SOME FIXED POINT ITERATION PROCEDURES

This paper provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. The author does not claim to provide complete coverage of the literature, and admits to certain biases in the theorems that are cited herein. In spite of these shortcomings, however, this paper should be a useful reference for those persons vishing to become better acquainted with the area.

The literature abounds with papers which establish fixed points for maps satisfying a variety of contractive conditions.In most cases the contractive definition is strong enough, not only to guarantee the existence of a unique fixed point, but also to obtain that fixed point by repeated iteration of the function.However, for certain kinds of maps, such as nonexpansive maps, repeated function iteration need not converge to a fixed point.A nonexpansive map satisfies the condition IITx Tyll <_ ]Ix Yll for each pair of points x, y in the space.A simple example is the following.Define T(x) 1-x for 0 _< x _< 1.Then T is a nonexpansive selfmap of [0,1] with a unique fixed point at x 1/2, but, if one chooses as a starting point the value z a,a 1/2, then repeated iteration of T yields the sequence {1 -a,a, 1 -a,a,...).
In 1953 W.R. Mann [32] defined the following iteration procedure.Let A be a lower triangular matrix with nonnegative entries and row sums 1. Define xn+l T(vr,), where Vn E ankXk" k=O The most interesting cases of the Mann iterative process are obtained by choosing matrices A such that a,+l,k (1-a,+,,+)a,t,,k 0,1,...,n;n 0,1,2,..., and either a,, 1 for all n or a,, < 1 for all n > 0. Thus, if one chooses any sequence {c,} satisfying (i) co 1, (ii) 0 _< c,, < 1 for n > 0, and (iii) c, c, then the entries of A become atn C/t ank=ck H (1-cjl, k<n.(1.1) j=k+l and A is a regular matrix.(A regular matrix is a bounded linear operator on go such that A is limit preserving for convergent sequences.) The above representation for A allows one to write the iteration scheme in the following form: x,+, (1 c,.,)x,., + c,., T(x, ).
One example of such matrices is the Cesaro matrix, obtained by choosing c,, 1/(n + 1).
Another is c. 1 for all n >_ O, which corresponds to ordinary function iteration, commonly called Picard iteration.
Let C be a closed, bounded, and convex subset of a uniformly convex Banach space, T a nonexpansive selfmap of C. Then T has a fixed point.
Unfortunately the proofs of above theorem are not constructive.A number of papers have been written to obtain some kind of sequential convergence for nonexpansive maps.Most such theorems are valid only under some additional hypothesis, such as compactness, and converge only weakly.Some examples of theorems of this type appear in [8].Halpern  [18] obtained two algorithms for obtaining fixed points for nonexpansive maps on Hilbert spaces.In his dissertation, Humphreys [23] constructed an algorithm which can be applied to obtain fixed points for nonexpansive maps on uniformly convex Banach spaces.
A nonexpansive mapping is said to be asymptotically regular if, for each point x in the space, lim(T"+lx T"z) 0. In 1966 Browder and Petryshyn [7] established the following result.
THEOREM I. ([7, Theorem 6]).Let X be a Banach space, T a nonexpansive asymptot- ically regular selfmap of X. Suppose that T has a fixed point, and that I-T maps bounded closed subsets of X into closed subsets of X.Then, for each z0 E X, {Tnx0 converges to a fixed point of T in X.In 1972 Groetsch [17] established the following theorem, which removes the hypothesis that T be asymptotically regular.THEOREM 2. ([17, Corollary 3]).Suppose T is a nonexpansive selfmap of a closed convex subset E of X which has at least one fixed point.If I-T maps bounded closed subsets of E into closed subsets of E, then the Mann iterative procedure, with {c,} satisfying conditions (i), (ii), and (iv) c,,(1 c,,) oo, converges strongly to a fixed point of T.
