In this paper, we consider several classes of mappings related to the class of $\alpha$-$ϝ$-contraction mappings by introducing a convexity condition and establish some fixed-point theorems for such mappings in complete metric spaces. The present result extends and generalizes the well-known results of $\alpha$-admissible and convex contraction mapping and many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results. Finally, we derive the existence and uniqueness of a solution to an integral equation to support our main result and give a numerical example to validate the application of our obtained results.

In the current study, a novel multi-term mixed sub-diffusion and wave-diffusion model was considered. The new model has a unique time-space coupled derivative in addition to having the diffusion-wave and sub-diffusion terms concurrently. Typically, an elliptic equation in the space variable is obtained by applying a finite difference time-stepping procedure. The severe stability restrictions are the main disadvantage of the finite difference method in time. It has been demonstrated that the Laplace transform is an excellent choice for solving diffusion problems and offers a substitute to the finite difference approach. In this paper, a method based on Laplace transform coupled with the pseudospectral method was developed for the novel model. The proposed method has three main steps: First, the model was reduced to a time-independent model via Laplace transform; second, the pseudospectral method was employed for spatial discretization; and finally, the inverse Laplace transform was applied to transform the obtained solution in Laplace transform domain back into a real domain. We also presented the numerical scheme's stability and convergence analysis. To demonstrate our method's efficacy, four problems were examined.

In this paper, we modify various contractive conditions (C.C.)s such as Ciric type (C.C.), Rhoades type (C.C.), Seghal type (C.C.), Bianchini type (C.C.), and Berinde type (C.C.) for two self-mappings, considering that the contractive property plays a major role in establishing a fixed circle (F.C.) on both metric spaces (M-s) and $\mathbb{S}$-(M-s) where the symmetry condition is satisfied, and we utilize them to establish a common (F.C.). We prove new (F.C.) results on both (M-s) and $\mathbb{S}$-(M-s) with illustrative examples. Finally, we provide an application to activation functions such as rectified linear unit activation functions and parametric rectified linear unit activation functions.

The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping: $F_c$-contractive mapping and $F_c$-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were.

In this paper we introduce the concept of complex-valued double controlled metric like spaces. These new results generalize and extend the corresponding results about complex-valued double controlled metric type spaces. We prove some complex-valued fixed point theorems in this new complex-valued metric like spaces and, as application, we give an existence and uniqueness of the solution of a Fredholm type integral equation result. Moreover, some examples are also presented in favor of our given results.

In this paper, we find all static, cylindrically symmetric spacetime metrics admitting homothetic symmetries. For this purpose, first we analyze the homothetic symmetry equations by an algorithm developed in Maple which gives all possible static, cylindrically symmetric metrics that may possess proper homothetic symmetry. After that, we have solved the homothetic symmetry equations for all these metrics to get the final form of homothetic symmetry vector fields. Comparing the obtained results with those of direct integration technique, it is observed that the Rif tree approach not only recovers the metrics already found by direct integration technique, but it also produces some new metrics.