Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 3791506, 10 pages http://dx.doi.org/10.1155/2016/3791506
Research Article On Faster Implicit Hybrid Kirk-Multistep Schemes for Contractive-Type Operators O. T. Wahab1 and K. Rauf2 1
Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria Department of Mathematics, University of Ilorin, Ilorin 240001, Nigeria
2
Correspondence should be addressed to K. Rauf;
[email protected] Received 21 March 2016; Revised 9 July 2016; Accepted 26 July 2016 Academic Editor: Lianwen Wang Copyright Š 2016 O. T. Wahab and K. Rauf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The purpose of this paper is to prove strong convergence and T-stability results of some modified hybrid Kirk-Multistep iterations for contractive-type operator in normed linear spaces. Our results show through analytical and numerical approach that the modified hybrid schemes are better in terms of convergence rate than other hybrid Kirk-Multistep iterative schemes in the literature.
1. Introduction In the recent years, numerous papers have been published on the strong convergence and T-stability of various iterative approximations of fixed points for contractive-type operators. See [1â9]. The Picard iterative scheme defined for đĽ0 â đ đĽđ = đđĽđâ1 , đ ⼠1,
đĽđ = (1 â đźđ ) đĽđâ1 + đźđ đđŚđâ1 đŚđâ1 = (1 â đ˝đ ) đĽđâ1 + đ˝đ đđĽđâ1 ,
(1)
was the first iteration to be proved by Banach [10] for a selfmap đ in a complete metric space (đ, đ) satisfying đ (đđĽ, đđŚ) ⤠đđ (đĽ, đŚ)
(2)
(called strict contraction), for all đĽ, đŚ â đ and đ â (0, 1). Picard iteration (1) which obeys (2) is said to have a fixed point in đšđ , where đšđ is the set of all fixed points. The Picard iteration will no longer converge to a fixed point of the operator if contractive condition (2) is weaker. Hence, there is a need to consider other iterative procedures. Mann [11] defined a more general iteration in a Banach space đ¸ satisfying quasi-nonexpansive operators. For đĽ0 â đ¸, the Mann iteration is given by đĽđ = (1 â đźđ ) đĽđâ1 + đźđ đđĽđâ1 ,
A double Mann iteration, called Ishikawa iteration, was introduced by Ishikawa [12]. It is defined for đĽ0 â đ¸ as
đ ⼠1,
(3)
where {đźđ } is a sequence of positive numbers in [0, 1]. Putting đźđ = 1 in (3) yields Picard iteration (1).
(4) đ ⼠1,
where {đźđ } and {đ˝đ } are sequences of positive numbers in [0, 1]. The three-step iteration, which is more general than Mann and Ishikawa iterations, was defined by Noor [13]. For đĽ0 â đ¸, the Noor iteration is given as đĽđ = (1 â đźđ ) đĽđâ1 + đźđ đđŚđâ1 đŚđâ1 = (1 â đ˝đ ) đĽđâ1 + đ˝đ đđ§đâ1 đ§đâ1 = (1 â đžđ ) đĽđâ1 + đžđ đđĽđâ1 ,
(5)
đ ⼠1, where {đźđ }, {đ˝đ }, and {đžđ } are sequences in [0, 1] with â đźđ = â.
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Rhoades and Soltuz [14] defined a multistep iteration in a normed linear space đ¸ as follows. For đĽ0 â đ¸, (1) đĽđ = (1 â đźđ ) đĽđâ1 + đźđ đđŚđâ1 (đ) đ+1 đŚđâ1 = (1 â đ˝đ(đ) ) đĽđâ1 + đ˝đ(đ) đđŚđâ1 ,
scheme of fixed points for different types of contractivelike operators in various spaces. See [3, 6, 18, 19]. The stability results of explicit and SP-iterative schemes have been discussed in [2, 4, 9, 20, 21]. Kirkâs iterative procedure was defined by Kirk [22]. For đĽ0 â đ¸, where đ¸ is a Banach space and đ is a self-map of đ¸,
đ = 1, 2, . . . , đ â 2 (6) (đâ1) đŚđâ1
đ
đĽđ+1 = âđźđ đđ đĽđ , đ = 0, 1, 2, . . . ,
= (1 â đ˝đ ) đĽđâ1 + đ˝đ đđĽđâ1 ,
(10)
đ=0
đ ⼠1, where {đźđ } and {đ˝đ(đ) }, đ = 1, 2, . . . , đ â 1, are sequences of positive numbers in [0, 1] with â đźđ = â. Iteration (6) generalized (3), (4), and (5); for example, if đ = 3 in (6), we recover the form (5); if đ = 2, we have (4); on putting đ = 2 and đ˝đ(đ) = 0 for each đ, we have (3). Another two-step scheme, which is independent of (4), was introduced by Thianwan [15]. Let đĽ0 â đ¸; the sequence {đĽđ } â đ¸ is defined as đĽđ = (1 â đźđ ) đŚđâ1 + đźđ đđŚđâ1
where đ is a fixed integer with đ ⼠1, đźđ ⼠0 for each đ and đ âđ=0 đźđ = 1. In order to reduce the cost of computations, Olatinwo [8] introduced two hybrid schemes, namely, Kirk-Mann and Kirk-Ishikawa iterative schemes in a normed linear space. For đĽ0 â đ¸, đ
