scholarly journals Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

scholarly journals Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 630
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Fractional Differential Inclusions ◽  
Nonlinear Alternative ◽  
Contractive Map ◽  
Fractional Differential ◽  
Definition Of

In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results.

scholarly journals Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Manzoor Ahmad ◽  
Jiqiang Jiang ◽  
Akbar Zada ◽  
Syed Omar Shah ◽  
Jiafa Xu
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Coupled System ◽  
Uniqueness Of Solutions ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

scholarly journals Nonexistence Results for Some Classes of Nonlinear Fractional Differential Inequalities

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Differential Inequalities ◽  
Test Function ◽  
Fractional Differential ◽  
Test Function Method

We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

scholarly journals Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 322
Author(s):  
Ricardo Almeida ◽  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Lyapunov Functions ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Hopfield Neural Network ◽  
Quadratic Lyapunov Functions ◽  
Fractional Differential ◽  
The Stability ◽  
Theoretical Results

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

scholarly journals Nonexistence of Global Solutions of Systems of Time Fractional Differential equations posed on the Heisenberg group

2021 ◽  
Author(s):  
Mokhtar Kirane ◽  
alrazi abdeljabbar
Keyword(s):  
Heisenberg Group ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Positive Real ◽  
Nonlinear Capacity ◽  
Linear Damping ◽  
Fractional Differential ◽  
Time Fractional Differential Equations ◽  
Capacity Method

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.

Boundary Value Problems for Fractional Differential Equations

2009 ◽  
Vol 16 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Mouffak Benchohra ◽  
Samira Hamani
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Fractional Differential

Abstract The sufficient conditions are established for the existence of solutions for a class of boundary value problems for fractional differential equations involving the Caputo fractional derivative.

scholarly journals Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation

2016 ◽  
Vol 12 (11) ◽  
pp. 6807-6811
Author(s):  
Haribhau Laxman Tidke
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Initial Condition ◽  
Fractional Differential

We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.

scholarly journals A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehboob Alam ◽  
Akbar Zada ◽  
Ioan-Lucian Popa ◽  
Alireza Kheiryan ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Integral Boundary Conditions ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential

AbstractIn this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

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