In this paper, a modified implicit Kirk-multistep iteration scheme and a strong convergence result for a general class of maps in a normed linear space was established. It was also shown that the convergence of this iteration scheme is equivalent to the convergency of some other implicit Kirk-type iteration (implicit Kirk-Noor, implicit Kirk-Ishikawa and implicit Kirk-Mann iterations) for the same class of maps. Some numerical examples were considered to show that the equivalence of convergence results to the fixed point is true. The results unify most equivalence results in literature.
In 1971, Kirk introduced the Kirk iterative scheme as follows: Let ( E , ‖ . ‖ ) be a normed linear space and D a non-empty, convex, closed subset of E and T : D → D be a selfmap of D, let x 0 ∈ E , the sequence { x n } n = 1 ∞ is defined by
x n + 1 = ∑ i = 0 k α i T i x n , n ≥ 0 , ∑ i = 0 k α i = 1 (1)
Various authors have written inspiring papers on Kirk-type iterative schemes. Worthy to mention are the following: the explicit Kirk-Mann, Olatinwo [
In 2014, Akewe, Okeke and Olayiwola [
The main aim of this work is in three folds: firstly, to develop a modified implicit Kirk-multistep scheme and prove strong convergence results for a general class of mapping introduced by Bosede and Rhoades [
Let ( E , ‖ . ‖ ) be a normed linear space and D a non-empty, convex, closed subset of E and T : D → D be a selfmap of D, let x 0 ∈ D . Then, the sequence { x n } n = 0 ∞ defined by
x n + 1 = α n , 0 x n 1 + ∑ i = 1 q 1 α n , i T i x n + 1 , ∑ i = 0 q 1 α n , i = 1 x n j = β n , 0 j x n j + 1 + ∑ i = 1 q j + 1 β n , i j T i x n j , ∑ i = 0 q j + 1 β n , i j = 1 , j = 1 , 2 , ⋯ , k − 2 , x n k − 1 = β n , 0 k − 1 x n + ∑ i = 1 q k β n , i k − 1 T i x n k − 1 , ∑ i = 0 q k β n , i k − 1 = 1 , k ≥ 2 , n ≥ 1 } (2)
where q 1 ≥ q 2 ≥ q 3 ≥ ⋯ ≥ q k , for each j, α n , i ≥ 0 , α n ,0 ≠ 0 , β n , i j ≥ 0 , β n ,0 j ≠ 0 , for each j, α n , i , β n , i j ∈ [ 0,1 ] for each j and q 1 , q j are fixed integers (for each j). (2) is called implicit Kirk-multistep iterations.
Equation (2) serves as a general formula for obtaining other implicit Kirk-type iterations. Infact, if k = 3 in (2), we obtain a three step (implicit Kirk-Noor) iteration as follows:
x n + 1 = α n , 0 x n 1 + ∑ i = 1 q 1 α n , i T i x n + 1 , ∑ i = 0 q 1 α n , i = 1 x n 1 = β n , 0 1 x n 2 + ∑ i = 1 q 2 β n , i 1 T i x n 1 , ∑ i = 0 q 2 β n , i 1 = 1 , x n 2 = β n , 0 2 x n + ∑ i = 1 q 3 β n , i 2 T i x n 2 , ∑ i = 0 q 3 β n , i 2 = 1 , } (3)
where q 1 ≥ q 2 ≥ q 3 , α n , i ≥ 0 , α n ,0 ≠ 0 , β n , i 1 ≥ 0 , β n ,0 1 ≠ 0 , β n , i 2 ≥ 0 , β n ,0 2 ≠ 0 , α n , i , β n , i 1 , β n , i 2 ∈ [ 0,1 ] and q 1 , q 2 and q 3 are fixed integers.
If k = 2 in (2), we obtain a two step (implicit Kirk-Ishikawa) iteration as follows:
x n + 1 = α n , 0 x n 1 + ∑ i = 1 q 1 α n , i T i x n + 1 , ∑ i = 0 q 1 α n , i = 1 x n 1 = β n , 0 1 x n + ∑ i = 1 q 2 β n , i 1 T i x n 1 , ∑ i = 0 q 3 β n , i 1 = 1 , } (4)
where q 1 ≥ q 2 , α n , i ≥ 0 , α n ,0 ≠ 0 , β n , i 1 ≥ 0 , β n ,0 1 ≠ 0 , α n , i , β n , i 1 ∈ [ 0,1 ] and q 1 and q 2 are fixed integers.
