In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function α ( x , y ) of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + α ( z , y ) d ( z , y ) , f o r a l l x , y , z ∈ X . Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension.

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions α ( x , y ) and μ ( x , y ) on the right-hand side of the b - triangle inequality, that is, d ( x , y ) ≤ α ( x , z ) d ( x , z ) + μ ( z , y ) d ( z , y ) , for all x , y , z ∈ X . Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and ϕ -nonlinear type contractions in the setting of double controlled metric type spaces.

In this paper, we introduce the concept of the rectangular M-metric spaces, along with its topology and we prove some fixed-point theorems under different contraction principles with various techniques. The obtained results generalize some classical fixed-point results such as the Banach's contraction principle, the Kannan's fixed-point theorem and the Chatterjea's fixed-point theorem. Also we give an application to the fixed-circle problem.

In this paper, we consider a new distance structure, extended Branciari b-distance, to combine and unify several distance notions and obtain fixed point results that cover several existing ones in the corresponding literature. As an application of our obtained result, we present a solution for a fourth-order differential equation boundary value problem.

This work is devoted to establish a modified population model of susceptible and infected (SI) compartments to predict the spread of the infectious disease COVID-19 in Pakistan. We have formulated the model and derived its boundedness and feasibility. By using next generation matrices method we have derived some results for the global and local stability of different kinds of equilibrium points. Also, by using fixed point approach some results of existence of at least one solution are studied. Furthermore, the numerical simulations for various values of isolation parameters corresponding to different fractional order are investigated by using modified Euler’s method.