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In this paper, we derive existence and uniqueness results for a nonlinear Caputo&ndash, Riemann&ndash, Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel&rsquo, skiĭ&rsquo, s fixed point theorems. Examples are included for the illustration of the obtained results.

This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann&ndash, Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented.

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov&rsquo, s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

This paper involves complex valued versions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation. Under various suitable assumptions the results are established in the setting of complex valued double controlled metric space. Thereafter, by making consequent use of the fixed point method, short and simple proofs are obtained for solutions of Riemann-Liouville integral, complex valued Atangana-Baleanu integral operator and non-linear Telegraph equation.

In this paper, we investigate the existence of extremal solutions for fractional differential systems involving the p-Laplacian operator and Riemann-Liouville integral boundary conditions. We derive our results based on the monotone iterative technique, combined with the method of upper and lower solutions. An example is added to illustrate the main result.

In this paper we study the existence and uniqueness of a solution for fractional impulsive differential equations with Riemann-Liouville integral boundary condition, using a fixed point theorem. Also, we discuss Ulamstability for these equations via some generalized singular Gronwall inequalities.

We use a generalization of the Hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval-valued functions. The properties of Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative and Caputo fractional derivative for interval-valued functions are investigated. Several examples are presented to illustrate the concepts and results.

In recent years, prey---predator models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance. The paper presents the solutions of time-fractional Lotka---Volterra models with the help of analytical method of nonlinear problem called homotopy perturbation method (HPM). By using initial values, the explicit solutions of time-fractional prey and predator populations for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The dynamic behavior of the system investigated from the point of view of local stability. We also carry out a detailed analysis on stability of equilibrium.

In this paper, we introduce a new type of fractional derivative, which we called truncated $${\mathcal {V}}$$ -fractional derivative, for $$\alpha $$ -differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated $${\mathcal {V}}$$ -fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated $${\mathcal {V}}$$ -fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the $${\mathcal {V}}$$ -fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the $${\mathcal {V}}$$ -fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated $${\mathcal {V}}$$ -fractional derivative and the truncated $${\mathcal {V}}$$ -fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter $$\alpha $$ lies between 0 and 1 ( $$0

First of all, in this paper, we prove the convergence of the nabla h-sum to the Riemann–Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Second, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.

We study the existence and uniqueness of positive solutions for a sum-type singular fractional integro-differential equation with m-point boundary condition. Also, we provide an example to illustrate our main result. Publisher's Version

In the paper certain method for constructing exact solutions of a class of linear differential equations of fractional order is considered. Algorithms for constructing solutions of the explicit form are developed for homogeneous and inhomogeneous differential equations of fractional order. This method is based on construction of normalized systems associated with fractional differentiation operator . 0 - normalized and f - normalized systems are built concerning to the pair of operators connected with the considered equation. Using 0 - normalized systems, linearly independent solutions of the homogeneous equation are constructed. Similarly, with the help of f - normalized systems partial solutions of the inhomogeneous equation are built in the case, where the right side is a quasi-polynomial, analytic function and an arbitrary function from the class of continuous functions.

In this paper, two novel and computationally efficient, zero-phase filtering techniques are proposed based on the Riemann-Liouville integral. Thanks to the reverse phase characteristics of backward filtering, an overall zero-phase effect can be achieved by cascading a fractional forward filtering with a fractional backward filtering, and vice versa. The fractional zero-phase filtering can not only effectively suppress the phase distortion in the filtering process but also better enhance the compromise capability between signal denoising and signal information retention than the conventional filtering methods do. The proposed methods are evaluated on Electrocardiogram (ECG) signal, by adding disturbance, random, and white Gaussian noises to visually clean ECG record, and studying SNR and MSE of the filter outputs. The results of the study demonstrate superior performances compared with conventional signal denoising methods, such as Riemann-Liouville integral filtering, Grunwald-Letnikov integral filtering, zero-phase Butterworth filtering, and zero-phase average window filtering.