Finally, if k = 2 and q 2 = 0 in (2), we obtain a one step (implicit Kirk-Mann) iteration as follows:
x n + 1 = α n , 0 x n 1 + ∑ i = 1 q 1 α n , i T i x n + 1 , ∑ i = 0 q 1 α n , i = 1 , (5)
where α n , i ≥ 0 , α n ,0 ≠ 0 , α n , i ∈ [ 0,1 ] and q 1 is a fixed integer.
Equations (3)-(5) will be rewritten in the following forms to help us prove our equivalence result:
Let ( E , ‖ . ‖ ) be a normed linear space and D a non-empty, convex, closed subset of E and T : D → D be a selfmap of D, let y 0 ∈ D . Then, the implicit Kirk-Noor scheme is a sequence { y n } n = 0 ∞ defined by
y n + 1 = α n , 0 y n 1 + ∑ i = 1 q 1 α n , i T i y n + 1 , ∑ i = 0 q 1 α n , i = 1 y n 1 = β n , 0 1 y n 2 + ∑ i = 1 q 2 β n , i 1 T i y n 1 , ∑ i = 0 q 2 β n , i 1 = 1 y n 2 = β n , 0 2 y n + ∑ i = 1 q 3 β n , i 2 T i x n 2 , ∑ i = 0 q 3 β n , i 2 = 1 } (6)
where q 1 ≥ q 2 ≥ q 3 , α n , i ≥ 0 , α n ,0 ≠ 0 , β n , i 1 ≥ 0 , β n ,0 1 ≠ 0 , β n , i 2 ≥ 0 , β n ,0 2 ≠ 0 , α n , i , β n , i 1 , β n , i 2 ∈ [ 0,1 ] and q 1 , q 2 and q 3 are fixed integers.
Also, for z 0 ∈ D , the two step (implicit Kirk-Ishikawa) iteration scheme is a sequence { z n } n = 0 ∞ defined as follows:
z n + 1 = α n , 0 z n 1 + ∑ i = 1 q 1 α n , i T i z n + 1 , ∑ i = 0 q 1 α n , i = 1 z n 1 = β n , 0 1 z n + ∑ i = 1 q 2 β n , i 1 T i z n 1 , ∑ i = 0 q 2 β n , i 1 = 1 } (7)
where q 1 ≥ q 2 , α n , i ≥ 0 , α n ,0 ≠ 0 , β n , i 1 ≥ 0 , β n ,0 1 ≠ 0 , α n , i , β n , i 1 ∈ [ 0,1 ] and q 1 and q 2 are fixed integers.
Finally, for u 0 ∈ D , the implicit Kirk-Mann iteration scheme is a sequence { u n } n = 0 ∞ defined by:
u n + 1 = α n , 0 u n 1 + ∑ i = 1 q 1 α n , i T i u n + 1 , ∑ i = 0 q 1 α n , i = 1 (8)
where α n , i ≥ 0 , α n ,0 ≠ 0 , α n , i ∈ [ 0,1 ] and q 1 is a fixed integer. We shall now consider some of the contractive mappings useful in proving our main results.
Let E be a normed linear space and D a non-empty, convex, closed subset of E and T : D → D be a selfmap of D. There exists a real number a ∈ [ 0,1 ) and all x , y ∈ D such that
‖ x − T y ‖ ≤ a ‖ x − y ‖ (9)
Zamfirescu [
‖ T x − T y ‖ ≤ δ ‖ x − y ‖ + 2 δ ‖ x − T x ‖ (10)
where δ ∈ [ 0,1 ) . Inequality (10) becomes (9) if x is a fixed point of T.
Osilike [
‖ T x − T y ‖ ≤ a ‖ x − y ‖ + L ‖ x − T x ‖ (11)
In 2003, Imoru and Olatinwo [
‖ T x − T y ‖ ≤ δ ‖ x − y ‖ + φ ( ‖ x − T x ‖ ) . (12)
In 2010, Bosede and Rhoades [
In 2014, Chidume and Olaleru [
We shall need the following lemma in proving our result.
Lemma 1.2 [
u n + 1 ≤ δ u n + ϵ n ; n = 0 , 1 , 2 , ⋯ (13)
we have lim n → ∞ u n = 0 .