đ
đ=0
đ=0
đĽđ+1 = âđźđ,đ đđ đĽđ ; âđźđ,đ = 1, đ = 0, 1, 2, . . . ,
đŚđâ1 = (1 â đ˝đ ) đĽđâ1 + đ˝đ đđĽđâ1 ,
(7)
đ
đ
đ=1
đ=0
đĽđ+1 = đźđ,0 đĽđ + âđźđ,đ đđ đŚđ , âđźđ,đ = 1,
đ ⼠1, where {đźđ } and {đ˝đ } are sequences of positive numbers in [0, 1] with â đźđ = â. The three-step iteration of (7) called SP iteration was introduced by Phuengrattana and Suantai [16] and it was defined as follows. For đĽ0 â đ¸,
đ
đŚđ = âđ˝đ,đ đđ đĽđ , đ=0
đ
âđ˝đ,đ = 1; đ=0
đ = 0, 1, 2 . . . ,
(1) (1) đĽđ = (1 â đźđ ) đŚđâ1 + đźđ đđŚđâ1 (1) (2) (2) đŚđâ1 = (1 â đ˝đ ) đŚđâ1 + đ˝đ đđŚđâ1
(8)
(2) đŚđâ1 = (1 â đžđ ) đĽđâ1 + đžđ đđĽđâ1 ,
đ ⼠1, where {đźđ }, {đ˝đ }, {đžđ } â [0, 1] with â đźđ = â. The two-step iteration of (8) can be easily obtained when đžđ = 0. In [17], G¨ursoy et al. defined a generalized scheme of forms (7) and (8) in a Banach space. For đĽ0 â đ¸, đĽđ = (1 â
(1) đźđ ) đŚđâ1
+
(1) đźđ đđŚđâ1
(11)
respectively, where đ and đ are fixed integers with đ ⼠đ and {đźđ,đ } and {đ˝đ,đ } are sequences in [0, 1] satisfying đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ ⼠0, and đ˝đ,0 ≠ 0. The Kirk-Noor iteration was introduced by Chugh and Kumar [23] as follows: for đĽ0 â đ¸, đ
đĽđ+1 = đźđ,0 đĽđ + âđźđ,đ đđ đŚđ , đ=1
đ
âđźđ,đ = 1, đ=0
đ
đ
đ=1
đ=0
đŚđ = đ˝đ,0 đĽđ + âđ˝đ,đ đđ đ§đ , âđ˝đ,đ = 1,
(đ) (đ+1) (đ+1) đŚđâ1 = (1 â đ˝đ(đ) ) đŚđâ1 + đ˝đ(đ) đđŚđâ1 ,
đ = 1, 2, 3, . . . , đ â 2 (9) (đâ1) = (1 â đ˝đ(đâ1) ) đĽđâ1 + đ˝đ(đâ1) đđĽđâ1 , đŚđâ1
đ
đ
đ=0
đ=0
(12)
đ§đ = âđžđ,đ đđ đĽđ , âđžđ,đ = 1; đ = 0, 1, 2 . . . ,
đ ⼠1, where {đźđ }, {đ˝đ(đ) }, â [0, 1], đ = 1, 2, . . . , đ â 1, with â đźđ = â. Several results have been proved for the strong convergence of the explicit iteration as well as the SP-iterative
where đ, đ, and đ are fixed integers with đ ⼠đ ⼠đ , and {đźđ,đ }, {đ˝đ,đ }, and {đžđ,đ } are sequences in [0, 1] satisfying đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ ⼠0, đ˝đ,0 ≠ 0, đžđ,đ ⼠0, and đžđ,0 ≠ 0.
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In an attempt to generalize (11) and (12), G¨ursoy et al. [17] introduced the Kirk-Multistep iteration in an arbitrary Banach space đ¸. For đĽ0 â đ¸, đ1
đ1
đ=1
đ=0
đĽđ+1 = đźđ,0 đĽđ + âđźđ,đ đđ đŚđ , âđźđ,đ = 1, đđ+1
(đ) (đ) đ (đ+1) đĽđ + â đ˝đ,đ đ đŚđ , đŚđ(đ) = đ˝đ,0 đ=1
đđ+1
(đ) = 1, đ = 1 (1) đ â 2 â đ˝đ,đ
(13)
đđ
đ=0
đ=0
đŚđâ1 = đ˝đ đĽđâ1 + (1 â đ˝đ ) đđŚđâ1 ,
(đâ1) đ (đâ1) đ đĽđ , âđ˝đ,đ = 1; đ ⼠2, đŚđ(đâ1) = âđ˝đ,đ
where đ1 , đ2 , đ3 , . . . , đđ are fixed integers with đ1 ⼠đ2 ⼠â
â
â
⼠(đ) â đđ ; {đźđ,đ }â đ=1 and {đ˝đ,đ }đ=1 are sequences in [0, 1] satisfying (đ) (đ) đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ ⼠0, and đ˝đ,0 ≠ 0 for each đ. The Kirk-SP iteration was defined by Hussain et al. [24] as follows: for đĽ0 â đ¸, đ
đ=1
đ=0
đĽđ = đźđ đŚđâ1 + (1 â đźđ ) đđĽđ đŚđâ1 = đ˝đ đ§đâ1 + (1 â đ˝đ ) đđŚđâ1 đ§đâ1 = đžđ đĽđâ1 + (1 â đžđ ) đđ§đâ1 ,
đŚđ = đ˝đ,0 đ§đ + âđ˝đ,đ đ đ§đ , âđ˝đ,đ = 1, đ=1
đ=0
đ
đ
đ=0
đ=0
(14)
đ§đ = âđžđ,đ đđ đĽđ , âđžđ,đ = 1; đ = 0, 1, 2 . . . , where đ, đ, and đ are fixed integers with đ ⼠đ ⼠đ and {đźđ,đ }, {đ˝đ,đ }, and {đžđ,đ } are sequences in [0, 1] satisfying đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ ⼠0, đ˝đ,0 ≠ 0, đžđ,đ ⼠0, and đžđ,0 ≠ 0. The KirkThianwan iteration can be obtained if đ = 0 in (14). The Kirk-Multistep-SP, which generalized both Kirk-SP and Kirk-Thianwan schemes, was introduced by Akewe et al. [1] and it was defined as follows: for đĽ0 â đ¸, đ1
đĽđ+1 = đźđ,0 đŚđ(1) + âđźđ,đ đđ đŚđ(1) , đ=1
đ1
âđźđ,đ = 1,
đđ+1
đđ
đđ
đ=0
đ=0
đ1 : đ (đđĽ, đđŚ) ⤠đđ (đĽ, đŚ) đ2 : đ (đđĽ, đđŚ) ⤠đ [đ (đĽ, đđĽ) + đ (đŚ, đđŚ)]
= 1, đ = 1 (1) đ â 2
(đâ1) đ (đâ1) đ đĽđ , âđ˝đ,đ = 1; đ ⼠2, đŚđ(đâ1) = âđ˝đ,đ
đ = 0, 1, 2 . . . ,
(19)
đ3 : đ (đđĽ, đđŚ) ⤠đ [đ (đĽ, đđŚ) + đ (đŚ, đđĽ)] , where đ, đ, and đ are nonnegative constants satisfying đ â [0, 1), đ, đ ⤠1/2. The equivalence form of (19) is
1 [đ (đĽ, đđĽ) + đ (đŚ, đđŚ)] , 2
đ=1
(đ) â đ˝đ,đ đ=0
where {đźđ }, {đ˝đ }, and {đžđ } are sequences in [0, 1] with â(1 â đźđ ) = â. The implicit Noor iteration (18) is more general than (16) and (17). The most generalized Banach operator used by several authors is the one proved by Zamfirescu [25]. Let đ be a complete metric space and let đ be a self-map of đ. The operator đ is Zamfirescu operator if for each pair of points đĽ, đŚ â đ, at least one of the following is true:
