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We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter γ = 0 and γ = 1 , respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer's sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to γ = 0 and γ = 1 , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability for Hilfer fractional stochastic hemivariational inequalities of order 1 < μ < 2 in Hilbert spaces. Initially, we define the concept of a mild solution for our problem in terms of fractional calculus, cosine families, stochastic analysis, and generalized Clarke subdifferential. Then, the existence and approximate controllability for Hilfer fractional stochastic evolution hemivariational inequalities are formulated and proven under appropriate conditions using fixed point theorems for multivalued maps. Finally, an example is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This article explores the existence and asymptotic stability in the p-th moment of mild solutions to a class of Hilfer fractional neutral stochastic differential equations with infinite delay in Hilbert spaces. To prove our main results, we use fractional calculus, stochastic analysis, semigroup theory, and the Krasnoselskii-Schaefer type fixed point theorem. Moreover, a set of novel sufficient conditions is derived for achieving the required result. Following that, we extend the given system to the Sobolev type and provided the existence results of the considered system. After that, we provided an example to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

The main innovation of this article is to establish the existence and approximate controllability of Hilfer fractional differential inclusions of order 1<μ<2$$ 1<\mu <2 $$ using Clarke's subdifferential type in Hilbert spaces. By applying the fixed‐point theorem for multivalued maps, fractional calculus, generalized Clarke's subdifferential, and cosine families, we derive a novel existence result for the mild solutions of the Hilfer fractional differential inclusion system. Furthermore, we develop and demonstrate a new set of sufficient conditions for the approximate controllability of Hilfer fractional differential systems. These criteria are based on the assumption that the associated linear part of the system is approximately controllable. Finally, an example is provided to show the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 {\frac{1}{2}<\alpha<1} . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our result is provided. [ABSTRACT FROM AUTHOR]

In studying boundary value problems and coupled systems of fractional order in (1 , 2 ] , involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1 , 2 ] , involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach's fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results. [ABSTRACT FROM AUTHOR]

This paper describes a new class of boundary value fractional-order differential equations of the q-Hilfer and q-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented. [ABSTRACT FROM AUTHOR]

The focus of this article lies in the existence of global mild solutions for Hilfer fractional stochastic evolution equations (HFSEEs) on the positive semi-axis (0,+∞) with a derivative order greater than 32 and less than 2. A significant challenge, which also presents novelty here, is to extend the generalized Ascoli–Arzela (A–A) theorem established in Zhou and He (2022) to the stochastic case under the associated sine family {S(t)}t≥0 not necessary compact. Then, by relying on our newly established generalized A–A theorem, standard stochastic analysis theory and Schauder’s fixed point principle, we derive the existence of global mild solutions for the addressed system. Finally, we demonstrate the feasibility of our obtained results through an illustrative example.

The main aim of this paper is to study the existence and uniqueness solutions for the nonlinear Hilfer pantograph fractional differential equations. This paper initiates with the persistence of the nonlinear Hilfer pantograph fractional differential equation. Also, it extended to the fractional integrodifferential equation. The premises are attained by using the fixed-point theorem. Ultimately, numerical examples are furnished to demonstrate our outcomes. [ABSTRACT FROM AUTHOR]

This paper describes a new class of boundary value fractional-order differential equations of the $ q $-Hilfer and $ q $-Caputo types, with separated boundary conditions. The presented problem is converted to an equivalent integral form, and fixed-point theorems are used to prove the existence and uniqueness of solutions. Moreover, several special cases demonstrate how the proposed problems advance beyond the existing literature. Examples are provided to support the analysis presented.

