scholarly journals Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions

2022 ◽  
Vol 5 (1) ◽  
pp. 12-28
Author(s):  
Saleh REDHWAN ◽  
Sadikali SHAİKH ◽  
Mohammed ABDO
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Fractional Differential

Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Uniqueness Theorems ◽  
Parametric Functions ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

AbstractIn the present manuscript we analyze non-linear multi-order fractional differential equation $$L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), t \in \left[ {0,T} \right], T > 0,$$ where $$L\left( D \right) = \lambda _n ^c D^{\alpha _n } + \lambda _{n - 1} ^c D^{\alpha _{n - 1} } + \cdots + \lambda _1 ^c D^{\alpha _1 } + \lambda _0 ^c D^{\alpha _0 } ,\lambda _i \in \mathbb{R}\left( {i = 0,1, \cdots ,n} \right),\lambda _n \ne 0, 0 \leqslant \alpha _0 < \alpha _1 < \cdots < \alpha _{n - 1} < \alpha _n < 1,$$ and c D α denotes the Caputo fractional derivative of order α. We find the Greens functions for this equation corresponding to periodic/anti-periodic boundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for these fractional boundary value problems.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Positive Solutions for BVP of Fractional Differential Equation with Integral Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Min Li ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Integral Boundary Conditions ◽  
Monotone Iterative Method ◽  
Nonlinear Fractional Differential Equation ◽  
Integral Boundary ◽  
Zero Function ◽  
Fractional Differential

In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.

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