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This document is an editorial for a special issue of the journal "Symmetry" focused on qualitative studies of fractional differential equations, fractional integro-differential equations, and related topics. The editorial provides an overview of the published articles in the special issue, summarizing their contents and contributions. The articles cover a range of topics, including numerical simulations of fractional differential equations, analysis of cancer models using stem cells and chemotherapy, analysis of image segmentation algorithms, and control of bioreactors using fractional order models. The editorial concludes by listing the authors and titles of the contributions. [Extracted from the article]

This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution x ∈ C [ 0 , T ] to the problem. Moreover, we outline the necessary demands for the solution's uniqueness. In addition, the continuous dependence of the solution and the Hyers–Ulam stability of the problem are analyzed. In order to illustrate our results, we provide some particular cases and instances. [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L 1 [ 0 , T ] and, secondly, the existence of nonincreasing solutions in unbounded interval L 1 (R +) . Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ ∈ L 1 [ 0 , T ] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ ∈ L 1 (R +) on some parameters. Finally, some examples are provided to verify our investigation. [ABSTRACT FROM AUTHOR]

Here, we discuss the solvability of a class of hybrid functional integral equations by applying Darbo's fixed point theorem and the technique of the measure of noncompactness (MNC). This study has been located in space BC (R +) . Furthermore, we prove the asymptotic stability of the solution of our problem on R +. We introduce the idea of asymptotic dependency of the solutions on some parameters for that class. Moreover, general discussion, examples, and remarks are demonstrated. [ABSTRACT FROM AUTHOR]

Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters will be examined. Finally, the problems of conjugate orders and integer orders, and some other problems and remarks will be discussed and presented. [ABSTRACT FROM AUTHOR]

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in B C (R +). Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes. [ABSTRACT FROM AUTHOR]

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions BC (R +) , by applying the technique of measure of noncompactness on R + . Furthermore, we establish sufficient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results. [ABSTRACT FROM AUTHOR]

Quadratic integro-differential equations have been discussed in many works, for instance. Some analytic results on the existence and the uniqueness of problem solutions to quadratic integro-differential equations have been investigated in different classes. Various techniques have been applied such as measure of noncompactness, Schauder's fixed point theorem and Banach contraction mapping. Here, we shall investigate quadratic functional integro-differential equations with delay. To prove the existence of solutions of the quadratic integro-differential equations, we use the technique of De Blasi measure of noncompactness. Moreover, we study some uniqueness results and continuous dependence of the solution on the initial condition and on the delay function. Some examples are presented to verify our results. [ABSTRACT FROM AUTHOR]

Symmetry analysis has been applied to solve many differential equations, although determining the symmetries can be computationally intensive compared to other solution methods. In this work, we study some operators which keep the set of solutions invariant. We discuss the existence of solutions for two initial value problems of a delay quadratic functional integro-differential equation of arbitrary (fractional) orders and its corresponding integer orders equation. The existence of the maximal and the minimal solutions is proved. The sufficient condition for the uniqueness of the solutions is given. The continuous dependence of the unique solution on some data is studied. The continuation of the arbitrary (fractional) orders problem to the integer order problem is investigated. [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of positive nondecreasing solutions for a multi-term fractional-order functional differential equations. We consider Cauchy boundary problems with: nonlocal conditions, two-point boundary conditions, integral conditions, and deviated arguments. [ABSTRACT FROM AUTHOR]

Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved. [ABSTRACT FROM AUTHOR]

In this work, we are discussing the solvability of an implicit hybrid delay nonlinear functional integral equation. We prove the existence of integrable solutions by using the well known technique of measure of noncompactnes. Next, we give the sufficient conditions of the uniqueness of the solution and continuous dependence of the solution on the delay function and on some functions. Finally, we present some examples to illustrate our results. [ABSTRACT FROM AUTHOR]