Hadamard-type Fractional Differential Equations Research Articles

Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section.

Analysis of p‐Laplacian Hadamard fractional boundary value problems with the derivative term involved in the nonlinear term

This paper investigates a new class of p‐Laplacian boundary value problems of Hadamard type fractional differential equations. We emphasize that few works on Hadamard fractional boundary value problems include p‐Laplacian Hadamard type fractional differential equations with nonlinear terms involving Hadamard fractional derivatives of unknown functions. By means of the five functional fixed point theorem, multiple positive solutions are established for the p‐Laplacian Hadamard fractional boundary value problem. An example is constructed for illustration of the main result.

Stability of positive solutions for a class of nonlinear Hadamard type fractional differential equations with $p$-Laplacian

Stability of positive solutions for a class of nonlinear Hadamard type fractional differential equations with $p$-Laplacian

An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation

In this paper, we are concerned with the eigenvalue problem of Hadamard-type singular fractional differential equations with multi-point boundary conditions. By constructing the upper and lower solutions of the eigenvalue problem and using the properties of the Green function, the eigenvalue interval of the problem is established via Schauder’s fixed point theorem. The main contribution of this work is on tackling the nonlinearity which possesses singularity on some space variables.

A coupled non-separated system of Hadamard-type fractional differential equations

In this article, we discuss the existence and uniqueness of solutions of a coupled non-separated system forfractional differential equations involving a Hadamard fractional derivative. The existence and uniquenessresults obtained in the present study are not only new but also cover some results corresponding to specialvalues of the parameters involved in the Caputo problems. These developed results are obtained by apply-ing Banach's fixed point theorem and Leray-Schauder's nonlinear alternative. An example is presented toillustrate our main results.

A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

On the Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order via Kuratowski MNC Technique

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.

New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

This paper is concerned with the existence of multiple positive solutions for a nonlinear Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. We first transform the given problem into an equivalent fixed point problem on cone, and then solve it by using the generalized Avery–Henderson fixed point theorem. Finally, an example is constructed to illustrate the application of the obtained results.

Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations.

Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval

We consider the existence of solutions for the following Hadamard-type fractional differential equations: {DαHu(t)+q(t)f(t,u(t),HDβ1u(t),HDβ2u(t))=0,1<t<+∞,u(1)=0,Dα−2Hu(1)=∫1+∞g1(s)u(s)dss,Dα−1Hu(+∞)=∫1+∞g2(s)u(s)dss,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} {}^{H}D^{\\alpha }u(t)+q(t)f(t,u(t), {}^{H}D^{\\beta _{1}}u(t),{}^{H}D^{ \\beta _{2}}u(t))=0,\\quad 1< t< +\\infty , \\\\ u(1)=0, \\\\ {}^{H}D^{\\alpha -2}u(1)=\\int ^{+\\infty }_{1}g_{1}(s)u(s)\\frac{ds}{s}, \\\\ {}^{H}D^{\\alpha -1}u(+\\infty )=\\int ^{+\\infty }_{1}g_{2}(s)u(s) \\frac{ds}{s}, \\end{cases} $$\\end{document} where 2<alpha leq 3, 0<beta _{1}leq alpha -2<beta _{2}leq alpha -1, f:J times mathbb{R}^{3}rightarrow mathbb{R} satisfies the q-Carathéodory condition, q,g_{1},g_{2}:Jrightarrow mathbb{R}^{+} are nonnegative, where J=[1,+infty ). Nonlinear term f is dependent on the fractional derivative of lower order beta _{1}, beta _{2}, which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with {}^{H}D^{beta _{2}}uin C(1,+infty ) at the left endpoint t=1 (if beta _{2}<alpha -1).

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACTThis paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.

Positive Solutions for a System of Hadamard-Type Fractional Differential Equations with Semipositone Nonlinearities

In this paper, we use the fixed-point index and nonnegative matrices to study the existence of positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities.

Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions

We establish the existence of positive solutions to a class of a singular nonlinear Hadamard-type fractional differential equations with infinite-point boundary conditions (BCs) or integral BCs. Our analysis is based on Leray–Schauder type continuation. Several examples are given to illustrate our results.

Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval

In this paper, we investigate the existence of solutions for several higher-order integral boundary value problems of Hadamard-type fractional differential equations on an infinite interval by using the monotone iterative technique and Mawhin’s continuation theorem. The results enrich and extend some known conclusions of Hadamard-type fractional boundary value problems. Moreover, we give two concrete examples to illustrate the theoretical results.

COMPARISON PRINCIPLES FOR HADAMARD-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

The aim of this paper is to establish the comparison principles for differential equations involving Hadamard-type fractional derivatives. First, the continuous dependence of solutions on the right-hand side functions of Hadamard-type fractional differential equations (HTFDEs) is proposed. Then, we state and prove the first and second comparison principles for HTFDEs, respectively. The corresponding examples are provided as well.

Nonlocal initial value problems for Hadamard-type fractional differential equations and inclusions

In this paper, we study initial value problems of fractional differential equations and inclusions of Hadamard type, supplemented with nonlocal conditions. Some new existence and uniqueness results are obtained by using fixed point theorems for single valued and multi-valued maps. Examples illustrating the main results are also presented.

Boundary Value Problems for Nonlinear Implicit Caputo–Hadamard-Type Fractional Differential Equations with Impulses

In this paper, the authors establish the existence and uniqueness of solutions for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and Caputo–Hadamard type fractional derivatives. The stability of this problem is also studied. The arguments are based on the Banach contraction principle and Schaefer’s fixed point theorem. Two examples are presented to show the applicability of the results.

Boundary Value Problems of Hadamard-Type Fractional Differential Equations and Inclusions with Nonlocal Conditions

In this paper, we obtain some new existence and uniqueness results for boundary value problems of Hadamard type fractional differential equations and inclusions with nonlocal boundary conditions. The contraction mapping principle and a fixed point theorem due to O’Regan are the main tools of our study for the single-valued case, while the multi-valued case is based on the nonlinear alternative for contractive maps. The results are well illustrated with the aid of examples.

ON HADAMARD FRACTIONAL CALCULUS

This paper is devoted to the investigation of the Hadamard fractional calculus in three aspects. First, we study the semigroup and reciprocal properties of the Hadamard-type fractional operators. Then, the definite conditions of certain class of Hadamard-type fractional differential equations (HTFDEs) are proposed through the Banach contraction mapping principle. Finally, we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of HTFDEs on the derivative order and the perturbation terms along with the proposed initial value conditions. The illustrative examples are presented as well.

Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems of Hadamard-type fractional functional differential equations and inclusions involving both retarded and advanced arguments. We make use of the standard tools of fixed point theory to obtain the main results.