THEOREM 3. Let D be a closed subset of a Banach space X and let T be a nonexpansive map from D into a compact subset of X.Then T has a fixed point in D and the Mann iterative process with {c,} satisfying conditions (i)-(iii), and 0 _< c, < b < 1 for all n, converges to a fixed point of T.
For spaces of dimension higher than one, continuity is not adequate to guarantee con- vergence to a fixed point, either by repeated function iteration, or by some other iteration procedure.Therefore it is necessary to impose some kind of growth condition on the map.If the contractive condition is strong enough, then the map will have a unique fixed point, which can be obtained by repeated iteration of the function.If the contractive condition is slightly weaker, then some other iteration scheme is required.Even if the fixed point can be obtained by function iteration, it is not without interest to determine if other iteration procedures converge to the fixed point.
A generalization of a nonexpansive map with at least one fixed point that of a quasi- nonexpansive map.A function T is a quasi-nonexpansive map if it has at least one fixed point, and, for each fixed point p, [[Tx -p[[ _< [Ix -p[[.The following is due to Dotson [12].THEOREM 4. Let E be a strictly convex Banach space, C a closed convex subset of E, T a continuous quasi-nonexpansive selfmap of C such that T(C) C K C C, where K is compact.Let x0 E C and consider a Mann iteration process such that {c,} clusters at some point in (0,1).Then the sequences {x,,}, {v,} converge strongly to a fixed point of T.
A contractive definition which is included in the class of quasi-contractive maps is the following, due to Zamfirescu [52].A map satisfies condition Z if, for each pair of points x, y in the space, at least one of the following is true: where a,/3, 3' are real nonnegative constants satisfying a < 1,/3, 3' < 1/2.As shown in [52], T has a unique fixed point, which can be obtained by repeated iteration of the function.The following result appears in [42].
THEOREM 5. ([42, Theorem 4]).Let X be a uniformly convex Banach space, E a closed convex subset of X, T a selfmap of E satisfying condition Z. Then the Mann iterative process with {c,} satisfying conditions (i), (ii), and (iv) converges to the fixed point of T. A generalization of definition Z was made by Ciric [11].A map satisfies condition C if there exists a constant k satisfying 0 _< k < 1 such that, for each pair of points x, y in the space, IlTx Tyll < k max{llx ll, Ilx TxII, Ilu TyII, II Tull, II In [42] the author proved the following for Hilbert spaces.THEOREM 6. ([42, Theorem 7]).Let H be a Hilbert space, T a selfmap of H satisfying condition C. Then the Mann iterative process, with {c,} satisfying conditions (i)-(iii) and limsup c, < 1 k 2 converges to the fixed point of T.
As noted earlier, if T is continuous, then, if the Mann iterative process converges, it must converge to a fixed point of T. If T is not continuous, there is no guarantee that, even if the Mann process converges, it will converge to a fixed point of T. Consider, for example, the map T defined by TO T1 0, Tz 1, 0 < z < 1.Then T is a selfmap of [0,1], with a fixed point at x 0. However, the Mann iteration scheme, with cn 1/(n + 1), 0 < 0 < 1, converges to 1, which is not a fixed point of T.
A Inap T is said to be strictly-pseudocontractive if there exists a constant k, 0 _< k < 1 such that, for all points x, y in the space, [ITx Tyll <_ Ilx y]l + kll(I T)z -(I T)yII .
We shall call denote the class of all such maps by P2.Clearly P2 mappings contain the nonex- pansive mappings, but the classes P2, C, and quasi-nonexpansive mappings are independent.THEOREM 7. ([42, Theorem 8]).Let H be a Hilbert space, E a compact, convex subset of H, T a P2 selfmap of E. Then the Mann iteration scheme with {c, } satisfying conditions (i)-(iii) and limsup c, c < 1 k, converges strongly to a fixed point of T.
