The concept of fixed points serves as an effective and essential tool in analyzing nonlinear phenomena. This study investigates the existence and uniqueness of solutions for a class of Basset-type fractional differential equations with boundary conditions involving the Caputo–Fabrizio fractional derivative. These equations emerge from the generalized Basset force describing the motion of a sphere settling in a viscous fluid. Darbo’s fixed point theorem, combined with the measure of noncompactness, is applied to establish the existence of solutions. Uniqueness is ensured via Banach’s fixed point theorem. Additionally, stability analysis is performed using Ulam–Hyers and Ulam–Hyers–Rassias concepts. An illustrative example, supported by tables and figures, demonstrates the applicability of the theoretical results.

In this manuscript, we provide an in-depth analysis of existence and uniqueness results, along with stability assessments associated with the κ-Mittag-Leffler-Ulam-Hyers type, specifically focusing on a newly formulated category of hybrid boundary value problems (BVPs) that incorporate fractional derivatives. Our study leverages the properties of tempered (κ,𝜛)-Hilfer fractional operators to explore the mathematical underpinnings of the problem, which is characterized by implicit nonlinear fractional differential equations. To derive the results, we employ Banach’s fixed point theorem, which facilitates the demonstration of the existence of solutions under certain contractive conditions. We also utilize a generalized Gronwall inequality to establish bounds and stability criteria for the solutions, thereby ensuring their robustness under perturbations. Moreover, we underscore the practical applicability of our theoretical findings by presenting several illustrative examples. These examples not only help demonstrate the effectiveness of our approach but also highlight the relevance of the results in addressing real-world scenarios where fractional dynamics are pertinent.

This research manuscript aims to study a novel implicit differential equation in the non-singular fractional derivatives sense, namely Atangana-Baleanu-Caputo (ABC) of arbitrary orders belonging to the interval (2, 3] with respect to another positive and increasing function. The major results of the existence and uniqueness are investigated by utilizing the Banach and topology degree theorems. The stability of the Ulam-Hyers (UH) type is analyzed by employing the topics of nonlinear analysis. Finally, two examples are constructed and enhanced with some special cases as well as illustrative graphics for checking the influenceof major outcomes.

The research is focused on establishing the existence and uniqueness of solutions for a specific set of Caputo–Fabrizio fractional differential equations under periodic boundary conditions (PBCs). To achieve this, the study employs the fractional derivative within the Caputo–Fabrizio framework and utilizes the proposed variation of the parameter method. This method is utilized to simplify the second-order fractional differential equation, transforming it into a second-order nonlinear ordinary differential equation. The research findings are rooted in the fixed points of the Schauder fixed point and Banach fixed point theorems. Furthermore, the study delves into stability analysis using the Hyers–Ulam stability concept. Theoretical examples are included to illustrate and demonstrate the implications of the established theorems.

The focal point of this investigation is the exploration of solutions for Caputo-Hadamard fractional differential equations with boundary conditions, and it follows the initial formulation of a model that is intended to address practical problems. The research emphasizes resolving the challenges associated with determining precise solutions across diverse scenarios. The application of the Burton-Kirk fixed-point theorem and the Kolmogorov compactness criterion in Lp-spaces ensures the existence of the solution to our problem. Banach's theory is crucial for the establishment of solution uniqueness, and it is complemented by utilizing the Hölder inequality in integral analysis. Stability analyses from the Ulam-Hyers perspective provide key insights into the system's reliability. We have included practical examples, tables, and figures, thereby furnishing a comprehensive and multifaceted examination of the outcomes.

This research contributes to the understanding of nonlinear systems of differential equations with operators, specifically in the context of generalizing Volterra and Fredholm integral equations. The use of the Picard approximation method, Banach fixed point theorem, and stability analysis further enhances the analysis of the solutions. The examples provided help to solidify the theoretical findings and highlight their applicability. The outcomes illustrate the efficacy of these tools in envisaging and modelling complex social phenomena. To tackle multifaceted societal challenges, future research in this field should prioritize interdisciplinary collaborations. It is crucial to incorporate empirical data into nonlinear models to validate theoretical findings and enhance practical relevance.

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (ℂℍ)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed ℂℍ-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (𝔽 ℙ𝕋 𝑠), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (𝕌ℍ) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

We investigate the existence of a unique weak solution to a boundary value problem for a non-linear parabolic integro-differential equation. This equation can model heat diffusion phenomena in the case when a nonlinear dependence on a memory term is assumed.The proof of existence relies on a regularization – fixed point – passage to the limit scheme, whereas uniqueness is proved via contractive estimates.

