# Estimation of Newly Established Iterative Scheme for Generalized Nonexpansive Mappings

Authors: Aftab Hussain; Nawab Hussain; Danish Ali
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We introduce a new iterative method in this article, called theD iterative approach for fixed point approximation. Analytically, and also numerically, we demonstrate that our established D I.P is faster than the well-known I.P of the prior art. Finally, in a uniformly convex Banach space environment, we present weak as well as strong convergence theorems for Suzuki’s generalized nonexpansive maps. Our findings are an extension, refinement, and induction of several existing iterative literatures.


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## Body

## 1. Introduction and Preliminaries

In many branches of mathematics and various sciences, the existence of a fixed point is crucial. A completely fine synthesis of analysis, geometry, and topology is the fixed point theory. In particular, fixed point techniques are used in economics, biology, engineering, biochemistry, game theory, and physics. So, once the presence of a fixed point is found, it is difficult to find its value; that is why we use an I.P to calculate them. A variety of I.Ps have been introduced and it is not possible to discuss all I.Ps. The famous Banach contraction theorem uses the Picard I.P to approximate fixed points. Other well-known I.Ps can be found in references [1–8, 15–18]. The approximation speed of the I.P has a major role, and an I.P tends to be chosen in another iterative process. In [2], the writer believes that the approximation rate of the Agarwal I.P is like that of the Picard I.P and is faster than the contraction mapping of the Mann I.P. In [9], the writers help by numerical examples to prove for nonexpansive mapping, that the approximation rate of the Picard-S I.P is better than existing literature. They proved that the convergence rate was good. In [6], the creators illustrate that the approximation rate of the M∗ I.P is better than existing literature. Recently, in [7], another I.P, namely, M I.P, was developed, and it is shown that its approximation rate is better than existing literature. In [10], one more I.P called K I.P was developed, in which they have shown that the approximation rate is better than previous literature. By hypothesis, we introduce a new I.P, namely, the D iterative process.Numerically, the approximation rate for the latest I.P is contrasted with the Agarwal I.P, Picard-S I.P, M I.P, M∗ I.P, and K I.P. We present the weak as well as strong convergence theorems of Suzuki generalized nonexpansive maps and contraction map for our newly developed I.P. We first remember those definitions, ideas, and lemmas which we have to use in the upcoming two sections. A Banach space X is referred to as uniformly convex [11] if ∀ε∈0,2∃δ>0 s.t for ξ,η∈X,
