# Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications

Authors: Jie Jia; Khurram Shabbir; Khushdil Ahmad; Nehad Ali Shah; Thongchai Botmart
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the article, we have proposed a new type of hybrid iterative scheme which is a hybrid of Picard and Thakur et al. repetitive schemes. This new hybrid iterative scheme converges faster than all leading schemes like Picard-S∗ hybrid, Picard-S, Picard-Ishikawa hybrid, Picard-Mann hybrid, Thakur et al. and Abbas and Nazir, S-iterative, Ishikawa and Mann iterative schemes for contraction mapping. By using the Picard-Thakur hybrid iterative scheme, we can find the solution of delay differential equations and also prove some convergence results for nonexpansive mapping in a uniformly convex Banach space.


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## Body

## 1. Introduction

In this article, the set of all positive integers is denoted byI+. Let N denote the nonempty convex subset of a normed space and S be its convex subset, and V:S⟶S is called contraction mapping if ∥Vj−Vk∥≤δ∥j−k∥ for all j,k∈S and δ∈0,1. If δ=1, then, the mapping V is called nonexpansive mapping. An element j∈S is said to be a fixed point of V if Vj=j, and the set of fixed points of V is denoted by FV.In 1890, Picard [1] presented an iterative scheme for approximating the fixed point which is defined by the sequence jn as
(1)j1=j∈S,jn+1=Vjn,n∈I+.The Krasnoselskii [2] iterative sequence un is defined as
(2)u1=u∈S,un+1=1−μun+μVun,n∈I+,where μ∈0,1.In 1953, Mann [3] proposed an iterative scheme which is defined as
(3)v1=v∈S,vn+1=1−θnvn+θnVvn,n∈I+,where θn∈0,1.In 1974, Ishikawa [4] gave the concept of the two-step iterative scheme and the sequence wn of this iterative is defined as
(4)w1=w∈S,wn+1=1−θnwn+θnVtn,tn=1−ϑnwn+ϑnVwn,n∈I+,where θn,ϑn∈0,1.In 2007, Agarwal et al. [5] introduced a more generalized form of the Ishikawa iterative scheme and they called it the S-iterative scheme and the sequence pn of the iterative scheme is defined as
(5)p1=p∈S,pn+1=1−θnVpn+θnVqn,qn=1−ϑnpn+ϑnVpn,n∈I+,where θn,ϑn∈0,1.In 2016, Sahu et al. [6] and Thakur et al. [7] proposed a new scheme which converges faster than all the existing schemes. The iterative sequence kn of this scheme is defined as
(6)k1=k∈S,kn+1=1−θnVmn+θnVln,ln=1−ϑnmn+ϑnVmn,mn=1−σnkn+σnVkn,n∈I+,where θn,ϑn, and σn∈0,1.Thakur et al. [7] proposed another iterative scheme which converges faster than all the above schemes and the iterative sequence jn of Thakur et al. is defined as
(7)j1=j∈S,jn+1=Vkn,kn=V1−θnjn+θnVln,ln=1−ϑnjn+ϑnVjn,n∈I+,where θn,ϑn∈0,1.Recently, Lamba and Panwar [8] introduced a new three-step iteration process for Susuzki’s nonexpansive mapping and called it the Ap iterative scheme whose rate of convergence is faster than those of the leading schemes. The sequence of the Ap iterative scheme is defined as
