Some new convergence and stability results for Jungck generalized pseudo-contractive and Lipschitzian type operators using hybrid iterative techniques in the Hilbert space

The convergence of generalized pseudo-contractive operators were studied with regard to a unique fixed point in the article (Verma in Proc Roy Irish Acad Soct A, 97(1):83–86, 1997). In this present article, we introduce Jungck generalized pseudo-contractive and Lipschitzian type operators in the prehilbertian space and in the Hilbert space settings, establish the existence and the uniqueness of a common fixed point for the operator S and a sequence of operators {Ti:D→D}i=1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{T_i:D\rightarrow D\}_{i=1}^k$$\end{document} using a new Jungck–Kirk–Mann type fixed point iterative algorithm as well as the general Kirk–Mann type iterative algorithm. The strong convergence of these fixed point iterative algorithms to a unique common fixed point is also investigated for the same set of operators considered. Moreover, some stability results are established for the Jungck–Kirk–Mann type iterative algorithm as well as the general Kirk–Mann type iterative algorithm. In addition, some examples are given to support our arguments. The results are new, original as well as generalizing some existing ones in the literature.


Introduction
Verma [42], it was proved that Schaefer's fixed point iterative process converges to the unique fixed point of a generalized pseudo-contractive and Lipschitzian operator in the Hilbert space. Nevertheless, to the best of our knowledge, convergence of an iterative process to the unique common fixed point of such operator has not been considered in the literature. Theorem 3.6 of Berinde [5] was the first of both convergence results proved in [42] (which is Theorem 2.1 in [42]). We refer to Browder and Petryshyn [7,8] as well as the papers [43][44][45][46] for a detailed and germane study concerning convergence of iterative processes to fixed points of nonlinear mappings. The interested readers can also consult the recent articles in [6,15,[27][28][29][35][36][37][38][39][40][41] for an excellent study on the foregoing convergence notion.
Our purpose in this article is to examine the existence and the uniqueness of a common fixed point for the operator S and a sequence of operators {T i ∶ D → D} k i=1 using a new Jungck-Kirk-Mann type fixed point iterative algorithm as well as the general Kirk-Mann type iterative algorithm. The strong convergence of these fixed point iterative algorithms to a unique common fixed point is also investigated when each T i is Jungck generalized pseudo-contractive and Lipschitzian type and when each T i is generalized pseudo-contractive and Lipschitzian. The new iterative algorithm is also considered when each T i is Jungck generalized pseudo-contractive and nonexpansive type. We support our arguments concerning the new type of operators with some examples.
In addition, a new notion of stability is given in Sect. 4 to establish some stability results for the family of operators. Our results generalize and extend the existing ones in the literature. For an excellently detailed study on the stability of fixed point iterative algorithms and various contractive conditions, we refer to the articles [9,24,25,[30][31][32]34] and the papers of the author [18][19][20].

Preliminaries
The following definitions shall be required in the sequel: Definition 2.1 Let Y be a normed space, and S, T ∶ Y → Y. Then, T is said to be Lipschitzian type if there exists L > 0 such that
Also, we introduce a class of contractive operators in the Hilbert space setting satisfying the following contractive inequality condition:

Definition 2.3
Let H be a real prehilbertian space and ⟨., .⟩ an inner product on H. Also let ‖ ⋅ ‖ be the norm induced by the inner product. An operator T ∶ H → H is said to be Jungck generalized pseudocontractive, or, a Jungck generalized pseudocontraction if there exist r > 0 and an operator S ∶ H → H such that ∀ u, v ∈ H, Now, from (2) above, we have by using the fact that the norm is induced by the inner product that which reduces to that is, thus, showing that Inequality condition (2) is equivalent to that in (Δ).
Also, subtracting each side of Inequality (Δ) from ⟨Su − Sv, Su − Sv⟩ and using the fact that the norm is induced by the inner product yield that is, which proves that Inequality condition (2) is equivalent to that in (ΔΔ) too.
Applying the Cauchy-Schwarz inequality in (Δ) yields that is,
In a complete metric space, for two commuting operators S, U ∶ X → X and for any x 0 ∈ X, the sequence {Sx n } ∞ n=0 ⊂ X defined by is the iterative process associated to the result of Jungck [11].