đ (đđĽ, đđŚ) ⤠đ max {đ (đĽ, đŚ) ,
đ=0
(đ) (đ+1) (đ) đ (đ+1) đŚđ + â đ˝đ,đ đ đŚđ , đŚđ(đ) = đ˝đ,0 đđ+1
(18)
đ ⼠1,
đ
đ
(17)
where {đźđ } and {đ˝đ } are sequences in [0, 1]. The implicit Noor iteration was defined by Chugh et al. [5] as follows: for đĽ0 â đś
đĽđ+1 = đźđ,0 đŚđ + âđźđ,đ đđ đŚđ , âđźđ,đ = 1, đ
(16)
đ ⼠1,
đ = 0, 1, 2, . . . ,
đ
đĽđ = đźđ đĽđâ1 + (1 â đźđ ) đđĽđ , đ ⼠1, đĽđ = đźđ đŚđâ1 + (1 â đźđ ) đđĽđ
đ=0
đđ
where đ1 , đ2 , đ3 , . . . , đđ are fixed integers with đ1 ⼠đ2 ⼠â
â
â
⼠(đ) â đđ ; {đźđ,đ }â đ=1 and {đ˝đ,đ }đ=1 are sequences in [0, 1] satisfying (đ) (đ) ⼠0, and đ˝đ,0 ≠ 0 for each đ. đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ Another modified form of explicit iteration is the implicit iteration. The implicit Mann iteration and implicit Ishikawa ´ c et al. [7] and Xue and Zhang iteration were discussed by Ciri´ [19], respectively. For đĽ0 â đś, đś being a closed subset of normed linear space, the implicit Mann and implicit Ishikawa iterations are, respectively,
(15)
(20)
1 [đ (đĽ, đđŚ) + đ (đŚ, đđĽ)]} 2 for đĽ, đŚ â đ and đ â [0, 1). Berinde [2] observed that condition (20) implies đ (đđĽ, đđŚ) ⤠2âđ (đĽ, đđĽ) + âđ (đĽ, đŚ) , where â = max{đ, đ/(2 â đ)}.
(21)
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In [26], Rhoades used a more general contractive condition than (21): for đĽ, đŚ â đ, there exists đ â [0, 1) such that đ (đđĽ, đđŚ) ⤠đ max {đ (đĽ, đŚ) , 1 [đ (đĽ, đđĽ) + đ (đŚ, đđŚ)] , đ (đĽ, đđŚ) , đ (đŚ, đđĽ)} . 2
(22) đ˘đ+1 ⤠đżđ˘đ + đđ , âđ â N
Osilike [20] extended and generalized the contractive condition (22): for đĽ, đŚ â đ, there exists đ â [0, 1) and đż ⼠0 such that đ (đđĽ, đđŚ) ⤠đżđ (đĽ, đđĽ) + đđ (đĽ, đŚ) .
(23)
Imoru and Olatinwo [21] employed a more general class of operators đ than (23) satisfying the following contractive conditions: đ (đđĽ, đđŚ) ⤠đđ (đĽ, đŚ) + đ (đ (đĽ, đđĽ))
for đĽ, đŚ â đ, (24)
where đ â [0, 1) and đ : R+ â R+ is a monotone increasing function with đ(0) = 0. The equivalence form of (24) in a normed linear space is óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠđđĽ â đđŚóľŠóľŠóľŠ ⤠đ óľŠóľŠóľŠđĽ â đŚóľŠóľŠóľŠ + đ (âđĽ â đđĽâ)
for đĽ, đŚ â đ (25)
We will need the following definitions and lemmas to prove our main results. Definition 1 (see [9]). Let (đ, đ) be a metric space and đ : đ â đ a self-mapping. Suppose that đšđ = {đ â đ : đđ = đ} is the set of fixed points of đ. Let {đĽđ }â đ=0 â đ be the sequence generated by an iterative procedure involving đ which is defined by đĽ
đĽđ+1 = đđ,đźđ đ ,
đ ⼠0,
Lemma 3 (see [2]). Let đż be a real number such that đż â [0, 1) and {đđ }â đ=0 is a sequence of nonnegative numbers such that limđââ đđ = 0; then, for any sequence of positive numbers {đ˘đ }â đ=0 satisfying
(26) đĽ
where đĽ0 â đ is the initial approximation and đđ,đźđ đ is a function such that đźđ â [0, 1]. Suppose that {đĽđ } converges to a fixed point đ of đ. Let {đŚđ }â đ=0 â đ and set đđ = đŚ đ(đŚđ+1 , đđ,đźđ đ ), đ = 0, 1, 2, . . .. Then, iterative procedure (26) is said to be đ-stable or stable with respect to đ if and only if limđââ đđ = 0 implies limđââ đŚđ = đ. â Definition 2 (see [2]). Let {đđ }â đ=0 and {đđ }đ=0 be two nonnegative real sequences which converge to đ and đ, respectively. Let óľ¨óľ¨óľ¨đ â đóľ¨óľ¨óľ¨ óľ¨; (27) đ = lim óľ¨óľ¨óľ¨ đ đââ óľ¨đ â đóľ¨óľ¨óľ¨ óľ¨đ óľ¨ â (1) if đ = 0, then {đđ }â đ=0 converges to đ faster than {đđ }đ=0 to đ; â (2) if 0 < đ < â, then both {đđ }â đ=0 and {đđ }đ=0 have the same convergence rate; â (3) if đ = â, then {đđ }â đ=0 converges to đ faster than {đđ }đ=0 to đ.
(28)
we have limđââ đ˘đ = 0. Lemma 4 (see [8]). Let (đ¸, â â
â) be a normed linear space and đ : đ¸ â đ¸ a map satisfying (25). Let đ : R+ â R+ be a subadditive, monotone increasing function such that đ(0) = 0, đ(đżđ˘) = đżđ(đ˘), for đ˘ â R+ , đż ⼠0. Then, for all đ â N, đż ⼠0 and for all đĽ, đŚ â đ¸ óľŠ óľŠóľŠ đ óľŠóľŠđ đĽ â đđ đŚóľŠóľŠóľŠ óľŠ óľŠ đ đ óľŠ óľŠ â¤ â ( ) đđâđ đđ (âđĽ â đđĽâ) + đđ óľŠóľŠóľŠđĽ â đŚóľŠóľŠóľŠ . đ đ=1
(29)
Note that đ â [0, 1) in (29).
