(\lambda )$ of the equation

We introduce the inexact Steffensen-Aitken-type method

where by $x_0$$x_0$x_(0)x_{0}$x_0$ we mean $x_0(\lambda )$$x_0(\lambda )$x_(0)(lambda)x_{0}(\lambda)$x_0(\lambda )$. That is, $x_0$$x_0$x_(0)x_{0}$x_0$ depends on the $\lambda$$\lambda$lambda\lambda$\lambda$ used in (2). $A(x,\lambda )\in L(E\times \mathrm{\Lambda },E)$$A(x,\lambda )\in L(E\times \mathrm{\Lambda },E)$A(x,lambda)in L(E xx Lambda,E)A(x, \lambda) \in L(E \times \Lambda, E)$A(x,\lambda )\in L(E\times \mathrm{\Lambda },E)$ and is given by

where $[x(\lambda ),y(\lambda );F]$$[x(\lambda ),y(\lambda );F]$[x(lambda),y(lambda);F][x(\lambda), y(\lambda) ; F]$[x(\lambda ),y(\lambda );F]$ (or $[x(\lambda ),y(\lambda );G]$$[x(\lambda ),y(\lambda );G]$[x(lambda),y(lambda);G][x(\lambda), y(\lambda) ; G]$[x(\lambda ),y(\lambda );G]$ ) denotes divided difference of order one on $F$$F$FF$F$ (or $G$$G$GG$G$ ) at the points $x(\lambda ),y(\lambda )\in D$$x(\lambda ),y(\lambda )\in D$x(lambda),y(lambda)in Dx(\lambda), y(\lambda) \in D$x(\lambda ),y(\lambda )\in D$, satisfying

for all $x(\lambda )\ne y(\lambda ),\lambda \in U({\lambda }_0,S)$$x(\lambda )\ne y(\lambda ),\lambda \in U\left({\lambda }_0,S\right)$x(lambda)!=y(lambda),lambda in U(lambda_(0),S)x(\lambda) \neq y(\lambda), \lambda \in U\left(\lambda_{0}, S\right)$x(\lambda )\ne y(\lambda ),\lambda \in U({\lambda }_0,S)$, and

if $F(x(\lambda ),\lambda )$$F(x(\lambda ),\lambda )$F(x(lambda),lambda)F(x(\lambda), \lambda)$F(x(\lambda ),\lambda )$ is Fréchet-differentiable at $x(\lambda )$$x(\lambda )$x(lambda)x(\lambda)$x(\lambda )$ for all $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$. The operator $z:D\times U({\lambda }_0,S)\to E$$z:D\times U\left({\lambda }_0,S\right)\to E$z:D xx U(lambda_(0),S)rarr Ez: D \times U\left(\lambda_{0}, S\right) \rightarrow E$z:D\times U({\lambda }_0,S)\to E$ is chosen so that iteration $\{x_n(\lambda )\}(n\ge$$\left\{x_n(\lambda )\right\}(n\ge${x_(n)(lambda)}(n >=\left\{x_{n}(\lambda)\right\}(n \geq$\{x_n(\lambda )\}(n\ge$ 0 ) generated by (2) converges to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$. The operator $g^1,g^2,g^3,g^4:D\times U({\lambda }_0,S)\to E$$g^1,g^2,g^3,g^4:D\times U\left({\lambda }_0,S\right)\to E$g^(1),g^(2),g^(3),g^(4):D xx U(lambda_(0),S)rarr Eg^{1}, g^{2}, g^{3}, g^{4}: D \times U\left(\lambda_{0}, S\right) \rightarrow E$g^1,g^2,g^3,g^4:D\times U({\lambda }_0,S)\to E$ are continuous.

The importance of studying inexact Steffensen-Aitken methods comes from the fact that many commonly used variants can be considered procedures of this type. Indeed approximation (2) characterizes any iterative process in which corrections are taken as approximate solutions of Steffensen-Aitken equations. Moreover we note that if for example an equation on the real line is solved $F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda )\ge 0$$F\left(x_n(\lambda ),\lambda \right)+G\left(x_n(\lambda ),\lambda \right)\ge 0$F(x_(n)(lambda),lambda)+G(x_(n)(lambda),lambda) >= 0F\left(x_{n}(\lambda), \lambda\right)+ G\left(x_{n}(\lambda), \lambda\right) \geq 0$F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda )\ge 0$ and $A(x_n(\lambda ),\lambda )$$A\left(x_n(\lambda ),\lambda \right)$A(x_(n)(lambda),lambda)A\left(x_{n}(\lambda), \lambda\right)$A(x_n(\lambda ),\lambda )$ overestimates the derivative $x_n-A{(x_n(\lambda ),\lambda )}^{-1}(F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda ))$$x_n-A{\left(x_n(\lambda ),\lambda \right)}^{-1}\left(F\left(x_n(\lambda ),\lambda \right)+G\left(x_n(\lambda ),\lambda \right)\right)$x_(n)-A(x_(n)(lambda),lambda)^(-1)(F(x_(n)(lambda),lambda)+G(x_(n)(lambda),lambda))x_{n}- A\left(x_{n}(\lambda), \lambda\right)^{-1}\left(F\left(x_{n}(\lambda), \lambda\right)+G\left(x_{n}(\lambda), \lambda\right)\right)$x_n-A{(x_n(\lambda ),\lambda )}^{-1}(F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda ))$ is always "larger" than the corresponding Steffensen-Aitken iterate. In such cases a positive $z(x_n(\lambda ),\lambda )(n\ge 0)$$z\left(x_n(\lambda ),\lambda \right)(n\ge 0)$z(x_(n)(lambda),lambda)(n >= 0)z\left(x_{n}(\lambda), \lambda\right)(n \geq 0)$z(x_n(\lambda ),\lambda )(n\ge 0)$ correction term is appropriate.

