On Rate of Convergence of Jungck-Type Iterative Schemes

Abstract

We introduce a new iterative scheme called Jungck-CR iterative scheme and study the stability and strong convergence of this iterative scheme for a pair of nonself-mappings using a certain contractive condition. Also, convergence speed comparison and applications of Jungck-type iterative schemes will be shown through examples.

1. Introduction and Preliminaries

Let 𝑋 be a Banach space, 𝑌 an arbitrary set, and 𝑆,𝑇∶𝑌→𝑋 such that 𝑇(𝑌)⊆𝑆(𝑌). For 𝑥₀∈𝑌, consider the following iterative scheme: 𝑆𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛=0,1,…. (1) This scheme is called Jungck iterative scheme and was essentially introduced by Jungck [1] in 1976 and it becomes the Picard iterative scheme when 𝑆=𝐼_𝑑 (identity mapping) and 𝑌=𝑋.

For 𝛼_𝑛∈[0,1], Singh et al. [2] defined the Jungck-Mann iterative scheme as 𝑆𝑥_𝑛+1=1−𝛼_𝑛𝑆𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛. (2) For 𝛼_𝑛,𝛽_𝑛,𝛾_𝑛∈[0,1], Olatinwo defined the Jungck-Ishikawa [3] (see also [4, 5]) and Jungck-Noor [6] iterative schemes as 𝑆𝑥_𝑛+1=1−𝛼_𝑛𝑆𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛽_𝑛𝑆𝑥_𝑛+𝛽_𝑛𝑇𝑥_𝑛,𝑆𝑥_𝑛+1=1−𝛼_𝑛𝑆𝑥_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛽_𝑛𝑆𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑆𝑧_𝑛=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛, (3)(4) respectively.

Chugh and Kumar [7] defined the Jungck-SP iterative scheme as 𝑆𝑥_𝑛+1=1−𝛼_𝑛𝑆𝑦_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛽_𝑛𝑆𝑧_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑆𝑧_𝑛=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛, (5) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are sequences of positive numbers in [0,1].

Remark 1. If 𝑋=𝑌and 𝑆=𝐼_𝑑 (identity mapping), then the Jungck-SP (5), Jungck-Noor (4), Jungck-Ishikawa (3), and the Jungck-Mann (2) iterative schemes, respectively, become the SP [8], Noor [9], Ishikawa [10] and the Mann [11] iterative schemes.

Jungck [1] used the iterative scheme (1) to approximate the common fixed points of the mappings 𝑆 and 𝑇 satisfying the following Jungck contraction: 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑆𝑥,𝑆𝑦,0≤𝛼<1. (6)

Olatinwo [3] used the following more general contractive definition than (6) to prove the stability and strong convergence results for the Jungck-Ishikawa iteration process: there exists a real number 𝑎∈[0,1) and a monotone increasing function 𝜙: 𝑅⁺→𝑅⁺ such that 𝜙(0)=0 and for all 𝑥,𝑦∈𝑌, we have 𝑇𝑥−𝑇𝑦≤𝜙(‖𝑆𝑥−𝑇𝑥‖)+𝑎𝑆𝑥−𝑆𝑦. (7) Olatinwo [6] used the convergences of Jungck-Noor iterative scheme (4) to approximate the coincidence points (not common fixed points) of some pairs of generalized contractive-like operators with the assumption that one of each of the pairs of maps is injective.

Motivated by the above facts, for 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛∈[0,1], we introduce the following iterative scheme: 𝑆𝑥_𝑛+1=1−𝛼_𝑛𝑆𝑦_𝑛+𝛼_𝑛𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛽_𝑛𝑇𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛,𝑆𝑧_𝑛=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛 (JCR) and call it Jungck-CR iterative scheme.

Remark 2. Putting 𝛼_𝑛=0 and 𝛼_𝑛=0, 𝛽_𝑛=1 in Jungck-CR iterative scheme, we get Jungck versions of Agarwal et al. [12] and Sahu and Petruşel [13] iterative schemes, respectively, as defined below: 𝑆𝑥_𝑛+1=1−𝛽_𝑛𝑇𝑥_𝑛+𝛽_𝑛𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,𝑆𝑥_𝑛+1=𝑇𝑦_𝑛,𝑆𝑦_𝑛=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛. (JA)(JS)

We will need the following definitions and lemma.

Definition 3 (see [14]). Let {𝑢_𝑛} and {𝑣_𝑛} be two fixed-point iteration procedures that converge to the same fixed point 𝑝 on a normed space 𝑋 such that the error estimates 𝑢_𝑛−𝑝≤𝑎_𝑛,𝑣_𝑛−𝑝≤𝑏_𝑛 (8) are available, where {𝑎_𝑛} and {𝑏_𝑛} are two sequences of positive numbers (converging to zero). If {𝑎_𝑛} converge faster than {𝑏_𝑛}, then we say that {𝑢_𝑛} converges faster to 𝑝 than {𝑣_𝑛}.

Definition 4 (see [15]). Suppose that {𝑎_𝑛} and {𝑏_𝑛} are two real convergent sequences with limits 𝑎 and 𝑏, respectively. Then, {𝑎_𝑛} is said to converge faster than {𝑏_𝑛} if lim_𝑛→∞𝑎_𝑛−𝑎𝑏_𝑛−𝑏=0. (9)

Definition 5 (see [16, 17]). Let 𝑓 and 𝑔 be two self-maps on 𝑋. A point 𝑥 in 𝑋 is called (1) a fixed point of 𝑓 if 𝑓(𝑥)=𝑥; (2) coincidence point of a pair (𝑓,𝑔) if 𝑓𝑥=𝑔𝑥; (3) common fixed point of a pair (𝑓,𝑔) if 𝑥=𝑓𝑥=𝑔𝑥. If 𝑤=𝑓𝑥=𝑔𝑥 for some 𝑥 in 𝑋, then 𝑤 is called a point of coincidence of 𝑓 and 𝑔. A pair (𝑓,𝑔) is said to be weakly compatible if 𝑓 and 𝑔 commute at their coincidence points.

