Fixed Point Results for F-Contractions in Cone Metric Spaces over Topological Modules and Applications to Integral Equations

In this paper, the concept of F-contraction was generalized for cone metric spaces over topological left modules and some fixed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being a fractional integral equation.

Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem

The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of β−G, ψ−G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points.

Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory

A new class of fractional differential equations with exponential memory was recently defined in the space A C δ n [ a , b ] $A{C}_{\delta }^{n}\left[a,b\right]$ . In order to use the famous predictor–corrector method, a new quasi-linear interpolation with a non-equidistant partition is suggested in this study. New Euler and Adams–Moulton methods are proposed for the fractional integral equation. Error estimates of the generalized fractional integral and numerical solutions are provided. The predictor–corrector method for the new fractional differential equation is developed and numerical solutions of fractional nonlinear relaxation equation are given. It can be concluded that the non-equidistant partition is needed for non-standard fractional differential equations.

On a fractional cantilever beam model in the q-difference inclusion settings via special multi-valued operators

The fundamental goal of the study under consideration is to establish some of the existence criteria needed for a particular fractional inclusion model of cantilever beam in the setting of quantum calculus using new arguments of existence theory. In this way, we investigate a fractional integral equation that corresponds to the aforementioned boundary value problem. In a more concrete sense, we design new multi-valued operators based on this integral equation, which belong to the certain subclasses of functions, called α-admissible and α-ψ-contractive multi-functions, in combination with the AEP-property. Also, we use some inequalities such as Ω-inequality and set-valued version inequalities. Moreover, we add a simulative example for a numerical analysis of our results obtained in this study.

On a Hilfer Type Fractional Differential Equation with Nonlinear Right-Hand Side

In this article we consider the questions of one-valued solvability and numerical realization of initial value problem for a nonlinear Hilfer type fractional differential equation with maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation. The theorem of existence and uniqueness of the solution of given initial value problem in the segment under consideration is proved. For numerical realization of solution the generalized Jacobi–Galerkin method is applied. Illustrative examples are provided.

On solutions of fractional multi-term sequential problems via some special categories of functions and (AEP)-property

The main intention of this article is that new techniques of existence theory are used to derive some required criteria pertinent to a given fractional multi-term problem and its inclusion version. In such an approach, we do our research on a fractional integral equation corresponding to the mentioned BVPs. In more precise words, by virtue of this integral equation, we construct new operators which belong to a special category of functions named α-admissible and α-ψ-contraction maps coupled with operators having (AEP)-property. Next, by considering some new properties on the existing Banach space having properties (B) and $(C_{\alpha })$ ( C α ) , our argument for ensuring the existence of solutions is completed. In addition, we also add two simulative examples to review our findings by a numerical view.

A Generalized ML-Hyers-Ulam Stability of Quadratic Fractional Integral Equation

An interesting quadratic fractional integral equation is investigated in this work via a generalized Mittag-Leffler (ML) function. The generalized ML–Hyers–Ulam stability is established in this investigation. We study both of the Hyers–Ulam stability (HUS) and ML–Hyers–Ulam–Rassias stability (ML-HURS) in detail for our proposed differential equation (DEq). Our proposed technique unifies various differential equations’ classes. Therefore, this technique can be further applied in future research works with applications to science and engineering.

Existence and stability of solutions of $ \psi $-Hilfer fractional functional differential inclusions with non-instantaneous impulses

<abstract><p>In this paper, we prove two existence results of solutions for an $ \psi $-Hilfer fractional non-instantaneous impulsive differential inclusion in the presence of delay in an infinite dimensional Banah spaces. Then, by using the multivalued weakly Picard operator theory, we study the stability of solutions for the considered problem in the sense of $ \psi $-generalized Ulam-Hyers. To achieve our aim, we present a relation between any solution of the considered problem and the corresponding fractional integral equation. The given problem here is new because it contains a delay and non-instantaneous impulses effect. Examples are given to clarify the possibility of applicability our assumptions.</p></abstract>