(1)∥ξ∥≤1∥η∥≤1≤1∥ξ−η∥>ε>ε⇒∥ξ+η2∥≤δ.X is referred to obey the Opial property [4] if ∀ξn∈X, approaching weakly to ξ∈X, we have
(2)limn→∞sup∥ξn−ξ∥<limn→∞sup∥ξn−η∥,(3)∀η∈Xs.tη≠ξ.IfFp=p, TF denotes the set of all fixed points of F; a point p is referred to as the fixed point of mapping F.LetC≠ϕ⊆X, and F:C→C is referred as contraction if ∃β∈0,1 s.t ∥Fξ−Fη∥≤β∥ξ−η∥, ∀ξ,η∈C.F:C→C is referred to as nonexpansive if ∥Fξ−Fη∥≤∥ξ−η∥∀ξ,η∈C, and quasinonexpansive if ∀ξ∈Candp∈TF, so ∥Fξ−p∥≤∥ξ−p∥. In [12], F:C→C is referred to satisfy condition M if ∀ξ,η∈C, we have
(4)12∥ξ−Fξ∥≤∥ξ−η∥⇒∥Fξ−Fη∥≤∥ξ−η∥.In [12], the author approved that the mapping holding condition M is weaker than the nonexpansive mapping and stronger than the quasinonexpansive mapping. The mapping which holds condition M is called the Suzuki generalized nonexpansive mapping. Reference [12] includes the fixed point theorem and convergence theorem for Suzuki’s generalized nonexpansive mapping. Recently, the fixed point theorem of the Suzuki generalized nonexpansive mapping has been studied by many authors [5–7]. Now, we define properties of Suzuki generalized nonexpansive mappings.Proposition 1.
LetC≠ϕ⊆X and F:C→C. Then,
(i)
([12], Proposition 1) If F is nonexpansive, so F must be a Suzuki generalized nonexpansive(ii)
([12], Proposition 2) If F is a Suzuki generalized nonexpansive having a fixed point, then F must be a quasinonexpansive(iii)
([12], Lemma 7) If F is a Suzuki generalized nonexpansive, then ∥ξ−Fη∥≤3∥Fξ−ξ∥+∥ξ−η∥∀ξ,η∈C.Lemma 2 ([12], Proposition 3).
LetF:C→C which satisfies the Opial property. Suppose that F is a Suzuki generalized nonexpansive. If ξn approaches weakly to t and limn→∞∥Fξn−ξn∥=0, then Ft=t. So, I-F is demiclosed at zero.Lemma 3 ([12], Theorem 5).
LetF:C→C and F is a Suzuki generalized nonexpansive. Then F must have a fixed point.Lemma 4 ([13], Lemma 1.3).
Lettn be any real sequence in X s.t 0<p≤tn≤q<1∀n≥1. Let ξn and ηn be any two sequences of X s.t limsupn→∞∥ξn∥≤r, limsupn→∞∥ηn∥≤r, and limsupn→∞∥tnξn+1−tnηn∥=r hold for some r≥0. Then, limn→∞∥ξn−ηn∥=0.
LetC≠ϕ⊆X, and let ξn be a bounded sequence ∈X. For ξ∈X, we set
(5)rξ,ξn=limsupn→∞∥ξn−ξ∥.The asymptotic radius ofξn relative to C is given by
(6)rC,ξn=infrξ,ξn:ξ∈C,and the asymptotic center of ξn relative to C is defined as
(7)AC,ξn=ξ∈C:rξ,ξn=rC,ξn.Definition 5 (see [2]).
Letξnn=0∞ and ηnn=0∞ be two fixed point I.P sequences that approach the single fixed point p and ∥ξn−p∥≤un and ∥ηn−p∥≤vn,∀n≥0. If the sequence unn=0∞ and vnn=0∞ approaches to u and v,respectively, and limn→∞∥un−u∥/∥vn−v∥=0, then ξnn=0∞ approaches faster as ηnn=0∞ to p.