Lemma 1.3 [
Theorem 2.1. Let ( E , ‖ . ‖ ) be a normed linear space, D a non-empty, convex, closed subset of E and T : D → D , a self map satisfying the inequality:
‖ T i x − T p ‖ ≤ a i ‖ x − p ‖ (14)
where a i ∈ [ 0,1 ) and p ∈ F ( T ) . For x 0 ∈ D , let { x n } be the implicit Kirk-multistep iteration scheme defined by (2) with ∑ n = 1 ∞ ( 1 − α n , 0 ) = ∞ . Then
i) the fixed point p of T defined by (14) is unique;
ii) the implicit Kirk-multistep iteration scheme converges strongly to the unique fixed point p of T.
Proof:
i) The first thing is to establish that the mapping T satisfying the contractive condition (14) has a unique fixed point.
Suppose there exist p 1 , p 2 ∈ F T , and that p 1 ≠ p 2 , with ‖ p 1 − p 2 ‖ > 0 , then,
( 1 − a i ) ‖ p 1 − p 2 ‖ ≤ 0. (15)
Since a i ∈ [ 0,1 ) , then 1 − a i > 0 and ‖ p 1 − p 2 ‖ ≤ 0 . Since norm is nonnegative we have that ‖ p 1 − p 2 ‖ = 0 . That is, p 1 = p 2 = p (say). Thus, T has a unique fixed point p.
ii) Next, we prove that (2) converges strongly to p. In veiw of (2) and (14),
‖ x n + 1 − p ‖ ≤ α n , 0 ‖ x n ( 1 ) − p ‖ + ∑ i = 1 q 1 α n , i ‖ T i x n + 1 − T i p ‖ ≤ α n , 0 ‖ x n ( 1 ) − p ‖ + ∑ i = 1 q 1 α n , i [ a i ‖ x n + 1 − p ‖ ] ≤ α n , 0 1 − ∑ i = 1 q 1 α n , i a i ‖ x n ( 1 ) − p ‖ (16)
Also, using (2) and (14), we have:
‖ x n ( 1 ) − p ‖ ≤ β n , 0 ( 1 ) ‖ x n ( 2 ) − p ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ T i x n ( 1 ) − T i p ‖ ≤ β n , 0 ( 1 ) ‖ x n ( 2 ) − p ‖ + ∑ i = 1 q 2 β n , i ( 1 ) [ a i ‖ x n ( 1 ) − p ‖ ] ≤ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ x n ( 2 ) − p ‖ (17)
Again, using (2) and (14), we have:
‖ x n ( 2 ) − p ‖ ≤ β n , 0 ( 2 ) ‖ x n ( 3 ) − p ‖ + ∑ i = 1 q 3 β n , i ( 2 ) ‖ T i x n 2 − T i p ‖ ≤ β n , 0 ( 2 ) ‖ x n ( 3 ) − p ‖ + ∑ i = 1 q 3 β n , i ( 2 ) [ a i ‖ x n ( 2 ) − p ‖ ] ≤ β n , 0 ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ‖ x n ( 3 ) − p ‖ (18)
Continuing the process using (2) and (14), we have
‖ x n ( k − 2 ) − p ‖ ≤ β n , 0 ( k − 2 ) ‖ x n ( k − 1 ) − p ‖ + ∑ i = 1 q k − 1 β n , i ( k − 2 ) ‖ T i x n k − 2 − T i p ‖ ≤ β n , 0 ( k − 2 ) ‖ x n ( k − 1 ) − p ‖ + ∑ i = 1 q k − 1 β n , i ( k − 2 ) [ a i ‖ x n ( k − 2 ) − p ‖ ] ≤ β n , 0 ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ‖ x n ( k − 1 ) − p ‖ (19)
Finally, using (2) and (14) for ( k − 1 ) , we have:
‖ x n ( k − 1 ) − p ‖ ≤ β n , 0 ( k ) ‖ x n ( k ) − p ‖ + ∑ i = 1 q k β n , i ( k − 1 ) ‖ T i x n k − 1 − T i p ‖ ≤ β n , 0 ( k − 1 ) ‖ x n ( k ) − p ‖ + ∑ i = 1 q k β n , i ( k − 1 ) [ a i ‖ x n ( k − 1 ) − p ‖ ] ≤ β n , 0 ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ‖ x n ( k ) − p ‖ (20)
