arXiv subjects
Publications and source records attributed to J. Vanterler da C. Sousa.
In this paper, we investigate the existence of mild solutions to Hilfer fractional equation of semi-linear evolution with non-instantaneous impulses, using the concepts of equicontinuous $C_{0}$-semigroup and Kuratowski measure of non-compactness in Banach space $Ω$.
In this paper, by means of Banach fixed point theorem, we investigate the existence and Ulam--Hyers--Rassias stability of the non-instantaneous impulsive integrodifferential equation by means of $ψ$-Hilfer fractional derivative. In this sense, some examples are presented, in order to consolidate the results obtained.
In this paper, we present a new class of inequality of the Gronwall type and discuss some particular cases. In this sense, we investigate the uniqueness and $δ$-Ulam-Hyers-Rassias stability of the mild solution for the fractional differential equation with non-instantaneous impulses in a P$δ$-normed Banach space.
In this paper, we present the Leibniz rule for the $Ψ-$Hilfer ($Ψ-$H) fractional derivative in two versions, the first in relation to $Ψ-$RL fractional derivative and the second in relation to the $Ψ-$H fractional derivative. In this sense, we present some particular cases of Leibniz rules and Leibniz type rules from the investigated case.
Motivated by the ${\rm Ψ}$-Riemann-Liouville $({\rm Ψ-RL})$ fractional derivative and by the ${\rm Ψ}$-Hilfer $({\rm Ψ-H})$ fractional derivative, we introduced a new fractional operator the so-called $\rmΨ-$fractional integral. We present some important results by means of theorems and in particular, that the $\rmΨ-$fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag-Leffler $({\rm M-L})$ function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear $Ψ$-fractional Volterra integral equation (${\rm VIE}$) using $β-$distance functions.
The present paper plans to examine the existence, uniqueness and data dependence of the solution of the fractional functional differential equation with the abstract operator of Volterra, in the context of the Picard operators. We present an example, in order to illustrate the results obtained. We present an application, with the end goal to illustrate the results obtained.
In this paper, we consider the new class of the fractional differential equation involving the abstract Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of Ulam--Hyers on the compact interval $Δ=[a,b]$ and on the infinite interval $I=[a,\infty )$. Our analysis is based on the application of the Banach fixed point theorem and the Gronwall inequality involving generalized $Ψ$-fractional integral. At last, we performed out an application to elucidate the outcomes got.
We consider the nonlinear Cauchy problem for $ Ψ$- Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $ Ψ$- Hilfer fractional derivative and an estimation have been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formula for the solution of linear Cauchy problem for $ Ψ$-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas-Saigo) Mittag-Leffler function.
In this paper, we present a study on the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solution of the fractional functional differential equation using the Banach fixed point theorem.
In this paper, we investigate the sufficient conditions for existence and uniqueness of solutions and δ-Ulam-Hyers-Rassias stability of an impulsive fractional differential equation involving $ψ$-Hilfer fractional derivative. Fixed point approach is used to obtain our main results.
By means of two fractional order integral inequalities we investigate the existence and uniqueness of the solutions of the fractional nonlinear Volterra integral equation and a fractional nonlinear integrodifferential equation in Banach space $C_ξ$, using an adequate norm, $||\cdot||_{ξ,\infty}$. We estimate the solutions and investigate their continuous dependence.
Using Gronwall inequality we will investigate the Ulam-Hyers and generalized Ulam-Hyers-Rassias stabilities for the solution of a fractional order pseudoparabolic partial differential equation.
In this paper, by means of the Gronwall inequality, the ψ-Riemann-Liouville fractional partial integral and the ψ-Hilfer fractional partial derivative are introduced and some of its particular cases are recovered. Using these results, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the solutions of a fractional partial differential equation of hyperbolic type in a Banach space (B, |.|), real or complex.
By means of the recent $ψ$-Hilfer fractional derivative and of the Banach fixed-point theorem, we investigate stabilities of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias on closed intervals $[a,b]$ and $[a,\infty)$ for a particular class of fractional integro-differential equations.
In this paper, using the Riemann-Liouville fractional integral with respect to another function and the $ψ-$Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the $α$-resolvent operator in the Sobolev space $W^{1,1}(\mathbb{R}_{+},\mathbb{C})$.
We present the validation of a recent fractional mathematical model for erythrocyte sedimentation proposed by Sharma et al. \cite{GMR}. The model uses a Caputo fractional derivative to build a time fractional diffusion equation suitable to predict blood sedimentation rates. This validation was carried out by means of erythrocyte sedimentation tests in laboratory. Data on sedimentation rates (percentages) were analyzed and compared with the analytical solution of the time fractional diffusion equation. The behavior of the analytical solution related to each blood sample sedimentation data was described and analyzed.
Using the $ψ-$Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of continuous functions, in particular the Comparison's principle, and others that allow, the study of the limitation of fractional nonlinear differential equations.