In this manuscript, we introduce a new notion: a Berinde type $(\alpha ,\psi )$ -contraction mapping. Thereafter, we investigate not only the existence, but also the uniqueness of a fixed point of such mappings in the setting of right-complete quasi-metric spaces. The result, presented here, not only generalizes a number of existing results, but also unifies several ones on the topic in the literature. An application of nonlinear fractional differential equations is given.

We study the split fixed point problem with multiple output sets in nonlinear spaces, particularly in CAT(0) spaces. We modify the existing self-adaptive algorithm for solving the split common fixed point problem with multiple output sets in the settings of generalized structures. We also present the consequences of our main theorem in terms of the split feasibility problem and the split common fixed point problem.

In this article, we consider a wider class of nonexpansive mappings (locally related quasi-nonexpansive) than monotone nonexpansive mappings. We obtained the convergence of fixed point for quasi -preserving locally related quasi-nonexpansive mappings in hyperbolic space. An iterative process is also used to obtain the convergence results for this mapping. Fixed point is approximated numerically in a nontrivial example by using Matlab.

In this article, we consider an extensive class of monotone nonexpansive mappings and introduce a new iteration algorithm to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the framework of hyperbolic space. Faster convergence and stability results are proved for that iteration; also, fixed point is approximated numerically in a nontrivial example by using MATLAB.

In this paper, we explore some extensions of multiple fixed point results for various distance spaces such as s-distance space, $\left(s,q\right)$-distance space, and balanced distance space. Some examples are also discussed for the elaboration of these generalized structures. An application of our result that demonstrates the existence of a unique solution of a system of integral equations is also provided.