In this paper, some fixed point results for multivalued contractions are established in setting $G$-complete extended fuzzy $b$-metric spaces. An example is furnished to demonstrate the validity of results. An application of integral type inclusion is given to authenticate the theorems. Our results extend and generalize many existing results in literature.

This manuscript is devoted to presenting some convergence results of a three-step iterative scheme under the Chatterjea–Suzuki–C ((CSC), for short) condition in the setting of a Banach space. Also, an example of mappings satisfying the (CSC) condition with a unique fixed point is provided. This example proves that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Finally, our main results will be applied to find a solution to functional and fractional differential equations (FDEs) as an application.

The goal of this manuscript is to introduce the JK iterative scheme for the numerical reckoning of fixed points in generalized contraction mappings. Also, weak and strong convergence results are investigated under this scheme in the setting of Banach spaces. Moreover, two numerical examples are given to illustrate that the JK iterative scheme is more effective than some other iterative schemes in the literature. Ultimately, as an application, the JK iterative scheme is applied to solve a discrete composite functional differential equation of the Volterra-Stieljes type.

The goal of this manuscript is to introduce a new class of generalized nonexpansive operators, called $(\alpha ,\beta ,\gamma )$-nonexpansive mappings. Furthermore, some related properties of these mappings are investigated in a general Banach space. Moreover, the proposed operators utilized in the $K$-iterative technique estimate the fixed point and examine its behavior. Also, two examples are provided to support our main results. The numerical results clearly show that the $K$-iterative approach converges more quickly when used with this new class of operators. Ultimately, we used the $K$-type iterative method to solve a variational inequality problem on a Hilbert space.

In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.