We consider a potential-type integral operator with oscillating kernel in the space of functions representable as Riesz potentials. We obtain estimates of its norm in the Lebesgue spaces which depend on the oscillation parameter to a negative power.

This paper is concerned with Riemann–Liouville abstract fractional Cauchy Problems with damping. The notion of Riemann–Liouville fractional ( α , β , c ) resolvent is developed, where 0 β α ≤ 1 . Some of its properties are obtained. By combining such properties with the properties of general Mittag-Leffler functions, existence and uniqueness results of the strong solution of Riemann–Liouville abstract fractional Cauchy Problems with damping are established. As an application, a fractional diffusion equation with damping is presented.

A sufficient condition for the nonexistence of blowing-up solutions to evolution functional-differential equations in Banach spaces with the Riemann-Liouville integrals in their right-hand sides is proved. The linear part of such type of equations is an infinitesimal generator of a strongly continuous semigroup of linear bounded operators. The proof of the main result is based on a desingularization method applied by the author in his papers on integral inequalities with weakly singular kernels. The result is illustrated on an example of a scalar equation with one Riemann-Liouville integral.

In this paper, we discuss the existence of solutions for a boundary value problem of fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Our results include the cases when the multivalued map involved in the problem is (i) convex valued, (ii) lower semicontinuous with nonempty closed and decomposable values and (iii) nonconvex valued. In case (i) we apply a nonlinear alternative of Leray-Schauder type, in the second case we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo, while in the third case we use a fixed point theorem for multivalued contractions due to Covitz and Nadler.

In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and H\"older continuous random processes $X$ and $Y$. We discuss a notion of sufficient variability for the process $X$ which ensures that the paths of the composite process $t \mapsto f(X_t)$ are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann-Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new estimates on the accuracy of numerical approximations of such integrals, together with a change of variables formula for integrals of the form $\int f(X_t) \, dX_t$., Comment: The 2nd version contains lots of new examples illustrating applications of the main results to various stochastic processes

The article proposes a generalized polynomial projection method for solving linear integral equations with a fractional Riemann - Liouville integral in the main part. Fractional-integral equations have numerous applications in various applied problems, in particular, in problems of plasma diagnostics, optical and X-ray diffraction, in ultrasonic measurements in moving media, in metallurgy, biology, microscopy, seismology, in astrophysics and other fields of science.The equation we are considering is an equation of the first kind and therefore, generally speaking, refers to incorrectly set by Hadamard equations. The latter circumstance is connected with the fact that in known function spaces the fractional integral Riemann-Liouville operator is completely continuous. All this imposes its own characteristics on the construction and study of approximate methods for solving integral equations with a fractional integral operator in the main part. It should be noted that the operator of the projection method is not necessarily projective, which allows the construction of computational schemes using the methods of summation of Fourier series and interpolation polynomials. A theoretical and functional substantiation of the proposed projection method was carried out, based on the correct formulation of the equation in a pair of specially selected different Holder spaces. In particular, it proved the unique solvability of the system of linear algebraic equations of the method and the convergence of the constructed approximations to the exact solution in the norm of the space of Holder functions and, as a consequence, in the uniform metric. This also implies the substantiation of specific well-known projection methods such as the Galerkin methods, colocations, and subdomains.

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s, m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.

We obtain bounds for integro-differential inequalities involving derivatives of orders between 0 and 2. We give applications and examples demonstrating the use of these bounds in analyzing the existence and asymptotic behavior of solutions for a class of singular fractional differential equations.

By using two fundamental fractional integral identities, we deduce some new Hermite–Hadamard-type inequalities for differentiable r-convex functions and twice-differentiable r-convex functions involving Riemann–Liouville fractional integrals. Iз використанням двох фундаментальних дробових iнтегральних тотожностей отримано новi нерiвностi типу Ермiта – Адамара для диференцiйовних r-опуклих функцiй та двiчi диференцiйовних r-опуклих функцiй, що мiстять дробовi iнтеграли Рiмана – Лiувiлля.