## 2. TheD Iteration Process

In this section, letn≥0 and θn and ϑn are real sequences ∈0,1 and C≠ϕ⊆X. For detail on I.P, please see [14].Following is two step Agrawal I.P orS I.P as
(8)ξ0∈C,ηn=1−ϑnξn+ϑnFξn,ξn+1=1−θnFξn+θnFηn.In [9], the authors develop a new I.P called Picard-S I.P as given below:
(9)ξ0∈C,ωn=1−ϑnξn+ϑnFξn,ηn=1−θnFξn+θnTωn,ξn+1=Fηn.They have demonstrated that with Picard-S I.P, we approximate the fixed point of contraction mapping. Also, by a numerical example, they proved that the Picard-S I.P have a better approximation rate than all previously developed I.P.In [7], the authors introduced a new I.P, namely, M I.P, which is given below:
(10)ξ0∈C,ωn=1−θnξn+θnFξn,ηn=Fωn,ξn+1=Fηn.In [6], the authors introduced one new I.P, namely, M∗ I.P, which is defined as
(11)ξ0∈C,ωn=1−ϑnξn+ϑnFξn,yn=F1−θnξn+θnFωn,ξn+1=Fηn.They proved that each of (10) and (11) is moving faster as compared to (8) and (9).In [10], the authors introduced a three-step I.P, called KI.P, which is given below:
(12)ξ0∈C,ωn=1−ϑnξn+ϑnFξn,ηn=F1−θnFξn+θnFωn,ξn+1=Fηn.They insisted that the new I.P converged quickly. By providing an example, they showed that theK I.P has a better approximating rate than the S I.P., Picard-S I.P, M I.P, and M∗ I.P.In this competition, we developed following (new) three-step I.P, namely,D I.P, defined by
(13)ξ0∈C,ωn=F1−ϑnξn+ϑnFξn,ηn=F1−θnFξn+θnFωn,ξn+1=Fηn.In order to prove that our new I.P (13) have a better approximation rate as compared to (8),(9), (10), (11), and (12), first, we generally prove that our I.P strongly converges to unique fixed point, and then it is supported with a numerical example.Theorem 6.
Let the contraction mapF:C→C, where C≠ϕ⊆X. Assume ξnn=0∞ to be an iterative sequence generated by D I.P, where θnn=0∞ and ϑnn=0∞∈01 are real sequences satisfying Σn=0∞θn=∞ or Σn=0∞ϑn=∞. Then, ξnn=0∞ approaches strongly to a unique fixed point of F.Proof.
AsF is a contraction in X, so F must have a unique fixed point in C. Assume p is a particular fixed point of F. From D I.P, we get
(14)∥ωn−p∥=∥F1−ϑnξn+ϑnFξn−Fp∥≤k∥1−ϑnξn+ϑnFξn−p∥≤k∥1−ϑnξn−p+βnFξn−p∥≤k1−ϑn∥ξn−p∥+ϑn∥Fξn−p∥≤k1−ϑn∥ξn−p∥+kϑn∥ξn−p∥≤k1−ϑn1−k∥ξn−p∥.
Now,(15)∥ηn−p∥=∥F1−θnFξn+θnFωn−Fp∥≤k1−θn∥Fξn−p∥+θn∥Fωn−p∥≤k1−θnk∥ξn−p∥+θnk∥ωn−p∥≤k21−θn∥ξn−p∥+θn∥ωn−p∥≤k21−θn∥ξn−p∥+θnk1−ϑn1−k∥ξn−p∥≤k21−θn+kθnϑn1−k∥ξn−p∥.
Then,(16)∥ξn+1−p∥=∥Fηn−Fp∥≤k∥ηn−p∥≤k31−θn+kθnϑn1−k∥ξn−p∥.After repetition, we get