A pseudocontractive mapping is a P2 mapping with k 1.Let P3 denote the family of pseudocontractive mappings.Hicks and Kubicek [22] gave an exaxaple of a P3 mapping with a fixed point such that the Mann iterative procedure, with c,., 1/(n + 1), does not converge to the fixed point.Ishikawa [24] defined the following iterative procedure.Given any x0 in the space, define x,+, a,T[nTx, + (I ft,)x,.,]+ (I where {a, }, {/3, } are sequences of positive numbers satisfying the conditions 0 < c,, _</3.<_ 1, lim/3. 0,c,,13.oo.He then established the following result.THEOREM 8. Let E be a convex, compact subset of a Hilbert space H, T a lipschitzian P selfmap of E. Then {x.} converges strongly to a fixed point of T.
Qihou [38] extended the above result to lipschitzian hemicontractive maps.A hemicon- tractive map is a pseudocontractive map with respect to a fixed point; i.e., if p is any fixed point of T, and x is any point in the space, then T satisfies IITx pll <_ IIx p]l + ]Ix Tx]] .
Neither the proof of Qihou nor Ishikawa can be used to establish a similar result for the Mann iterative technique.
In 1968 Kannan [27] introduced a contractive definition which did not require continuity of the map.There then followed a spate of fixed point papers dealing with fixed points for a variety of maps which were not necessarily continuous.For example, maps of type C and Z need not be continuous.In an earlier paper the author [45] partially ordered many of these definitions.As noted above, it has been shown that the Mann iteration scheme converges for some of these.For other contractive definitions it is not possible to determine whether or not the Mann or Ishikawa iteration processes converge.However, in some eases it is possible to show that, if they do converge, then they must converge to a fixed point of T. The following theorems illustrate this idea.THEOREM 9. ([42, Theorems 5,6]).Let X be a Banach space, T a selfmap of X, T E C or T E P2.Let {c,,} satisfy the conditions (i), (ii), and bounded away from zero.Then, if the corresponding Mann iteration process converges, it converges to a fixed point of T. THEOREM 10. ([46, Theorem 1]).Let X be a closed convex subset of a normed space, T a selfmap of X satisfying the property that there exist constants c, k > 0, 0 < k < 1 such that, for each x, y X, IITx Ty[[ <_ k max{cllx ttll, [ll Txll / I[Y TYH], Ill x TyII + Ily TII]}.
Let {c,,} satisfy conditions (i), (ii), and limc, > 0. If the Mann iteration scheme converges, then it converges to a fixed point of T.
A contractive definition which is independent of that of Ciric is the following, due to Pal and Maiti [37].For each pair of points in the space, at least one of the following conditions is satisfied: Using this demition the author established the following result.
TI-ImOIM ll.([46, Theorem 2]).Let X be a manaeh space, a selfmap of X satisfying the bove contractive definition.Then if the Mann iteration scheme, with {cn} satisfying conditions (i), (ii), and lim , > 0, converges, it converges to a fixed point of T.
Similar results have been established for the Ishikawa procedure.In its original form the Ishikawa procedure does not include the Mann iteration process because of the condition 0 <_ cn _</3n _< 1. For, if the Ishikawa process were to include the Mann process as a special case, then one would have to assign each/3n to be zero, forcing then each cn to be zero.In an effort to have an Ishikwa type iteration scheme which does include the Mann iterative process as a special case, some authors have modified the inequality condition to real 0 _< n,/3n _< 1.This change is reflected in the following two theorems.THFOREM 12. ([35, Theorem 1.2]).Let X be a normed linear space, (7 a closed convex subset of X.Let T be a seffmap o C satisfying the contractive condition: there exists a constant k, 0 _< k < I, such that, for each pair of points m, y in X, If the Ishikawa iteration scheme, with the an bounded away from zero, converges to a point p, then p is a fixed point of T. THEOREM 13. ( [40).Let E be a closed convex subset of a Banach space, X, T a selfmap of X satisfying the condition: there exists constants c, k _> 0, 0 _< k < 1, such that, for each pair of points z, y in X, ][Tx Tyl[ k max{cllx ll, [}lx Txll / Ily TYl]],[]I x TyII / Ily-Txll]).