2. Main Results We present our main results as follows. Let đ¸ be an arbitrary Banach space and đ : đ¸ â đ¸ a selfmap. Let đĽ0 â đ¸; we define the following iteration, namely, implicit hybrid Kirk-Multistep iterative scheme, as follows: đ1
(1) đĽđ = đźđ,0 đĽđâ1 + âđźđ,đ đđ đĽđ , đ=1
đ1
âđźđ,đ = 1 đ=0
đđ+1
(đ) (đ) (đ+1) (đ) đ (đ) = đ˝đ,0 đĽđâ1 + â đ˝đ,đ đ đĽđâ1 , đĽđâ1 đ=1
đđ+1
(đ) = 1, đ = 1 (1) đ â 2 â đ˝đ,đ đ=0
(30)
đđ
(đâ1) (đâ1) (đâ1) đ (đâ1) = đ˝đ,0 đĽđâ1 + âđ˝đ,đ đ đĽđâ1 , đĽđâ1 đ=1
đđ
(đâ1) = 1, đ ⼠2, âđ˝đ,đ đ=0
đ ⼠1, where đ1 , đ2 , đ3 , . . . , đđ are fixed integers with đ1 ⼠đ2 ⼠(đ) â đ3 ⼠â
â
â
⼠đđ ; {đźđ,đ }â đ=1 and {đ˝đ,đ }đ=1 are sequences in [0, 1] (đ) (đ) ≠ 0 for each đ satisfying đźđ,đ ⼠0, đźđ,0 ≠ 0, đ˝đ,đ ⼠0, and đ˝đ,0 â with âđ=1 (1 â đźđ,0 ) = â.
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If we let đ = 3 in (30), we obtain the implicit Kirk-Noor iteration defined by đ1
đ1
đ=1
đ=0
This implies that đźđ,0 óľŠ óľŠóľŠ óľŠóľŠóľŠđĽ(1) â đóľŠóľŠóľŠ . óľŠóľŠđĽđ â đóľŠóľŠóľŠ â¤ óľŠ óľŠóľŠ đ1 1 â âđ=1 đźđ,đ đđ óľŠ đâ1
(1) đĽđ = đźđ,0 đĽđâ1 + âđźđ,đ đđ đĽđ , âđźđ,đ = 1 đ2
đ2
đ=1
đ=0
đ3
đ3
Also, from (30), we have
(1) (1) (2) (1) đ (1) (1) = đ˝đ,0 đĽđâ1 + âđ˝đ,đ đ đĽđâ1 , âđ˝đ,đ =1 đĽđâ1
(2) đĽđâ1
=
(2) đ˝đ,0 đĽđâ1
+
(2) đ (2) đ đĽđâ1 , âđ˝đ,đ đ=1
(2) âđ˝đ,đ đ=0
(31)
= 1,
(1) óľŠ óľŠóľŠđĽ(2) â đóľŠóľŠóľŠ ⤠đ˝đ,0 óľŠóľŠ đâ1 óľŠóľŠ
Theorem 5. Let (đ¸, â â
â) be a normed linear space. Assume đ is self-map of đ¸ satisfying the contractive condition (29) with đšđ ≠ đ. Then, for đĽ0 â đ¸, the sequence {đĽđ } defined by (30) with ââ đ=1 (1 â đźđ,0 ) = â converges strongly to the fixed point đ â đšđ . Proof. Let đĽ0 â đ and đ â đšđ ; then using (29) and (30) we have đ
1 óľŠ (1) óľŠ đ óľŠ óľŠ óľŠóľŠ đ óľŠ óľŠóľŠđĽđ â đóľŠóľŠóľŠ ⤠đźđ,0 óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ + âđźđ,đ óľŠóľŠóľŠóľŠđ đĽđ â đ đóľŠóľŠóľŠóľŠ
đ=1
óľŠ óľŠ + âđźđ,đ [ â ( ) đ đ (óľŠóľŠóľŠđ â đđóľŠóľŠóľŠ) đ đ=1 [đ=1 đâđ đ
đ
1 óľŠ (1) óľŠ óľŠ óľŠ óľŠ + đđ óľŠóľŠóľŠđĽđ â đóľŠóľŠóľŠ] = đźđ,0 óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ + âđźđ,đ đđ óľŠóľŠóľŠđĽđ đ=1 ]
óľŠ â đóľŠóľŠóľŠ .
(34)
óľŠ (1) óľŠ (1) óľŠ óľŠóľŠđĽ(2) â đóľŠóľŠóľŠ + đđ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ] = đ˝đ,0 óľŠóľŠ đâ1 óľŠóľŠ ] đ
2 óľŠ (1) đ óľŠ (1) + âđ˝đ,đ đ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ .
đ=1
This becomes đ˝đ(1) óľŠóľŠ (1) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ â¤ óľŠ óľŠ 1 â âđ2 đ˝(1) đđ đ=1 đ,đ
óľŠóľŠ (2) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ . óľŠ óľŠ
(35)
From (30) again, we have đ
3 óľŠóľŠ (2) óľŠ (2) óľŠ óľŠóľŠđĽ(3) â đóľŠóľŠóľŠ + âđ˝(2) óľŠóľŠóľŠđđ đĽ(2) â đđ đóľŠóľŠóľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ ⤠đ˝đ,0 óľŠ óľŠ óľŠóľŠ đ,đ óľŠ óľŠ óľŠ đâ1 óľŠ đâ1 óľŠ óľŠ
đ=1
(2) óľŠ óľŠóľŠđĽ(3) â đóľŠóľŠóľŠ ⤠đ˝đ,0 óľŠóľŠ đâ1 óľŠóľŠ đ3 đ đ óľŠ óľŠ (2) [ + âđ˝đ,đ â ( ) đđâđ đđ (óľŠóľŠóľŠđ â đđóľŠóľŠóľŠ) đ=1 [đ=1 đ
(36)
óľŠ (2) óľŠ (2) óľŠ óľŠóľŠđĽ(3) â đóľŠóľŠóľŠ + đđ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ] = đ˝đ,0 óľŠóľŠ đâ1 óľŠóľŠ ]
óľŠ (1) óľŠ â¤ đźđ,0 óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ đ
đ đ óľŠ óľŠ (1) [ + âđ˝đ,đ â ( ) đđâđ đđ (óľŠóľŠóľŠđ â đđóľŠóľŠóľŠ) đ=1 [đ=1 đ đ2
By setting đ = 2 in (30), we have the implicit Kirk-Ishikawa iteration and we can also obtain the implicit Kirk-Mann iteration when đ = 2 and đ2 = 0 in (30). If đ1 = đ2 = đ3 = â
â
â
= đđ = 1 in (30), then we obtain the (đ) implicit multistep iteration (19) with đźđ,1 = đźđ , đ˝đ,1 = đ˝đ(đ) . If đ = 3, đ1 = đ2 = đ3 = 1, and đ4 = đ5 = â
â
â
= đđ = 0 in (30), we have the implicit Noor scheme (18) with đźđ,1 = đźđ , (1) (2) = đ˝đ(1) , and đ˝đ,1 = đ˝đ(2) . đ˝đ,1 If đ = 2, đ1 = đ2 = 1, and đ3 = đ4 = â
â
â
= đđ = 0 in (30), we have the implicit Ishikawa scheme (17) with đźđ,1 = đźđ , (1) = đ˝đ(1) . đ˝đ,1 If đ = 2, đ1 = 1, and đ2 = đ3 = đ4 = â
â
â
= đđ = 0 in (30), we have the implicit Mann scheme (16) with đźđ,1 = đźđ . Throughout, the operator đ will be assumed as fixed and a fixed point đ â đšđ with the condition (29) is unique.