In this article, we discuss the optimal feedback control of nonlocal Hilfer fractional state‐dependent delay inclusion of order 1<α<2$$ 1&lt;\alpha &lt;2 $$ with Clarke's subdifferential. Firstly, we establish the existence of mild solutions for this class of equations by employing the Krasnoselskii's fixed‐point theorem, without using Lipschitz conditions. Subsequently, we develop set of sufficient assumptions, utilizing Cesari property and Filippov theorem, to ensure the existence of feasible pairs for the feedback control systems. Eventually, using the Mazur theorem and Fatou's lemma, we derive the optimal state–control pair. The study also introduces and explores the problem of time‐optimal feedback control. In conclusion, an application is provided to enhance the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a system of fully nonlinear Hilfer fractional differential equations on the half‐line is studied. Firstly, considering that the half‐line as the domain has the problem of insufficient compactness, we defined the space of weighting functions and normed the convergence at infinity. Secondly, by constructing appropriate operators in these Banach spaces and using the extension for the continuous theorem, some sufficient conditions for the solvability of the problem are obtained. Finally, an example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result. [ABSTRACT FROM AUTHOR]

This paper studies the integro-differential equations of Hilfer fractional (HF) neutral stochastic evolution on an infinite interval with almost sectorial operators and their attractive solutions. We use semigroup theory, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. We establish the existence and attractivity theorems for mild solutions by considering the fact that the almost sectorial operator is both compact and noncompact. Example that highlight the key findings are also provided. [ABSTRACT FROM AUTHOR]

This article explores the Hilfer fractional derivative within the context of fractional differential equations and investigates a mathematical model formulated as a three-point boundary value problem (BVP). The primary focus is on the application of these models to analyze the jet flow of the Antarctic Circumpolar Current. The study establishes the existence of stream functions using Schaefer's fixed point theorem under the assumption of the continuity of the vorticity function Φ . Furthermore, the article delves into the existence and uniqueness results of the stream functions by employing the Banach fixed point theorem. This analysis is conducted under the condition that the vorticity function Φ is Lipschitz continuous with respect to the stream function. Additionally, the stability of the stream functions of the BVP is explored through Ulam–Hyers and generalized Ulam–Hyers stability analyses. In contrast to the foundational results presented for the three-point BVP, the article includes illustrative examples aimed at validating the findings. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is the concept of controllability criteria for the impulsive Hilfer fractional integro-differential equation (IHFrIDE) of order 0≤ϖ≤1 and 1

This paper studies the integro-differential equations of Hilfer fractional (HF) neutral stochastic evolution on an infinite interval with almost sectorial operators and their attractive solutions. We use semigroup theory, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. We establish the existence and attractivity theorems for mild solutions by considering the fact that the almost sectorial operator is both compact and noncompact. Example that highlight the key findings are also provided.

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, $ k $-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators.

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples. [ABSTRACT FROM AUTHOR]

A survey of results on Grüss-type inequalities associated with a variety of fractional integral and differential operators is presented. The fractional differential operators includes, Riemann-Liouville fractional integral operators, Riemann-Liouville fractional integrals of a function with respect to another function, Katugampola fractional integral operators, Hadamard's fractional integral operators, k-fractional integral operators, Raina's fractional integral operators, tempered fractional integral operators, conformable fractional integrals operators, proportional fractional integrals operators, generalized Riemann-Liouville fractional integral operators, Caputo-Fabrizio fractional integrals operators, Saigo fractional integral operators, quantum integral operators, and Hilfer fractional differential operators. [ABSTRACT FROM AUTHOR]

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Abstract In this work, a system of nonlinear, switched, coupled, implicit, impulsive Langevin equations with two Hilfer fractional derivatives is introduced. The suitable conditions and results are established to discuss existence, uniqueness, and Ulam-type stability results of the mentioned model, with the help of nonlinear functional analysis techniques and Banach’s fixed-point theorem. Furthermore, we examine our results with the help of example.

The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.

A review on Hermite-Hadamard type inequalities connected with a different classes of convexities and fractional differential operators is presented. In the various classes of convexities it includes, classical convex functions, quasi-convex functions, p-convex functions, strongly-m-convex functions, strongly-(θ,m)-convex functions, (s, m)-convex functions, (θ, h- m)-convex functions, strongly (θ, h- m)-convex functions, (h, m)-convex functions of the second type, m-convex functions, h-convex functions, (h,m)-convex functions, relative-convex functions, exponentially (θ, h- m)-convex functions, harmonically h-convex functions and geometric-arithmetically s-convex functions. In the fractional differential operators it includes, Caputo fractional derivative, k-Caputo fractional derivative and Hilfer fractional derivative.