(8)j1=j∈S,jn+1=Vkn,kn=V1−θnVjn+θnVln,ln=V1−ϑnjn+ϑnVjn,n∈I+,where θn,ϑn∈0,1.Many physical problems of engineering and applied sciences are mostly constructed in the form of fixed point equations. In the existence of a fixed point equation involving an operator,V is guaranteed but the exact solution is not possible. We can only approximate the solution which becomes very relevant and this necessitated various iterative schemes [9–14]. Also, the iterative schemes are used for solving different problems like minimization, equilibrium, viscosity approximation, and many more problems in different spaces [15–18].The Picard iterative scheme is the simplest iteration to estimate the solution of a fixed point equation. Chidume [19] introduced some basic results on this iterative scheme. Chidume generalized and improved the existing results of the fixed point equation in [20]. Okeke and Abbas [21] proved the convergence and almost V-stability of Mann-type and Ishikawa-type random iterative schemes.In 2013, Khan [22] proposed the Picard-Mann hybrid iterative scheme. The sequence rn of this scheme is defined as
(9)r1=r∈S,rn+1=Vsn,sn=1−θnrn+θnVrn,n∈I+,where θn∈0,1.In 2017, Okeke and Abbas [23] proposed the Picard-Krasnoselskii hybrid iterative scheme and the sequence rn of this iterative scheme is defined as
(10)r1=r∈S,rn+1=Vsn,sn=1−νrn+νVrn,n∈I+,where ν∈0,1.In 2019, Okeke [24] proposed the Picard-Ishikawa hybrid iterative scheme and the sequence fn of this iteration defined as
(11)f1=f∈S,fn+1=Vgn,gn=1−θnfn+θnVhn,hn=1−ϑnfn+ϑnVfn,n∈I+,where θn and ϑn∈0,1.Recently, Srivastava [25] introduced a new type of hybrid iterative scheme from Picard and S-iteration (Picars-S hybrid iterative scheme). The sequence an of the scheme is defined as
(12)a1=a∈S,an+1=Vbn,bn=1−θnVan+θnVcn,cn=1−ϑnan+ϑnVan,n∈I+,where θn and ϑn∈0,1.Also Lamba and Panwar [26] introduced another hybrid scheme from Picard and S∗-iteration (Picard-S∗ hybrid iterative scheme) and the sequence an of the scheme is defined as
(13)a1=a∈S,an+1=Vbn,bn=1−θnVan+θnVcn,cn=1−ϑnVan+ϑnVdn,dn=1−σnan+σnVan,n∈I+,where θn,ϑn, and σn∈0,1.With the motivation towards the usage of hybridization of iterative schemes, we proposed another type of hybrid scheme which is the Picard-Thakur hybrid iterative scheme, defined by the sequencejn as
(14)j1=j∈S,jn+1=Vkn,kn=1−θnVmn+θnVln,ln=1−ϑnmn+ϑnVmn,mn=1−σnjn+σnVjn,n∈I+,where θn,ϑn and σn∈0,1.Rhoades [27] commented on the convergence of two iterative schemes that converges to a certain fixed point is as follows:Letan and bn be the two fixed point iterative schemes that converge to a certain fixed point j∗ of a given operator V. The sequence an is better than bn if
(15)an−j∗≤bn−j∗∀n∈I+.


