scholarly journals Some qualitative properties of nonlinear fractional integro-differential equations of variable order

2021 ◽  
Vol 11 (3) ◽  
pp. 68-78
Author(s):  
Ahmed Refice ◽  
Mohammed Said Souid ◽  
Ali Yakar
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Banach Fixed Point Theorem ◽  
Variable Order ◽  
Qualitative Properties ◽  
Piecewise Constant ◽  
Fractional Differential ◽  
The Stability ◽  
Uniqueness Criteria

The existence-uniqueness criteria of nonlinear fractional integro-differential equations of variable order with multiterm boundary value conditions are considered in this work. By utilizing the concepts of generalized intervals combined with the piecewise constant functions, we transform our problem into usual Caputo’s fractional differential equations of constant order. We develop the necessary criteria for assuring the solution's existence and uniqueness by applying Schauder and Banach fixed point theorem. We also examine the stability of the derived solution in the Ulam-Hyers-Rassias (UHR) sense and provide an example to demonstrate the credibility of the results.

scholarly journals The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu ◽  
Ravi P. Agarwal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Infinite Delay ◽  
Banach Fixed Point Theorem ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem inΩ={y:(−∞,b]→ℝ:y|(−∞,0]∈ℬ}such thaty|[0,b]is continuous andℬis a phase space.

scholarly journals Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jinhua Wang ◽  
Hongjun Xiang ◽  
Zhigang Liu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Positive Solution ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Coupled System ◽  
Banach Fixed Point Theorem ◽  
Boundary Values ◽  
Nonzero Boundary ◽  
Fractional Differential

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

scholarly journals Existence and Uniqueness of Solutions for the Cauchy-Type Problems of Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Chunhai Kou ◽  
Jian Liu ◽  
Yan Ye
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Uniqueness Of Solutions ◽  
Step Method ◽  
Fractional Differential

By using the Banach fixed point theorem and step method, we study the existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations. Meanwhile, by citing some counterexamples, it is pointed out that there exist a few defects in the proofs of the known results.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

scholarly journals On Fuzzy Extended Hexagonal b-Metric Spaces with Applications to Nonlinear Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2032
Author(s):  
Sumaiya Tasneem Zubair ◽  
Kalpana Gopalan ◽  
Thabet Abdeljawad ◽  
Bahaaeldin Abdalla
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Feasible Solution ◽  
Banach Fixed Point Theorem ◽  
Nonlinear Fractional Differential Equations ◽  
Research Article ◽  
Fractional Differential

The focus of this research article is to investigate the notion of fuzzy extended hexagonal b-metric spaces as a technique of broadening the fuzzy rectangular b-metric spaces and extended fuzzy rectangular b-metric spaces as well as to derive the Banach fixed point theorem and several novel fixed point theorems with certain contraction mappings. The analog of hexagonal inequality in fuzzy extended hexagonal b-metric spaces is specified as follows utilizing the function b(c,d): mhc,d,t+s+u+v+w≥mhc,e,tb(c,d)∗mhe,f,sb(c,d)∗mhf,g,ub(c,d)∗mhg,k,vb(c,d)∗mhk,d,wb(c,d) for all t,s,u,v,w>0 and c≠e,e≠f,f≠g,g≠k,k≠d. Further to that, this research attempts to provide a feasible solution for the Caputo type nonlinear fractional differential equations through effective applications of our results obtained.

scholarly journals Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type

2019 ◽  
Vol 52 (1) ◽  
pp. 283-295 ◽  
Author(s):  
Manzoor Ahmad ◽  
Akbar Zada ◽  
Jehad Alzabut
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Coupled System ◽  
Ulam Stability ◽  
Fractional Differential System ◽  
Different Types ◽  
Fractional Differential

AbstractIn this paper, existence and uniqueness of solution for a coupled impulsive Hilfer–Hadamard type fractional differential system are obtained by using Kransnoselskii’s fixed point theorem. Different types of Hyers–Ulam stability are also discussed.We provide an example demonstrating consistency to the theoretical findings.

scholarly journals Existence and Uniqueness of Solutions for Coupled Impulsive Fractional Pantograph Differential Equations with Antiperiodic Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Karim Guida ◽  
Lahcen Ibnelazyz ◽  
Khalid Hilal ◽  
Said Melliani
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
Banach’S Contraction Principle

In this paper, we investigate the solutions of coupled fractional pantograph differential equations with instantaneous impulses. The work improves some existing results and contributes toward the development of the fractional differential equation theory. We first provide some definitions that will be used throughout the paper; after that, we give the existence and uniqueness results that are based on Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Two examples are given in the last part to support our study.

scholarly journals Positive solutions for generalized two-term fractional differential equations with integral boundary conditions

2020 ◽  
Vol 1 (1) ◽  
pp. 47-63
Author(s):  
Hanan A. Wahash ◽  
Satish K. Panchal
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Upper And Lower Solutions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Fractional Differential

In this paper, we consider a class of boundary value problems for nonlinear two-term fractional differential equations with integral boundary conditions involving two $\psi $-Caputo fractional derivative. With the help of the properties Green function, the fixed point theorems of Schauder and Banach, and the method of upper and lower solutions, we derive the existence and uniqueness of positive solution of a proposed problem. Finally, an example is provided to illustrate the acquired results.

scholarly journals On the Theory of ψ-Hilfer Nonlocal Cauchy Problem

2021 ◽  
pp. 161-177
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Constant Coefficient ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Representation Formula ◽  
Qualitative Properties ◽  
Fractional Differential ◽  
Nonlocal Cauchy Problem

In this paper, we derive the representation formula of the solution for ψ-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard’s successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze Eα -Ulam-Hyers stability. Finally, one example to illustrate the obtained results

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