In this research, we investigate Hilfer-type FDDEs of both linear and nonlinear nature, with orders α and β. We studied some Riemann-Liouville, Caputo-Liouville, and Hilfer fractional derivatives properties; and then used them to convert our fractional delay problem into an equivalent fractional Volterra delay problem. After that, for the resulting fractional Volterra delay problem, using the CMT, existence-uniqueness is demonstrated. For numerical solutions, we studied the orthogonal shifted Legendre polynomials and used them as the basis for the Galerkin algorithm to find an approximation solution to the corresponding FDDE. By employing this algorithm, the provided FDDE is converted into a series of algebraic systems. By solving this system, the approximated solutions for the equation are obtained. The main algorithm advantage is that the number of needed iterations is low compared with the soft gained results. Some linear and nonlinear numerical applications together with several figures and tables are utilized to prove the performance and accuracy of the algorithm. The outcomes results are considered according to the given analysis and the numerical method was offered in the last section with several hints to guide future work.

In studying boundary value problems and coupled systems of fractional order in (1,2], involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in (1,2], involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results.

In this article, we focus on the new class of neutral Hilfer fractional differential equations on infinite interval in Banach space. The results are obtained with the help of the theory of fractional calculus, mild solutions, the fixed point theorem, semigroup theory, and infinite interval. First, we introduce a mild solution for neutral Hilfer fractional differential equations. Then, we prove the existence of a mild solution for the neutral Hilfer fractional differential equations. Finally, an example is provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Calcium is an essential element in our body and plays a vital role in moderating calcium signalling. Calcium is also called the second messenger. Calcium signalling depends on cytosolic calcium concentration. In this study, we focus on cellular calcium fluctuations with different buffers, including calcium-binding buffers, using the Hilfer fractional advection-diffusion equation for cellular calcium. Limits and start conditions are also set. By combining with intracellular free calcium ions, buffers reduce the cytosolic calcium concentration. The buffer depletes cellular calcium and protects against toxicity. Association, dissociation, diffusion, and buffer concentration are modelled. The solution of the Hilfer fractional calcium model is achieved through utilizing the integral transform technique. To investigate the influence of the buffer on the calcium concentration distribution, simulations are done in MATLAB 21. The results show that the modified calcium model is a function of time, position, and the Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a richer physical explanation than the classical calcium model. [ABSTRACT FROM AUTHOR]

The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system. [ABSTRACT FROM AUTHOR]

The paper sheds light on Hilfer’s controllability facts of neutral fractional system. Originally, the mild solution is derived using semigroup theory and the Laplace transform approach. Controllability of the Hilfer fractional system in non-dense domain using integral contractor which employs the sequence technique with the advantage that the nonlinear function does not meet the Lipschitz condition. To support the computed results, an appropriate examples are discussed. [ABSTRACT FROM AUTHOR]

This manuscript is concerned with the trajectory controllability of Hilfer fractional neutral stochastic differential equation having deviated arguments and mixed fractional Brownian motion with the Hurst parameter H ∈ (1 / 2 , 1). The proposed Hilfer fractional system's solvability in Hilbert space is acquired by employing fractional calculus, stochastic analysis, semigroup theory, and Krasnoselskii's fixed point theorem. Furthermore, under some suitable assumptions, the considered system's trajectory controllability is established using generalized Gronwall's inequality. Finally, an example is delivered to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω ∈ (1 , 2) for the RLC model. Based on Schaefer's fixed point theorem and Banach's contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples. [ABSTRACT FROM AUTHOR]

In this research, we investigate Hilfer-type FDDEs of both linear and nonlinear nature, with orders α and β. We studied some Riemann-Liouville, Caputo-Liouville, and Hilfer fractional derivatives properties; and then used them to convert our fractional delay problem into an equivalent fractional Volterra delay problem. After that, for the resulting fractional Volterra delay problem, using the CMT, existence-uniqueness is demonstrated. For numerical solutions, we studied the orthogonal shifted Legendre polynomials and used them as the basis for the Galerkin algorithm to find an approximation solution to the corresponding FDDE. By employing this algorithm, the provided FDDE is converted into a series of algebraic systems. By solving this system, the approximated solutions for the equation are obtained. The main algorithm advantage is that the number of needed iterations is low compared with the soft gained results. Some linear and nonlinear numerical applications together with several figures and tables are utilized to prove the performance and accuracy of the algorithm. The outcomes results are considered according to the given analysis and the numerical method was offered in the last section with several hints to guide future work. [ABSTRACT FROM AUTHOR]