It can easily be shown by induction on $n$$n$nn$n$ that under the above hypotheses $F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda )$$F\left(x_n(\lambda ),\lambda \right)+G\left(x_n(\lambda ),\lambda \right)$F(x_(n)(lambda),lambda)+G(x_(n)(lambda),lambda)F\left(x_{n}(\lambda), \lambda\right)+G\left(x_{n}(\lambda), \lambda\right)$F(x_n(\lambda ),\lambda )+G(x_n(\lambda ),\lambda )$ belong to the domain of $A{(x_n(\lambda ),\lambda )}^{-1}$$A{\left(x_n(\lambda ),\lambda \right)}^{-1}$A(x_(n)(lambda),lambda)^(-1)A\left(x_{n}(\lambda), \lambda\right)^{-1}$A{(x_n(\lambda ),\lambda )}^{-1}$ for all $n\ge 0$$n\ge 0$n >= 0n \geq 0$n\ge 0$.

Therefore, if the inverses exist (as it will be shown later in the theorem), then the iterates $\{x_n(\lambda )\}$$\left\{x_n(\lambda )\right\}${x_(n)(lambda)}\left\{x_{n}(\lambda)\right\}$\{x_n(\lambda )\}$ can be computed for all $n\ge 0$$n\ge 0$n >= 0n \geq 0$n\ge 0$. The iterates generated when $P=I$$P=I$P=IP=I$P=I$ (identity operator on $E$$E$EE$E$ ) cannot easily be computed in infinite dimensional spaces since the inverses may be too difficult or impossible to find. It is easy to see, however, that the solution of equations (2) reduces to solving certain operator equations in the space $EP$$EP$EPE P$EP$. If, moreover, $EP$$EP$EPE P$EP$ is a finite dimensional space of dimension $N$$N$NN$N$, we obtain a system of linear algebraic equations of at most order $N$$N$NN$N$. Special choices of the operators introduced above reduce our iteration (2) to earlier considered methods. Indeed we can have: for $g^1(x(\lambda ),\lambda )=g^2(x(\lambda ),\lambda )=x(\lambda ),g^3(x(\lambda ),\lambda )=g^4(x(\lambda ),\lambda )=$$g^1(x(\lambda ),\lambda )=g^2(x(\lambda ),\lambda )=x(\lambda ),g^3(x(\lambda ),\lambda )=g^4(x(\lambda ),\lambda )=$g^(1)(x(lambda),lambda)=g^(2)(x(lambda),lambda)=x(lambda),g^(3)(x(lambda),lambda)=g^(4)(x(lambda),lambda)=g^{1}(x(\lambda), \lambda)=g^{2}(x(\lambda), \lambda)=x(\lambda), g^{3}(x(\lambda), \lambda)=g^{4}(x(\lambda), \lambda)=$g^1(x(\lambda ),\lambda )=g^2(x(\lambda ),\lambda )=x(\lambda ),g^3(x(\lambda ),\lambda )=g^4(x(\lambda ),\lambda )=$ 0, $z=0$$z=0$z=0z=0$z=0$ we obtain Newton methods considered in [3], [4], [5]; for $P=I$$P=I$P=IP=I$P=I$, no $\lambda ,g^1(x)=g^2(x)=x(x\in D),g^3\left(x_n\right)=x_{n-1}(n\ge 1)$$\lambda ,g^1(x)=g^2(x)=x(x\in D),g^3\left(x_n\right)=x_{n-1}(n\ge 1)$lambda,g^(1)(x)=g^(2)(x)=x(x in D),g^(3)(x_(n))=x_(n-1)(n >= 1)\lambda, g^{1}(x)=g^{2}(x)=x(x \in D), g^{3}\left(x_{n}\right)=x_{n-1}(n \geq 1)$\lambda ,g^1(x)=g^2(x)=x(x\in D),g^3\left(x_n\right)=x_{n-1}(n\ge 1)$, $g^4\left(x_n\right)=x_n(n\ge 0),z_n=0(n\ge 0)$$g^4\left(x_n\right)=x_n(n\ge 0),z_n=0(n\ge 0)$g^(4)(x_(n))=x_(n)(n >= 0),z_(n)=0(n >= 0)g^{4}\left(x_{n}\right)=x_{n}(n \geq 0), z_{n}=0(n \geq 0)$g^4\left(x_n\right)=x_n(n\ge 0),z_n=0(n\ge 0)$ we obtain Catinas method [4]; for $P=I$$P=I$P=IP=I$P=I$, no $\lambda ,G(x)=0(x\in D),z_n=0(n\ge 0),g^3(x)=g^4(x)=0$$\lambda ,G(x)=0(x\in D),z_n=0(n\ge 0),g^3(x)=g^4(x)=0$lambda,G(x)=0(x in D),z_(n)=0(n >= 0),g^(3)(x)=g^(4)(x)=0\lambda, G(x)=0(x \in D), z_{n}=0(n \geq 0), g^{3}(x)=g^{4}(x)=0$\lambda ,G(x)=0(x\in D),z_n=0(n\ge 0),g^3(x)=g^4(x)=0$, $g^2(x)=g^1(F(x))(x\in D)$$g^2(x)=g^1(F(x))(x\in D)$g^(2)(x)=g^(1)(F(x))(x in D)g^{2}(x)=g^{1}(F(x))(x \in D)$g^2(x)=g^1(F(x))(x\in D)$, we obtain methods considered by Pǎvǎloiu in [3], [4], [6], [7]. Our choices of the operators since they include all previous methods allow us to consider a wider class of problems.