Remark 2.7
(i) We shall employ both Lemmas 2.5 and 2.6 in the Banach space setting while recalling that the metric is induced by the norm. Therefore, we have d(x, y) = ‖x − y‖, x, y ∈ X. (ii) Lemma 2.6 is more general than Lemma 2.5 in the sense that we recover Lemma 2.5 from Lemma 2.6 if S = I = identity operator in (6). (iii) Several authors have generalized and extended Banach's Fixed Point Theorem [1,2,5] as well as Jungck's Theorem [11] in different directions. In particular, interested readers can consult the recent articles of the authors [21][22][23] and a host of others in the literature.

3
We introduce and employ the iterative technique below: Let X be a normed space. Consider a sequence of operators {T i ∶ X → X} k i=0 and the operator S ∶ X → X. For any x 0 ∈ X, the sequence {Sx n } ∞ n=0 ⊂ X is defined by where k is a fixed integer and T 0 = I = identity operator. The iterative process defined in (7) shall be nomenclated as Jungck-Kirk-Mann type iterative algorithm for a sequence of operators.

Remark 2.8
(i) With T i = T i and S = I = Identity operator; then, Jungck-Kirk-Mann type iterative algorithm (7) reduces to the Kirk-Mann iterative algorithm of Olatinwo [18]. (ii) If T i = T i , the algorithm (7) becomes the Jungck-Kirk-Mann iterative algorithm defined in Olatinwo [19]. (iii) The iterative processes of Kirk [12], Krasnoselskij [13], Mann [14], Picard [26] and Schaefer [33] are also some of the special cases of Jungck-Kirk-Mann type iterative algorithm. (iv) If k = 1 in (7), we obtain the Jungck-Mann iterative algorithm of Singh et al. [34] as a special case. (v) Indeed, if in (7), S = I = Identity operator, then we obtain the following general Kirk-Mann type iterative algorithm: For any x 0 ∈ X, the sequence {x n } ∞ n=0 ⊂ X is defined by where k is a fixed integer and T 0 = I = identity operator. The general Kirk-Mann type iterative algorithm defined in (8) is contained in Olatinwo [17].
We shall employ both algorithms defined in (7) and (8) to obtain our results in the present article. Some examples of operators satisfying our contractive condition are given later at the end of each section. Lemma 2.9 [5,[17][18][19] Let be a real number such that 0 ≤ < 1 and { n } ∞ n=0 a sequence of positive numbers such that lim n→∞ n = 0. Then, for any sequence of positive numbers {u n } ∞ n=0 satisfying we have lim n→∞ u n = 0.
Throughout, we shall assume that H is a real Hilbert space and ⟨., .⟩ an inner product on H. Also, let ‖ ⋅ ‖ be the norm induced by the inner product.
Sections 3 and 4 are devoted to our main results: 3 Convergence of iterative algorithms to common fixed points Theorem 3.1 Let D be a nonempty closed convex subset of a real Hilbert space H and let the operator is a sequence of Jungck generalized pseudo-contractive and Lipschitzian type operators with corresponding constants r i and L i such that 0 < r i < L i < 1 (i = 1, 2, … , k). Then: Let U and S be commuting mappings such that U(D) ⊂ S(D). Since D is a convex subset of H, we have U(D) ⊂ D for each n,i ∈ (0, 1). Since D is a closed subset of a Hilbert space, D is a complete metric space.
. Then, we have by using (19) again that Thus, combining the ranges of validity of in both cases yields 0 < < 1. That is, we have ∈ (0, 1). Therefore, this completes the proof of the second part of the claim given in (11). Next, since ∈ (0, 1), U and S commute with U(D) ⊂ S(D) and that S is continuous, then we have by Lemma 2.6 (Jungck's fixed point theorem) again that U and S satisfying Inequality (11) have a unique common fixed point. Again, by extension, S and each T i have a unique common fixed point since S and each T i commute, S is continuous and Indeed, the unique common fixed point of S and U is computed by the Jungck-Kirk-Mann's iterative algorithm {Sx n } ∞ n=0 defined by Hence, the unique common fixed point of S and T is simultaneously and automatically computed too. Therefore, we now prove (ii) of the result using Jungck-Kirk-Mann type iterative algorithm {Sx n } ∞ n=0 ⊂ D as follows: We have that if x ⋆ is a common fixed point of S and Again, using (16) and (17) in (20) yield Hence, the Jungck-Kirk-Mann type iterative algorithm {Sx n } ∞ n=0 converges strongly to the unique common fixed point ◻