đ
đ
2 óľŠóľŠ (1) óľŠ (1) óľŠ óľŠóľŠđĽ(2) â đóľŠóľŠóľŠ + âđ˝(1) óľŠóľŠóľŠđđ đĽ(1) â đđ đóľŠóľŠóľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ ⤠đ˝đ,0 óľŠ óľŠ óľŠóľŠ đâ1 đ,đ óľŠ óľŠ óľŠ óľŠ đâ1 óľŠ óľŠ
đ=1
đ ⼠1.
đ1
(33)
đ
(32)
3 óľŠ (2) đ óľŠ (2) + âđ˝đ,đ đ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ
đ=1
which implies đ˝đ(2) óľŠóľŠ (2) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ â¤ óľŠ óľŠ 1 â âđ3 đ˝(2) đđ đ=1 đ,đ
óľŠóľŠ (3) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ . óľŠ óľŠ
(37)
6
International Journal of Analysis
Continuing this way up to đ â 1 in (30), we have
Similarly, we can easily obtain the following from (40): (1) đ˝đ,0
đđ
óľŠóľŠ (đâ1) óľŠ (đâ1) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ + âđ˝(đâ1) óľŠóľŠóľŠóľŠđđ đĽ(đâ1) óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ ⤠đ˝đ,0 đ,đ óľŠ óľŠ óľŠ óľŠ đâ1 óľŠ
đ
(1) đ 2 1 â âđ=1 đ˝đ,đ đ
đ=1
(2) đ˝đ,0
óľŠ (đâ1) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ â đ đóľŠóľŠóľŠóľŠ ⤠đ˝đ,0 óľŠ óľŠ đ
1â
đđ đ đ óľŠ óľŠ (đâ1) [ + âđ˝đ,đ â ( ) đđâđ đđ (óľŠóľŠóľŠđ â đđóľŠóľŠóľŠ) đ đ=1 [đ=1
đ3 âđ=1
(2) đ đ˝đ,đ đ
(3) đ˝đ,0
(38)
1â
đ4 âđ=1
(3) đ đ˝đ,đ đ
đ2
đ2
đ=1
đ=1
đ3
đ3
đ=1
đ=1
đ4
đ4
đ=1
đ=1
(1) đ (1) ⤠âđ˝đ,đ đ < âđ˝đ,đ =1
(2) đ (2) ⤠âđ˝đ,đ đ < âđ˝đ,đ =1
(3) đ (3) ⤠âđ˝đ,đ đ < âđ˝đ,đ =1
(43)
.. .
óľŠ (đâ1) óľŠ (đâ1) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ + đđ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ] = đ˝đ,0 óľŠ óľŠ ]
(đâ1) đ˝đ,0 đ
(đâ1) đ đ 1 â âđ=1 đ˝đ,đ đ
đ
đ óľŠ (đâ1) đ óľŠ (đâ1) + âđ˝đ,đ đ óľŠóľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠóľŠ
đđ
đđ
đ=1
đ=1
(đâ1) đ (đâ1) ⤠âđ˝đ,đ đ < âđ˝đ,đ = 1.
Applying (42) and (43) in (40) and letting đđ ⤠đ < 1 for each đ, we have
đ=1
đ1
implying that
óľŠóľŠ óľŠ óľŠ óľŠ đ óľŠóľŠđĽđ â đóľŠóľŠóľŠ ⤠(âđźđ,đ đ + đźđ,0 ) óľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ đ=1
óľŠóľŠ (đâ1) óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ â¤ óľŠ óľŠ 1â
đ˝đ(đâ1) đđ (đâ1) đ đ˝đ,đ đ âđ=1
óľŠóľŠ óľŠ óľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ .
(39)
óľŠ óľŠ = [1 â (1 â đźđ,0 ) (1 â đ)] óľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ
Substituting (35)â(39) into (33) it becomes
⤠[1 â (1 â đźđ,0 ) (1 â đ)] [1 â (1 â đźđâ1,0 ) (1 â đ)]
đ˝đ(2) đ
(2) đ 3 1 â âđ=1 đ˝đ,đ đ
Ă â
â
â
(
1â
)
.. . (40)
đ˝đ(đâ1) óľŠ ) óľŠóľŠóľŠđĽđâ1 đđ (đâ1) đ âđ=1 đ˝đ,đ đ
óľŠ â đóľŠóľŠóľŠ .
đ óľŠ óľŠ â¤ đâ(1âđ) âđ=1 (1âđźđ,0 ) óľŠóľŠóľŠđĽ0 â đóľŠóľŠóľŠ .
As đ â â, ââ đ=1 (1 â đźđ,0 ) = â. Hence, limđââ âđĽđ â đâ = 0. Therefore, implicit Kirk-Multistep scheme (30) converges strongly to đ â đšđ .
đ
1 â đđ = 1 â
đ
1 1 â âđ=1 đźđ,đ đđ
đ
=
đ
óľŠ óľŠ â¤ â [1 â (1 â đźđ,0 ) (1 â đ)] óľŠóľŠóľŠđĽ0 â đóľŠóľŠóľŠ đ=1
1 Let đ đ = đźđ,0 /(1 â âđ=1 đźđ,đ đđ ); then
đźđ,0
(44)
óľŠ óľŠ â
óľŠóľŠóľŠđĽđâ2 â đóľŠóľŠóľŠ
đźđ,0 đ˝đ(1) óľŠóľŠ óľŠ ) ( ) óľŠóľŠđĽđ â đóľŠóľŠóľŠ ⤠( đ1 đ2 (1) đ 1 â âđ=1 đźđ,đ đđ 1 â âđ=1 đ˝đ,đ đ â
(
óľŠ óľŠ â¤ [(1 â đźđ,0 ) đ + đźđ,0 ] óľŠóľŠóľŠđĽđâ1 â đóľŠóľŠóľŠ
1 đźđ,đ đđ â đźđ,0 1 â âđ=1
đ
1 1 â âđ=1 đźđ,đ đđ
(41)
đ1
⼠1 â (âđźđ,đ đđ + đźđ,0 ) .