Lemma 6 (see [18]). If 𝛿 is a real number such that 0≤𝛿<1 and {∈_𝑛}^∞_𝑛=0 is a sequence of positive numbers such that lim_𝑛→∞∈_𝑛=0, then for any sequence of positive numbers {𝑢_𝑛}^∞_𝑛=0 satisfying 𝑢_𝑛+1≤𝛿𝑢_𝑛+∈_𝑛,𝑛=0,1,2,… (10) one has lim_𝑛→∞𝑢_𝑛=0.

Definition 7 (see [2]). Let 𝑆, 𝑇∶𝑌→𝑋 be non-self-operators for an arbitrary set 𝑌 such that 𝑇(𝑌)⊆𝑆(𝑌) and 𝑝 a point of coincidence of 𝑆 and 𝑇. Let {𝑆𝑥_𝑛}^∞_𝑛=0⊂𝑋, be the sequence generated by an iterative procedure 𝑆𝑥_𝑛+1=𝑓𝑇,𝑥_𝑛,𝑛=0,1…, (11) where 𝑥₀∈𝑋 is the initial approximation and 𝑓 is some function. Suppose that {𝑆𝑥_𝑛}^∞_𝑛=0 converges to 𝑝. Let {𝑆𝑦_𝑛}^∞_𝑛=0⊂𝑋 be an arbitrary sequence and set 𝜀_𝑛=𝑑(𝑆𝑦_𝑛,𝑓(𝑇,𝑦_𝑛)),𝑛=0,1,…. Then, the iterative procedure (11) is said to be (𝑆,𝑇)-stable or stable if and only if lim_𝑛→∞∈_𝑛=0 implies lim_𝑛→∞𝑆𝑦_𝑛=𝑝.

The purpose of this paper is to study the stability and strong convergence of Jungck-CR (JCR) iterative scheme for nonself-mappings in an arbitrary Banach space by employing the contractive conditions (7) and then to compare convergence rates of Jungck-type iterative schemes. Moreover, applications of Jungck-type iterative schemes in recurrent neural networks (RNN) analysis will be discussed.

2. Strong Convergence in an Arbitrary Banach Space

Theorem 8. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let {𝑆𝑥_𝑛}^∞_𝑛=0 be the Jungck-CR iterative scheme defined by (JCR), where {𝛼_𝑛}, {𝛽_𝑛},{𝛾_𝑛} are sequences of positive numbers in [0,1] with {𝛼_𝑛} satisfying ∑^∞_𝑛=0𝛼_𝑛=∞.Then, the Jungck-CR iterative scheme {𝑆𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝. Also, 𝑝 will be the unique common fixed point of 𝑆, 𝑇 provided that 𝑌=𝑋, and 𝑆 and 𝑇 are weakly compatible.

Proof. First, we prove that Jungck-CR iterative scheme {𝑆𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝.
It follows from (JCR) and (7) that 𝑆𝑥_1𝑛+−𝑝=1−𝛼_𝑛𝑆𝑦_𝑛+𝛼_𝑛𝑇𝑦_𝑛−1−𝛼_𝑛+𝛼_𝑛𝑝≤1−𝛼_𝑛𝑆𝑦_𝑛−𝑝+𝛼_𝑛𝑇𝑦_𝑛=−𝑝1−𝛼_𝑛𝑆𝑦_𝑛−𝑝+𝛼_𝑛𝑇𝑧−𝑇𝑦_𝑛≤1−𝛼_𝑛𝑆𝑦_𝑛−𝑝+𝛼_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧−𝑆𝑦_𝑛=1−𝛼_𝑛𝑆𝑦_𝑛−𝑝+𝑎𝛼_𝑛𝑆𝑦_𝑛=−𝑝1−𝛼_𝑛(1−𝑎)𝑆𝑦_𝑛−𝑝. (12) Now, we have the following estimates: 𝑆𝑦_𝑛−𝑝=1−𝛽_𝑛𝑇𝑥_𝑛+𝛽_𝑛𝑇𝑧_𝑛−1−𝛽_𝑛+𝛽_𝑛𝑝≤1−𝛽_𝑛𝑇𝑥_𝑛−𝑝+𝛽_𝑛𝑇𝑧_𝑛≤−𝑝1−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑧+𝛽_𝑛𝑇𝑧_𝑛≤−𝑇𝑧1−𝛽_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑥_𝑛−𝑆𝑧+𝛽_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧_𝑛≤−𝑆𝑧1−𝛽_𝑛𝑎𝑆𝑥_𝑛−𝑝+𝛽_𝑛𝑎𝑆𝑧_𝑛−𝑝,𝑆𝑧_𝑛−𝑝=1−𝛾_𝑛𝑆𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛−1−𝛾_𝑛+𝛾_𝑛𝑝≤1−𝛾_𝑛𝑆𝑥_𝑛−𝑝+𝛾_𝑛𝑇𝑥_𝑛≤−𝑇𝑧1−𝛾_𝑛𝑆𝑥_𝑛−𝑝+𝛾_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑥_𝑛=−𝑆𝑧1−𝛾_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (13) It follows from (13) that 𝑆𝑦_𝑛−𝑝≤1−𝛽_𝑛𝑎𝑆𝑥_𝑛−𝑝+𝛽_𝑛𝑎1−𝛾_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (14) Using (1−𝛽_𝑛)𝑎≤(1−𝛽_𝑛) and 𝛽_𝑛𝑎(1−𝛾_𝑛(1−𝑎))≤𝛽_𝑛𝑎, inequality (14) yields 𝑆𝑦_𝑛−𝑝≤1−𝛽_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (15) It follows from (15) and (12) that 𝑆𝑥_1𝑛+−𝑝≤1−𝛼_𝑛(1−𝑎)1−𝛽_𝑛(1−𝑎)𝑆𝑥_𝑛≤−𝑝1−𝛼_𝑛(1−𝑎)𝑆𝑥_𝑛≤−𝑝_𝑛𝑘=01−𝛼_𝑘(1−𝑎)𝑆𝑥₀−𝑝≤𝑒^{−(1−𝑎)∑∞𝑘=0𝛼𝑘}𝑆𝑥₀−𝑝. (16) Since 0≤𝑎<1,𝛼_𝑘∈[0,1] and ∑^∞_𝑛=0𝛼_𝑛=∞, so 𝑒^{−(1−𝑎)∑𝑛𝑘=0𝛼𝑘}→0 as 𝑛→∞.
Hence, it follows from (16) that lim_𝑛→∞‖𝑆𝑥_𝑛+1−𝑝‖=0. Therefore, {𝑆𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝.
Now, we prove that 𝑝 is unique common fixed point of 𝑆 and 𝑇.
Let there exist another point of coincidence say 𝑝^∗. Then, there exists 𝑞^∗∈𝑋 such that 𝑆𝑞^∗=𝑇𝑞^∗=𝑝^∗. But from (7), we have 0≤𝑝−𝑝^∗=𝑇𝑞−𝑇𝑞^∗≤𝜙𝑆𝑞−𝑇𝑞+𝑎𝑆𝑞−𝑆𝑞^∗=𝑎𝑝−𝑝^∗, (17) which implies that 𝑝=𝑝^∗ as 0≤𝑎<1.
Now, as 𝑆 and 𝑇 are weakly compatible and 𝑝=𝑇𝑞=𝑆𝑞, so 𝑇𝑝=𝑇𝑇𝑞=𝑇𝑆𝑞=𝑆𝑇𝑞 and hence 𝑇𝑝=𝑆𝑝. Therefore, 𝑇𝑝 is a point of coincidence of 𝑆, 𝑇 and since the point of coincidence is unique then 𝑝=𝑇𝑝. Thus, 𝑇𝑝=𝑆𝑝=𝑝, and therefore 𝑝 is unique common fixed point of 𝑆 and 𝑇.