Theorem 3.2 Let D be a nonempty closed convex subset of a real Hilbert space H and let the operator
is a sequence of Jungck generalized pseudo-contractive and nonexpansive type operators with corresponding constants r i such that 0 < r i < 1 (i = 1, 2, … , k). Then: Proof The operator U is as defined in (19) in Theorem 3.1. The line of argument in the proof of Theorem 3.2 is similar to that of Theorem 3.1 except that in the present case L = 1 since each T i is nonexpansive type in addition to being Jungck-generalized pseudo-contractive. Therefore, we have It follows from (30) that with We now show that ∈ (0, 1) using the fact that n,0 ∈ (0, 1) as follows: So, we have Also, we have the following cases: Case 1 Given that n,0 > 0. Then, using (23) gives Since 2 > 0 as in Case 1 of Theorem 2.1, then we conclude that > 0. Case 2 Given that n,0 < 1. Then, we have by using (23) again that from which we obtain 2 < 1. That is, ( + 1)( − 1) < 0, which is the same inequality at which we arrived in Case 2 of Theorem 3.1. Hence, we obtain ∈ (0, 1). Therefore, with ∈ (0, 1), we have by the application of Lemma 2.6 again that S and T i (for each i = 1, 2, … , k) have a unique common fixed point. The unique common fixed point of S and T i (for each i = 1, 2, … , k) is computed by the Jungck-Kirk-Mann type iterative algorithm as in Theorem 3.1.
We also prove (ii) of the result using Jungck-Kirk-Mann type iterative algorithm {Sx n } ∞ n=0 ⊂ D as follows: we have that if x ⋆ is a common fixed point of S and each T i , then Sx ⋆ = T i x ⋆ = x ⋆ . Then, as in Theorem 3.1, we have that from which we obtain that ∈ (0, 1). We therefore have from (24) that since ∈ (0, 1). Hence, the Jungck-Kirk-Mann iterative algorithm {Sx n } ∞ n=0 converges strongly to x ⋆ which is the unique common fixed point of S and T i (for each i = 1, 2, … , k). ◻

Theorem 3.3 Let D be a nonempty closed convex subset of a real Hilbert space H and let
be a sequence of generalized pseudo-contractive and Lipschitzian operators with corresponding constants q i and L i such that 0 < q i < L i < 1. Then: (i) all T i (i = 1, 2, … , k) have a unique common fixed point x ⋆ ∈ D; (ii) for each x 0 ∈ D, the general Kirk-Mann type iterative algorithm {x n } ∞ n=0 defined by (8) converges strongly to x ⋆ , ∀ n,0 ∈ (0, 1) satisfying Proof When S = I = Identity operator in the statement of Theorem 3.1 and its proof, then we obtain the proof of Theorem 3.3. However, Lemma 2.5 (Banach's fixed point theorem [1,2,5]) is invoked here rather than Jungck's fixed point theorem [11]. ◻ [42], while Theorem 3.2 is also an extension of Theorem 3.1 of Verma [42] (which is itself Theorem 3.6 of Berinde [5]).
. Then, we obtain which implies that T 2 is Jungck generalized pseudo-contractive.
is a sequence of Jungck generalized pseudo-contractive and Lipschitzian type mappings. ◻

Stability results for Jungck-Kirk-Mann type iterative algorithm and general Kirk-Mann type iterative algorithm
The rest of this article is devoted to the study of stability of Jungck-Kirk-Mann type iterative algorithm and general Kirk-Mann type iterative algorithm. We now state the following definition: Suppose that (X, d) is a complete metric space, {T i ∶ X → X} n i=1 a sequence of mappings and S ∶ X → X a mapping. Denote by F(S ∩ T i ) the set of all the common fixed points of S and {T i } n i=1 , which is given by , be mappings defined on a complete metric space (X, d). Let x ⋆ be a common fixed point of S and T i , that is, For any x 0 ∈ X (initial approximation), let the sequence {Sx n } ∞ n=0 ⊂ X generated by the iterative algorithm converge to x ⋆ , where h is some function. Let {Sy n } ∞ n=0 ⊂ X be an arbitrary sequence and set Then, the iterative procedure (Δ⋆) will be called (S, {T i } k i=1 )-stable, or, stable with respect to S and the sequence of mappings {T i } k i=1 whenever lim n→∞ n = 0 if and only if lim n→∞ Sy n = x ⋆ .