Corollary 6. Let (đ¸, ââ
â) be a normed linear space. Assume đ is self-map of đ¸ satisfying the contractive condition (29) with đšđ ≠ đ. Then, for đĽ0 â đ¸, the implicit Kirk-Noor, the implicit KirkIshikawa, and the implicit Kirk-Mann schemes with ââ đ=1 (1 â đźđ,0 ) = â converge strongly to the fixed point đ â đšđ . Remark 7. The strong convergence results for implicit Noor, implicit Ishikawa, and implicit Mann schemes are obvious from Theorem 5.
đ=1
Therefore, đ1
đ1
đ1
đ=1
đ=0
đ=0
đ đ ⤠âđźđ,đ đđ + đźđ,0 = âđźđ,đ đđ < âđźđ,đ = 1.
(42)
Theorem 8. Let (đ¸, â â
â) be a normed linear space and đ is a self-map of đ¸ satisfying contractive condition (29) with đšđ ≠ đ. Then, for đĽ0 â đ¸ and đ â đšđ, the sequence {đĽđ } defined by (30) is T-stable.
International Journal of Analysis
7
Proof. Let {đŚđ } â đ¸ be an arbitrary sequence and let đđ = đ1 (1) âđŚđ â đźđ,0 đ§đâ1 â âđ=1 đźđ,đ đđ đŚđ â, where đ2
(1) (1) (2) (1) đ (1) đ§đâ1 = đ˝đ,0 đ§đâ1 + âđ˝đ,đ đ đ§đâ1 , đ=1
đ2
(1) = 1, âđ˝đ,đ đ=0
đđ+1
(đ) (đ) (đ+1) (đ) đ (đ) = đ˝đ,0 đ§đâ1 + â đ˝đ,đ đ đ§đâ1 , đ§đâ1
Conversely, suppose âđŚđ â đâ = 0 for đ â đšđ ; then óľŠóľŠ óľŠóľŠ đ1 óľŠóľŠ óľŠóľŠ (1) â âđźđ,đ đđ đŚđ óľŠóľŠóľŠ đđ = óľŠóľŠóľŠđŚđ â đźđ,0 đ§đâ1 óľŠóľŠ óľŠóľŠ đ=1 óľŠ óľŠ óľŠóľŠ óľŠóľŠ đ1 óľŠóľŠ óľŠ óľŠ óľŠóľŠ (1) ⤠óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ + óľŠóľŠóľŠđ â (đźđ,0 đ§đâ1 + âđźđ,đ đđ đŚđ )óľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ đ=1 óľŠ óľŠ đ
đ=1
1 óľŠ (1) óľŠ óľŠ óľŠ óľŠ óľŠ â¤ óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ + đźđ,0 óľŠóľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ + âđźđ,đ óľŠóľŠóľŠóľŠđđ đŚđ â đđ đóľŠóľŠóľŠóľŠ (51)
(45)
đđ+1
đ=1
(đ) = 1, đ = 1 (1) đ â 2 â đ˝đ,đ
đ1
đ=0
đđ
(đâ1) (đâ1) (đ) đ (đâ1) = đ˝đ,0 đŚđâ1 + âđ˝đ,đ đ đ§đâ1 , đ§đâ1 đ=1
óľŠ óľŠ óľŠ óľŠ óľŠ óľŠ â¤ óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ + đźđ,0 óľŠóľŠóľŠđŚđâ1 â đóľŠóľŠóľŠ + âđźđ,đ đđ óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ
đđ
đ=1
(đâ1) = 1. âđ˝đ,đ
đ1
đ=0
óľŠ óľŠ óľŠ óľŠ â¤ (1 + âđźđ,đ đđ ) óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ + đźđ,0 óľŠóľŠóľŠđŚđâ1 â đóľŠóľŠóľŠ .
Suppose limđââ đđ = 0 and đ â đšđ ; by (29) we have óľŠóľŠ óľŠóľŠ đ1 óľŠ óľŠóľŠ óľŠóľŠóľŠ óľŠóľŠ (1) đ óľŠ óľŠóľŠđŚđ â đóľŠóľŠ ⤠óľŠóľŠđŚđ â đźđ,0 đ§đâ1 â âđźđ,đ đ đŚđ óľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ đ=1 óľŠ óľŠ óľŠóľŠ óľŠóľŠ đ1 óľŠóľŠ óľŠóľŠ (1) + óľŠóľŠóľŠđźđ,0 đ§đâ1 + âđźđ,đ đđ đŚđ â đóľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ đ=1 óľŠ óľŠ đ
1 óľŠ (1) óľŠ óľŠ óľŠ â¤ đđ + đźđ,0 óľŠóľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ + âđźđ,đ óľŠóľŠóľŠóľŠđđ đŚđ â đóľŠóľŠóľŠóľŠ
đ=1
Since âđŚđ â đâ â 0, then limđââ đđ = 0. Therefore, iterative scheme (30) is T-stable.
(46)
đ=1
Corollary 9. Let (đ¸, â â
â) be a normed linear space and đ is a self-map of đ¸ satisfying the contractive condition (29) with đšđ ≠ đ. Then, for đĽ0 â đ¸ and đ â đšđ , the sequence {đĽđ } defined by implicit Kirk-Mann, implicit Kirk-Ishikawa, and implicit KirkNoor schemes are T-stable. Remark 10. The stability results for implicit Mann, implicit Ishikawa, and implicit Noor schemes with contractive condition (29) are special cases of Corollary 9.
đ1
óľŠ (1) óľŠ óľŠ óľŠ â¤ đđ + đźđ,0 óľŠóľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ + âđźđ,đ đđ óľŠóľŠóľŠđŚđ â đóľŠóľŠóľŠ . đ=1
2.1. Comparison of Several Iterative Schemes. We compare our iterative schemes with others by using the following example.