Corollary 9. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and, 𝑆, 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let {𝑆𝑥_𝑛}^∞_𝑛=0 be the iterative scheme defined by (JA), where {𝛼_𝑛},{𝛽_𝑛} are sequences of positive numbers in [0,1] with {𝛼_𝑛} satisfying ∑^∞_𝑛=0𝛼_𝑛=∞. Then the Jungck-Agarwal iterative scheme {𝑆𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝. Also, 𝑝 will be the unique common fixed point of 𝑆, 𝑇 provided that 𝑌=𝑋, and 𝑆 and 𝑇 are weakly compatible.

Proof. Putting 𝛼_𝑛=0 and 𝛽_𝑛=𝛼_𝑛, in iterative scheme (JCR), convergence of iterative scheme (JA) can be proved on the same lines as in Theorem 8.

Corollary 10. Let (𝑋,‖⋅‖) be an arbitrary Banach space and 𝑆, and let 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let {𝑆𝑥_𝑛}^∞_𝑛=0 be the Jungck-S iterative scheme defined by (JS), where {𝛼_𝑛},{𝛽_𝑛} are sequences of positive numbers in [0,1] with {𝛼_𝑛} satisfying ∑^∞_𝑛=0𝛼_𝑛=∞. Then the Jungck-S iterative scheme {𝑆𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝. Also, 𝑝 will be the unique common fixed point of 𝑆, 𝑇 provided that 𝑌=𝑋, and 𝑆 and 𝑇 are weakly compatible.

Proof. Putting 𝛼_𝑛=0 and 𝛾_𝑛=𝛼_𝑛,𝛽_𝑛=1 in iterative scheme (JCR), convergence of iterative scheme (JS) can be proved on the same lines as in the Theorem 8.

The following examples reveal the validity of our results.

Example 11. Let 𝑋=𝑌=[0,1]. Define 𝑇 and 𝑆 by 𝑇(𝑥)=[)10,𝑥∈0,12,𝑥=1,𝑆𝑥=𝑥²,𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1√,𝜙(𝑡)2𝑛+4=2𝑎𝑡. (18) It is clear that 𝑇 and 𝑆 are quasicontractive operators satisfying (7) but do not satisfy contractive condition (6), with a unique common fixed point 0.

Using computer programming in C++ with initial approximation 𝑥₀=1, convergence of Jungck-CR iterative scheme to the common fixed point 0 is shown in Table 1.

Table 1 

Example 12. Let 𝑌=𝑋=[0,1]. Define 𝑇 and 𝑆 by 𝑇(𝑥)=(1/2)(1/2+𝑥), 𝑆(𝑥)=1−𝑥,𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/√2𝑛+4, and 𝜙(𝑡)=2𝑎𝑡. It is clear that 𝑇 and 𝑆 are weakly compatible quasicontractive operators satisfying (7) with a unique common fixed point 0.5.

Using computer programming in C++ with initial approximation 𝑥₀=0.8, convergence of Jungck-CR iterative scheme to the common fixed point 0.5 is shown in Table 2.

Table 2 

Theorem 13. Let (𝑋,‖⋅‖) be an arbitrary Banach space and 𝑆, and let 𝑇∶𝑌→𝑋 be nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), S(Y) is a complete subspace of 𝑋, and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌 and 𝛼∈(0,1), let {𝑆𝑥_𝑛}^∞_𝑛=0 be the Jungck-CR iterative scheme (JCR) converging to 𝑝, where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are sequences in [0,1] with {𝛼_𝑛} satisfying 𝛼≤𝛼_𝑛 for all 𝑛. Then, the Jungck-CR iterative scheme is (𝑆,𝑇)-stable.