(17)∥ξn−p∥≤k31−θn−1+kθn−1ϑn−11−k∥ξn−1−p∥∥ξn−1−p∥≤k31−θn−2+kθn−2ϑn−21−k∥ξn−2−p∥⋮∥ξn−2−p∥≤k31−θn−3+kθn−3ϑn−31−k∥ξn−3−p∥∥ξ1−p∥≤k31−θ0+kθ0ϑ01−k∥ξ0−p.Therefore, we obtain∥ξn+1−p∥≤k3n+1∥ξ0−p∥∏i=0n1−θi+kθiϑi1−k. Now, k<1 so 1−k>0 and θn,ϑn≤1∀n∈N. Thus, we get 1−θi+kθiϑi1−k<1 and ∀n∈N. As 1−x≤e−x, ∀ξ∈01. So we have
(18)∥ξn+1−p∥≤k3n+1∥ξ0−p∥e−1−k∑i=0nθi+kθiϑi.Taking the limitsn→∞ both sides, we get limn→∞∥ξn−p∥=0.Theorem 7.
LetC≠ϕ⊆X, and let contraction map F:C→C holding condition M, having a fixed point p. For a given ξ0=η0, let ξnn=0∞ and ηnn=0∞ be iterative sequences developed by D I.P and Picard-S I.P as in [11], respectively, where θnn=0∞, ϑnn=0∞, and ηnn=0∞∈0;1 are real sequences satisfying θ≤θn<1 and ϑ≤ϑn<1, for some θ,ϑ>0 and ∀n∈N. Then, ξnn=0∞ approaches to p firstly rather than ηnn=0∞..Proof.
By Theorem 2.5 in [11], we have
(19)∥ηn+1−p∥≤k2n+1∥η0−p∥∏i=0n1−θiϑi1−k.
Sinceθ≤θn and for all n∈N, we obtain ∥ηn+1−p∥≤k2n+1∥η0−p∥1−θiϑi1−kn+1. Let an=k2n+1∥η0−p∥1−θiϑi1−kn+1.
Now, from Theorem6, we get
(20)∥ξn+1−p∥≤k3n+1∥ξ0−p∥∏i=0n1−θi+kθiϑi1−k.Againθ≤θn for all n∈N gives
(21)∥ξn+1−p∥≤k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1.Letbn=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1. Then,
(22)bnan=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1k2n+1∥η0−p∥1−θiϑi1−k.Thus, taking the limit asn→∞limn→∞bn/an=0. Hence, the result follows.Next, we present a result which defines the better approximation rate between theD and M I.P [7].Theorem 8.
LetC≠ϕ⊆X, also let contraction map F:C→C holding condition M, having a fixed point p. For a given ξ0=η0, let ξnn=0∞ and unn=0∞ be iterative sequences developed by D I.P and M I.P as in [7], respectively, where θnn=0∞, ϑnn=0∞, and ηnn=0∞∈0;1 are real sequences satisfying θ≤θn<1 and ϑ≤ϑn<1, for some θ,ϑ>0 and ∀n∈N. Then, ξnn=0∞ approaches to p firstly rather than unn=0∞.Proof.
By Theorem6, we have
(23)∥ξn+1−p∥≤k3n+1∥ξ0−p∥∏i=0n1−θi+kθiϑi1−k.
Sinceθ≤θn and for all n∈N, we obtain
(24)∥ξn+1−p∥≤k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1.
Letan=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1.Then, for the M iteration, we have
(25)u0∈C,wn=1−ϑnun+ϑnFun,vn=Fwn,un+1=Fvn,∥wn−p∥=∥1−ϑnun+ϑnFun−Fp∥≤∥1−ϑnun+ϑnFun−p∥≤∥1−ϑnun−p+ϑnFun−p∥≤1−ϑn∥un−p∥+ϑn∥Fun−p∥≤1−ϑn∥un−p∥+kϑn∥un−p∥≤k1−ϑn1−k∥un−p∥.
Now,(26)∥vn−p∥≤∥Fwn−p∥≤k∥wn−p∥≤k1−ϑn1−k∥un−p∥.
Therefore, we get∥un+1−p∥≤∥Fvn−p∥≤k∥vn−p∥≤k21−ϑn1−k∥un−p∥.
After repetition,(27)∥un−p∥≤k21−ϑn−11−k∥un−1−p∥,∥un−1−p∥≤k21−ϑn−21−k∥un−2−p∥,∥un−2−p∥≤k21−ϑn−31−k∥un−3−p∥,⋮∥u1−p∥≤k21−ϑ01−k∥u0−p∥.
Thus, we have∥un+1−p∥≤k2n+1∥u0−p∥∏i=0n1−ϑi1−k. Now, since ϑ≤ϑn and for all n∈N, we obtain ∥un+1−p∥≤k2n+1∥u0−p∥1−ϑθi1−kn+1. Let bn=k2n+1∥u0−p∥1−ϑθi1−kn+1.
