An Analysis and Study of Iteration Procedures

In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n-th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed; notwithstanding, heuristic-based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions.


INTRODUCTION
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 none expansive map satisfies the condition ||Tx-Ty||≤||x-y|| 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 none expansive self map of [0,1] with a unique fixed point at x = 1/2, but, if one chooses as a starting point the value x = a,a a≠ 1/2, then repeated iteration of T yields the sequence {1 -a, a, 1 -a, a,...}.
In 1953 W.R. Mann defined the following iteration procedure. Let A be a lower triangular matrix with nonnegative entries and row sums 1. Define where The most interesting cases of the Mann iterative process are obtained by choosing matrices A such that k = 0,1,... ,n;n = 0,1,2,..., and either ann=1 for all for all n>0. Thus, if one chooses any sequence {cn} satisfying (i) C0 =1, (ii) 0 ≤cn < 1 for n > 0, and (iii) ΣCn=∞ then the entries of A become ann=cn ……. (1.1) and A is a regular matrix. (A regular matrix is a bounded linear operator on £°° 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: One example of such matrices is the Cesaro matrix, obtained by choosing Another is cn= 1 for all n ≥ 0, which corresponds to ordinary function iteration, commonly called Picard iteration.
Pichard iteration of the function S1/2=(1+T)/2 is equivalent to the Mann iteration scheme with This matrix is the Euler matrix of order 1, and the transformation S1/2 has been investigated by Edelstein and Krasnoselskii [30]. Krasnoselskii showed that, if X is a uniformly convex Banach space, and T is a nonexpansive selfmap of X, then S1/2 converges to a fixed point of T. Edelstein showed that the condition of uniform convexity could be weakened to that of strict convexity. Pichard iteration of the function Sλ = λI + (1 -λ)T, 0 < λ < 1, for any function T, homogeneous of degree 1, is equivalent to the  2. Theorems: 2.1 THEOREM 1:Let X be a Banach space, T a nonexpansive asymptotically 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 x0 ∈ X, {TnxQ} converges to a fixed point of T in X.
In 1972 Groetsch established the following theorem, which removes the hypothesis that T be asymptotically regular.

THEOREM 2.
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 {cn} satisfying conditions (i), (ii), and (iv) Σcn(l -cn) = ∝, converges strongly to a fixed point of T. Ishikawa established the following theorem.

THEOREM 3.
Let D be a closed subset of a Banach space X and let T be a no expansive map from D into a compact subset of X. Then T has a fixed point in D and the Mann iterative process with {cn} satisfying conditions (i) -(iii), and (0 < cn < 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 convergence 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|| < ||x -p||. The following is due to Dotson.

THEOREM 4.
Let E be a strictly convex Banach space, C a closed convex subset of E, T a continuous quasinonexpansive selfmap of C such that T(C) ⊂K ⊂ C, where K is compact. Let x0∈ C and consider a Mann iteration process such that {cn} clusters at some point in (0,1). Then the sequences {xn}, {vn} converge strongly to a fixed point of T.A contractive definition which is included in the class of quasicontractive maps is the following, due to Zamfirescu . A map satisfies condition Z if, for each pair of points x,y in the space, at least one of the following is true: 2.6 THEOREM 6. Let H be a Hilbert space, T a selfmap of H satisfying condition C. Then the Mann iterative process, with {cn} satisfying conditions (i)-(iii) and limsup cn < 1 -k2 converges to the fixed point of T. Chidume [10] has extended the above result to lp spaces, p ≥ 2, under the conditions k2(p -1) < 1 and limsupc" < (p -1)_1 -k2.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 = Tl = 0, Tx = 1,0 < x < 1. Then T is a selfmap of [0,1], with a fixed point at x = 0. However, the Mann iteration scheme, with cn = l/(n + 1),0 < x0 < 1, converges to 1, which is not a fixed point of T. A map T is said to be strictly-pseudo contractive if there exists a constant k, 0 < k < 1 such that, for all points x, y in the space, ||Tx -Ty||2 < ||x -y||2 + k|| (I -T)x -(I -T)y||2. 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.

STABILITY.
We shall now discuss the question of stability of iteration processes, adopting the definition of stability that appears in .Let X be a Banach space, T a selfmap of X, and assume that xn+1 = f(T, xn) defines some iteration procedure involving T.
For example, f(T, xn) = Txn. Suppose that {xn} converges to a fixed point p of T. Let {yn}be an arbitrary sequence in X and define en = ||yn+1 -f(Ty yn)|| for n = 0,1,2, If limn = 0 implies that limn y" = p, then the iteration procedure xn+i = /(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 the authors show that function iteration is stable for a variety of contractive definitions. Their best result for function iteration is the following.  For the iteration method of Kirk , they have the following result.

Conclusion:
Iteration is the redundancy of a procedure so as to create a (perhaps unbounded) succession of results. The grouping will approach some end point or end esteem. Every redundancy of the procedure is a solitary iteration, and the result of every iteration is then the beginning stage of the following iteration. In mathematics and software engineering, iteration (alongside the related system of recursion) is a standard component of calculations. In algorithmic circumstances, recursion and iteration can be utilized to a similar impact. The essential distinction is that recursion can be utilized as an answer without earlier learning about how often the activity should rehash, while an effective iteration necessitates that premonition.