We provide sufficient conditions for the convergence of iteration (2) to a locally unique solution $x^{\ast }(\lambda )$$x^{\ast }(\lambda )$x^(**)(lambda)x^{*}(\lambda)$x^{\ast }(\lambda )$ of equation (1) as well as several
error bounds on the distances $\Vert x_{n+1}(\lambda )-x_n(\lambda )\Vert$$‖x_{n+1}(\lambda )-x_n(\lambda )‖$||x_(n+1)(lambda)-x_(n)(lambda)||\left\|x_{n+1}(\lambda)-x_{n}(\lambda)\right\|$\Vert x_{n+1}(\lambda )-x_n(\lambda )\Vert$ and $\Vert x_n(\lambda )-x^{\ast }(\lambda )\Vert$$‖x_n(\lambda )-x^{\ast }(\lambda )‖$||x_(n)(lambda)-x^(**)(lambda)||\left\|x_{n}(\lambda)-x^{*}(\lambda)\right\|$\Vert x_n(\lambda )-x^{\ast }(\lambda )\Vert$ ( $n\ge 0$$n\ge 0$n >= 0n \geq 0$n\ge 0$ ).

We can now state and prove the following semilocal convergence result:

Theorem. Let $F,G,P,Q$$F,G,P,Q$F,G,P,QF, G, P, Q$F,G,P,Q$ be as in the introduction. Assume:
(a) there exist $x_0(\lambda )\in D,{\lambda }_0\in \mathrm{\Lambda }$$x_0(\lambda )\in D,{\lambda }_0\in \mathrm{\Lambda }$x_(0)(lambda)in D,lambda_(0)in Lambdax_{0}(\lambda) \in D, \lambda_{0} \in \Lambda$x_0(\lambda )\in D,{\lambda }_0\in \mathrm{\Lambda }$ such that $C:=C(\lambda )=A(x_0(\lambda ),{\lambda }_0)$$C:=C(\lambda )=A\left(x_0(\lambda ),{\lambda }_0\right)$C:=C(lambda)=A(x_(0)(lambda),lambda_(0))C:=C(\lambda)=A\left(x_{0}(\lambda), \lambda_{0}\right)$C:=C(\lambda )=A(x_0(\lambda ),{\lambda }_0)$ is invertible. Set $B=C^{-1}$$B=C^{-1}$B=C^(-1)B=C^{-1}$B=C^{-1}$;
(b) there exist nonnegative numbers $a_i,R,S,i=1,2,\dots ,15$$a_i,R,S,i=1,2,\dots ,15$a_(i),R,S,i=1,2,dots,15a_{i}, R, S, i=1,2, \ldots, 15$a_i,R,S,i=1,2,\dots ,15$ such that:
(6) $\Vert BP([x,y;F]-[v,w;F])\Vert \le a_1(\Vert x-v\Vert +\Vert y-w\Vert )$$\Vert BP([x,y;F]-[v,w;F])\Vert \le a_1(\Vert x-v\Vert +\Vert y-w\Vert )$quad||BP([x,y;F]-[v,w;F])|| <= a_(1)(||x-v||+||y-w||)\quad\|B P([x, y ; F]-[v, w ; F])\| \leq a_{1}(\|x-v\|+\|y-w\|)$\Vert BP([x,y;F]-[v,w;F])\Vert \le a_1(\Vert x-v\Vert +\Vert y-w\Vert )$,
(7) $\Vert x-g^1(x,\lambda )\Vert \le a_2\Vert A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )\Vert$$‖x-g^1(x,\lambda )‖\le a_2‖A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )‖$||x-g^(1)(x,lambda)|| <= a_(2)||A(x,lambda)^(-1)(F(x,lambda)+G(x,lambda))-z(x,lambda)||\left\|x-g^{1}(x, \lambda)\right\| \leq a_{2}\left\|A(x, \lambda)^{-1}(F(x, \lambda)+G(x, \lambda))-z(x, \lambda)\right\|$\Vert x-g^1(x,\lambda )\Vert \le a_2\Vert A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )\Vert$