Proof. Suppose that {𝑆𝑦_𝑛}^∞_𝑛=0⊂𝑋 be an arbitrary sequence, 𝜀_𝑛=‖𝑆𝑦_𝑛+1−(1−𝛼_𝑛)𝑆𝑏_𝑛−𝛼_𝑛𝑇𝑏_𝑛‖, 𝑛=0,1,2,3…, where 𝑆𝑏_𝑛=(1−𝛽_𝑛)𝑇𝑦_𝑛+𝛽_𝑛𝑇𝑐_𝑛,𝑆𝑐_𝑛=(1−𝛾_𝑛)𝑆𝑦_𝑛+𝛾_𝑛𝑇𝑦_𝑛 and let lim_𝑛→∞∈_𝑛=0.
Then, for Jungck-CR iterative scheme (JCR), we have 𝑆𝑦_1𝑛+−𝑝≤𝑆𝑦_1𝑛+−1−𝛼_𝑛𝑆𝑏_𝑛−𝛼_𝑛𝑇𝑏_𝑛+1−𝛼_𝑛𝑆𝑏_𝑛+𝛼_𝑛𝑇𝑏_𝑛−1−𝛼_𝑛+𝛼_𝑛𝑝≤𝜀_𝑛+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝛼_𝑛𝑇𝑏_𝑛−𝑝=𝜀_𝑛+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝛼_𝑛𝑇𝑧−𝑇𝑏_𝑛≤𝜀_𝑛+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝛼_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧−𝑆𝑏_𝑛=𝜀_𝑛+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝛼_𝑛𝜙(‖0‖)+𝑎𝑆𝑧−𝑆𝑏_𝑛=1−𝛼_𝑛(1−𝑎)𝑆𝑏_𝑛−𝑝+𝜀_𝑛. (19) Now, we have the following estimates: 𝑆𝑏_𝑛−𝑝=1−𝛽_𝑛𝑇𝑦_𝑛+𝛽_𝑛𝑇𝑐_𝑛−1−𝛽_𝑛+𝛽_𝑛𝑝≤1−𝛽_𝑛𝑇𝑦_𝑛−𝑝+𝛽_𝑛𝑇𝑐_𝑛=−𝑝1−𝛽_𝑛𝑇𝑦_𝑛−𝑇𝑧+𝛽_𝑛𝑇𝑧−𝑇𝑐_𝑛≤1−𝛽_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧−𝑆𝑦_𝑛+𝛽_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧−𝑆𝑐_𝑛≤1−𝛽_𝑛𝑎𝑝−𝑆𝑦_𝑛+𝛽_𝑛𝑎𝑝−𝑆𝑐_𝑛,𝑆𝑐_𝑛−𝑝=1−𝛾_𝑛𝑆𝑦_𝑛+𝛾_𝑛𝑇𝑦_𝑛−1−𝛾_𝑛+𝛾_𝑛𝑝≤1−𝛾_𝑛𝑆𝑦_𝑛−𝑝+𝛾_𝑛𝑇𝑦_𝑛=−𝑝1−𝛾_𝑛𝑆𝑦_𝑛−𝑝+𝛾_𝑛𝑇𝑧−𝑇𝑦_𝑛≤1−𝛾_𝑛𝑆𝑦_𝑛−𝑇𝑧+𝛾_𝑛𝜙(‖𝑆𝑧−𝑇𝑧‖)+𝑎𝑆𝑧−𝑆𝑦_𝑛=1−𝛾_𝑛(1−𝑎)𝑆𝑦_𝑛−𝑝. (20) It follows from (19), (20) that 𝑆𝑦_1𝑛+−𝑝≤1−𝛼_𝑛(1−𝑎)𝑆𝑦_𝑛−𝑝+𝜀_𝑛. (21) Using 0<𝛼≤𝛼_𝑛 and 𝑎∈[0,1), we have [1−𝛼_𝑛(1−𝑎)]<1.
Hence using Lemma 6, (21) yields lim_𝑛→∞𝑆𝑦_𝑛+1=𝑝.
Conversely, let lim_𝑛→∞𝑆𝑦_𝑛+1=𝑝. Then, using contractive condition (7) and the triangle inequality, we have 𝜀_𝑛=𝑆𝑦_𝑛+1−1−𝛼_𝑛𝑆𝑏_𝑛−𝛼_𝑛𝑇𝑏_𝑛≤𝑆𝑦_𝑛+1−𝑝+1−𝛼_𝑛+𝛼_𝑛𝑝−1−𝛼_𝑛𝑆𝑏_𝑛−𝛼_𝑛𝑇𝑏_𝑛≤𝑆𝑦_𝑛+1−𝑝+1−𝛼_𝑛𝑝−𝑆𝑏_𝑛+𝛼_𝑛𝑝−𝑇𝑏_𝑛=𝑆𝑦_𝑛+1−𝑝+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝛼_𝑛𝑇𝑧−𝑇𝑏_𝑛≤𝑆𝑦_𝑛+1−𝑝+1−𝛼_𝑛𝑆𝑏_𝑛−𝑝+𝑎𝛼_𝑛𝑆𝑧−𝑆𝑏_𝑛=𝑆𝑦_𝑛+1−𝑝+1−𝛼_𝑛(1−𝑎)𝑆𝑏_𝑛−𝑝. (22) By using estimates (20), (22), yields 𝜀_𝑛≤1−𝛼_𝑛(1−𝑎)𝑆𝑦_𝑛−𝑝+𝑆𝑦_1𝑛+−𝑝. (23) Hence, lim_𝑛→∞𝜀_𝑛=0.
Therefore, the JCR iterative scheme is (𝑆,𝑇) stable.

3. Results on Direct Comparison of Jungck-Type Iterative Schemes

Various authors [7, 13–15, 19–22] have worked on convergence speed of iterative schemes. In [14], Berinde showed that Picard iteration is faster than Mann iteration for quasicontractive operators. In [15], Qing and Rhoades by taking an example showed that Ishikawa iteration is faster than Mann iteration for a certain quasicontractive operator. In [20], Hussain et al. provided an example of a quasicontractive operator for which the iterative scheme due to Agarwal et al. is faster than Mann and Ishikawa iterative schemes. Recently, Chugh and Kumar [19] showed that SP iterative scheme with error terms converges faster than Ishikawa and Noor iterative schemes for accretive-type mappings. For recent work in this direction, we refer the reader to [23–27] and references therein.

Theorem 14. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of X and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-Mann iterative scheme be defined by (JM) and Jungck-Ishikawa iterative scheme be defined by (JI), with 𝛼_𝑛∈[0,1/(1+(1+2/𝑚)𝑎)], 𝛽_𝑛≤1−𝛼_𝑛(1−𝑎), for some 𝑚>0 and 𝑛∈𝑁 satisfying ∑^∞_𝑛=0𝛼_𝑛=∞.
Then, the Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to 𝑝.