Then,(28)anbn=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1k2n+1∥u0−p∥1−ϑθi1−kn+1.
Thus, taking the limit asn→∞limn→∞bn/an=0. Hence, the result follows.
Next, we prove thatD I.P is faster than that of the M∗ I.P [6].Theorem 9.
LetC≠ϕ⊆X; also, let a contraction mapping F:C→C holding condition M, having a particular fixed point p. For a given ξ0 = η0, let ξnn=0∞ and unn=0∞ be iterative sequences developed by D I.P and M I.P as in [6], respectively, where θnn=0∞, ϑnn=0∞, and ηnn=0∞∈0;1 are real sequences satisfying θ≤θn<1 and ϑ≤ϑn<1, for some θ,ϑ>0 and ∀n∈N. Then, ξnn=0∞ approaches to p firstly rather than unn=0∞.Proof.
By result 2.1, we have(29)∥ξn+1−p∥≤k3n+1∥ξ0−p∥∏i=0n1−θi+kθiϑi1−k.
Sinceθ≤θn and for all n∈N, we obtain
(30)∥ξn+1−p∥≤k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1.
Letan=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1.Then, for the M∗iteration, we have
(31)u0∈C,wn=1−ϑnun+ϑnFun,vn=F1−θnun+θnFwn,un+1=Fvn,∥wn−p∥=∥1−ϑnun+ϑnFun−Fp∥≤∥1−ϑnun+ϑnFun−p∥≤∥1−ϑnun−p+ϑnFun−p∥≤1−ϑn∥un−p∥+ϑn∥Fun−p∥≤1−ϑn∥un−p∥+kϑn∥un−p∥≤1−ϑn1−k∥un−p∥.Now,(32)vn−p=F1−θnun+θnFwn−Fp≤k1−θnun+θnFwn−p≤k1−θnun−p+θnFwn−p≤k1−θnun−p+θnFwn−p≤k1−θnun−p+kθnwn−p≤k1−θnun−p+kθn1−ϑn1−kun−p≤k1−θn+kθn−kθnϑn1−kun−p≤k1−1−kθn−kθnϑn1−kun−p≤k1−θn1−k1−kϑnun−pun+1−p≤Fvn−p≤k21−θn1−k1−kϑnun−p.After repetition,
(33)∥un−p∥≤k21−ϑn−11−k∥un−1−p∥,∥un−1−p∥≤k21−ϑn−21−k∥un−2−p∥,∥un−1−p∥≤k21−ϑn−21−k∥un−2−p∥,⋮∥u1−p∥≤k21−ϑ01−k∥u0−p∥.Thus,∥un+1−p∥≤k2n+1∥u0−p∥∏i=0n1−ϑi1−k. Now, since θ≤θnϑ≤ϑn and for all n∈N, we obtain ∥un+1−p∥≤k2n+1∥u0−p∥1−ϑθi1−kn+1. Let bn=k2n+1∥u0−p∥1−ϑθi1−kn+1.Then,(34)anbn=k3n+1∥ξ0−p∥1−θi+kθiϑi1−kn+1k2n+1∥u0−p∥1−θi1−k1−kϑin+1=kn+11−θi1−kn+11−θi1−k1−kϑin+1.Thus, taking the limit asn→∞limn→∞bn/an=0. Hence, the result follows.Now, we prove by numerical example that ourD I.P has a better approximation rate than existing I.Ps in literature.Example 10.
Let a contraction mappingF:R→R be defined by ξ=4ξ+2/5. Let θn=2n/3n+1 and ϑn=3n/4n+1. The iterative values for ξ0=3.5 are defined in Table 1. Figures 1(a) and 1(b) present the convergence graph.Table 1
Sequence generated byD, K, M∗, M, Picard-S, and S I.P for mapping F of Example 10.