(8) $\Vert x-g^2(x,\lambda )\Vert \le a_3\Vert A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )\Vert$$‖x-g^2(x,\lambda )‖\le a_3‖A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )‖$||x-g^(2)(x,lambda)|| <= a_(3)||A(x,lambda)^(-1)(F(x,lambda)+G(x,lambda))-z(x,lambda)||\left\|x-g^{2}(x, \lambda)\right\| \leq a_{3}\left\|A(x, \lambda)^{-1}(F(x, \lambda)+G(x, \lambda))-z(x, \lambda)\right\|$\Vert x-g^2(x,\lambda )\Vert \le a_3\Vert A(x,\lambda {)}^{-1}(F(x,\lambda )+G(x,\lambda ))-z(x,\lambda )\Vert$,
$\Vert B(A(x_{n+1},\lambda )\left(z(x_{n+1},\lambda )\right)-A(x_n,\lambda )\left(z(x_n,\lambda )\right)\Vert \le a_7\Vert x_{n+1}-x_n\Vert$$\Vert B\left(A\left(x_{n+1},\lambda \right)\left(z\left(x_{n+1},\lambda \right)\right)-A\left(x_n,\lambda \right)\left(z\left(x_n,\lambda \right)\right)‖\le a_7‖x_{n+1}-x_n\Vert \right.$||B(A(x_(n+1),lambda)(z(x_(n+1),lambda))-A(x_(n),lambda)(z(x_(n),lambda))|| <= a_(7)||x_(n+1)-x_(n)||:}\| B\left(A\left(x_{n+1}, \lambda\right)\left(z\left(x_{n+1}, \lambda\right)\right)-A\left(x_{n}, \lambda\right)\left(z\left(x_{n}, \lambda\right)\right)\left\|\leq a_{7}\right\| x_{n+1}-x_{n} \|\right.$\Vert B(A(x_{n+1},\lambda )\left(z(x_{n+1},\lambda )\right)-A(x_n,\lambda )\left(z(x_n,\lambda )\right)\Vert \le a_7\Vert x_{n+1}-x_n\Vert$

and

for all $v,w,x,y\in U(x_0,R),\lambda \in U(\lambda ,S);$$v,w,x,y\in U\left(x_0,R\right),\lambda \in U(\lambda ,S);$v,w,x,y in U(x_(0),R),lambda in U(lambda,S);v, w, x, y \in U\left(x_{0}, R\right), \lambda \in U(\lambda, S) ;$v,w,x,y\in U(x_0,R),\lambda \in U(\lambda ,S);$
(c) the sequence $\left\{z(x_n(\lambda ),\lambda )\right\}(n\ge 0)$$\left\{z\left(x_n(\lambda ),\lambda \right)\right\}(n\ge 0)${z(x_(n)(lambda),lambda)}(n >= 0)\left\{z\left(x_{n}(\lambda), \lambda\right)\right\}(n \geq 0)$\left\{z(x_n(\lambda ),\lambda )\right\}(n\ge 0)$ is null for all $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$;
(d) for each fixed $\lambda \in U({\lambda }_0,S)$$\lambda \in U\left({\lambda }_0,S\right)$lambda in U(lambda_(0),S)\lambda \in U\left(\lambda_{0}, S\right)$\lambda \in U({\lambda }_0,S)$ there exists a minimum nonnegative number $r^{\ast }:=r_{\lambda }^{\ast }$$r^{\ast }:=r_{\lambda }^{\ast }$r^(**):=r_(lambda)^(**)r^{*}:=r_{\lambda}^{*}$r^{\ast }:=r_{\lambda }^{\ast }$ satisfying

with $r:=r(\lambda )$$r:=r(\lambda )$r:=r(lambda)r:=r(\lambda)$r:=r(\lambda )$,

where