Discrete fractional boundary value problems and inequalities

2021 ◽  
Vol 24 (6) ◽  
pp. 1777-1796
Author(s):  
Martin Bohner ◽  
Nick Fewster-Young
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Integral Representation ◽  
Existence Theorem ◽  
Boundary Value ◽  
A Priori ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

Abstract In this paper, a general nonlinear discrete fractional boundary value problem is considered, of order between one and two. The main result is an existence theorem, proving the existence of at least one solution to the boundary value problem, subject to validity of a certain key inequality that allows unrestricted growth in the problem. The proof of this existence theorem is accomplished by using Brouwer's fixed point theorem as well as two other main results of this paper, namely, first, a result showing that the solutions of the boundary value problem are exactly the solutions to a certain equivalent integral representation, and, second, the establishment of solutions satisfying certain a priori bounds provided the key inequality holds. In order to establish the latter result, several novel discrete fractional inequalities are developed, each of them interesting in itself and of potential future use in different contexts. We illustrate the usefulness of our existence results by presenting two examples.

scholarly journals Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjie Liu ◽  
Xiao Fu ◽  
Liangping Qi
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

scholarly journals Blow-Up Solutions for a Singular Nonlinear Hadamard Fractional Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Imed Bachar ◽  
Said Mesloub
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Blow Up ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Fractional Boundary Value Problems

We consider singular nonlinear Hadamard fractional boundary value problems. Using properties of Green’s function and a fixed point theorem, we show that the problem has positive solutions which blow up. Finally, some examples are provided to explain the applications of the results.

scholarly journals Existence Results for a Fully Fourth-Order Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yongxiang Li ◽  
Qiuyan Liang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fourier Analysis ◽  
Growth Condition ◽  
Fourth Order ◽  
Boundary Value ◽  
Existence Of Solution ◽  
Existence Results ◽  
Analysis Method

We discuss the existence of solution for the fully fourth-order boundary value problemu(4)=f(t,u,u′,u′′,u′′′),0≤t≤1,u(0)=u(1)=u′′(0)=u′′(1)=0. A growth condition onfguaranteeing the existence of solution is presented. The discussion is based on the Fourier analysis method and Leray-Schauder fixed point theorem.

scholarly journals Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Thanin Sitthiwirattham ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Difference Equations ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Fractional Difference ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Nonlinear Alternative

We consider a discrete fractional boundary value problem of the formΔαu(t)=f(t+α-1,u(t+α-1)),  t∈[0,T]ℕ0:={0,1,…,T},  u(α-2)=0,  u(α+T)=Δ-βu(η+β),where1<α≤2,β>0,η∈[α-2,α+T-1]ℕα-2:={α-2,α-1,…,α+T-1}, andf:[α-1,α,…,α+T-1]ℕα-1×ℝ→ℝis a continuous function. The existence of at least one solution is proved by using Krasnoselskii's fixed point theorem and Leray-Schauder's nonlinear alternative. Some illustrative examples are also presented.

scholarly journals The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yuanyuan Pan ◽  
Zhenlai Han ◽  
Shurong Sun ◽  
Yige Zhao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Continuous Functions ◽  
Fractional Boundary Value Problems ◽  
Schäuder Fixed Point Theorem

We study the existence of solutions for the boundary value problem-Δνy1(t)=f(y1(t+ν-1),y2(t+μ-1)),-Δμy2(t)=g(y1(t+ν-1),y2(t+μ-1)),y1(ν-2)=Δy1(ν+b)=0,y2(μ-2)=Δy2(μ+b)=0, where1<μ,ν≤2,f,g:R×R→Rare continuous functions,b∈N0. The existence of solutions to this problem is established by the Guo-Krasnosel'kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.

scholarly journals Some New Existence Results of Positive Solutions to an Even-Order Boundary Value Problem on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yanbin Sang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Existence Results ◽  
Point Boundary ◽  
Even Order

We consider a high-order three-point boundary value problem. Firstly, some new existence results of at least one positive solution for a noneigenvalue problem and an eigenvalue problem are established. Our approach is based on the application of three different fixed point theorems, which have extended and improved the famous Guo-Krasnosel’skii fixed point theorem at different aspects. Secondly, some examples are included to illustrate our results.

scholarly journals Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential Problems Using Dhage’s Approach

2021 ◽  
Vol 6 (1) ◽  
pp. 17
Author(s):  
Muhammad Yaseen ◽  
Sadia Mumtaz ◽  
Reny George ◽  
Azhar Hussain
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Differential

In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve these goals, we utilize the well-known fixed point theorems attributed to Dhage for both BVPs. Moreover, we present two numerical examples to validate our analytical findings.

scholarly journals Positive Solutions for Boundary Value Problem of Nonlinear Fractional -Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Farah M. Al-Askar
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Fractional Difference ◽  
Fractional Difference Equation

We investigate the existence of multiple positive solutions to the nonlinear -fractional boundary value problem , , by using a fixed point theorem in a cone.

scholarly journals Multiple Positive Solutions for a Fractional Boundary Value Problem with Fractional Integral Deviating Argument

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
A. Guezane-Lakoud ◽  
R. Khaldi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Fractional Integral ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Fractional Boundary Value Problem ◽  
Deviating Argument

This work is devoted to the existence of positive solutions for a fractional boundary value problem with fractional integral deviating argument. The proofs of the main results are based on Guo-Krasnoselskii fixed point theorem and Avery and Peterson fixed point theorem. Two examples are given to illustrate the obtained results, ending the paper.

scholarly journals On solutions of a class of three-point fractional boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zhanbing Bai ◽  
Yu Cheng ◽  
Sujing Sun
Keyword(s):  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Existence Results ◽  
Conformable Fractional Derivative ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Alternative ◽  
Fractional Boundary Value Problems

AbstractExistence results for the three-point fractional boundary value problem $$\begin{aligned}& D^{\alpha}x(t)= f \bigl(t, x(t), D^{\alpha-1} x(t) \bigr),\quad 0< t< 1, \\& x(0)=A, \qquad x(\eta)-x(1)=(\eta-1)B, \end{aligned}$$ Dαx(t)=f(t,x(t),Dα−1x(t)),0<t<1,x(0)=A,x(η)−x(1)=(η−1)B, are presented, where $A, B\in\mathbb{R}$A,B∈R, $0<\eta<1$0<η<1, $1<\alpha\leq2$1<α≤2. $D^{\alpha}x(t)$Dαx(t) is the conformable fractional derivative, and $f: [0, 1]\times\mathbb{R}^{2}\to\mathbb{R}$f:[0,1]×R2→R is continuous. The analysis is based on the nonlinear alternative of Leray–Schauder.

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