SPicard-SMM∗KDξ03.53.53.53.53.53.5ξ13.22.962.962.962.7682.768ξ22.90242.577542.552962.523472.369622.33502ξ32.666922.341462.313452.276312.174832.14147ξ42.489212.200382.176532.143872.082082.05893ξ52.357372.11712.099082.074342.038372.02436ξ62.260372.068252.055482.038232.017892.01002ξ72.189352.039712.031022.01962.008332.00028ξ82.137522.023072.017332.010022.003872.00168ξ92.099772.013392.009672.005122.00182.00005ξ102.072332.007772.005392.002612.000832.00028ξ112.05242.00452.003012.001332.000392.00011ξ122.037942.002612.001682.000682.000182.00005ξ132.027462.001512.000932.000342.000082.00002ξ142.019872.000872.000522.000172.000042.00001ξ152.014372.000512.000292.000092.000022ξ162.010392.000292.000162.000042.000012ξ172.007512.000172.000092.0000222ξ182.005432.00012.000052.0000122ξ192.003922.000062.000032.0000122ξ202.002832.000032.00002222Figure 1
(a, b) Convergence ofD, K, M∗, M, Picard-S, and S I.P to the fixed point 2 for F of Example 10.
(a)(b)By Figures1(a) and 1(b) and Table 1, it is clear that the new D I.P has a better approximation rate then the K, M∗, M, Picard-S, and S I.P.Example 11.
Let a contraction mappingF:0,50→0,50 by Fξ=4ξ−3. Let θn=2n/3n+1 and ϑn=3n/4n+1; the iterative values for the initial guess ξ0=1.5 are given in Table 2. Figures 2(a) and 2(b) represent the convergence graph. We can see that our new I.P (13) has a better approximation rate as compared to S, Picard-S, M, M∗, and K I.P.Table 2
Sequence generated byD, K, M∗, M, Picard-S, and S I.P for F of Example 11.
SPicard-SMM∗KDξ01.51.51.51.51.51.5ξ11.7320508081.9819695341.9819695341.981972.2198822.219882ξ22.0227433622.4533926892.4821098442.5063892.7273222.75395ξ32.3024625972.7476576642.7766302962.8013812.9200852.936761ξ42.5303737442.8921128252.9109174652.9265542.9779312.984781ξ52.696618372.9554901572.9656670232.9738092.9940262.99642ξ62.8093844442.981935922.9869644912.9908122.9983852.999166ξ72.8823894072.9927262822.9950851122.9968042.9995712.999807ξ82.9282773542.9970831592.9981536082.9988932.9998852.999955ξ92.9565890222.9988331162.99993078532.9996182.9999692.99999ξ102.9738527782.999533922.9997409152.9998692.9999922.999998ξ112.9843015262.9998140512.9999031232.9999552.9999982.999999ξ122.9905951942.9999258772.9999638062.9999852.9999993ξ132.9943741262.9999704772.9999864882.99999533ξ142.9966382812.9999882472.9999949582.99999833ξ152.9979928352.9999953232.999998122.99999933ξ162.9988023392.999998142.999999298333ξ172.9992857222.999999262.999999738333ξ182.9995741852.9999997052.999999902333ξ192.9997462412.9999998832.999999963333ξ202.9998488222.9999999532.999999985333Figure 2
(a, b) Convergence of D, K,M∗, M, Picard-S, S I.P to the fixed point 3 for F of Example 11.
(a)(b)By Figures2(a) and 2(b) and Table 2, it is proven that the new D iteration has a better approximation rate than the K, M∗, M, Picard-S, and S I.P.


