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.

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In this paper, we give the existence and uniqueness result for the fractional order Langevin equation with modified argument by using the Bielecki norm. After, we consider a special form of this equation (delayed form) and then apply Burton's method to this special form to prove that there is a unique solution under weaker conditions than the other result. Further, we derive the Ulam–Hyers stability for this equation. Finally, two examples are given to illustrate our main results.

We introduce the condition of being Cauchy for a sequence in cone b-metric spaces (cbms) over Banach algebras. Based on this result, we extend almost mappings in cone metric spaces over Banach algebras to cbms over Banach algebras and prove the related fixed point theorem. In addition, we apply our results to some applications to illustrate their usability.

In this paper, we first consider Nadler type contractions with the generalized Lipschitz constant holding () < 1 instead of () < 1 where () is the spectral radius of and ≥ 1 is the coefficient of the underlying cone-metric spaces over Banach algebras. Then, we prove the corresponding fixed point theorem for such mappings. Finally, we compare our result with one obtained by the case () < 1 by introducing some proper examples.