This implies that đđ óľŠóľŠ óľŠ óľŠóľŠđŚđ â đóľŠóľŠóľŠ ⤠đ1 1 â âđ=1 đźđ,đ đđ đźđ,0
óľŠóľŠ (1) óľŠ óľŠóľŠđ§đâ1 â đóľŠóľŠóľŠ . + đ1 đ óľŠ óľŠ 1 â âđ=1 đźđ,đ đ
(47)
Example 11. Let đ : [0, 1] â [0, 1] and đđĽ = đĽ/2 with đĽ0 ≠ 0 (đ) = 1 â 2/âđ, đźđ,đ = and fixed point đ = 0 using đźđ,0 = đ˝đ,0 (đ) đ˝đ,đ = 4/âđ, for each đ, đ ⼠25, and đ1 = đ2 = đ3 = đ4 = 2. For the implicit Kirk-Mann iteration (IKM), we have
From inequalities (42) and (43), one can easily obtain the following: óľŠóľŠ (1) óľŠ óľŠ (2) óľŠ óľŠ (3) óľŠ óľŠóľŠđ§đâ1 â đóľŠóľŠóľŠ ⤠óľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ ⤠óľŠóľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ ⤠â
â
â
óľŠ óľŠ óľŠ óľŠ (đâ1) óľŠ óľŠ óľŠ â¤ óľŠóľŠóľŠóľŠđ§đâ1 â đóľŠóľŠóľŠóľŠ ⤠óľŠóľŠóľŠđŚđâ1 â đóľŠóľŠóľŠ .
2
đĽđ = đźđ,0 đĽđâ1 + âđźđ,đ đđ đĽđ đ=1
2 2 1 ) đĽđâ1 + đĽđ + đĽ . = (1 â âđ âđ âđ đ
(48)
This implies that
Then, inequality (47) becomes đđ óľŠóľŠ óľŠ óľŠóľŠđŚđ â đóľŠóľŠóľŠ ⤠đ1 1 â âđ=1 đźđ,đ đđ đźđ,0
óľŠóľŠ óľŠ + óľŠđŚđâ1 â đóľŠóľŠóľŠ . đ1 1 â âđ=1 đźđ,đ đđ óľŠ
đĽđ (IKM) = (49)
đ 2âđ â 4 2âđ â 4 )đĽ . đĽđâ1 = â ( 2âđ â 3 2âđ â3 0 đ=25
đ
(50)
(53)
Also, for implicit Kirk-Ishikawa iteration (IKI), we have đĽđ =
1 Letting đż = đźđ,0 /(1 â âđ=1 đźđ,đ đđ ) < 1 and by Lemma 3, we have
óľŠ óľŠóľŠ óľŠóľŠđŚđ â đóľŠóľŠóľŠ = 0.
(52)
2âđ â 4 (1) đĽ 2âđ â 3 đâ1
(54)
2âđ â 4 đĽ . 2âđ â 3 đâ1
(55)
with (1) đĽđâ1 =
8
International Journal of Analysis
Hence,
Similarly, using Definition 2, we have that 2
đĽđ (IKI) = (
óľ¨ óľ¨óľ¨ óľ¨đĽ (IKI) â 0óľ¨óľ¨óľ¨ =0 lim óľ¨óľ¨óľ¨ đ đââ óľ¨đĽ (IKM) â 0óľ¨óľ¨óľ¨ óľ¨ đ óľ¨
2
đ 2âđ â 4 2âđ â 4 ) đĽđâ1 = â ( ) đĽ0 . (56) 2âđ â 3 2âđ â3 đ=25
Similarly, implicit Kirk-Noor iteration (IKN) implies 3
đĽđ (IKN) = (
3
đ 2âđ â 4 2âđ â 4 ) đĽđâ1 = â ( ) đĽ0 (57) 2âđ â 3 2âđ â3 đ=25
while the implicit multistep Kirk iteration (IMK) gives
which implies that the implicit Kirk-Ishikawa iteration đĽđ (IKI) converges faster than the implicit Kirk-Mann iteration đĽđ (IKM). For the Kirk-Mann iteration (KM), we have the following estimate: 2
đĽđ = đźđ,0 đĽđâ1 + âđźđ,đ đđ đĽđâ1
đ
đ
đĽđ (IMK) = â ( đ=25
2âđ â 4 ) đĽ0 . 2âđ â 3
(58)
Now, using Definition 2, we compare the implicit Kirk type iterations as follows: for đ ⼠4, we have óľ¨óľ¨ óľ¨ đâ3 đ óľ¨óľ¨đĽđ (IMK) â 0óľ¨óľ¨óľ¨ 2âđ â 4 = ) ( â óľ¨óľ¨óľ¨đĽđ (IKN) â 0óľ¨óľ¨óľ¨ óľ¨ đ=25 2âđ â 3 óľ¨ (59) đâ3 đ 1 ) = â (1 â 2âđ â 3 đ=25
đ=1
2 2 1 )đĽ + đĽ + đĽ . = (1 â âđ đâ1 âđ đâ1 âđ đâ1
đĽđ (KM) = (1 +
đ âđ âđ ) đĽđâ1 = â (1 + ) đĽ0 . đ đ đ=25
2
đ
đâ3 1 0 ⤠lim â (1 â ) đââ 2âđ â 3 đ=25
đĽđ (KT) = â (1 + đ=25
đĽđ (KSP) = â (1 + đ=25
(60)
đĽđ (KMSP) = â (1 + đ=25
24 đâ3 ) = 0, âđ ⼠4. đââ đ
Remark 12. The implicit multistep Kirk iteration (IMK) converges faster than the implicit Kirk-Noor iteration (IKN) for đ = 4, 5, . . ..