In some theorems of this type, the contractive definition is weak enough that the a priori existence of a fixed point for T cannot be assured.Thus the convergence of the iteration procedure also yields the existence of a fixed point.
The literature also contains theorems of this type for other contractive definitions.In 1988 the author [47] proved that, for many contractive definitions, even though the function need not be continuous everywhere, it is always continuous in a neighborhood of a fixed point.Therefore, assuming the convergence of the Mann procedure is tantamount, in some cases, to assuming the convergence to a fixed point, in light of the early result of Mann [32].
We now return to nonexpansive maps.THEOREM 14. ( [1]).Let K be a closed bounded convex subset of a Hilbert space H, T a nonexpansive selfmap of K.For each point e of K, the Cesaro transform of {T'e} converges weakly to a fixed point of T.
If T is also an odd map, then Baillon [2] showed that the convergence of the Cesaro transform of {The} is strong.He also extended the above theorm to L ' spaces.
We now mention some results for other matrix transforms of iterates of T. THEOREM 15. ([5, Theorem 1]).Let T be a Hilbert space, C a closed bounded convex subset of H, T a nonexpansive selfmap of C. Suppose that A is an infinite matrix with zero column limits and satisfies lim,, k(a,,,k+l a,,k) + 0.Then, for each x in C, { a,.,,T'x} converges weakly to a fixed point of T.
A sequence x is said to be almost convergent to a limit q if lim,[x,, + x,,+l +... + x,.,+,_ ]/n q, uniformly in p.An infinite matrix A is said to be strongly regular if A assigns a limit to each almost convergent sequence.Necessary and sufficient conditions for A to be strongly regular were established by Lorentz [31].These conditions are that A be regular and also satisfy lim, E la-,*+ a-l 0.
Let {u,,} be a bounded sequence in a closed convex subset C of X. Define rm(x) sup{llu.zlln >_ m}, d denote by c,,, the unique point in C with the property that r,,,(c,,,) inf{rm(x)'x E C}.Then limcm c, and c is called the asymptotic center of {u,,}. (See, e.g.[41) Bruck proved the following theorem.THEOREM 16. ([9, Theorem 1.1]).Suppose that C is a closed convex subset of a real Hilbert space and T is a nonexpansive selfmap of C with a fixed point.Then for each x in C, A a strongly regular matrix, the A-transform of T"x converges weakly to a fixed point p, which is the asymptotic centcr of {T'x}.
Schoenberg [51] obtained the following characterization for nonexpansive maps on a Hilbert space.
THEOREM 17. ( [51]).Let T be a nonexpansive of a Hilbert space H,A be a nonnegative matrix with row sums one and zero column limits, k----0 Then the following are equivalent; (a) {S,} is weakly convergent to the asymptotic center z of {T"x}.(b) {,.,c is weakly convergent to a fixed point for T. (c) If V is a weak limit point of {S,}, then lim,...oo(Re(T"z T"+*z,z Ty)) 0 (d) If y is a weak limit point of {S,}, then (e) Each weak limit point of {S,} is a fixed point of T.
THEOREM 19.([48, Theorem 2]).Let 1 < p < oo, X e',K a nonempty closed, bounded, convex subset of X, T an asymptotically regular selfmap of K satisfying (1.2) and (1.3) with b b'.Then each weak limit point of {s } is a fixed point of T. THEOREM 20. ([48, Theorem 3]).Let 1 < p < o,X gp, K a noneinpty closed, bounded, convex subset of X, T a quasi-nonexpansive selfmap of Ix'.If each subsequential weak limit of {s is a fixed point of T, then {s converges weakly to a fixed point of T.
The special case of the above theorems in which A is the Cesaro matrix and T is nonex- pansive appear in Beauzamy and Enflo [4].