## 3. Convergence Results of a Sequence Generated byD I.P

Here, we prove some strong and weak convergence result of sequence generated byD I.P for the Suzuki generalized nonexpansive mapping in uniformly convex Banach spaces.Lemma 12.
LetC≠ϕ⊆X and let contraction mapping F:C→C be a holding condition M and also TF≠∅. For randomly chosen ξ0∈C, let the sequence ξn be generated by (13), then limn→∞∥ξn−p∥ exists for any p∈TF.Proof.
Letp∈TF and ω∈C. As F holds the M condition, so
(35)12∥p−Fp∥=0≤∥p−ω∥⇒∥Fp−Fω∥≤∥p−ω∥.So from 1.1(ii), we have
(a)
ωn−p=1−ϑnξn+ϑnFξn−p≤1−ϑnξn−p+ϑnFξn−p≤1−ϑnξn−p+ϑnξn−p=ξn−p.
So by using (a), we get(b)
ηn−p=F1−θnFξn+θnFωn−p≤1−θnFξn+θnFωn−p≤1−θnFξn−p+θnFωn−p≤1−θnξn−p+θnωn−p≤1−θnξn−p+θnξn−p=ξn−p.
So using (b), we have(c)
ξn+1−p=Fηn−p≤ηn−p≤ξn−p,which means that ξn−p is nonincreasing and bounded, ∀p∈TF. So limn→∞∥ξn−p∥ exists, hence proven.Theorem 13.
LetC≠ϕ⊆X, and let contraction mapping F:C→C holding condition M. For randomly chosen ξ0∈C, let ξn be the sequence generated by (13) ∀n≥1 where θn and ϑn are sequences of real numbers ∈01. Then, TF≠∅ iff ξn is bounded and limn→∞∥Fξn−ξn∥=0.Proof.
AssumeFT≠ϕ and consider p∈FT. So by result 3.1, limn→∞∥ξn−p∥ exists and ξn is bounded put
(d)
limn→∞∥ξn−p∥=r.From (a) and (d), we have
(e)
limn→∞sup∥ωn−p∥≤limn→∞sup∥ξn−p∥=r.
By result 1.1(ii), we have(f)
limn→∞sup∥Tξn−p∥≤limn→∞sup∥ξn−p∥=r.
On the other hand,(36)∥ξn+1−p∥=∥Fηn−p∥≤∥ηn−p∥=∥F1−θnFξn+θnFωn−p∥≤∥1−θnFξn+θnFωn−p∥≤1−θn∥Fξn−p∥+θn∥Fωn−p∥≤1−θn∥ξn−p∥+θn∥ωn−p∥≤∥ξn−p∥−θn∥ξn−p∥+θn∥ωn−p∥.This implies that(37)∥ξn+1−p∥−∥ξn−p∥θn≤∥ωn−p∥−∥ξn−p∥.So(38)∥ξn+1−p∥−∥ξn−p∥≤∥ξn+1−p∥−∥ξn−p∥θn≤∥ωn−p∥−∥ξn−p∥,implies that
(39)∥ξn+1−p∥≤∥ωn−p∥.Therefore,(g)
r≤limn→∞inf∥ωn−p∥.From (e) and (g), we get(h)
r=limn→∞∥ωn−p∥=limn→∞∥1−ϑnξn+ϑnFξn−p∥=limn→∞∥ϑnFξn−p+1−ϑnξn−p.From (d), (f), and (h) together with Lemma4, we have
(40)limn→∞∥Fξn−ξn∥=0.On the other hand, assume thatξn is a bounded sequence and limn→∞∥Fξn−ξn∥=0. Also, p∈AC,ξn. So by Proposition 1,
(41)rFp,ξn=limn→∞sup∥ξn−Fp∥≤limn→∞sup3∥Fξn−ξn∥+∥ξn−p∥≤limn→∞sup∥ξn−p∥=rp,ξn.Hence,Fp∈AC,ξn. As X is uniformly convex, and AC,ξn is a singleton set, therefore Fp=p. Hence, F≠∅.