## 2. Preliminaries

Berinde and Takens [10] gave the following definitions.Definition 1 (see [10]).
Lettn and wn be the two sequences of the real number converging to t and w, respectively. Suppose that
(16)limn⟶∞tn−twn−w=k.(i)
Ifk=0, then, tn⟶t faster than wn⟶w(ii)
If0<k<∞, then, the rate of convergence of both sequences are the sameDefinition 2 (see [10]).
Lettn and wn be the two sequences of a fixed point iterative scheme, both converges to a fixed point ξ for a given operator V and pn,qn are two sequences of positive numbers. Suppose that the error estimates,
(17)tn−ξ≤pn∀n∈I+,wn−ξ≤qn∀n∈I+,are available and pn,qn converge to zero. If pn converges faster than qn, then, tn converges faster than wn⟶ξ. Most of the literature on the iterative schemes deals with the convergence rate and some analyzes its stability [28–34]. For proving the results, we need the following lemma.Lemma 3 (see [35]).
Letrn∈ℝ+ be a sequence with rn+1≤1−τnrn. If τn⊂0,1 and ∑n=1∞=∞, then, limn⟶∞rn=0.Definition 4 (see [36]).
LetS be a subset of Banach space B which is nonempty closed and convex. A mapping V:S⟶S is demiclosed w.r.t. b∈B, if for each sequence jn in S and a∈S, jn converges weakly at a and Vjn converges strongly at b⇒Va=b.Definition 5 (see [37]).
A Banach spaceB is said to satisfy Opial’s condition if for any sequence jn∈B,jn⇀a, implies that
(18)liminfn⟶∞jn−a≤liminfn⟶∞jn−b,for all b∈B with a≠b.Lemma 6 (see [38]).
LetB be a uniformly convex Banach space and 0<x≤ρn≤y<1∀n∈I+. Let jn, kn be the two sequences such that limsupn⟶∞jn≤l, limsupn⟶∞kn≤l, and limsupn⟶∞1−σnjn+σnkn=l hold for some l≥0, then limn⟶∞jn−kn=0.Lemma 7 (see [36]).
LetV:S⟶S be a nonexpansive mapping with Opial’s property. If jn⇀a and limn⟶∞jn−Vjn=0, then, Va=a, i.e., I−V is demiclosed at zero, where I is the identity mapping on B.Proposition 8 (see [39]).
LetS be a subset of Banach space B and V:S⟶S a nonexpansive mapping. Then, for all p,q∈S(19)p−Vq≤3p−Vp+p−q.Senter and Dotson [40] introduced the concept of condition (I) which is defined asDefinition 9.
LetV be a self-mapping on S which is said to satisfy condition (I), if there is an increasing function Z:0,∞⟶0,∞ with Z0=0 and Zt>0, for all t>0 such that
(20)dj,Vj≥Zdj,FV,∀j∈S,where dj,FV=infdj,j∗:j∗∈FV.In this article, we proposed a new hybrid iterative scheme which converges faster than Mann [3], Ishikawa [4], S-iteration [5], Abbas et al. [9], Thakur et al. [7], Picard-Mann hybrid [22], Picard-Krasnoselskii [23], Picard-Ishikawa [24], and Picard-S hybrid iterative schemes [25]. Recently, Srivastava [25] already proved that the Picard-S hybrid iterative scheme converges faster than all of the above iterative schemes. Therefore, we show that our hybrid iterative scheme converges faster than all the leading schemes. We find the solution of delay differential equations using our proposed hybrid iterative scheme while in last section, we prove some results of this scheme for nonexpansive mapping in the uniformly convex Banach space.