We shall now examine two other iteration methods for obtaining fixed points.The first of these is due to Kirk [29] and is defined as follows.Let {x,,} be any sequence of points in the space.Then the iteration procedure is defined by the operator S a,T', (1.6) '-0 where k is a fixed integer, k >_ 1,a, >_ 0, 0, 1,2,...,a > 0 and =0 a, 1.
For T any nonexpansive selfmap of a convex set, S and T have the same fixed points.(See [29].) Let T be a selfmap of a Banach space X satisfying, for each x, y in X, IIT: Tull <_ mx{ll: ulI, [11: T:II + Ilu TulI]/., [11: Tull + Ilu T:II]/2}. (1.7)In [41] it is shown that, if S satisfies (1.6), then S and T have the same fixed points.The following result is then proved.
THEOREM 21. ([41, Theorem 2]).Let X be a uniformly convex Banach space, K a bounded closed convex subset of X, T a selfmap of K which satisfies condition (7), S defined by (6).If I-S maps bounded closed subsets of K into closed sets, then, for each x0 E K the sequence {S"x0 } converges to a fLxed point of T in K.
Then {S"x converges to the only fixed point of T.

RATE OF CONVERGENCE.
There has been no systematic study of the rate of convergence for these iteration proce- dures, and it is doubtful if any global statements can be made, since there is nothing about these iteration procedures to cause their analysis to be different from that of other approximation methods.
The author [44] obtained some evidence on the behavior of the Mann iteration procedure for the decreasing functions f(x) 1 xrn,g(x) (1 z) for 1 _< rn <: 6 and c,, [(n + 1)(n + 2)]-1/',3 _< k _< 8.The fixed point of each function was first found by the bisection method, accurate to 10 places.Then both the Mann iteration and Newton-Raphson methods were employed to find each fixed point to within 8 places, using initial guesses of x0 .1,.2,..., .9.The author [43] also examined the functions g for m 7, 8,..., 29, with an initial guess of x0 .9, and a, , (n + 1) -1/2.In each case the Mann iterative procedure converged to a fixed point (accurate to eight places) in 9-12 iterations, whereas the Ishikawa method required from 38 to 42 iterations for the same degree of accuracy.For increasing functions the Ishikawa method is better than the Mann process, but ordinary function iteration is the best of all three for increasing functions.
An examination of the printout showed that the Newton-Raphson method converges faster than the Mann scheme.This is not surprising, since the Newton-Raphson method converges quadratically, whereas the Mann process converges linearly.However, whereas the rate of convergence of the Newton-Raphson method is very sensitive to the starting point, the rate of convergence for the Mann process appears to be independent of the initial guess.

STABILITY.
We shall now discuss the question of stability of iteration processes, adopting the definition of stability that appears in [19].
Let X be a Banach space, T a selfmap of X, and assume that xn+l f(T,x,) defines some iteration procedure involving T. For example, f(T, Xn) Tz Suppose that {x, converges to a fixed point p of T. Let {y,}be an arbitrary sequence in X and define for n 0,1,2, If lim, 0 implies that lim, y, p, then the iteration procedure f(T, x) is said to be T-stable.The first result on T-stable mappings was proved by Ostrovski for the Banach contraction principle.In [20] the authors show that function iteration is stable for a variety of contractive definitions.Their best result for function iteration is the following.
THEOREM 23. ([20, Theorem 2]).Let X be a complete metric space, T a selfmap of X satisfying the contractive condition of Zamfirescu.Let p be the fixed point of For the Mann iteration procedure their best result is the following.
THEOREM 27. ([49, Theorem 2]).Let (X, I1" II) be a Banach space, T a selfmap of X satisfying (3.1), p the fixed point of T. Let {z,} denote the Mann iterative process with the {c,} satisfying (i)-(iii), where it is understood that the product is 1 when j n.Then hm /,,=p if and only if lim e,=0.
, p if and only if lim,, 0.