Proof. For Jungck-Mann iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≥1−𝛼_𝑛𝑆𝑥_𝑛−𝑝−𝛼_𝑛𝑇𝑥_𝑛≥−𝑝1−𝛼_𝑛𝑆𝑥_𝑛−𝑝−𝛼_𝑛𝑎𝑆𝑥_𝑛≥−𝑝1−𝛼_𝑛(1+𝑎)𝑆𝑢_𝑛−𝑝. (24) Also, for Jungck-Ishikawa iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≤1−𝛼_𝑛𝑆𝑥_𝑛−𝑝+𝑎𝑇𝑦_𝑛≤−𝑝1−𝛼_𝑛𝑆𝑥_𝑛−𝑝+𝛼_𝑛𝑎𝑆𝑦_𝑛−𝑝. (25) But 𝑆𝑥_1𝑛+−𝑝≤1−𝛽_𝑛𝑆𝑥_𝑛−𝑝+𝛽_𝑛𝑇𝑥_𝑛≤−𝑝1−𝛽_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (26) Hence, 𝑆𝑥_1𝑛+−𝑝≤1−𝛼_𝑛(1−𝑎)−𝛼_𝑛𝛽_𝑛𝑎(1−𝑎)𝑇𝑥_𝑛−𝑝. (27) Using (24) and (27), we have JI_𝑛+1JM_𝑛+1≤^𝑛_𝑖=01−𝛼_𝑖(1−𝑎)−𝛼_𝑖𝛽_𝑖𝑎(1−𝑎)1−𝛼_𝑖().1+𝑎 (28) But we observe that 1−𝛼_𝑖(1−𝑎)1−𝛼_𝑖(1+𝑎)≤1+𝑚∀𝑖=0,1,2,…. (29) Using (29) together with 𝛽_𝑛≤1−𝛼_𝑛(1−𝑎), we have ^𝑛_𝑖=01−𝛼_𝑖(1−𝑎)−𝛼_𝑖𝛽_𝑖𝑎(1−𝑎)1−𝛼_𝑖()≤()1+𝑎1+𝑚1−𝛼_𝑖(1−𝑎)≤(1+𝑚)𝑒^{−𝛼𝑖(1−𝑎)}. (30) As ∑^∞_𝑛=0𝛼_𝑛=∞, so (28) yields lim_𝑛→∞‖(JI_𝑛+1−𝑝)/(JM_𝑛+1−𝑝)‖=0.
Therefore, by Definition 4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to 𝑝.

Theorem 15. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋, and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-Noor iterative scheme be defined by (JN) and Jungck-Ishikawa iterative scheme defined by (JI), with 𝛼_𝑛∈[0,1/(1+(1+2/𝑚)𝑎)], 𝛽_𝑛≤1−𝛼_𝑛(1−𝑎),for some 𝑚>0 and 𝑛∈𝑁 satisfying ∑^∞_𝑛=0𝛼_𝑛=∞. Then, the Jungck-Noor iterative scheme converges faster than Jungck-Ishikawa iterative scheme to 𝑝.

Proof. For Jungck-Ishikawa iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≥1−𝛼_𝑛(1+𝑎)𝑆𝑢_𝑛−𝑝. (31) Also, for Jungck-Noor iterative scheme, we have 𝑆𝑥_1𝑛+≤−𝑝1−𝛼_𝑛(1−𝑎)−𝛼_𝑛𝛽_𝑛𝑎(1−𝑎)−𝛼_𝑛𝛽_𝑛𝛾_𝑛𝑎²()×1−𝑎𝑇𝑥_𝑛−𝑝. (32) Using (31) and (32), we haveJN_𝑛+1JI_𝑛+1≤^𝑛_𝑖=01−𝛼_𝑖(1−𝑎)−𝛼_𝑖𝛽_𝑖𝑎(1−𝑎)−𝛼_𝑛𝛽_𝑛𝛾_𝑛𝑎²(1−𝑎)1−𝛼_𝑖()≤1+𝑎_𝑛𝑖=01−𝛼_𝑖(1−𝑎)−𝛼_𝑖𝛽_𝑖𝑎(1−𝑎)1−𝛼_𝑖().1+𝑎 (33)Making the same calculations as in Theorem 14, (33) yields lim_𝑛→∞JN_𝑛+1−𝑝JI_𝑛+1−𝑝=0. (34) By Definition 4, Jungck-Noor iterative scheme converges faster than Jungck-Ishikawa iterative scheme to 𝑝.

Theorem 16. Let (𝑋,‖⋅‖) be an arbitrary Banach space and 𝑆, 𝑇∶𝑌→𝑋 be nonself operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-Noor iterative scheme be defined by (JN) and Jungck-SP iterative scheme defined by (JSP), with 𝛼_𝑛∈[0,1/(1+(1+2/𝑚)𝑎)], for some 𝑚>0 satisfying ∑^∞_𝑛=0𝛽_𝑛=∞. Then, the Jungck-SP iterative scheme converges faster than Jungck-Noor iterative scheme to 𝑝.

Proof. For Jungck-Noor iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≥1−𝛼_𝑛(1+𝑎)𝑆𝑢_𝑛−𝑝. (35) Also, for Jungck-SP iterative scheme, we have 𝑆𝑥_1𝑛+≤−𝑝1−𝛼_𝑛(1−𝑎)1−𝛽_𝑛()×1−𝑎1−𝛾_𝑛(1−𝑎)𝑇𝑥_𝑛−𝑝. (36) Using (35) and (36), we have JSP_𝑛+1−𝑝JN_𝑛+1≤−𝑝_𝑛𝑖=01−𝛼_𝑛(1−𝑎)1−𝛽_𝑛(1−𝑎)1−𝛾_𝑛(1−𝑎)1−𝛼_𝑖()≤1+𝑎_𝑛𝑖=01−𝛼_𝑛(1−𝑎)1−𝛽_𝑛(1−𝑎)1−𝛼_𝑖().1+𝑎 (37) We observe that 1−𝛼_𝑖(1−𝑎)1−𝛼_𝑖(1+𝑎)≤1+𝑚∀𝑖=0,1,2,…. (38) Using (38) together with ∑^∞_𝑛=0𝛽_𝑛=∞, (37) yields lim_𝑛→∞JSP_𝑛+1−𝑝JN_𝑛+1−𝑝=0. (39) Therefore, by Definition 4, Jungck-SP iterative scheme converges faster than Jungck-Noor iterative scheme 𝑝.