Substituting (20) in (19), (19) in (18), (18) in (17) and (17) in (16), we obtain:
‖ x n + 1 − p ‖ ≤ [ α n , 0 1 − ∑ i = 1 q 1 α n , i a i ] [ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ] [ β n , 0 ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ] ⋯ ⋅ [ β n , 0 ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ] [ β n , 0 ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ] ‖ x n ( k ) − p ‖ (21)
Note that
1 − α n , 0 1 − ∑ i = 1 q 1 α n , i a i = 1 − [ ∑ i = 1 q 1 α n , i a i + α n , 0 ] 1 − ∑ i = 1 q 1 α n , i a i ≥ 1 − [ ∑ i = 1 q 1 α n , i a i + α n , 0 ] (22)
hence
α n , 0 1 − ∑ i = 1 q 1 α n , i a i ≤ ∑ i = 1 q 1 α n , i a i + α n , 0
Let a i < a < 1 , then
∑ i = 1 q 1 α n , i a i + α n , 0 ≤ ( 1 − α n , 0 ) a + α n , 0 (23)
That is,
α n , 0 1 − ∑ i = 1 q 1 α n , i a i ≤ ( 1 − α n , 0 ) a + α n , 0 (24)
Therefore,
‖ x n + 1 − p ‖ ≤ [ ( 1 − α n ,0 ) a + α n ,0 ] [ ( 1 − β n ,0 ( 1 ) ) a + β n ,0 ( 1 ) ] [ ( 1 − β n ,0 ( 2 ) ) a + β n ,0 ( 2 ) ] ⋯ ⋅ [ ( 1 − β n ,0 ( k − 2 ) ) a + β n ,0 ( k − 2 ) ] [ ( 1 − β n ,0 ( k − 1 ) ) a + β n ,0 ( k − 1 ) ] ‖ x n − p ‖ ≤ [ 1 − ( 1 − α n ,0 ) ( 1 − a ) ] ‖ x n − p ‖ (25)
Hence, using Lemma 1.2 in (25), then lim n → ∞ ‖ x n − p ‖ = 0 This ends the proof.
Theorem 2.1 leads to the following corollary:
Corollary 2.2. Let ( E , ‖ . ‖ ) be a normed linear space, D a non-empty, convex, closed subset of E and T : D → D , with p ∈ F ( T ) , such that:
‖ T i x − p ‖ ≤ a i ‖ x − p ‖ (26)
where a i ∈ [ 0,1 ) . For y 0 = z 0 = u 0 ∈ D , let { y n } { z n } { u n } be the implicit Kirk-Noor, implicit Kirk-Ishikawa and implicit Kirk-Mann iteration scheme respectively defined by (6), (7) and (8) with ∑ n = 1 ∞ ( 1 − α n , 0 ) = ∞ , ( 1 − β n , 0 1 ) = ∞ , ( 1 − β n , 0 2 ) = ∞ . Then
i) T defined by (26) has a unique fixed point p;
ii) { y n } (6) converges strongly to the unique fixed point p of T;
iii) { z n } (7) converges strongly to the unique fixed point p of T;
iv) { u n } (8) converges strongly to the unique fixed point p of T.
Theorem 2.3. Let ( E , ‖ . ‖ ) be a normed linear space, D a non-empty, convex, closed subset of E and T : D → D an operator satisfying
‖ T i x − T p ‖ ≤ a i ‖ x − p ‖ (27)
where a i ∈ [ 0,1 ) and p ∈ F ( T ) . If u 0 = x 0 ∈ D , then the following are equivalent: the
i) implicit Kirk-Mann iterative scheme { u n } n = 0 ∞ (8) converges strongly to p;
ii) implicit Kirk-multistep iterative scheme { x n } n = 0 ∞ (2) converges strongly to p.
Proof:
We prove that (i) Þ (ii).