### 3.1. Weak and Strong Convergence Theorem

Theorem 14.
LetC≠ϕ⊆X, where X satisfies Opial condition, and let F:C→C holding condition M, where θn and ϑn are real sequences ∈01, such that TF≠∅. For randomly selected ξ0∈C, consider xn to be the sequence generated by (13) ∀n≥1. Then, the sequence generated by the D iterative process converges weakly to p∈TF.Proof.
AsTF≠ϕ, and from Theorem 13, ξn is a bounded sequence and limn→∞∥Fξn−ξn∥=0. Being uniformly convex X must be reflexive, so by Eberlin’s theorem ∃, a subsequence ξnj of ξn which approaches weakly to p1∈X. As C is closed and convex, by Mazur’s theorem, p1∈C. By Lemma 2, p1∈TF. Next, we prove that ξn approaches weakly to p1. Actually, if it is not true, so ∃ a subsequence ξnk for ξn s.t ξnk converges weakly to p2∈C and p2≠p1. By Lemma 2, p2∈TF. Since limn→∞∥ξn−p∥ defined ∀p∈TF. Thus, from Theorem 13 and Opial’s property,
(42)limn→∞∥ξn−p1∥=limj→∞∥ξnj−p1∥<limj→∞∥ξnj−p2∥=limn→∞∥ξn−p2∥=limk→∞∥ξnk−p2∥<limk→∞∥ξnk−p1∥=limn→∞∥ξn−p1∥,which is a contradiction to fact. So p1=p2⇒ξn converges weakly to a single fixed point of F.
Next, we prove the strong convergence theorem.Theorem 15.
LetC≠ϕ⊆X, and let contraction mapping F:C→C holding condition M, where θn and ϑn∈01 are real sequences, s.t TF≠∅. Then, ξn approaches strongly to p∈TF.Proof.
By Lemma 3,TF≠ϕ, and by Theorem 13, we have limn→∞∥Tξn−ξn∥=0. As C is compact, so ∃ a subsequence ξnj of ξn which approaches strongly to p where p∈C. By Proposition 1 (iii), we have ∥ξnk−Tp∥≤3∥Tξnk−ξnk∥+∥ξnk−p∥, ∀n≥1. Letting k→∞, we attain p∈TF. As, by Lemma 12, limn→∥ξn−p∥ has a defined value, for every p∈FT, so ξn converge strongly to p.


## 3.1. Weak and Strong Convergence Theorem

Theorem 14.
LetC≠ϕ⊆X, where X satisfies Opial condition, and let F:C→C holding condition M, where θn and ϑn are real sequences ∈01, such that TF≠∅. For randomly selected ξ0∈C, consider xn to be the sequence generated by (13) ∀n≥1. Then, the sequence generated by the D iterative process converges weakly to p∈TF.Proof.
AsTF≠ϕ, and from Theorem 13, ξn is a bounded sequence and limn→∞∥Fξn−ξn∥=0. Being uniformly convex X must be reflexive, so by Eberlin’s theorem ∃, a subsequence ξnj of ξn which approaches weakly to p1∈X. As C is closed and convex, by Mazur’s theorem, p1∈C. By Lemma 2, p1∈TF. Next, we prove that ξn approaches weakly to p1. Actually, if it is not true, so ∃ a subsequence ξnk for ξn s.t ξnk converges weakly to p2∈C and p2≠p1. By Lemma 2, p2∈TF. Since limn→∞∥ξn−p∥ defined ∀p∈TF. Thus, from Theorem 13 and Opial’s property,
(42)limn→∞∥ξn−p1∥=limj→∞∥ξnj−p1∥<limj→∞∥ξnj−p2∥=limn→∞∥ξn−p2∥=limk→∞∥ξnk−p2∥<limk→∞∥ξnk−p1∥=limn→∞∥ξn−p1∥,which is a contradiction to fact. So p1=p2⇒ξn converges weakly to a single fixed point of F.
Next, we prove the strong convergence theorem.Theorem 15.
LetC≠ϕ⊆X, and let contraction mapping F:C→C holding condition M, where θn and ϑn∈01 are real sequences, s.t TF≠∅. Then, ξn approaches strongly to p∈TF.Proof.
By Lemma 3,TF≠ϕ, and by Theorem 13, we have limn→∞∥Tξn−ξn∥=0. As C is compact, so ∃ a subsequence ξnj of ξn which approaches strongly to p where p∈C. By Proposition 1 (iii), we have ∥ξnk−Tp∥≤3∥Tξnk−ξnk∥+∥ξnk−p∥, ∀n≥1. Letting k→∞, we attain p∈TF. As, by Lemma 12, limn→∥ξn−p∥ has a defined value, for every p∈FT, so ξn converge strongly to p.


## 4. Conclusion

In this article, we present the new fastest iteration method to approximating fixed point of contraction mapping. First, we present theD iteration process and prove weak and strong convergence results. Also, we analytically and numerically proved that the D iteration process has a better approximation rate than existing iteration processes as defined in [1–4, 6, 7, 9].



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