Almost Contraction Mappings and $(S,T)$-Stability of Jungck Iteration in Cone Metric Spaces over Banach Algebras

Abstract

In this work, we first introduce almost contraction mappings for a pair of two mappings in cone metric spaces over Banach algebras (CMSBA). Then, we observe that the class of such mappings in this setting contains those of many well known mappings. Finally, we prove some fixed point theorems, and obtain $(S,T)$-stability results of Jungck iterations for some mappings in CMSBA.

Application of Fixed Point Theorems to Differential Equation in b-Multiplicative Metric Spaces

“…However, the converse is contradictory. A map T is said to hold the 𝑃 property if 𝐹 (𝑇 ) = 𝐹 (𝑇 𝑛 ) for each 𝑛 ∈ 𝑁 (4,5) . The implications of these findings in 𝑏 − 𝑀 𝑀 𝑆 are generalized in the following results.…”

“…Picard's iteration, whose stability holds an essential place in several fields, is the most crucial iteration procedure among them. However, for some mappings, the P properties of fixed points have drawn the attention of several authors (4,5) because proving that a map holds the property P, where 𝐹 (𝑇 ) = 𝐹 (𝑇 𝑛 ), is essential for understanding the behavior of the map under iteration, it tells us that applying the map multiple times (n times) has the same effect as applying it once. This understanding can be crucial in various mathematical and scientific contexts, such as in dynamical systems theory or optimization algorithms.…”