On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

2021 ◽  
pp. 2250011
Author(s):  
Zouaoui Bekri ◽  
Vedat Suat Erturk ◽  
Pushpendra Kumar
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Solution Existence ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Research Collection ◽  
The Given

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional [Formula: see text]-difference equation having the given form [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represents the Caputo-type nonclassical [Formula: see text]-derivative of order [Formula: see text]. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

scholarly journals The Existence of Solutions for Boundary Value Problem of Fractional Functional Differential Equations with Delay

2018 ◽  
Vol 228 ◽  
pp. 01005
Author(s):  
Mengrui Xu ◽  
Yanan Li ◽  
Yige Zhao ◽  
Shurong Sun
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Contraction Mapping Principle ◽  
Alternative Theorem ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Functional Differential ◽  
Fractional Functional Differential Equations ◽  
Fractional Functional

A class of boundary value problem for fractional functional differential equation with delay $ \left\{ {\begin{array}{*{20}c} {^{C} D^{\sigma } \omega (t) = f(t,\omega _{t} ),t \in [0,\zeta ],} \\ {\omega (0) = 0,\,\omega ^{\prime}(0) = 0,\,\omega ^{\prime\prime}(\zeta ) = 1,} \\ \end{array} } \right. $ is studied, where $ 2 < \sigma \le 3,\,\,^{c} D^{\sigma } $ devote standard Caputo fractional derivative. In this article, three new criteria on existence and uniqueness of solution are obtained by Banach contraction mapping principle, Schauder fixed point theorem and nonlinear alternative theorem.

scholarly journals Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Cheng-Min Su ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Fractional Differential

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: D0+qCut=ft,ut,  t∈0,1, u0=u′′0=0,  D0+σ1Cu1=λI0+σ2u1, where 2<q<3, 0<σ1≤1, σ2>0, and λ≠Γ2+σ2/Γ2-σ1. The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

scholarly journals On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdelatif Boutiara ◽  
Sina Etemad ◽  
Jehad Alzabut ◽  
Azhar Hussain ◽  
Muthaiah Subramanian ◽  
...  
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Operators ◽  
Stability Criteria ◽  
Uniqueness Of Solutions ◽  
Fractional Quantum ◽  
Condensing Operators

AbstractIn this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

scholarly journals Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Assia Guezane-Lakoud ◽  
Hamidane Nacira ◽  
Khaldi Rabah
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Integral Condition ◽  
Second Order ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Banach Contraction ◽  
Positivity Of Solutions ◽  
Uniqueness And Existence

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results.

scholarly journals Monotone-Iterative Method for Solving Antiperiodic Nonlinear Boundary Value Problems for Generalized Delay Difference Equations with Maxima

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Angel Golev ◽  
Snezhana Hristova ◽  
Svetoslav Nenov
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Unknown Function ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problem ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Nonlinear Boundary Value ◽  
The Given

A nonlinear generalized difference equation with both delays and the maximum value of the unknown function over a discrete past time interval are studied. A nonlinear boundary value problem of antiperiodic type for the given difference equation is set up. One of the main characteristics of the considered difference equation is the presence of the unknown function in both sides of the equation. It leads to impossibility for using the step method for explicit solving of the nonlinear difference equation. In this paper, an approximate method, namely, the monotone iterative technique, is applied to solve the problem. An important feature of the given algorithm is that each successive approximation of the unknown solution is equal to the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and an algorithm for its explicit solving is given. Also, each approximation is a lower/upper solution of the given nonlinear boundary value problem. The suggested scheme for approximate solving is computer realized, and it is applied to a particular example, which is a generalization of a model in population dynamics.

scholarly journals Positive Solution to a Fractional Boundary Value Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
A. Guezane-Lakoud ◽  
R. Khaldi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Positivity Of Solutions ◽  
The Green Function

A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel'skii fixed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.

scholarly journals Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Weidong Lv
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Equation Boundary

By means of Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness of solutions to a boundary value problem for a discrete fractional mixed type sum-difference equation with the nonlinear term dependent on a fractional difference of lower order. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded and two representative examples are presented to illustrate the effectiveness of the main results.

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