## 3. Convergence Analysis

This section deals with the rate of convergence of the Picard-Thakur hybrid iterative scheme (14) with Picard-S (12), Picard-Ishikawa (11), Picard-Mann (9), and Thakur et al. (6).Proposition 10.
Assume thatS be a nonempty closed convex subset of a normed space N and let V:S⟶S be a contraction mapping. Suppose that the iterative schemes (12), (11), (10), (9), and (6) converge to the same fixed point j∗ of V where θn,ϑn, and σn are sequences in 0,1 such that 0<μ≤θn,ϑn,σn<1,∀n∈I+, and for some μ and δ∈0,1. Then, the Picard-Thakur hybrid iterative scheme (14) converges faster than all the other schemes.Proof.
Letj∗ be a fixed point of an operator V. Using the definition of contractive mapping and the Thakur et al. iterative scheme (6), we have
(21)kn+1−j∗=1−θnVmn+θnVln−j∗≤1−θnVmn−j∗+θnVln−j∗≤1−θnδmn−j∗+θnδln−j∗≤1−θnδ1−1−δσnkn−j∗+δθn1−1−δϑn1−1−δσnkn−j∗≤δ1−1−δσn1−θn+ϑn1−1−δσnkn−j∗≤δ1−1−δσn1−1−1−δθnσnkn−j∗≤δ1−1−δσn−1−δθnϑn+1−δ2θnϑnσnkn−j∗≤δ1−1−δσn−1−δθnϑn+1−δθnϑnσnkn−j∗≤δ1−1−δσnkn−j∗.
Let(22)an=δn1−1−δσnk1−j∗.
Now, for (14),
(23)mn−j∗=1−σnjn+Vjn−j∗≤1−σnjn−j∗+σnδjn−j∗≤1−1−δσnjn−j∗,ln−j∗=1−ϑnmn+ϑnVmn−j∗≤1−ϑnmn−j∗+ϑnδmn−j∗≤1−1−δϑnmn−j∗≤1−1−δϑn1−1−δσnjn−j∗,kn−j∗=1−θnVmn+θnVln−j∗≤δ1−θnmn−j∗+δθnln−j∗=δ1−θn1−1−δσnjn−j∗+θn1−1−δϑn1−1−δσnjn−j∗=δ1−1−δσn1−θn+θn−1−δθnϑnjn−j∗=δ1−1−δσn−1−1−δσn1−δθnϑnjn−j∗=δ1−1−δσn−1−δθnϑn+1−δ2θnϑnσnjn−j∗≤δ1−1−δσn−1−δθnϑn+1−δθnϑnσnjn−j∗≤δ1−1−δσn+θnϑn−θnϑnσnjn−j∗.
Also,(24)jn+1−j∗=Vkn−j∗≤δkn−j∗≤δδ1−1−δσn+θnϑn−θnϑnσnjn−j∗≤δ21−1−δσn+θnϑn−θnϑnσnjn−j∗.
Let(25)bn=δ2n1−1−δσ+θϑ−θϑσnj1−j∗.
Then,(26)bnan=δ2n1−1−δσ+θϑ−θϑσnj1−j∗δn1−1−δσnk1−j∗=δn1−1−δσ+θϑ−θϑσnj1−j∗1−1−δσnk1−j∗⟶0,asn⟶∞.
Thus,jn converges faster than kn, i.e., the Picard-Thakur iterative scheme converges faster than the Thakur iterative scheme. Similarly, the inequality proved in Proposition 3.1 of the Picard-S hybrid iterative scheme [25] is as follows:
(27)cn=δ2n1−1−δθϑna1−j∗.
Then,(28)bnan=δ2n1−1−δσ+θϑ−θϑσnj1−j∗δ2n1−1−δθϑna1−j∗=1−1−δσ+θϑ−θϑσnj1−j∗1−1−δθϑna1−j∗⟶0,asn⟶∞.
Thus,jn converges faster than an., i.e., the Picard-Thakur iterative scheme converges faster than the Picard-S iterative scheme. Similarly, we can show that Picard-Thakur hybrid iterative scheme (14) converges faster than (11), (10), and (9).Next, we gave an example to show that the Picard-Thakur hybrid iterative scheme (14) converges faster than the Picard-S hybrid, Picard-Ishikawa hybrid, Picard-Mann hybrid, and Thakur iterative schemes.Example 11.
LetV:S⟶S where S=0,2⊂N=ℝ be an operator defined by
(29)Vj=1,ifj∈0,1,4−j23,ifj∈1,2.Chooseθn=n+2/n+6,ϑn=n2+1/n2+n+1,σn=n+1/2n+7, for each n∈I+ with an initial value j1=0.6.V is nonexpansive mapping. All the iterative schemes converge to the fixed point j∗=1. Clearly, in the Table 1 and Figure 1, we can see that the Picard-Thakur hybrid iterative scheme (14) converges faster than the schemes discussed above.Table 1
Convergence behavior of Thakur et al. (7), Ap (8), Picard-S (12), Picard-S∗ (13), and Picard-Thakur hybrid Iterative schemes (14).
StepsPicard-Ishikawa hybridThakur et al.Ap iterative schemePicard-S hybridPicard-S∗ hybridPicard-Thakur hybrid10.60000000000.60000000000.60000000000.60000000000.60000000000.600000000021.01729384941.00232629741.00339926881.00284851410.99420905970.999223364030.99910106160.99998961580.99996172940.99996702180.99987608560.999998841241.00004635441.00000004641.00000042991.00000038080.99999733480.999999998350.99999760870.99999999970.99999999520.99999999560.99999994270.999999999961.00000012341.00000000001.00000000011.00000000010.99999999880.999999999970.99999999360.99999999990.99999999990.99999999990.99999999991.000000000081.00000000031.00000000001.00000000001.00000000000.99999999991.000000000090.99999999991.00000000001.00000000001.00000000000.99999999991.0000000000101.00000000001.00000000001.00000000001.00000000001.00000000001.0000000000Figure 1
Convergence behavior of Thakur et al. (7), Ap (8), Picard-S (12), Picard-S∗ (13), and Picard-Thakur hybrid iterative schemes (14).


