scholarly journals Integral Inequalities and Differential Equations via Fractional Calculus

2020 ◽  
Author(s):  
Zoubir Dahmani ◽  
Meriem Mansouria Belhamiti
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Inequalities

Stability of fractional neutral stochastic partial integro-differential equations

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Liping Xu ◽  
Zhi Li
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Hilbert Spaces ◽  
Stochastic Analysis ◽  
Resolvent Operator ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Integral Inequalities

AbstractIn this paper, we are concerned with a class of fractional partial neutral stochastic integro-differential equations in Hilbert spaces. We assume that the linear part of this equation generates an α-resolvent operator and transform it into an integral equation. By the stochastic analysis and fractional calculus technique, and combining some integral inequalities, we obtain some sufficient conditions ensuring the exponential

scholarly journals The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities

2019 ◽  
Vol 52 (1) ◽  
pp. 204-212 ◽  
Author(s):  
Fuat Usta ◽  
Mehmet Zeki Sarıkaya
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Integral Inequalities ◽  
Fractional Operator ◽  
Van Der Pol ◽  
Global Existence Of Solutions ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Conformable Fractional Integral

AbstractIn this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

scholarly journals Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1503 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Artion Kashuri
Keyword(s):  
Fractional Calculus ◽  
Convex Function ◽  
Symmetric Function ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Important Direction ◽  
Increasing Function

There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities.

Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003
Author(s):  
Xiao Ma ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Stochastic Differential Equations ◽  
Periodic Solutions ◽  
Infinite Delay ◽  
Almost Periodic Solutions ◽  
Almost Periodic

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results.

scholarly journals The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Baojun Miao ◽  
Xuechen Li
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Asymptotic Estimates ◽  
Stability Of Solutions ◽  
Neutral Delay ◽  
Estimates Of Solutions ◽  
Fractional Delay ◽  
Asymptotic Behaviours

By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Power Law ◽  
Memory Function ◽  
Time Interval ◽  
Infinite Time ◽  
Suggested Approach ◽  
Infinite Time Interval ◽  
History Of ◽  
With Memory

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

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