đ
1 = â (1 â ) 2âđ â3 đ=25
âđ ) đĽ0 . đ
óľ¨ óľ¨óľ¨ đ óľ¨óľ¨đĽđ (IKM) â 0óľ¨óľ¨óľ¨ đ 2âđ â 4 = )( ) ( â óľ¨óľ¨óľ¨đĽđ (KM) â 0óľ¨óľ¨óľ¨ âđ â 3 +đ 2âđ óľ¨ đ=25 óľ¨ đ
= â( đ=25
(61)
2đ3/2 â 4đ ) â 3đ1/2 â đ 3 (đ â đ1/2 )
đ
đ=25
(67)
2đ3/2
= â (1 â
with
(66)
We compare Kirk-Mann (KM), Kirk-Thianwan (KT), KirkSP (KSP), and Kirk-Multistep-SP (KMSP) iterations with our iterative schemes as follows. Again, using Definition 2, we have
Also, óľ¨óľ¨ óľ¨ đ óľ¨óľ¨đĽđ (IKN) â 0óľ¨óľ¨óľ¨ 2âđ â 4 ) óľ¨óľ¨ óľ¨óľ¨ = â ( 2âđ â3 đĽ â 0 (IKI) óľ¨óľ¨ đ óľ¨óľ¨ đ=25
âđ ) đĽ0 , đ đ
đ
đâ3
âđ ) đĽ0 , đ 3
đ
đ
= ( lim
(65)
The estimates for Kirk-Thianwan (KT), Kirk-SP (KSP), and Kirk-Multistep-SP (KMSP) iterations are, respectively,
đ
24 25 đâ2 đâ1 = ( lim â
â
â
â
â
) đââ 25 26 đâ1 đ
(64)
This implies that
with
1 đâ3 ⤠lim â (1 â ) đââ đ đ=25
(63)
2đ3/2 â 3đ1/2 â đ
)
with đ
0 ⤠lim â (1 â đââ
= lim
đ=25
đ
1 1 ) ⤠lim â (1 â ) đââ đ 2âđ â 3 đ=25
24 25 24 đâ2 đâ1 â
â
â
â
â
= lim = 0. đââ 26 đâ1 đ đ
đ
(62)
đââ 25
Remark 13. The implicit Kirk-Noor iteration đĽđ (IKN) converges to đ = 0 faster than the implicit Kirk-Ishikawa iteration đĽđ (IKI) to đ = 0.
0 ⤠lim â (1 â đââ đ=25
3 (đ â đ1/2 ) 2đ3/2 â 3đ1/2 â đ
)
đ 24 25 1 đ â 2 đ â 1 (68) ⤠lim â (1 â ) = lim â
â
â
â
â
đââ đââ 25 26 đ đ â1 đ đ=25
= lim
đââ
24 = 0. đ
International Journal of Analysis
9
Remark 14. The implicit Kirk-Mann iteration đĽđ (IKM) converges to đ = 0 faster than the Kirk-Mann iteration đĽđ (KM) to đ = 0. For the comparison of implicit Kirk-Ishikawa iteration đĽđ (IKI) and Kirk-Thianwan iteration đĽđ (KT), we have óľ¨óľ¨ óľ¨ 2 đ óľ¨óľ¨đĽđ (IKI) â 0óľ¨óľ¨óľ¨ 2âđ â 4 đ )( )] óľ¨óľ¨ óľ¨óľ¨ = â [( âđ + đ óľ¨óľ¨đĽđ (KT) â 0óľ¨óľ¨ đ=25 2âđ â 3 đ
= [ â (1 â đ=25
3 (đ â đ1/2 ) 2đ3/2 â 3đ1/2 â đ
(69)
2
For the comparison of implicit Kirk-Multistep iteration đĽđ (IKM) and Kirk-Multistep-SP iteration đĽđ (KMSP), we have óľ¨ óľ¨óľ¨ đ đ óľ¨óľ¨đĽđ (IKM) â 0óľ¨óľ¨óľ¨ đ 2âđ â 4 = ) ( )] [( â óľ¨óľ¨óľ¨đĽđ (KMSP) â 0óľ¨óľ¨óľ¨ âđ + đ óľ¨ óľ¨ đ=25 2âđ â 3 (73) đ đ 3 (đ â đ1/2 ) = [ â (1 â 3/2 )] 2đ â 3đ1/2 â đ đ=25 with
)]
đ
0 ⤠[ lim â (1 â đââ đ=25
with
3 (đ â đ1/2 ) 2đ3/2 â 3đ1/2 â đ
đ
0 ⤠[ lim â (1 â đââ đ=25
2
3 (đ â đ1/2 )
đ
2đ3/2 â 3đ1/2 â đ
đ 1 ⤠[ lim â (1 â )] đââ đ đ=25
1 ⤠[ lim â (1 â )] đââ đ đ=25
)]
(74)
= 0. Remark 15. The implicit Kirk-Ishikawa iteration đĽđ (IKI) converges faster than the Kirk-Thianwan iteration đĽđ (KT). For the comparison of implicit Kirk-Noor iteration đĽđ (IKN) and Kirk-SP iteration đĽđ (KSP), we have óľ¨óľ¨ óľ¨ 3 đ óľ¨óľ¨đĽđ (IKN) â 0óľ¨óľ¨óľ¨ 2âđ â 4 đ = [( ) ( )] â óľ¨óľ¨óľ¨đĽđ (KSP) â 0óľ¨óľ¨óľ¨ âđ + đ óľ¨ đ=25 2âđ â 3 óľ¨ 3 (đ â đ1/2 ) 2đ3/2 â 3đ1/2 â đ
3
= 0. Remark 17. The implicit Kirk-Multistep iteration đĽđ (IKM) has better convergence rate than the Kirk-Multistep-SP iteration đĽđ (KMSP) for đ ⼠4.
24 25 24 2 đâ2 đâ1 2 â
â
â
â
â
) = ( lim ) đââ 25 26 đââ đ đâ1 đ
= ( lim
đ=25
đ
24 25 24 đ đâ2 đâ1 đ â
â
â
â
â
) = ( lim ) đââ 25 26 đââ đ đâ1 đ
(70)
= [ â (1 â
)]
= ( lim
2
đ
đ
(71)
)]
with
3. Conclusion We have established and proved strong convergence and Tstability results for implicit Kirk-Multistep, implicit KirkNoor, implicit Kirk-Ishikawa, and implicit Kirk-Mann iterative schemes of fixed points with contractive-type operators in normed linear spaces. These iterative schemes have better convergence rate when compared with other iterative schemes, namely, multistep Kirk-SP iteration, Kirk-Multistep scheme, Kirk-SP iteration, Kirk-Thianwan scheme, KirkNoor scheme, Kirk-Ishikawa scheme, Kirk-Mann scheme, implicit Noor iteration, implicit Ishikawa iteration, implicit Mann iteration, and many more iterative schemes of fixed point in the literature.
Disclosure đ
0 ⤠[ lim â (1 â đââ đ=25
2đ3/2 â 3đ1/2 â đ
đ 1 ⤠[ lim â (1 â )] đââ đ đ=25
= ( lim
3
3 (đ â đ1/2 )
Authors agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.
)]
3
(72) 3
24 25 24 đâ2 đâ1 â
â
â
â
â
) = ( lim ) đââ 26 đâ1 đ đ
3
đââ 25
= 0. Remark 16. The implicit Kirk-Noor iteration đĽđ (IKN) has better convergence rate than the Kirk-SP iteration đĽđ (KSP).
Competing Interests The authors hereby declare that there are no competing interests.
References [1] H. Akewe, G. A. Okeke, and A. F. Olayiwola, âStrong convergence and stability of Kirk-multistep-type iterative schemes for
10
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