The study of asymmetric structures and their applications in mathematics is interesting. One of the types of asymmetric structures on a metric space has been initiated by Kada et al. (1996) and is known as a w-distance. That lack of symmetry attracts many researchers in fixed point theory. In this manuscript, we introduce a new type of contraction named generalized $(\alpha ,\psi ,M_{\mathsf{\Omega }})$ -contractive mappings via w-distances, and then we prove some new related fixed point results, generalizing and improving the recent results of Lakzian et al. (2016) and others. At the end, we give some examples. To illustrate the usability of the new theory, we apply our obtained results to resolve a nonlinear Fredholm-integral-type equation.

Let $A$ and $B$ be unital Banach algebras‎, ‎$X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A‎, ‎B$ and $X$‎. The structure of some derivations on triangular Banach algebras was studied by some authors. ‏‎Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations‎, ‎there are fundamental differences between them‎. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we ‎completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$‎ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$‎‏.

In this paper, the relations between the topological centers of bounded bilinear mappings and some of their higher rank adjoints are investigated. Particularly, for a Banach algebra A, some results about the Banach A−modules and Arens regularity and strong Arens irregularity of module actions will be obtained.