## 4. Application: Delay Differential Equations

In this section, we can find the solution of the delay differential equation by using our proposed iterative scheme.Let the space of all continuous real-valued functions be denoted byCu,v on closed interval u,v endowed with the Chebyshev norm j−m∞ and defined as j−m∞=supr∈u,vjr−mr, and it is clear that in [41] that (Cu,v,.∞) is a Banach space. Now, consider the following delay differential equation
(30)j′r=ψr,jr,jr−γ,r∈r0,v,with initial condition
(31)jr=ζr,r∈r0−γ,r0.By the solution of the above delay differential equation, we mean a functionj∈Cr0−γ,v,ℝ∩C1r0,v,ℝ satisfying (30) and (31).Assume that the following conditions are satisfied.(1)
r0,v∈ℝ,γ>0(2)
ψ∈Cr0,v×ℝ2,ℝ(3)
ζ∈Cr0−γ,v,ℝ(4)
There existsLψ>0 such that(32)ψr,s1,s2−ψr,t1,t2≤LψΣi=12si−ti,∀si,ti∈ℝ,r∈r0,v(5)
2Lψv−r0<1Now, we construct (30) and (31) by the integral equation as
(33)jr=ζr,r∈r0−γ,v,ζr0+∫r0rψt,jt,jt−γdt,r∈r0,v.The following result is the generalization of the result of Coman et al. [42].Theorem 12.
Let the conditions∗1 to ∗5 be satisfied. Then, (30) and (31) have unique solution j∗∈Cr0−γ,ℝ∩C1r0v,ℝ and
(34)j∗=limn⟶∞Vnj,foranyj∈Cr0−γ,v,ℝ.Now, by using the Picard-Thakur hybrid iterative scheme (14), we prove the following result.Theorem 13.
Let the conditions∗1−∗5 be satisfied. Then, (30) and (31) have a unique solution j∗∈Cr0−γ,ℝ∩C1r0,v,ℝ and the Picard-Thakurb hybrid iterative scheme (14) converges to j∗.Proof.
Letjn be a sequence generated by the Picard-Thakur hybrid iterative scheme (14) for an operator V defined by
(35)Vjr=ζr,r∈r0−γ,v,ζr0+∫r0rψp,jp,jp−γdp,r∈r0,v.
Letj∗ be a fixed point of V. Now, we prove that jn⟶j∗ as n⟶∞. It is easy to see that jn⟶j∗ as n⟶∞ for each r∈r0−γ,r0.
Now, for eachr∈r0,v, we have
(36)jn+1−j∗∞≤Vkn−j∗∞≤supr0∈r0,vVkn−Vj∗≤supr0∈r0,vζr0+∫r0rψp,knp,knp−γdp−ζr0+∫r0rψp,j∗p,j∗p−γdp≤supr0∈r0,v∫r0rψp,knp,knp−γ−ψp,j∗p,j∗p−γdp≤supr0∈r0,v∫r0rLψknp−j∗p+knp−γ−j∗p−γdp≤∫r0rLψsupr0∈r0,vknp−j∗p+knp−γ−j∗p−γdp≤∫r0rLψkn−j∗∞+kn−j∗∞dp≤2Lψv−r0kn−j∗∞.
Now,(37)kn−j∗∞=1−θnVmn+θnVln−j∗∞≤1−θnVmn−j∗∞+θnVln−j∗∞,
As(38)Vln−j∗∞=Vln−Vj∗∞≤supr∈r0−γ,vζr0+∫r0rψp,lnp,lnp−γdp−ζr0+∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,lnp,lnp−γdp−∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,lnp,lnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,lnp,lnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rLψlnp−j∗p+lnp−γ−j∗p−γdp≤∫r0rLψsupr∈r0−γ,vlnp−j∗p+supr∈r0−γ,vlnp−γ−j∗p−γdp≤∫r0rLψln−j∗∞+ln−j∗∞dp≤2Lψr−r0ln−j∗∞≤2Lψv−r0ln−j∗∞,(39)ln−j∗∞=1−ϑnmn+ϑnVmn−j∗∞≤1−ϑnmn−j∗∞+ϑnVmn−j∗∞.
