Faster Multistep Iterations for the Approximation of Fixed Points Applied to Zamfirescu Operators

Definition 1 (see [4].)Suppose that {a_n} and {b_n} are two real convergent sequences with limits a and b, respectively. Then, {a_n} is said to converge faster than {b_n} if

Definition 2 (see [4].)Let {u_n} and {v_n} be two fixed-point iteration procedures which, both, converge to the same fixed point p, say, with error estimates,

where lim a_n = 0 = lim b_n. If {a_n} converges faster than {b_n}, then {u_n} is said to converge faster than {v_n}.

Theorem 3 (see [5].)Let (X, d) be a complete metric space, and let T : X → X be a mapping for which there exist real numbers a, b, and c satisfying a ∈ (0,1) and b, c ∈ (0, 1/2) such that, for each pair x, y ∈ X, at least one of the following is true:

•  

  (z1) d(Tx, Ty) ≤ ad(x, y),

•  

  (z2) d(Tx, Ty) ≤ b[d(x, Tx) + d(y, Ty)],

•  

  (z3) d(Tx, Ty) ≤ c[d(x, Ty) + d(y, Tx)].

Then, T has a unique fixed point p, and the Picard iteration {x_n} defined by

converges to p for any x₀ ∈ X.

Remark 4. An operator T, which satisfies the contraction conditions (z1)–(z3) of Theorem 3, will be called a Zamfirescu operator [4, 6, 7].

In [6, 7], Berinde introduced a new class of operators on a normed space E satisfying

for any x, y ∈ E,  0 ≤ δ < 1, and L ≥ 0.

He proved that this class is wider than the class of Zamfirescu operators.

The following results are proved in [6, 7].

Theorem 5 (see [7].)Let C be a nonempty closed convex subset of a normed space E. Let T : C → C be an operator satisfying (4). Let {x_n} be defined through the iterative process (Mn) and x₀ ∈ C. If F(T) ≠ ∅ and $$, then {x_n} converges strongly to the unique fixed point of T.

Theorem 6 (see [6].)Let C be a nonempty closed convex subset of an arbitrary Banach space E, and let T : C → C be an operator satisfying (4). Let {x_n} be defined through the iterative process (In) and x₀ ∈ C, where {b_n} and $$ are sequences of positive numbers in [0,1] with {b_n} satisfying $$. Then, {x_n} converges strongly to the fixed point of T.

Theorem 7. Let C be a closed convex subset of an arbitrary Banach space E. Let the Mann and Ishikawa iteration processes with real sequences {b_n} and $$ satisfy 0 ≤ b_n, $$, and $$. Then, (Mn) and (In) converge strongly to the unique fixed point of T. Let   T : C → C be a Zamfirescu operator, and, moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of T.

Theorem 8. Let C be a closed convex subset of an arbitrary Banach space E, and let T : C → C be a Zamfirescu operator. Let {y_n} be defined by (Mn) and y₀ ∈ C with a sequence {b_n} in [0,1] satisfying $$. Then, {y_n} converges strongly to the fixed point of T, and, moreover, the Picard iteration {x_n} converges faster than the Mann iteration.

Remark 9. In [9], Qing and Rhoades by taking a counterexample showed that the Mann iteration process converges more slowly than the Ishikawa iteration process for Zamfirescu operators.

Theorem 10. Let C be a nonempty closed convex subset of an arbitrary Banach space E, and let T : C → C be an operator satisfying (4). Let {x_n} be defined through the iterative process (RSn) and x₀ ∈ C, where {b_n} and $$,  i = 1,2, …, p − 2  (p ≥ 2), are sequences in [0,1] with $$. If F(T) ≠ ∅, then F(T) is a singleton, and the sequence {x_n} converges strongly to the fixed point of T.

Proof. Assume that F(T) ≠ ∅ and w ∈ F(T). Then, using (RSn), we have

Now, for x = w and $$, (4) gives

By substituting (6) in (5), we obtain

In a similar fashion, again by using (RSn), we can get

where i = 1,2, …, p − 2  (p ≥ 2) and It can be easily seen that, for i = 1,2, …, p − 2  (p ≥ 2), we have

Substituting (9) in (10) gives us

It may be noted that, for δ ∈ [0,1) and {η_n}∈[0,1], the following inequality is always true: From (11) and (12), we get By repeating the same procedure, finally from (7) and (10), we yield By (14), we inductively obtain Using the fact that 0 ≤ δ < 1,  0 ≤ b_n ≤ 1, and $$, it results that which, by (15), implies that Consequently, x_n → w ∈ F, and this completes the proof.

Example 11. Suppose that T : [0,1]→[0,1] is defined by Tx = (1/2)x;  $$, i = 1,2, …, p − 1  (p ≥ 2), and n = 1,2, …, 15;  $$,  i = 1,2, …, p − 1  (p ≥ 2), and n ≥ 16. It is clear that T is a Zamfirescu operator with a unique fixed point 0 and that all of the conditions of Theorem 10 are satisfied. Also, M_n = x₀ = I_n = RS_n,  n = 1,2, …, 15. Suppose that x₀ ≠ 0. For the Mann and Ishikawa iteration processes, we have

where i = 1,2, …, p − 2  (p ≥ 2) implies that Now, consider It is easy to see that Hence, Thus, the Mann iteration process converges more slowly than the multistep iteration process to the fixed point 0 of T.

Similarly,

with implies that Thus, the Ishikawa iteration process converges more slowly than the multistep iteration process to the fixed point 0 of T.