On the Correlation between Banach Contraction Principle and Caristi’s Fixed Point Theorem in b-Metric Spaces

We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article.

Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems

This paper is concerned with the existence and uniqueness of solutions for a new class of boundary value problems, consisting by Hilfer-Hadamard fractional differential equations, supplemented with nonlocal integro-multipoint boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

Langevin Equations with Generalized Proportional Hadamard–Caputo Fractional Derivative

We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard–Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres–Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.

A ROBUST COMPUTATIONAL DYNAMICS OF FRACTIONAL-ORDER SMOKING MODEL WITH RELAPSE HABIT

The social habit of smoking has affected the whole world in a social manner. It is the main cause of diseases like cancers, asthma, bad breath, etc., and a source of spreading of infectious diseases like COVID-19. This work is related to an existing smoking model with relapse habit converted in fractional order. First, formulation of fractional-order smoking model is presented and then the dynamics of proposed problem is analyzed. Fixed-point theory via Banach contraction and Schauder theorems is used to derive the existence and uniqueness of the model. At last, the adaptive predictor–corrector algorithm and Runge–Kutta fourth-order (RK4) strategy are used to perform simulation. To bolster the validity of the theoretical results, a set of numerical simulations are performed. A good agreement between hypothetical and numerical results is demonstrated via numerical simulations using MATLAB software.

Ulam Stability of n-th Order Delay Integro-Differential Equations

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

An extension system of sequential differential equations of arbitrary order

We are concerned with an extension of a coupled sequential differential system of fractional type. Using the Banach contraction principle, we establish new results for the existence and uniqueness of solutions. Then, we prove another existence result via Schaefer’s fixed point theorem. At the end, we illustrate one main result by an example.

Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

Hilfer–Hadamard Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions

This paper is concerned with the existence and uniqueness of solutions for a Hilfer–Hadamard fractional differential equation, supplemented with mixed nonlocal (multi-point, fractional integral multi-order and fractional derivative multi-order) boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying the fixed point theorems due to Krasnoselskiĭ and Schaefer and Leray–Schauder nonlinear alternatives. We demonstrate the application of the main results by presenting numerical examples. We also derive the existence results for the cases of convex and non-convex multifunctions involved in the multi-valued analogue of the problem at hand.

Comments on fixed point results in classes of function with values in a b–metric space

We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.

Qualitative Analysis of some Types of Neutral Delay Differential Equations

In this paper, we conduct some qualitative analysis that involves the global asymptotic stability (GAS) of the Neutral Differential Equation (NDE) with variable delay, by using Banach contraction mapping theorem, to give some necessary conditions to achieve the GAS of the zero solution.