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J. Vanterler da C. Sousa

Publications and source records attributed to J. Vanterler da C. Sousa.

At least 19 recordsLinked to original sources
arXiv2023

Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).

math.GM↗
arXiv2023

Existence of solutions for a singular double phase problem involving a $ψ$-Hilfer Fractional operator via Nehari Manifold

In this present paper, we investigate a new class of singular double phase $p$-Laplacian equation problems with a $ψ$-Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold.

math.GM↗
arXiv2023

Fractional Kirchhoff-type and method of sub-supersolutions

In the present paper, we are interested in investigating the existence of positive solutions of a new class of fractional Kirchhoff via the sub and supersolutions technique. For this, we first need to investigate two results through lemmas.

math.GM↗
arXiv2023

Fractional Kirchhoff-type systems via sub-supersolutions method in $\mathbb{H}^{α,β;ψ}_{p}(Ω)$

In the present paper, we first establish a version of the abstract lower and upper-solution method for our class of operators. In this sense, we investigated the main objective of this paper, that is, the existence of a positive solution for a new class of fractional systems of the Kirchhoff type with $ψ$-Hilfer operators via the method of sub and supersolutions in $ψ$-fractional space $\mathbb{H}^{α,β;ψ}_{p}(Ω)$

math.GM↗
arXiv2023

Fractional $p$-Laplacian equations with sandwich pairs

The main purpose of this paper is to consider new sandwich pairs and investigate the existence of solution for a new class of fractional differential equations with $p$-Laplacian via variational methods in $ψ$-fractional space $\mathbb{H}^{α,β;ψ}_{p}(Ω)$. The results obtained in this paper are the first to make use of the theory of $ψ$-Hilfer fractional operators with $p$-Laplacian.

math.GM↗
arXiv2023

Existence and multiplicity of solutions for fractional $κ(ξ)$-Kirchhoff-type equation

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of $κ(ξ)$-Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space $ψ$-fractional $\mathcal{H}^{μ,ν;\,ψ}_{κ(ξ)}(Λ)$. In this sense, we present a few special cases and remark on the outcomes explored.

math.GM↗
arXiv2023

Existence and multiplicity for fractional Dirichlet problem with $γ(ξ)$-Laplacian equation and Nehari manifold

This paper is divided in two parts. In the first part, we prove coercivity results and minimization of the Euler energy functional. In the second part, we focus on the existence and multiplicity of a positive solution of fractional Dirichlet problem involving the $γ(ξ)$-Laplacian equation with non-negative weight functions in $\mathcal{H}^{α,β;χ}_{γ(ξ)}(Λ,\mathbb{R})$ using some variational techniques and Nehari manifold.

math.GM↗
arXiv2023

Existence, uniqueness and controllability for Hilfer differential equations on times scales

We introduce a new version of $ψ$-Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated and, in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a $ψ$-Riemann-Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for particular choices: (i) of the time scale; (ii) of the order of differentiation; and/or (iii) function $ψ$; opening new directions of investigation.

math.OC↗
arXiv2020

On Coupled System of Nonlinear $Ψ$-Hilfer Hybrid Fractional Differential Equations

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of $Ψ$-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of $Ψ$-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided concrete examples to exhibit the effectiveness of our achieved results.

math.DS↗
arXiv2019

Mild and classical solutions for fractional evolution differential equation

Investigating the existence, uniqueness, stability, continuous dependence of data among other properties of solutions of fractional differential equations, has been the object of study by an important range of researchers in the scientific community, especially in fractional calculus. And over the years, these properties have been investigated more vehemently, as they enable more general and new results. In this paper, we investigate the existence and uniqueness of a class of mild and classical solutions of the fractional evolution differential equation in the Banach space $Ω$. To obtain such results, we use fundamental tools, namely: Banach contraction theorem, Gronwall inequality and the $β$-times integrated $β$-times integrated $α$-resolvent operator function of an $(α,β)$-resolvent operator function.

math.CA↗
arXiv2019

On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations

In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $ε$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.

math.CA↗
arXiv2019

Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator

This article investigates the existence and uniqueness of solutions to the second order Volterra integrodifferential equations with nonlocal and boundary conditions through its integral equivalent equations and fixed point of Banach. Further, utilising the Picard operator theory we obtain the dependency of solutions on the initial nonlocal data and on functions involved on the right hand side of the equations.

math.CA↗
arXiv2019

On the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in $L^{N} \left( \mathbb{R} ^{N}\right) ^{N} $

In this paper, we present the result of maximum regularity of the mild solution of the fractional Cauchy problem. As our main result, we investigate the uniqueness of mild solutions for time-fractional Navier-Stokes equations in class $C\left([0,\infty);L^{N}\left( \mathbb{R}^{N}\right)^{N}\right)$ by means of the estimates $L^{p}-L^{q}$ of Giga-Shor inequality and the Gronwall inequality.

math.AP↗
arXiv2019

Ulam-Hyers stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem

Since the main work on Ulam-Hyers dependable stabilities of differential equations to date, numerous significant and applicable papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we will investigate the Ulam--Hyers and Ulam--Hyers--Rassias stabilities of mild solutions of the fractional nonlinear abstract Cauchy problem on the intervals $[0,T]$ and $[0,\infty)$, by means of Banach fixed point theorem.

math.CA↗
arXiv2019

Mild and strong solutions for Hilfer evolution equation

In this paper, we investigate the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations in the Hilfer sense, by means of Banach fixed point theorem and the Gronwall inequality.

math.CA↗
arXiv2019

On the Nonlinear Impulsive $Ψ$--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $Ψ$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $Ψ$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.

math.DS↗
arXiv2018

A new class of mild and strong solutions of integro-differential equation of arbitrary order in Banach space

The motivation that the field of differential equations provide to several researchers for the challenges that have been challenging them over the decades has contributed to the strengthening of the area within mathematics. In this sense, investigating important properties of solutions of differential equations, in particular fractional, has been object of study due to the exponential growth of the fractional calculus. In this paper, we investigate the existence and uniqueness of a new class of mild and strong solution of the fractional integro-differential equations in the Hilfer fractional derivative sense in Banach space, by means of the continuously $C_{0}$-semigroup, Banach fixed point theorem and Gronwall inequality.

math.CA↗