Remark 4.2
(i) Since metric is induced by norm, then we have that whenever we are working in the normed linear space setting. (ii) Definition 4.1 reduces to that of Olatinwo [17] if in (Δ⋆), S = I = identity mapping. (iii) Again, Definition 4.1 becomes that of [10] if in (Δ⋆), k = 1 and S = I(identity mapping). (iv) Furthermore, Definition 4.1 is independent of the Definition 2.1 of stability given in [34].
The following are the stability results for the iterative algorithms aforementioned at the beginning of this section:

Theorem 4.3 Let D be a nonempty closed convex subset of a real Hilbert space H. Suppose that S ∶ D → D is continuous and {T
is a sequence of Jungck generalized pseudo-contractive and Lipschitzian type operators with corresponding constants r i , L i such that 0 < r i < L i < 1(i = 1, 2, … , k) and T i (D) ⊂ S(D), for each i. For each x 0 ∈ D, let {Sx n } ∞ n=0 be the Jungck-Kirk-Mann type iterative algorithm defined by (7).
Therefore, it follows from (26) that Using (27) in (25) gives Since ∈ (0, 1), then applying Lemma 2.9 in (28) gives lim n→∞ Sy n+1 = x ⋆ . Conversely, let lim n→∞ Sy n = x ⋆ . Then, we shall establish that lim n→∞ n = 0 by using (27) and the triangle inequality as follows: Hence, the Jungck-Kirk-Mann type iterative algorithm is T i -stable (for each i). ◻ The following is a stability result for the general Kirk-Mann type iterative algorithm for a sequence of generalized pseudo-contractive and Lipschitzian operators:

Theorem 4.4 Let D be a nonempty closed convex subset of a real Hilbert space H and let
be a sequence of generalized pseudo-contractive and Lipschitzian operators with corresponding constants q i and L i such that 0 < q i < L i < 1. For each x 0 ∈ D, let x n ∞ n=0 be the general Kirk-Mann type iterative algorithm defined by (8). We refer to Example 3.5 again to investigate the stability of the Jungck-Kirk-Mann type iterative algorithm defined in (7) as follows:

H = IR
(the real line) be endowed with the Euclidean inner product and norm. Let 3 4 ⊂ H and let S, T i ∶ X → X(i = 1, 2, … , k) be mappings such that Sx = x 2 , ∀x ∈ X. The common fixed point set of S and {T i } k i=1 is given by Define an arbitrary sequence {y n } ∞ n=0 ⊂ X by y n = 1 n 2 +1 (n = 0, 1, 2, …). Then, the Jungck-Kirk-Mann type iterative algorithm (28) is a sequence of Jungck generalized pseudo-contractive and Lipschitzian type mappings such that We observe that S is a continuous mapping in X. Again, the Jungck-Kirk-Mann type iterative algorithm is the following: for any x 0 ∈ X, the sequence {Sx n } ∞ n=0 ⊂ X is defined by Since n,0 = 1 4 = r, L = 9 32 , we have that With y n = 1 n 2 +1 , n = 0, 1, 2, … , then, the arbitrary sequence {Sy n } ∞ n=0 becomes Sy n = 1 (n 2 +1) 2 . Assume lim n→∞ n = 0. Then, we have from (28) that showing that lim n→∞ Sy n = x ⋆ = 0.
In a converse manner, let lim n→∞ Sy n = x ⋆ . Then, we have from the second part of the proof of Theorem 3.3 that Hence, the Jungck-Kirk-Mann type iterative algorithm is (S,

Remark 4.7
Indeed, the significance of this article is not only limited to generalizations and extensions of some results in the Hilbert space, or, Banach space setting, but also to introduce more classes of operators with which experts in the fixed point theory can now define and solve new problems. Also, it is our ardent hope that the applicability of our new operators by various experts will begin to emerge in due course.
when {T i } k i=1 is a sequence of Jungck generalized pseudo-contractive and nonexpansive type operators.
Moreover, in Theorem 3.3, we have established similar result for a general Kirk-Mann type iterative algorithm when {T i } k i=1 is a sequence of generalized pseudo-contractive and Lipschitzian operators.
In addition, two stability results have been established for both iterative algorithms employed in this article. Theorem 4.3 has illustrated the new stability notion given in Definition 4.1, while Theorem 4.4 improves, generalizes and extends the stability results contained in [17]. It has been shown in Theorem 4.3 that the Jungck-Kirk-Mann type iterative algorithm is (S, {T i } k i=1 )-stable, while the general Kirk-Mann type iterative algorithm is also shown to be {T i } k i=1 -stable in Theorem 4.4.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.