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The main purpose of this paper is to study the existence, uniqueness, Ea-Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the ps-Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of Ea-Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results.

In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ-Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of solutions for certain classes of nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with Dirichlet boundary conditions in Banach spaces. Moreover, our analysis is based on Darbo’s fixed point theorem in conjunction with the technique of Hausdorff measure of noncompactness. An example is also presented to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove the Ulam–Hyers stability of solutions for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the ψ-Caputo sense. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Next, we proceed to investigate the existence and uniqueness of the solution by applying Schauder and Banach fixed point theorems. Finally, we study the Ulam–Hyers stability criteria for the main fractional system. Illustrative examples are presented to demonstrate the validity of the obtained results. The results are new and provide extensions to some known results in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with the existence of extremal solutions for a novel class of nonlinear fractional p‐Laplacian differential equation in terms of ψ‐Caputo fractional derivative equipped with a new class of nonlinear boundary conditions. Initially, we focus on the linear problem and we give an explicit form of the solutions, from which we derive new comparison results which is crucial for the proof of the main theorem of this paper. Secondly, with the help of the monotone iterative method along with its accompanying upper and lower solutions, the existence of extremal solutions for the aforementioned problem is investigated. Then, based on the proposed method, we can construct two types of successive approximations which converge uniformly monotonically to minimal and maximal solutions of the given problem. Finally, we illustrate our results with an example. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of integrable solutions to fractional Langevin equations involving two fractional orders with initial value problems. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem. Examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the existence, uniqueness, Eα-Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the -Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of Eα-Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger's fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall's inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo's fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems. [ABSTRACT FROM AUTHOR]