Assume lim n → ∞ u n = p , then using (8), (2) and (27), we have
‖ u n + 1 − x n + 1 ‖ = ‖ α n , 0 ( u n − x n ( 1 ) ) + ∑ i = 1 q 1 α n , i T i u n + 1 − ∑ i = 1 q 1 α n , i T i x n + 1 ‖ ≤ α n , 0 ‖ u n − x n ( 1 ) ‖ + ∑ i = 1 q 1 α n , i ‖ T i u n + 1 − T i x n + 1 ‖ . (28)
Using condition (27) in (28), we have
‖ u n + 1 − x n + 1 ‖ ≤ α n , 0 ‖ u n − x n ( 1 ) ‖ + ∑ i = 1 q 1 α n , i a i ‖ u n + 1 − x n + 1 ‖ ≤ α n , 0 1 − ∑ i = 1 q 1 α n , i a i ‖ u n − x n ( 1 ) ‖ (29)
From (29),
‖ u n − x n ( 1 ) ‖ ≤ β n , 0 ( 1 ) ‖ u n − x n ( 2 ) ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ u n − T i u n + T i u n − T i x n ( 1 ) ‖ ≤ β n , 0 ( 1 ) ‖ u n − x n ( 2 ) ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ u n − T i u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ T i u n − T i x n ( 1 ) ‖ ≤ β n , 0 ( 1 ) ‖ u n − x n ( 2 ) ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ u n − T i u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ u n − x n ( 1 ) ‖ (30)
From (30),
‖ u n − x n ( 1 ) ‖ ≤ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ u n − x n ( 2 ) ‖ + ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ u n − T i u n ‖ (31)
But from (31),
‖ u n − T i u n ‖ = ‖ u n − p + T p − T i u n ‖ ≤ ‖ u n − p ‖ + ‖ T p − T i u n ‖ (32)
Applying condition (27) on (31), we get
‖ u n − T i u n ‖ ≤ ( 1 + a i ) ‖ u n − p ‖ (33)
Using (33) in (31), we have
‖ u n − x n ( 1 ) ‖ ≤ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ u n − x n ( 2 ) ‖ + ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ( 1 + a i ) ‖ u n − p ‖ (34)
Similarly,
‖ u n − x n ( 2 ) ‖ ≤ β n , 0 ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ‖ u n − x n ( 3 ) ‖ + ∑ i = 1 q 3 β n , i ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ( 1 + a i ) ‖ u n − p ‖ (35)
Also
‖ u n − x n ( 3 ) ‖ ≤ β n , 0 ( 3 ) 1 − ∑ i = 1 q 4 β n , i ( 3 ) a i ‖ u n − x n ( 4 ) ‖ + ∑ i = 1 q 4 β n , i ( 3 ) 1 − ∑ i = 1 q 4 β n , i ( 3 ) a i ( 1 + a i ) ‖ u n − p ‖ (36)
Continuing (k − 2) times, we have
‖ u n − x n ( k − 2 ) ‖ ≤ β n ,0 ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ‖ u n − x n ( k − 1 ) ‖ + ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ( 1 + a i ) ‖ u n − p ‖ (37)
Moving a step more, we have
‖ u n − x n ( k − 1 ) ‖ ≤ β n , 0 ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ‖ u n − x n ( k ) ‖ + ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ( 1 + a i ) ‖ u n − p ‖ (38)
Substituting (35) into (34), (34) into (33), (33) into (32) and (32) into (31) respectively, we obtain
Recall that
α n , 0 1 − ∑ i = 1 q 1 α n , i a i ≤ ∑ i = 1 q 1 α n , i a i + α n , 0 (40)
Let a i < a < 1 , then
∑ i = 1 q 1 α n , i a i + α n , 0 ≤ [ ( 1 − α n , 0 ) a + α n , 0 ] (41)
Using (36), (37) in (38), we have
‖ u n + 1 − x n + 1 ‖ ≤ [ ( 1 − α n , 0 ) a + α n , 0 ] [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] [ ( 1 − β n , 0 ( 2 ) ) a + β n , 0 ( 2 ) ] [ ( 1 − β n , 0 ( 3 ) ) a + β n , 0 ( 3 ) ] ⋯ [ ( 1 − β n , 0 ( k − 2 ) ) a + β n , 0 ( k − 2 ) ] [ ( 1 − β n , 0 ( k − 1 ) ) a + β n , 0 ( k − 1 ) ] ‖ u n − x n ‖ + { [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] [ ( 1 − β n , 0 ( 2 ) ) a + β n , 0 ( 2 ) ] [ ( 1 − β n , 0 ( 3 ) ) a + β n , 0 ( 3 ) ] ⋯ [ ( 1 − β n , 0 ( k − 2 ) ) a + β n , 0 ( k − 2 ) ] [ ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ] + [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ]