In this paper we deal with the problem of asymptotic integration of nonlinear higher order fractional differential equations of the Caputo’s type. We give some conditions under which all global solutions of these equations behave like linear functions as t -> ∞.

This paper presents the interpretation of the Caputo derivative with the help of the infinite state approach, also known as the fractional integrator approach. After presentation of the modified Laplace transform equations, definition of state variables is discussed, as well as its application to the characterization of fractional system energy. Then, the infinite state approach is used to invalidate system simulation based on usual Caputo's initial conditions.

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann–Liouville fractional integration operator has been obtained.

An existence and an uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated in this research work. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations.

We study the boundedness and compactness of Riemann–Liouville integral operators on the so-called Morrey spaces which are nonseparable spaces. There are no approximation or contractive skills in this kind of spaces. Moreover, unlike the use of dual or maximal point of view in integrable function spaces, the idea of our proof proceeds from the compactness of the truncated Riemann–Liouville fractional integrals by using a criterion for strongly pre-compact set. Constructing a truncated Marchaud fractional derivative function, we show the characterization of the solution to Abel’s equation in Morrey spaces. With the aid of fixed-point theorem, we establish the existence and uniqueness of solutions to a Cauchy type problem for fractional differential equations. We also give an example to illustrate the sufficiency of the conditions in our main result.

Initial value problem for a class of fractional order linear inhomogeneous equations in Banach spaces with a bounded operator at the unknown function is considered. The equation contains the Riemann–Liouville fractional derivative and the corresponding initial conditions are set for the fractional derivatives of a solution. The theorem of the problem unique solvability is proved. It is applied for studying of the solvability of initial boundary value problem for a filtration theory equation with Riemann–Liouville time-fractional order.

Here we study some important properties of right multidimensional Riemann–Liouville fractional integral operator, such as of continuity and boundedness.

First of all, in this paper, we prove the convergence of the nabla h-sum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.

This paper investigates the existence and uniqueness of solutions for a fractional boundary value problem involving four-point nonlocal Riemann-Liouville integral boundary conditions of different order. Our results are based on standard tools of fixed point theory and Leray-Schauder nonlinear alternative. Some illustrative examples are also discussed.

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument $$\left \{\begin{array}{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1 3 (n??), $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order ?,f:[0,?)?[0,?), h(t):[0,1]?(0,?) and ?:(0,1)?(0,1] are continuous functions. Some novel sufficient conditions are obtained for the existence of at least one or two positive solutions by using the Krasnosel'skii's fixed point theorem, and some other new sufficient conditions are derived for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams etc. In particular, the existence of at least n or 2n?1 distinct positive solutions is established by using the solution intervals and local properties. From the viewpoint of applications, two examples are given to illustrate the effectiveness of our results.

The well-known Liouville, Riemann–Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grunwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grunwald–Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann–Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann–Liouville and Caputo derivatives with the Grunwald–Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.

We give the explicit solutions of uncertain fractional differential equations (UFDEs) under Riemann–Liouville H-differentiability using Mittag-Leffler functions. To this end, Riemann–Liouville H-differentiability is introduced which is a direct generalization of the concept of Riemann–Liouville differentiability in deterministic sense to the fuzzy context. Moreover, equivalent integral forms of UFDEs are determined which are applied to derive the explicit solutions. Finally, some illustrative examples are given.

The main goal of this paper is to prove two Aronszajn type theorems for some initial value problems formulated in terms of fractional derivatives. Moreover, we are going to establish a theorem on the existence and uniqueness of positive solutions.

We consider the following equation for the Riesz potential of order one: Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost ever ...

Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann–Liouville and Erdelyi–Kober fractional integral operators. Formulas of compositions for such a generalized fractional integrals with Bessel function of the first kind are proved. Special cases of cosine and sine functions are given. The results are established in terms of generalized Wright function and generalized hypergeometric series. Corresponding assertions for Riemann–Liouville and Erdelyi–Kober fractional integral transforms are presented.