scholarly journals Positive solutions for eigenvalue problems of fractional q-difference equation with ϕ-Laplacian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jufang Wang ◽  
Changlong Yu ◽  
Boya Zhang ◽  
Si Wang
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Difference Equation ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Laplacian Operator ◽  
Existence And Nonexistence ◽  
Krasnoselskii Fixed Point Theorem ◽  
Eigenvalue Intervals ◽  
The Green Function

AbstractThe aim of this paper is to investigate the boundary value problem of a fractional q-difference equation with ϕ-Laplacian, where ϕ-Laplacian is a generalized p-Laplacian operator. We obtain the existence and nonexistence of positive solutions in terms of different eigenvalue intervals for this problem by means of the Green function and Guo–Krasnoselskii fixed point theorem on cones. Finally, we give some examples to illustrate the use of our results.

Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator

2020 ◽  
Vol 70 (1) ◽  
pp. 107-124
Author(s):  
Wengui Yang
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Eigenvalue Problems ◽  
Laplacian Operator ◽  
Hadamard Fractional Differential Equations ◽  
Existence And Nonexistence ◽  
Fractional Differential ◽  
Eigenvalue Intervals ◽  
The Green Function

AbstractThis paper is concerned with the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p-Laplacian operator. By applying the properties of the Green function and Guo-Krasnosel’skii fixed point theorem on cones, some existence and nonexistence results of positive solutions are obtained based on different eigenvalue intervals. Finally, some examples are presented to demonstrate the feasibility of our main results.

scholarly journals Existence of Positive Solutions for m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Moustafa El-Shahed ◽  
Wafa M. Shammakh
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential ◽  
The Green Function

We investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

scholarly journals Existence and Nonexistence of Positive Solutions for Mixed Fractional Boundary Value Problem with Parameter and p-Laplacian Operator

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Ying Wang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Fractional Boundary Value Problem ◽  
Existence And Nonexistence

This paper mainly studies a class of mixed fractional boundary value problem with parameter and p-Laplacian operator. Based on the Guo-Krasnosel’skii fixed point theorem, results on the existence and nonexistence of positive solutions for the fractional boundary value problem are established. An example is then presented to illustrate the effectiveness of the results.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function

2021 ◽  
Vol 19 (1) ◽  
pp. 990-1006
Author(s):  
Xueqin Cao ◽  
Chenghua Gao ◽  
Duihua Duan
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Existence Results ◽  
Point Boundary ◽  
Third Order ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
The Green Function

Abstract In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

scholarly journals Positive Solutions of a Fractional Thermostat Model with a Parameter

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 122 ◽  
Author(s):  
Xinan Hao ◽  
Luyao Zhang
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
Uniqueness Results ◽  
Fixed Point Index Theory ◽  
The Green Function ◽  
Thermostat Model

We study the existence, multiplicity, and uniqueness results of positive solutions for a fractional thermostat model. Our approach depends on the fixed point index theory, iterative method, and nonsymmetry property of the Green function. The properties of positive solutions depending on a parameter are also discussed.

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 Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

2018 ◽  
Vol 23 (5) ◽  
pp. 771-801 ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Existence Results ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

>We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

scholarly journals The Existence of Positive Solutions for Fractional Differential Equations with Sign Changing Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Weihua Jiang ◽  
Jiqing Qiu ◽  
Weiwei Guo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Generalized Boundary Conditions

We investigate the existence of at least two positive solutions to eigenvalue problems of fractional differential equations with sign changing nonlinearities in more generalized boundary conditions. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. Some examples are given for the illustration of main results.

ESTIMATES ON THE GREEN FUNCTION AND EXISTENCE OF POSITIVE SOLUTIONS FOR SOME NONLINEAR POLYHARMONIC PROBLEMS OUTSIDE THE UNIT BALL

2008 ◽  
Vol 06 (02) ◽  
pp. 121-150 ◽  
Author(s):  
IMED BACHAR ◽  
HABIB MÂAGLI ◽  
NOUREDDINE ZEDDINI
Keyword(s):  
Green Function ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Closed Ball ◽  
Dirichlet Boundary Conditions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
The Green Function ◽  
Class Of Functions

Let [Formula: see text] be the Green function of (-Δ)m, m ≥ 1, on the complementary D of the unit closed ball in ℝn, n ≥ 2, with Dirichlet boundary conditions [Formula: see text], 0 ≤ j ≤ m - 1. We establish some estimates on [Formula: see text] including the 3G-Inequality given by (1.3). Next, we introduce a polyharmonic Kato class of functions [Formula: see text] and we exploit the properties of this class to study the existence of positive solutions of some polyharmonic nonlinear elliptic problems.

scholarly journals Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Hua Luo ◽  
Chenghua Gao
Keyword(s):  
Fixed Point ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Solution Method ◽  
Lower Solution ◽  
Existence Results ◽  
Fourth Order Eigenvalue Problem ◽  
Existence And Nonexistence ◽  
Lower Solution Method

LetTbe a time scale anda,b∈T,a<ρ2(b). We study the nonlinear fourth-order eigenvalue problem onT,uΔ4(t)=λh(t)f(u(t),uΔ2(t)),t∈[a,ρ2(b)]T,u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0and obtain the existence and nonexistence of positive solutions when0<λ≤λ*andλ>λ*, respectively, for someλ*. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

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