Theorem 17. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-Agarwal’s et al. iterative scheme be defined by (JA) and Jungck-SP iterative scheme be defined by (JSP) with (i)∑^∞_𝑛=0𝛼_𝑛=∞, (ii)lim_𝑛→∞𝛼_𝑛=0, and (iii)lim_𝑛→∞𝛽_𝑛=0. Then, the Jungck-Agarwal iterative scheme converges faster than Jungck-SP iterative scheme to 𝑝.

Proof. For Jungck-SP iterative scheme, we have 𝑆𝑥_𝑛+1−𝑝≥1−𝛼_𝑛(1+𝑎)𝑆𝑥_𝑛−𝑝. (40) Also, for Jungck-Agarwal iterative scheme, we have 𝑆𝑥_𝑛+1−𝑝≤𝑎1−𝛼_𝑛𝛽_𝑛(1−𝑎)𝑇𝑥_𝑛−𝑝. (41) Using (40) and (41), we have JSP_𝑛+1−𝑝JA_𝑛+1−𝑝≤𝑎^𝑛𝑛_𝑖=01−𝛼_𝑛𝛽_𝑛(1−𝑎)1−𝛼_𝑖().1+𝑎 (42) Since 𝑎∈[0,1) and lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞𝛽_𝑛=0.
Hence from (42), we have lim_𝑛→∞JSP_𝑛+1−𝑝JA_𝑛+1−𝑝=0. (43) Therefore, by Definition 4, Jungck-SP iterative scheme converges faster than Jungck-Agarwal et al.’s iterative scheme to 𝑝.

Theorem 18. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself-operators on an arbitrary set 𝑌 satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-S iterative scheme be defined by (JS) and Jungck-Agarwal iterative scheme defined by (JA). Then, the Jungck-S iterative scheme converges faster than Jungck-Agarwal iterative scheme to 𝑝.

Proof. For Jungck-S iterative scheme, we have 𝑆𝑥_𝑛+1−𝑝≤𝑎1−𝛼_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (44) Also, for Jungck-Agarwal iterative scheme, we have 𝑆𝑥_𝑛+1−𝑝≤𝑎1−𝛼_𝑛𝛽_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (45) It is obvious that 𝑎1−𝛼_𝑛(1−𝑎)≤𝑎1−𝛼_𝑛𝛽_𝑛(1−𝑎)∀𝑛. (46) Hence by Definition 3, Jungck-S iterative scheme converges faster than Jungck-Agarwal iterative scheme.

Theorem 19. Let (𝑋,‖⋅‖) be an arbitrary Banach space, and let 𝑆, 𝑇∶𝑌→𝑋 be nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that 𝑇(𝑌)⊆𝑆(𝑌), 𝑆(𝑌) is a complete subspace of 𝑋 and 𝑆𝑧=𝑇𝑧=𝑝 (say). For 𝑥₀∈𝑌, let Jungck-S iterative scheme be defined by (JS) and Jungck-CR iterative scheme be defined by (JCR). Then, the Jungck-CR iterative scheme converges faster than Jungck-S iterative scheme to 𝑝.

Proof. For Jungck-S iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≤𝑎1−𝛼_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (47) Also, for Jungck-CR iterative scheme, we have 𝑆𝑥_1𝑛+−𝑝≤𝑎1−𝛼_𝑛(1−𝑎)1−𝛽_𝑛𝛾_𝑛(1−𝑎)𝑆𝑥_𝑛−𝑝. (48) It is obvious that 𝑎1−𝛼_𝑛(1−𝑎)1−𝛽_𝑛𝛾_𝑛(1−𝑎)≤𝑎1−𝛼_𝑛(1−𝑎)∀𝑛. (49) Hence by Definition 3, Jungck-CR iterative scheme converges faster than Jungck-S iterative scheme.

The following example supports the above results.

Example 20. Let 𝑌=[0,1], 𝑋=[0,1/2], 𝑆∶𝑌→𝑋=𝑥/2, 𝑇∶𝑌→𝑋=𝑥/4, 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=0, 𝑛=1,2…𝑛₀−1 for some 𝑛₀∈𝑁, and 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=4/√𝑛, 𝑛≥𝑛₀. It is clear that 𝑇 and 𝑆 are quasicontractive operators satisfying (7) with the unique common fixed point 0. Also, it is easy to see that Example 20 satisfies all the conditions of Theorem 8 and Theorems 14–19.