In this paper, we investigate a multi-order ϱ-Hilfer fractional pantograph implicit differential equation on unbounded domains (a,∞), a ≥ 0. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers (UH), Ulam-Hyers-Rassias (UHR) and semi- Ulam-Hyers-Rassias (s-UHR) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.

The purpose of this research is to investegate the existence and uniqueness of solutions for a new class of Atangana-Baleanu fractional differential equations of order α∈(2,3) with periodic boundary conditions. Our results are based on the fixed points of Schauder, and Banach. In addition, investigate the stability of the solution using the Hyers-Ulam stable. Finally, presented an example to satisfy all theorems studies

The nonlinear fractional differential equation (FDE) is discussed in this study. First, we will investigate the existence and uniqueness solution of the nonlinear differential equation to the Atangana-Baleanu fractional derivative in the sense of Caputo with the initial periodic condition and integral boundary condition by Krasnoselskii’s and Banach fixed point theorems. Then, we will study the Hyers-Ulam stability of our problem. Finally, we presented an example to demonstrate the use of our main theorems.

In this paper, we study a new approach to investigating of existence, uniqueness, and stability of the periodic solution of the nonlinear fractional integro-differential equation of type Caputo-Fabrizio fractional derivative with the initial condition, periodic boundary conditions, and integral boundary conditions by using successive approximations method and Banach fixed point theorem. Finally, some examples are presented to illustrate the theorems.

In this work, the existence and uniqueness theorems for integro-differential equation involving the Caputo fractional derivative are established under some sufficient conditions and example for application the results of the theorems are presented.

We give a new investigation of periodic solutions of nonlinear impulsive fractional integro-differential system with different orders of fractional derivatives with non-separated integral coupled boundary conditions. Uniformly Converging of the sequence of functions according to the main idea of the Numerical-analytic technique, from creating a sequence of functions. An example of impulsive fractional system is also presented to illustrate the theory.

Consider the following non-linear system of differential equation which has the form: 𝒅𝒙(𝒕)/𝒅𝒕+ 𝒂(𝒕)𝒉(𝒚(𝒕)) = 𝒇(𝒕, 𝒙(𝒕 − 𝝉(𝒕)), 𝒚(𝒕 − 𝝉(𝒕)), 𝒅𝟐𝒚(𝒕)/𝒅𝒕𝟐 + 𝒑(𝒕)𝒅𝒚(𝒕)/𝒅𝒕+ 𝒒(𝒕)𝑱(𝒙(𝒕)) = 𝒈(𝒕, 𝒙(𝒕 − 𝝉(𝒕)), 𝒚(𝒕 − 𝝉(𝒕)) The aim of this paper is to use Krasnoselskii's fixed point theorem to show the existence of a positive periodic solutions for the above system. To apply Krasnoselskii's fixed point theorem requirement to construct two mappings; one is compact and the other is contraction. Using the contraction mapping principle enables us to show the uniqueness of the periodic solution.

In this paper, we investigate the existence and uniqueness of solutions for a system of fractional integro-differential equations with non-separated integral coupled boundary conditions. Our results are based on the nonlinear alternative of Leray-Schauder type to study the existence of at least one continuous solution to fractional integro- differential system with non-separated integral coupled boundary conditions and uniqueness continuous solution using the Banach’s fixed-point theorem.The main results are well illustrated with the aid of an example.

We give a new approach of investigation of existence and approximate of periodic solutions for the system of nonlinear fractional integro-differential equations with different orders of fractional derivatives subject to non-separated integral coupled boundary conditions. According to the main idea of the Numerical-analytic technique, we construct a sequence of functions that it proved to be convergent. It is shown that the limit function of the constructed sequence satisfies a fractional integro-differential equations and periodic conditions Finally, we studies the stability of periodic solutions and an example of fractional system is also presented to illustrate the theory.

The article deals with approximate and uniqueness periodic solutions for nonlinear system of nonlinear integro-differential equations with parameter and with boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations for investigate the periodic solution of ordinary differential equations, which are given by A. M.Samoilenko. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function.

In this paper, we investigate the existence, uniqueness and stability of the periodic solution for the system of nonlinear integro-differential equations by using the numerical-analytic methods for investigate the solutions and the periodic solutions of ordinary differential equations, which are given by A. Samoilenko.

In this paper we investigate the existence and approximation of periodic solution for the system of nonlinear differential equations by using the numerical-analytic methods for investigate the periodic solutions of ordinary differential equations, which are given by A. Samoilenko.

In this paper, we study the existence and uniqueness solution for nonlinear Volterra integral equation, by using both methods ( Picard Approximation ) and (Banach Fixed Point Theorem). Also these methods could be developed and extended throughout the study.