⋯ [ ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ] + [ ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ] } [ ( 1 − α n , 0 ) a + α n , 0 ] ( 1 + a i ) ‖ u n − p ‖ ≤ [ 1 − ( 1 − α n , 0 ) ( 1 − a ) ] ‖ u n − x n ‖ + { [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] [ ( 1 − β n , 0 ( 2 ) ) a + β n , 0 ( 2 ) ] ⋅ [ ( 1 − β n , 0 ( 3 ) ) a + β n , 0 ( 3 ) ] ⋯ [ ( 1 − β n , 0 ( k − 2 ) ) a + β n , 0 ( k − 2 ) ] [ ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ] + [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] ⋯ [ ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ] + [ ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ] } ⋅ [ ( 1 − α n , 0 ) a + α n , 0 ] ( 1 + a i ) ‖ u n − p ‖ (42)
Let λ n = ( 1 − α n ,0 ) ( 1 − a ) and
e n = { [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] [ ( 1 − β n , 0 ( 2 ) ) a + β n , 0 ( 2 ) ] [ ( 1 − β n , 0 ( 3 ) ) a + β n , 0 ( 3 ) ] ⋯ [ ( 1 − β n , 0 ( k − 2 ) ) a + β n , 0 ( k − 2 ) ] [ ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ] + [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] ⋯ [ ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k β n , i ( k − 2 ) a i ] + [ ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ] } ⋅ [ ( 1 − α n , 0 ) a + α n , 0 ] ( 1 + a i ) ‖ u n − p ‖ (43)
Replacing (40) in (39), we have
‖ u n + 1 − x n + 1 ‖ ≤ ( 1 − λ n ) ‖ u n − x n ‖ + e n (44)
By Lemma 1.3 in (41), it follows that
lim n → ∞ ‖ u n − x n ‖ = 0 (45)
Since by the assumption, lim n → ∞ u n = p ,
then ‖ x n − p ‖ ≤ ‖ u n − x n ‖ + ‖ u n − p ‖ → 0 as n → ∞
Hence lim n → ∞ x n = p .
Next is to show that (ii) implies (i).
Assume, lim n → ∞ x n = p , then using (2), (8) and (27), we have
‖ x n + 1 − u n + 1 ‖ ≤ ‖ α n , 0 ( x n ( 1 ) − u n ) + ∑ i = 1 q 1 α n , i ( T i x n + 1 − T i u n + 1 ) ‖ ≤ α n , 0 ‖ x n ( 1 ) − u n ‖ + ∑ i = 1 q 1 α n , i a i ‖ T i x n + 1 − T i u n + 1 ‖ ≤ α n , 0 1 − ∑ i = 1 q 1 α n , i a i ‖ x n ( 1 ) − u n ‖ (46)
‖ x n ( 1 ) − u n ‖ ≤ ‖ β n , 0 ( 1 ) ( x n ( 2 ) − u n ) + ∑ i = 1 q 2 β n , i ( 1 ) ( T i x n ( 1 ) − u n ) ‖ ≤ β n , 0 ( 1 ) ‖ x n ( 2 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ∥ T i x n ( 1 ) − T i u n + T i u n − u n ∥ ≤ β n , 0 ( 1 ) ‖ x n ( 2 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ T i x n ( 1 ) − T i u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ T i u n − u n ‖ ≤ β n , 0 ( 1 ) ‖ x n ( 2 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ x n ( 1 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) ‖ T i u n − u n ‖ (47)
By simplifying (42), we obtain
‖ x n ( 1 ) − u n ‖ ≤ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ x n ( 2 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ T i u n − u n ‖ (48)
But,
‖ T i u n − u n ‖ ≤ ‖ T i u n − T i p + p − u n ‖ ≤ ‖ T i u n − T i p ‖ + ‖ p − u n ‖ ≤ a i ‖ u n − p ‖ + ‖ u n − p ‖ ≤ ( a i + 1 ) ‖ u n − p ‖ (49)
Substituting (44) in (43), we have
‖ x n ( 1 ) − u n ‖ ≤ β n , 0 ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ‖ x n ( 2 ) − u n ‖ + ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ( a i + 1 ) ‖ u n − p ‖ (50)