For(40)Vmn−j∗∞=Vmn−Vj∗∞≤supr∈r0−γ,vζr0+∫r0rψp,mnp,mnp−γdp−ζr0+∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,mnp,mnp−γdp−∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,mnp,mnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,mnp,mnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rLψmnp−j∗p+mnp−γ−j∗p−γdp≤∫r0rLψsupr∈r0−γ,vmnp−j∗p+supr∈r0−γ,vmnp−γ−j∗p−γdp≤∫r0rLψ∥mn−j∗mn−j∗∥∞+∣mn−j∗mn−j∗∥∞dp≤2Lψr−r0mn−j∗∞≤2Lψv−r0mn−j∗∞,(41)mn−j∗∞=1−σnjn+σnVjn−j∗∞≤1−σnjn−j∗∞+σnVjn−j∗∞,as
(42)Vjn−j∗=Vjn−Vj∗∞≤supr∈r0−γ,vζr0+∫r0rψp,jnp,jnp−γdp−ζr0+∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,jnp,jnp−γdp−∫r0rψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,jnp,jnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rψp,jnp,jnp−γ−ψp,j∗p,j∗p−γdp≤supr∈r0−γ,v∫r0rLψjnp−j∗p+jnp−γ−j∗p−γdp≤∫r0rLψsupr∈r0−γ,vjnp−j∗p+supr∈r0−γ,vjnp−γ−j∗p−γdp≤∫r0rLψjn−j∗∞+jn−j∗∞dp≤2Lψr−r0jn−j∗∞≤2Lψv−r0jn−j∗∞.
Putting (42) in (41), we get
(43)mn−j∗∞≤1−σnjn−j∗∞+σn2Lψv−r0jn−j∗∞≤1−1−2Lψv−r0σnjn−j∗∞.
Putting (43) in (40), we get
(44)Vmn−j∗∞≤2Lψv−r01−11−2Lψv−r0σnjn−j∗∞.
Putting (44) and (43) in (39), we get
(45)ln−j∗∞≤1−ϑn1−11−2Lψv−r0σnjn−j∗∞+ϑn2Lψv−r01−1−2Lψv−r0σnjn−j∗∞≤1−1−2Lψv−r0σn1−1−1−2Lψv−r0ϑnjn−j∗∞≤1−1−2Lψv−r0σn−1−1−2Lψv−r0σn1−2Lψv−r0ϑnjn−j∗∞≤1−1−2Lψv−r0σn−1−2Lψv−r0ϑn+1−2Lψv−r02ϑnσnjn−j∗∞≤1−1−2Lψv−r0σn−1−2Lψv−r0ϑn+1−2Lψv−r0ϑnσnjn−j∗∞≤1−1−2Lψv−r0σn−ϑn+ϑnσnjn−j∗∞.
Putting (45) in (38), we get
(46)∥Vln−j∗∥∞≤2Lψv−r01−1−2Lψv−r0σn−ϑn+ϑnσn∥jn−j∗∥∞.
Putting (46) and (40) in (37), we get
(47)kn−j∗∞≤1−θn2Lψv−r0mn−j∗∞+θn2Lψv−r0ln−j∗∞≤2Lψv−r01−θnmn−j∗∞+θnln−j∗∞≤2Lψv−r01−θn1−1−2Lψv−r0σnjn−j∗∞+θn1−1−2Lψv−r0σn−ϑn+ϑnσn∥jn−j∗∥∞≤2Lψv−r01−θn1−1−2Lψv−r0σn+θn1−1−2Lψv−r0σn−ϑn+ϑnσn∥jn−j∗∥∞≤2Lψv−r01−θn−1−2Lψv−r0σn+1−2Lψv−r0θnσn+θn−1−2Lψv−r0θnσn−ϑn+ϑnσn∥jn−j∗∥∞≤2Lψv−r01−1−2Lψv−r0σn−θnσn+θnσn+ϑn−ϑnσn∥jn−j∗∥∞≤2Lψv−r01−1−2Lψv−r0σn−θnσn+θnσn+θnϑn−θnϑnσn∥jn−j∗∥∞≤2Lψv−r01−1−2Lψv−r0σn+θnϑn−θnϑnσn∥jn−j∗∥∞.
Letσn+θnϑn−θnϑnσn=ρn, and by using condition ∗5, we have
(48)kn−j∗∞≤1−1−2Lψv−r0ρnjn−j∗∞.
Putting (48) in (36), we have
(49)jn+1−j∗∞≤2Lψv−r01−1−2Lψv−r0ρnjn−j∗∞.
Again, using condition∗5, we get
(50)jn+1−j∗∞≤1−1−2Lψv−r0ρnjn−j∗∞.
Let1−2Lψv−r0ρn=τn<1 and ∥jn−j∗∥∞=rn. So, the conditions of Lemma 3 are satisfied. Hence, limn⟶∞∥jn−j∗∥=0.


