In this manuscript, the mathematical model of COVID‐19 is considered with eight different classes under the fractional‐order derivative in Caputo sense. A couple of results regarding the existence and uniqueness of the solution for the proposed model is presented. Furthermore, the fractional‐order Taylor's method is used for the approximation of the solution of the concerned problem. Finally, we simulate the results for 50 days with the help of some available data for fractional differential order to display the excellency of the proposed model.

Here, we remarked that fractals or Housdarff derivatives are important tools to investigate various problems. However, fractals when combined with fractional orders of different kernels give birth to a big class of operators. Each operator has its own importance and applicability. One of the fractals-fractional operators is based on the exponential kernel which has also many important uses in modeling of different real-world problems. Reinfection is a worldwide issue nowadays and due to reinfection, people in our society are facing a lot of issues like economical, social, and a high rate of death. A fractal-fractional hybrid model with reinfection was investigated. The mentioned model was considered to deduce the qualitative theory and numerical aspects. Fundamental properties of the mentioned model were deduced including basic reproduction number, equilibrium points, and global and local stability of both equilibrium points. The qualitative analysis was consisted on the existence and stability in sense of Hyers-Ulam of the solutions. Interpolation technique was used to deduce a numerical scheme for the model under our study. Additionally, criteria for uniqueness of solution was proved by the Banach principle. In addition, sensitivity analysis was also included in the paper. Various numerical results were presented using different values of fractals fractional order. Some comparisons of simulated and real data was also given.

In this paper, we study the existence of solution for the following non-linear matrix equations: $$\begin{aligned}& X=Q+ \sum^{n}_{i=1} A^{*}_{i} X A_{i}- \sum ^{n}_{i=1} B^{*}_{i} X B_{i}, \\& X=Q+ \sum^{n}_{i=1} A^{*}_{i} \varUpsilon (X) A_{i}, \end{aligned}$$ where Q is a Hermitian positive definite matrix, \(A_{i}\), \(B_{i}\) are arbitrary \(m\times m\) matrices and \(\varUpsilon: \mathcal{H}(m)\rightarrow \mathcal{P}(m)\) is an order preserving continuous map such that \(\varUpsilon (0)=0\). To this aim, we establish several common fixed point theorems for two mapping satisfying a rational \(F_{\mathcal{R}}\)-contractive condition, where \(\mathcal{R}\) is a binary relation.

Owing to the notion of a complex-valued metric space, we prove fixed point results, which generalize some common fixed point results under contractive condition for rational expression in the context of complex-valued metric spaces. In application, we present a homotopy result to apply the results obtained herein.

This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results.