The primary focus of this article is to provide sufficient conditions for the controllability of Hilfer fractional neutral integrodifferential systems. By using the theories on fractional calculus, semigroup theory, and fixed point theorem method, the results are proved through integral contractors for the nonlinear neutral evolution equation in a non‐dense domain. In the end, the derived outcomes are validated through suitable illustrations. [ABSTRACT FROM AUTHOR]

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli–Arzelà theorem and Mönch's fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results. [ABSTRACT FROM AUTHOR]

This article focuses on deriving the averaging principle for Hilfer fractional stochastic evolution equations (HFSEEs) driven by Lévy noise. We show that the solutions of the averaged equations converge to the corresponding solutions of the original equations, both in the sense of mean square and of probability. Our results enable us to focus on the averaged system rather than the original, more complex one. Given that the existing literature on the averaging principle for Hilfer fractional stochastic differential equations has been established in finite-dimensional spaces, the novelty here is the derivation of the averaging principle for a class of HFSEEs in Hilbert space. Furthermore, an example is allotted to illustrate the feasibility and utility of our results. [ABSTRACT FROM AUTHOR]

In this manuscript, switched coupled system of nonlinear impulsive Langevin equations involving four Hilfer fractional-order derivatives is considered. Using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss the existence, uniqueness, and Ulam's type stability results of our proposed model, with the help of Schaefer's fixed point theorem. An example is provided at the end to illustrate our results. [ABSTRACT FROM AUTHOR]

In this work, the optimal control for a class of Hilfer fractional stochastic integrodifferential systems driven by Rosenblatt process and Poisson jumps has been discussed in infinite dimensional space involving the Hilfer fractional derivative. First, we study the existence and uniqueness of mild solution results are proved by the virtue of fractional calculus, successive approximation method and stochastic analysis techniques. Second, the optimal control of the proposed problem is presented by using Balder's theorem. Finally, an example is demonstrated to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Guaranteeing the uniqueness of the solution will simplify the analysis and provide a clear approximation of the considered problem. This article presents theoretical proof of the presence of a unique solution and leverages approximation for the time delay functions in integrodifferential models in the sense of the Hilfer fractional approach. Once the wellposedness discussion is done, our focus lies on utilizing the Galerkin pseudo-codes based on the OSLPs to generate an approximation by applying GSLM as follows: utilizing the OSLPs to replace the required functions in main Hilfer model, applying the Galerkin pseudo-codes, and transforming Hilfer model into an algebraic system of equations. Herein, the wellposedness is proven by utilizing the fixed point of Banach and the principle of contraction mapping. The proposed scheme is shown to provide efficient, flawless, and expedient solutions for the needed Hilfer fractional delay model. Whilst, the effectiveness of the approach is demonstrated by various delayed examples in comparison with the classical solution. Tabulated and plotted results of the study provide a novel reliable scheme for handling such a Hilfer delayed integrodifferential model. Eventually, valuable noteworthy points and suggestions for future endeavors have been offered.

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result.

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

This manuscript deals with the concept of Hilfer fractional neutral functional integro-differential equation with a nonlocal condition. The solution representation of a given system is obtained from the strongly continuous operator, linear operator and bounded operator, as well as the Wright type of function. The sufficient and necessary conditions for the existence of a solution are attained using the topological degree method. The uniqueness of the solution is attained by Gronwall's inequality. Finally, we employed some specific numerical computations to examine the effectiveness of the results.

This manuscript primarily focuses on the nonlocal controllability results of Hilfer neutral impulsive fractional integro-differential equations of order $ 0\leq w\leq1 $ and $ 0 < g < 1 $ in a Banach space. The outcomes are derived from the strongly continuous operator, Wright function, linear operator, and bounded operator. First, we explore the existence and uniqueness of the results of the mild solution of Hilfer's neutral impulsive fractional integro-differential equations using Schauder's fixed point theorem and an iterative process. In order to determine nonlocal controllability, the Banach fixed point technique is used. We employed some specific numerical computations and applications to examine the effectiveness of the results.