## 5. Convergence Results for Nonexpansive Mapping

Lemma 14.
LetS be a nonempty closed and convex subset of uniformly convex Banach space B and V:S⟶S be a nonexpansive mapping. If jn be a sequence generated by Picard-Thakur hybrid iterative scheme (14) and FV≠∅, then, limn⟶∞∥jn−j∗∥ exists.Proof.
Letj∗∈FV, and V is nonexpansive then
(51)∥mn−j∗∥=∥1−σnjn+σnVjn−j∗∥≤1−σn∥jn−j∗∥+σn∥Vjn−j∗∥≤1−σn∥jn−j∗∥+σn∥jn−j∗∥≤∥jn−j∗.
Also,(52)∥ln−j∗∥=∥1−ϑnmn+ϑnVmn−j∗∥≤1−ϑn∥mn−j∗∥+ϑn∥Vmn−j∗∥≤1−ϑn∥mn−j∗∥+ϑn∥mn−j∗∥≤∥mn−j∗.
Similarly,(53)∥kn−j∗∥=∥1−θnVmn+θnVln−j∗∥≤1−θn∥Vmn−j∗∥+θn∥Vln−j∗∥≤1−θn∥mn−j∗∥+θn∥ln−j∗∥≤1−θn∥mn−j∗∥+θn∥mn−j∗∥≤∥mn−j∗∥≤∥jn−j∗∥.
Now,(54)∥jn+1−j∗∥=∥Vkn−j∗∥≤∥kn−j∗∥≤∥jn−j∗∥.
This shows that∥jn−j∗∥ is a decreasing sequence and bounded below ∀j∗∈FV. Hence, limn⟶∞∥jn−j∗∥ exists.Lemma 15.
LetS and V:S⟶S be as in Lemma 14. Let jn be a sequence defined by Picard-Thakur hybrid iterative scheme (14) with FV≠∅. Then, limn⟶∞∥jn−Vjn∥=0.Proof.
As from the above Lemma14, limn⟶∞∥jn−j∗∥ exists for each j∗∈FV. Suppose that for some l≥0, we have
(55)limn⟶∞∥jn−j∗∥=l.
As from (53), (52), and (51), we have
(56)∥mn−j∗∥≤∥jn−j∗∥,(57)∥ln−j∗∥≤∥jn−j∗∥,(58)∥kn−j∗∥≤∥jn−j∗∥.
Takinglimsup as n⟶∞ of (58), (57), and (56), we get
(59)limsupn⟶∞mn−j∗≤l,(60)limsupn⟶∞ln−j∗≤l,(61)limsupn⟶∞kn−j∗≤l.
SinceV is nonexpansive, we have
(62)limsupn⟶∞Vjn−j∗≤l,(63)l=liminfn⟶∞∥jn+1−j∗∥=liminfn⟶∞∥Vkn−j∗∥≤liminfn⟶∞∥kn−j∗∥,
From (63) and (61), we get
(64)limn⟶∞∥kn−j∗∥=l.
Now, from (53), we have
(65)∥kn−j∗∥≤∥mn−j∗∥.
Takingliminf as n⟶∞, we have
(66)liminfn⟶∞∥kn−j∗∥≤liminfn⟶∞∥mn−j∗∥,(67)l≤liminfn⟶∞∥mn−j∗∥.
So, from (67) and (59), we have
(68)l=limn⟶∞∥mn−j∗∥=limn⟶∞∥1−σnjn+σnVjn−j∗∥=limn⟶∞∥1−σnjn−j∗+σnVjn−Vj∗∥.
From (68), (62), and (55) and applying Lemma 6, we get
(69)limn⟶∞∥jn−Vjn∥=0.Theorem 16.
LetS, V, jn be as in Lemma 14. Let B be the uniformly convex Banach space which satisfies Opial’s condition; then, jn converges weakly to a fixed point of V.Proof.
Letj∗∈FV; then, by Lemma 14, limn⟶∞∥jn−j∗∥ exists. Now, we show that jn has a unique weak subsequential limit in FV.
Letan and bn be two subsequences of jn and a, b be the weak limits of the subsequences of jn, respectively. From Lemma 15, limn⟶∞∥jn−Vjn∥=0 and I−V is demiclosed at zero. By Lemma 7.
Therefore, we getVa=a. For b∈FV, we follow the same manner.
From Lemma14, we know that limn⟶∞∥jn−j∗∥ exists.
For uniqueness, supposing thata≠b, then, by using Opial’s condition,
(70)limn⟶∞∥jn−a∥=limn⟶∞∥an−a∥<limn⟶∞∥an−b∥=limn⟶∞∥jn−b∥=limn⟶∞∥bn−b∥<limn⟶∞∥bn−a∥=limn⟶∞∥jn−a∥.