Similarly,
‖ x n ( 2 ) − u n ‖ ≤ β n ,0 ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ‖ x n ( 3 ) − u n ‖ + ∑ i = 1 q 3 β n , i ( 2 ) 1 − ∑ i = 1 q 3 β n , i ( 2 ) a i ( a i + 1 ) ‖ u n − p ‖ (51)
Also,
‖ x n ( 3 ) − u n ‖ ≤ β n , 0 ( 3 ) 1 − ∑ i = 1 q 4 β n , i ( 3 ) a i ‖ x n ( 4 ) − u n ‖ + ∑ i = 1 q 4 β n , i ( 3 ) 1 − ∑ i = 1 q 4 β n , i ( 3 ) a i ( a i + 1 ) ‖ u n − p ‖ (52)
Continuing the process upto (k − 2), we have
‖ x n ( k − 2 ) − u n ‖ ≤ β n , 0 ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ‖ x n ( k − 1 ) − u n ‖ + ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ( a i + 1 ) ‖ u n − p ‖ (53)
For (k − 1), we get
‖ x n ( k − 1 ) − u n ‖ ≤ β n , 0 ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ‖ x n − u n ‖ + ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ( a i + 1 ) ‖ u n − p ‖ (54)
Substituting accordingly from (41) to (48), we get
‖ x n + 1 − u n + 1 ‖ ≤ [ ( 1 − α n ,0 ) a + α n ,0 ] [ ( 1 − β n , 0 ( 1 ) ) a + β n , 0 ( 1 ) ] [ ( 1 − β n , 0 ( 2 ) ) a + β n , 0 ( 2 ) ] [ ( 1 − β n , 0 ( 3 ) ) a + β n , 0 ( 3 ) ] ⋯ [ ( 1 − β n ,0 ( k − 2 ) ) a + β n ,0 ( k − 2 ) ] [ ( 1 − β n ,0 ( k − 1 ) ) a + β n ,0 ( k − 1 ) ] ‖ x n − u n ‖
+ [ ( ( 1 − β n ,0 ( 1 ) ) a + β n ,0 ( 1 ) ) ( ( 1 − β n ,0 ( 2 ) ) a + β n ,0 ( 2 ) ) ⋯ ( ( 1 − β n ,0 ( k − 2 ) ) a + β n ,0 ( k − 2 ) ) ⋅ ( ∑ i = 1 q k β n , i ( k − 1 ) 1 − ∑ i = 1 q k β n , i ( k − 1 ) a i ) + ( ( 1 − β n ,0 ( 1 ) ) a + β n ,0 ( 1 ) ) ⋯ ( ∑ i = 1 q k − 1 β n , i ( k − 2 ) 1 − ∑ i = 1 q k − 1 β n , i ( k − 2 ) a i ) + ( ∑ i = 1 q 2 β n , i ( 1 ) 1 − ∑ i = 1 q 2 β n , i ( 1 ) a i ) ] ( ( 1 − α n ,0 ) a + α n ,0 ) ( a i + 1 ) ‖ u n − p ‖ (55)
(50) can be written as:
‖ x n + 1 − u n + 1 ‖ ≤ [ 1 − ( 1 − α n ,0 ) ( 1 − a ) ] ‖ x n − u n ‖ + [ 1 − ( 1 − α n ,0 ) a + α n ,0 ] ( a i + 1 ) ‖ u n − p ‖ (56)
Let
λ n = ( 1 − α n , 0 ) ( 1 − a ) e n = [ 1 − ( 1 − α n , 0 ) a + α n , 0 ] ( a i + 1 ) ‖ u n − p ‖
Therefore,
‖ x n + 1 − u n + 1 ‖ ≤ ( 1 − λ n ) ‖ x n − u n ‖ + e n (57)
It follows from Lemma 1.3 that: lim n → ∞ ‖ x n − u n ‖ = 0
Since by assumption, lim n → ∞ u n = p Then, ‖ u n − p ‖ ≤ ‖ x n − u n ‖ + ‖ x n − p ‖ → 0, n → ∞ .
This implies that lim n → ∞ u n = p .
Since (i) → (ii) and (ii) → (i), it shows that the convergence of implicit Kirk-Mann iterative scheme (8) is equivalent to the convergence of implicit Kirk-multistep iterative scheme (2) when applied to the general class of map (14). This ends the proof.
Theorem 2.3 leads to the following Corollaries:
Corollary 2.4. Let ( E , ‖ . ‖ ) be a normed linear space, D a non-empty, convex, closed subset of E and T : D → D , with p ∈ F ( T ) , satisfying
‖ T i − p ‖ ≤ a i ‖ x − p ‖ (58)
where a i ∈ [ 0,1 ) . If u 0 = z 0 = y 0 ∈ D , then the following are equivalent: the
a) (i) implicit Kirk-Mann iterative scheme { u n } n = 0 ∞ (8) converges strongly to p;
(ii) implicit Kirk-Ishikawa iterative scheme { z n } n = 0 ∞ (7) converges strongly to p.
b) (i) implicit Kirk-Mann iterative scheme { u n } n = 0 ∞ (8) converges strongly to p;
(ii) implicit Kirk-Noor iterative scheme { y n } n = 0 ∞ (6) converges strongly to p.