Proof . For JM, JI, JN, JA, JS, JSP, and JCR iterative schemes with initial approximation 𝑥₀≠0, we have JM_𝑛=^𝑛_𝑖=𝑛012−1√𝑖𝑥₀,JI_𝑛=^𝑛_𝑖=𝑛012−1√𝑖−2𝑖𝑥₀,JN_𝑛=^𝑛_𝑖=𝑛021−√𝑖−4𝑖−8𝑖^3/2𝑥₀,JSP_𝑛=^𝑛_𝑖=𝑛012−3√𝑖+6𝑖−4𝑖^3/2𝑥₀,JA_𝑛=^𝑛_𝑖=𝑛014−2𝑖𝑥₀,JS_𝑛=^𝑛_𝑖=𝑛014−2𝑖𝑥₀,JCR_𝑛=^𝑛_𝑖=𝑛014−12√𝑖−2𝑖+4𝑖^3/2𝑥₀. (50) Now, for 𝑛₀=16, consider JI_𝑛+1JM_𝑛+1=∏^𝑛_𝑖=161−2/√𝑖−4/𝑖𝑥₀∏^𝑛_𝑖=161−2/√𝑖𝑥₀=^𝑛_𝑖=161−4/𝑖1−2/√𝑖=^𝑛_𝑖=1641−𝑖−2√𝑖. (51) It is easy to see that 0≤lim^𝑛_{𝑛→∞𝑖=16}41−𝑖−2√𝑖≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0. (52) Hence, lim_𝑛→∞|JI_𝑛+1/JM_𝑛+1|=0.
Therefore, by Definition 4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
Similarly, for 𝑛₀=16, JN_𝑛JI_𝑛=∏^𝑛_𝑖=161−2/√𝑖−4/𝑖−8/𝑖^3/2𝑥₀∏^𝑛_𝑖=161−2/√𝑖−4/𝑖𝑥₀=^𝑛_𝑖=161−8/𝑖^3/21−2/√=𝑖−4/𝑖_𝑛𝑖=1681−𝑖^3/2−2𝑖−4√𝑖 (53) with 0≤lim^𝑛_{𝑛→∞𝑖=16}81−𝑖^3/2−2𝑖−4√𝑖≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0 (54) implies lim_𝑛→∞JN_𝑛JI_𝑛=0. (55) Therefore, by Definition 4, JN iterative scheme converges faster than JI iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
Again, similarly, for 𝑛₀=100, JSP_𝑛JN_𝑛=∏^𝑛_𝑖=1001−6/√𝑖+12/𝑖−8/𝑖^3/2𝑥₀∏^𝑛_𝑖=1001−2/√𝑖−4/𝑖−8/𝑖^3/2𝑥₀=^𝑛_𝑖=100√1−4/𝑖−16/𝑖1−2/√𝑖−4/𝑖−8/𝑖^3/2=^𝑛_𝑖=100√1−4𝑖−16𝑖𝑖^3/2−2𝑖−4√𝑖−8 (56) with 0≤lim^𝑛_{𝑛→∞𝑖=100}√1−4𝑖−16𝑖𝑖^3/2−2𝑖−4√𝑖−8≤lim^𝑛_{𝑛→∞𝑖=100}11−𝑖=lim_𝑛→∞99𝑛=0 (57) implies lim_𝑛→∞JSP_𝑛JN_𝑛=0. (58) Therefore, by Definition 4, JSP iterative scheme converges faster than JN iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
Again, similarly, for 𝑛₀=100, JA_𝑛JSP_𝑛=∏^𝑛_𝑖=100(1/2−4/𝑖)𝑥₀∏^𝑛_𝑖=1001−6/√𝑖+12/𝑖−8/𝑖^3/2𝑥₀=^𝑛_𝑖=100√1−1/2−6/𝑖+16/𝑖−8/𝑖^3/21−6/√𝑖+12/𝑖−8/𝑖^3/2=^𝑛_𝑖=100𝑖1−^3/2−12𝑖+32√𝑖−162𝑖^3/2−12𝑖+24√𝑖−16 (59) with 0≤lim^𝑛_{𝑛→∞𝑖=100}𝑖1−^3/2−12𝑖+32√𝑖−162𝑖^3/2−12𝑖+24√𝑖−16≤lim^𝑛_{𝑛→∞𝑖=100}11−𝑖=lim_𝑛→∞99𝑛=0 (60) implies lim_𝑛→∞JA_𝑛JSP_𝑛=0. (61) Therefore, by Definition 4, JA iterative scheme converges faster than JSP iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
Again, for 𝑛₀=16, JS_𝑛JA_𝑛=∏^𝑛_𝑖=161/2−1/√𝑖𝑥₀∏^𝑛_𝑖=16(1/2−4/𝑖)𝑥₀=^𝑛_𝑖=16√1−1/𝑖−4/𝑖=1/2−4/𝑖_𝑛𝑖=162√1−𝑖−8𝑖−8 (62) with 0≤lim^𝑛_{𝑛→∞𝑖=16}2√1−𝑖−8𝑖−8≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0 (63) implies lim_𝑛→∞JS_𝑛JA_𝑛=0. (64) Therefore, by Definition 4, JS iterative scheme converges faster than JA iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
Similarly, again, for 𝑛₀=16, JCR_𝑛JS_𝑛=∏^𝑛_𝑖=161/2−1/√𝑖−4/𝑖+8/𝑖^3/2𝑥₀∏^𝑛_𝑖=161/2−1/√𝑖𝑥₀=^𝑛_𝑖=161−4/𝑖−8/𝑖^3/21/2−1/√𝑖=^𝑛_𝑖=168√1−𝑖−16𝑖^3/2−2𝑖 (65) with 0≤lim^𝑛_{𝑛→∞𝑖=16}8√1−𝑖−16𝑖^3/2−2𝑖≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0 (66) implies lim_𝑛→∞JCR_𝑛JS_𝑛=0. (67) Therefore, by Definition 4, JCR iterative scheme converges faster than JS iterative scheme to the common fixed point 0 of 𝑇 and 𝑆.
From Example 20, we observe that the decreasing order of Jungck-type iterative schemes is as follows:
JCR, JS, JA, JSP, JN, JI, and JM.

4. Applications

4.1. Jungck-Type Iterative Schemes in RNN Analysis

Recurrent neural networks (RNNs) are a class of densely connected single-layer nonlinear networks of perceptrons. RNNs not only operate on an input space but also on an internal statespace. This is equivalent to a with-memory Iterated Function System [28]. The state space enables the representation (and learning) of temporally/sequentially extended dependencies over unspecified (and potentially infinite) intervals according to 𝑦(𝑡)(𝑠(𝑡𝑠(𝑡)(𝑠()(𝑡.=𝐺))=𝐹𝑡−1,𝑥)) (68) Because of the network’s nonlinearity, a number of undesirable local energy minima emerge from the learning procedure. This has been shown to significantly affect the network’s performance. The iterative schemes like Mann, Ishikawa and 𝐽-iteration may be used to estimate the number of iterations required to achieve a stable state in recurrent autoassociative neural networks.