This is a contradiction, soa=b. Hence, jn converges weakly to FV.Theorem 17.
LetS, V, jn be as in Lemma 14. Then, jn converges to a point of FV if and only if liminfn⟶∞djn,FV=0 or limsupn⟶∞jn,FV=0, where dan,FV=inf∥jn−j∗∥:j∗∈FV.Proof.
If the sequencejn⟶j∗∈FV, then, it is oblivious that liminfn⟶∞djn,FV=0 or limsupn⟶∞jn,FV=0.
Conversely, assume thatliminfn⟶∞djn,FV=0. From Lemma 14,
limn⟶∞∥jn−j∗∥ exists, ∀j∗∈FV. Therefore, by assumption,
(71)limn⟶∞djn,FV=0.
Now, to show, the sequencejn is cauchy in S. As limn⟶∞djn,FV=0, for given λ>0, there exists m0∈I+ such that ∀n≥m0,(72)djn,FV<λ2⇒inf∥jn−j∗∥:j∗∈FV<λ2.
Particularly,inf∥jn−j∗∥:j∗∈FV<λ/2. Therefore, there is j∗∈FV such that
(73)∥jm0−j∗∥<λ2.
Now, form,n≥m0,(74)∥jn+m−jn∥≤∥jm+n−j∗∥+∥jn−j∗∥≤∥jm0−j∗∥+∥jm0−j∗∥=2∥jm0−j∗∥<λ.
This shows that the sequencejn is cauchy in S. As S⊂B, so, p is a point in S such that limn⟶∞jn=p. Now, limn⟶∞djn,FV=0 gives that limn⟶∞djn,FV=0⇒p∈FV.Theorem 18.
LetS, V, jn be as in Lemma 14. Then, jn converges strongly to FV≠∅.Proof.
By Lemma15, we have
(75)limn⟶∞∥jn−Vjn∥=0.
Since,S is compact, then, let jnk be a subsequence of jn which converges strongly to j∗, for some j∗∈S. By Proposition 8, we have
(76)∥jnk−Vj∗∥≤3∥jnk−Vjnk∥+∥jnk−j∗∥∀k≥1.
Lettingk⟶∞, we get
(77)jnk⟶Vj∗⇒Vj∗=j∗,i.e.,j∗∈FV.
Also, by Lemma14, limn⟶∞∥jn−j∗∥ exists. Thus, jn converges strongly to j∗.Now, by using condition (I), we prove the strong convergence result.Theorem 19.
LetS, V be as in Lemma 14. Let B be a uniformly convex Banach space which is satisfying condition (I). Then, the sequence jn defined by the Picard-Thakur hybrid iterative scheme (14) converges strongly to FV≠∅.Proof.
As by Lemma15, we have
(78)limn⟶∞∥jn−Vjn∥=0.
By condition (I) and (78), we get
(79)0≤limn⟶∞Zdjn,FV≤limn⟶∞∥jn−Vjn∥⇒limn⟶∞Zdjn,FV=0.
SinceZ:0,∞⟶0,∞ is an increasing function satisfying Z0=0,Zt>0∀t>0.
Hence, we have(80)limn⟶∞djn,FV=0.
Since all the conditions of Theorem17 are satisfied, therefore, we can say that jn converges strongly to FV.


## 6. Conclusion

In this paper, we present a new hybrid scheme of Picard and Thakur et al. We discuss the convergence of this scheme to the iterative scheme of Mann, Ishikawa, Picard-Mann, Picard-Ishikawa, Picard-S, and Thakur et al. We showed the convergence of Picard-Thakur hybrid iterative with other iterative schemes on graphs and gave application to delay differential equations. We also generalize and extend various results for nonexpansive mapping in a uniformly convex Banach space including [7, 24, 25, 43].



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