Stability Results for Jungck-Kirk-Mann and Jungck-Kirk Hybrid Iterative Algorithms

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

“…(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.…”

“…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 ≤ 𝛿 < 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.…”

“…Case 1 Given that 𝛼 n,0 > 0. Then, using (19) gives So, we have 𝜇 2 > 0, from which either 𝜇 < 0, or, 𝜇 > 0. Since must be positive, we therefore, choose 𝜇 > 0.…”

“…Case 2 Given that 𝜆 < 2(L 2 −r) (1−2r+L 2 ) . Then, we have by using (19) again that from which we have 𝜇 2 < 1. That is, (𝜇 + 1)(𝜇 − 1) < 0.…”

“…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].…”