Proof. The proof of Corollary 2.4 is similar to that of Theorem 2.3.This ends the proof.
Corollary 2.5. Let ( E , ‖ . ‖ ) be a normed linear space, D a non-empty, convex, closed subset of E and
where
i) implicit Kirk-Mann iterative scheme
ii) implicit Kirk-Ishikawa iterative scheme
iii) implicit Kirk-Noor iterative scheme
iv) implicit Kirk-multistep iterative scheme
In this section, we use some examples to demonstrate the equivalence of convergence between implicit Kirk-multistep (IKMST) iterative scheme (2) and other implicit Kirk-type [implicit Kirk-Noor (IKN)(6), implicit Kirk-Ishikawa (IKI)(7) and implicit Kirk-Mann (IKM)(8)] iterative schemes with the help of computer programs in PYTHON 2.5.4. We shall consider our results for increasing and decreasing functions. The results are shown in
Let
Let
n | IKMSTP | IKN | IKI | IKM |
---|---|---|---|---|
0 | 7.000000 | 7.000000 | 7.000000 | 7.000000 |
1 | 6.000124 | 6.194906 | 6.336163 | 6.579796 |
2 | 6.000001 | 6.019386 | 6.072164 | 6.268634 |
3 | 6.000000 | 6.001241 | 6.011546 | 6.107454 |
4 | 6.000000 | 6.000057 | 6.001482 | 6.038496 |
5 | 6.000000 | 6.000002 | 6.000159 | 6.012624 |
6 | 6.000000 | 6.000000 | 6.000015 | 6.003844 |
7 | 6.000000 | 6.000000 | 6.000001 | 6.001098 |
8 | 6.000000 | 6.000000 | 6.000000 | 6.000297 |
9 | 6.000000 | 6.000000 | 6.000000 | 6.000076 |
10 | 6.000000 | 6.000000 | 6.000000 | 6.000019 |
11 | 6.000000 | 6.000000 | 6.000000 | 6.000004 |
12 | 6.000000 | 6.000000 | 6.000000 | 6.000001 |
13 | 6.000000 | 6.000000 | 6.000000 | 6.000000 |
14 | 6.000000 | 6.000000 | 6.000000 | 6.000000 |
n | IKMSTP | IKN | IKI | IKM |
---|---|---|---|---|
0 | 0.700000 | 0.700000 | 0.700000 | 0.700000 |
1 | 0.382001 | 0.382149 | 0.384209 | 0.409165 |
2 | 0.381966 | 0.381968 | 0.382091 | 0.388393 |
3 | 0.381966 | 0.381966 | 0.381972 | 0.383341 |
4 | 0.381966 | 0.381966 | 0.381966 | 0.382236 |
5 | 0.381966 | 0.381966 | 0.381966 | 0.382015 |
6 | 0.381966 | 0.381966 | 0.381966 | 0.381974 |
7 | 0.381966 | 0.381966 | 0.381966 | 0.381967 |
8 | 0.381966 | 0.381966 | 0.381966 | 0.381966 |
9 | 0.381966 | 0.381966 | 0.381966 | 0.381966 |
10 | 0.381966 | 0.381966 | 0.381966 | 0.381966 |
i) From
convergence of implicit Kirk-multistep iterative scheme (2) to the fixed point 6.000000 is equivalent to the convergence of other implicit Kirk-type [implicit Kirk-Noor (IKN) (6), implicit Kirk-Ishikawa (IKI) (7) and implicit Kirk-Mann (IKM) (8)] iterative schemes to the same fixed point 6.000000;
ii) from
The numerical examples considered in this paper justified our claim on the equivalence results obtained. These results show that our implicit Kirk-type hybrid iterative schemes have good potentials for further applications.
The authors declare no conflicts of interest regarding the publication of this paper.
Bosede, A.O., Akewe, H., Bakre, O.F. and Wusu, A.S. (2019) On the Equivalence of Implicit Kirk-Type Fixed Point Iteration Schemes for a General Class of Maps. Journal of Applied Mathematics and Physics, 7, 123-137. https://doi.org/10.4236/jamp.2019.71011