scholarly journals Existence and Uniqueness of Uncertain Fractional Backward Difference Equations of Riemann–Liouville Type

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Fahd Jarad ◽  
Yu-Ming Chu
Keyword(s):  
Difference Equations ◽  
Existence And Uniqueness ◽  
Lipschitz Constant ◽  
Analytic Solutions ◽  
Fractional Operators ◽  
Existence And Uniqueness Theorem ◽  
Liouville Type ◽  
Successive Iteration ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

In this article, we consider the analytic solutions of the uncertain fractional backward difference equations in the sense of Riemann–Liouville fractional operators which are solved by using the Picard successive iteration method. Also, we consider the existence and uniqueness theorem of the solution to an uncertain fractional backward difference equation via the Banach contraction fixed-point theorem under the conditions of Lipschitz constant and linear combination growth. Finally, we point out some examples to confirm the validity of the existence and uniqueness of the solution.

scholarly journals A Generalized Uncertain Fractional Forward Difference Equations of Riemann-Liouville Type

2019 ◽  
Vol 11 (4) ◽  
pp. 43 ◽  
Author(s):  
Pshtiwan Othman Mohammed
Keyword(s):  
Uniqueness Theorem ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Analytic Solutions ◽  
Existence And Uniqueness Theorem ◽  
Contraction Mapping Theorem ◽  
Forward Difference ◽  
Banach Contraction ◽  
Definition Of ◽  
Banach Contraction Mapping

In this paper, we firstly recall the definition of an uncertain fractional forward difference equation with Riemann-Liouvillelike forward difference. After that analytic solutions to a generalized uncertain fractional difference equations are solved by using the Picard successive iteration method. Moreover, the existence and uniqueness theorem of the solutions are proved by applying Banach contraction mapping theorem. Finally, two examples are presented to illustrate the validity of the existence and uniqueness theorem.

scholarly journals On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 476
Author(s):  
Jiraporn Reunsumrit ◽  
Thanin Sitthiwirattham
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Akbar Zada
Keyword(s):  
Fractional Derivative ◽  
Positive Solutions ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Relaxation Equation ◽  
Monotone Iterative ◽  
Banach Contraction ◽  
Uniqueness Of The Solution ◽  
Fractional Relaxation ◽  
The Banach Contraction Principle

Abstract This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

scholarly journals Existence and Uniqueness Theorem for Stochastic Differential Equations with Self-Exciting Switching

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Guixin Hu ◽  
Ke Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Sufficient Condition ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of The Solution

We introduce a new kind of equation, stochastic differential equations with self-exciting switching. Firstly, we give some preliminaries for this kind of equation, and then, we get the main results of our paper; that is, we gave the sufficient condition which can guarantee the existence and uniqueness of the solution.

scholarly journals Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

2020 ◽  
Vol 8 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed
Keyword(s):  
Fractional Calculus ◽  
Uniqueness Theorem ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Comparison Theorems ◽  
Existence And Uniqueness Theorem ◽  
Discrete Fractional Calculus ◽  
Uncertain Variables ◽  
Forward Difference ◽  
The Relationship

We consider the comparison theorems for the fractional forward h-difference equations in the context of discrete fractional calculus. Moreover, we consider the existence and uniqueness theorem for the uncertain fractional forward h-difference equations. After that the relations between the solutions for the uncertain fractional forward h-difference equations with symmetrical uncertain variables and their α-paths are established and verified using the comparison theorems and existence and uniqueness theorem. Finally, two examples are provided to illustrate the relationship between the solutions.

scholarly journals Fixed Point Theorems for Mizoguchi-Takahashi Type Contraction in Bicomplex-Valued Metric Spaces and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fuli He ◽  
Z. Mostefaoui ◽  
M. Abdalla
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlinear Integral Equation ◽  
Metric Spaces ◽  
Banach Contraction ◽  
Uniqueness Of The Solution ◽  
Contractive Maps ◽  
The Banach Contraction Principle ◽  
Type Contraction

The main aim of this paper is to study and establish some new fixed point theorems for contractive maps that satisfied Mizoguchi-Takahashi’s condition in the setting of bicomplex-valued metric spaces. These new results improve and generalize the Banach contraction principle and some well-known results in the literature. Finally, as applications of our results, we give the existence and uniqueness of the solution of a nonlinear integral equation.

scholarly journals Ulam Stability of n-th Order Delay Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3029
Author(s):  
Shuyi Wang ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Mathematical Induction ◽  
Ulam Stability ◽  
Existence And Uniqueness Theorem ◽  
Integro Differential Equation ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

A Global Existence and Uniqueness Theorem for Ordinary Differential Equations

1976 ◽  
Vol 19 (1) ◽  
pp. 105-107 ◽  
Author(s):  
W. Derrick ◽  
L. Janos
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Contraction Principle ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.

Sign in / Sign up

Export Citation Format

Share Document