Solving an Integral Equation by Using Fixed Point Approach in Fuzzy Bipolar Metric Spaces

The purpose of this manuscript is to obtain some fixed point results under mild contractive conditions in fuzzy bipolar metric spaces. Our results generalize and extend many of the previous findings in the same approach. Moreover, two examples to support our theorems are obtained. Finally, to examine and strengthen the theoretical results, the existence and uniqueness of the solution to a nonlinear integral equation was studied as a kind of applications.

Existence and controllability of higher-order nonlinear fractional integrodifferential systems via resolvent operator

This work analyzes the existence of solution and approximate controllability for higher order non-linear fractional integro-differential systems with Riemann-Liouville derivatives in Banach spaces. Firstly, the definition of mild solution for the system is derived. Then a set of sufficient conditions for the existence of mild solution and approximate controllability of the system is obtained. The discussions are based on fixed point approach, and the theory of convolution and fractional resolvent. To illustrate the feasibility of developed theory, an example is given.

Hyers–Ulam stability of impulsive Volterra delay integro-differential equations

This paper discusses different types of Ulam stability of first-order nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory effects represented by the delay and the kernel of the used fractional integral operator. Our analysis is based on Pachpatte’s inequality and the fixed point approach represented by the Picard operators. Applications are provided to illustrate the stability results obtained in the case of a finite interval.

A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

In this paper, we consider pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.

Analysis of the isotropic and anisotropic Grad–Shafranov equation

Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad–Shafranov equation; such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad–Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad–Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space $H_0^1$ to the Grad–Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad–Shafranov equation are also constructed.

A Fractional-Order Sequential Hybrid System with an Application to a Biological System

With the help of Banach’s fixed-point approach and the Leray–Schauder alternative theorem, we produced existence results for a general class of fractional differential equations in this paper. The proposed problem is more comprehensive and applicable to real-life situations. As an example of how our problem might be used, we have created a fractional-order COVID-19 model whose solution is guaranteed by our results. We employed a numerical approach to solve the COVID-19 model, and the results were compared for different fractional orders. Our numerical results for fractional orders follow the same pattern as the classical example of order 1, indicating that our numerical scheme is accurate.