4.1.1. Decreasing Function (1−𝑥)⁹

In order to solve this function by Jungck-type iterative schemes, we write it in the form 𝑆𝑥=𝑇𝑥, where the functions 𝑇, 𝑆∶[0,1]→[0,2] are defined as 𝑇(𝑥)=(1−𝑥)⁹ and 𝑆𝑥=𝑥, respectively. By taking initial approximation 𝑥₀=0.8 and 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/√⁴𝑛+1, the obtained results are listed in Table 3 showing convergence of different Jungck-type schemes to 𝑝=0.175699=𝑇0.175699=𝑆0.175699.

Table 3 

Decreasing function.

4.1.2. Increasing Function 𝑥²−2𝑥−3

In order to solve this function by Jungck-type iterative schemes, we write it in the form 𝑆𝑥=𝑇𝑥, where the functions 𝑇, 𝑆∶[3,4]→[9,16] are defined as 𝑇𝑥=2𝑥+3 and 𝑆𝑥=𝑥², respectively. By taking initial approximation 𝑥₀=4 and 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/√⁴𝑛+1, the obtained results are listed in Table 4 showing convergence of different Jungck-type schemes to 𝑝=9=𝑇3=𝑆3.

Table 4 

Increasing function.

4.1.3. Oscillating Function 1/𝑥

In order to solve this function by Jungck-type iterative schemes, we write it in the form 𝑆𝑥=𝑇𝑥, where the functions 𝑇, 𝑆∶[0.5,2]→[0.25,4] are defined as 𝑇𝑥=1/𝑥 and 𝑆𝑥=𝑥², respectively. By taking initial approximation 𝑥₀=2 and 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/√⁴𝑛+1, the obtained results are listed in Table 5 showing convergence of different Jungck type schemes to 𝑝=1=𝑇1=𝑆1.

Table 5 

Oscillatory function.

4.1.4. Biquadratic Equation 𝑥⁴−36𝑥²−52𝑥+87=0

In order to solve this equation, we rewrite it in the form 𝑆𝑥=𝑇𝑥, where the functions 𝑇, 𝑆∶[0.5,1.5]→[9,81] are defined as 𝑇𝑥=𝑥⁴−52𝑥+87 and 𝑆𝑥=36𝑥², respectively. Taking initial approximation 𝑥₀=0.5 and 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/√²𝑛+1, the obtained results are listed in Table 6 showing convergence of different Jungck-type schemes to 𝑝=36=𝑇1=𝑆1.

Table 6 

Biquadratic equation.

For detailed study, these programs are again executed after changing the parameters, and some observations are given as below.

Decreasing Function(1)Taking initial guess 𝑥_𝑜=0.3 (near common fixed point), Jungck-Noor iterative scheme converges in 14 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, and Jungck-S iterative scheme converges in 25 iterations while Jungck-Mann iterative scheme shows strange constant behavior.(2)Taking 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/(1+𝑛)^1/6 and 𝑥_𝑜=0.8, we observe that Jungck-Noor iterative scheme converges in 13 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 11 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 8 iterations, and Jungck-S iterative scheme converges in 27 iterations while Jungck-Mann iterative scheme shows strange constant behavior.

Increasing Functions(1)Taking initial guess 𝑥_𝑜=3.2 (near coincidence point), Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 13 iterations.(2)Taking 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/(1+𝑛)^1/6 and 𝑥_𝑜=4, we observe that Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 14 iterations.

Oscillatory Function(1)Taking initial guess 𝑥_𝑜=1.3 (near common fixed point), Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 6 terations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, Jungck-S iterative scheme converges in 11 iterations while Jungck-Mann iterative scheme converges in 19 iterations.(2)Taking 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/(1+𝑛)^1/6 and 𝑥_𝑜=2, we observe that Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 9 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, Jungck-S iterative scheme converges in 12 iterations while Jungck-Mann iterative scheme converges in 21 iterations.

Biquadratic Equation(1)Taking initial guess 𝑥_𝑜=0.8 (near coincidence point), Jungck-Noor iterative scheme converges in 11 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 7 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 4 iterations, and Jungck-S iterative scheme converges in 18 iterations while Jungck-Mann iterative scheme converges in 35 iterations.(2)Taking 𝛼_𝑛=𝛽_𝑛=𝛾_𝑛=1/(1+𝑛)^1/4 and 𝑥_𝑜=0.5, we observe that Jungck-Noor iterative scheme converges in 12 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 19 iterations while Jungck-Mann iterative scheme converges in 37 iterations.

5. Conclusions

The speed of iterative schemes depends on 𝛼_𝑛, 𝛽_𝑛, and 𝛾_𝑛. From Tables 3–6 and observations made above, we make the following conjectures.

5.1. Decreasing Function

(1)Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, and Jungck-S iterative scheme.(2)For initial guess near to common fixed point, Jungck-CR (Jungck-SP), Jungck-Noor, and Jungck-S iterative schemes show a decrease while Jungck-Agarwal (Jungck-Ishikawa) iterative scheme shows no change in the number of iterations to converge.

5.2. Increasing Functions

(1)Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-S (Jungck-Noor), Jungck-Agarwal (Jungck-Ishikawa), and Jungck Mann iterative scheme.(2)For initial guess near to the coincidence point, all Jungck-type iterative schemes show a decrease in the number of iterations to converge.

5.3. Oscillatory Functions

(1)Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.(2)For initial guess near to the common fixed point, Jungck-Mann and Jungck-S iterative schemes show a decr ease while Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), and Jungck-Noor iterative schemes show no change in the number of iterations to converge.

5.4. Biquadratic Equation

(1)Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.(2)For initial guess near to the coincidence point, all Jungck-type iterative schemes show a decrease in the number of iterations to converge.

Remark 21. In each case mentioned above, Jungck-CR and Jungck-SP iterative schemes have better convergence rate as compared to other iterative schemes and hence have a good potential for further applications.

Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The first and third authors acknowledge with thanks DSR, KAU, Saudi Arabia, for financial support.