scholarly journals Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Thanin Sitthiwirattham ◽  
Jessada Tariboon
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Order Difference ◽  
Existence Of Positive Solutions ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Fixed Positive Integer

By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problemΔ2u(t-1)+a(t)f(u(t))=0,t∈{1,2,…,T},u(0)=β∑s=1ηu(s),u(T+1)=α∑s=1ηu(s), wherefis continuous,T≥3is a fixed positive integer,η∈{1,2,...,T-1},0<α<(2T+2)/η(η+1),0<β<(2T+2-αη(η+1))/η(2T-η+1),andΔu(t-1)=u(t)-u(t-1). We show the existence of at least one positive solution iffis either superlinear or sublinear.

scholarly journals Existence of positive solutions for a semipositone discrete boundary value problem

2019 ◽  
Vol 24 (4) ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Boundary ◽  
Order Difference ◽  
Existence Of Positive Solutions ◽  
Second Order Difference Equation ◽  
Order Difference Equation

We investigate the existence of positive solutions for a nonlinear second-order difference equation with a linear term and a sign-changing nonlinearity, supplemented with multi-point boundary conditions. In the proof of our main results, we use the Guo–Krasnosel'skii fixed point theorem.  

scholarly journals Positive Solutions of a Fractional Boundary Value Problem with Changing Sign Nonlinearity

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yongqing Wang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Fractional Boundary Value Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

We discuss the existence of positive solutions of a boundary value problem of nonlinear fractional differential equation with changing sign nonlinearity. We first derive some properties of the associated Green function and then obtain some results on the existence of positive solutions by means of the Krasnoselskii's fixed point theorem in a cone.

scholarly journals Existence of Positive Solutions for Fourth-Order Boundary Value Problems with Sign-Changing Nonlinear Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Xingfang Feng ◽  
Hanying Feng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions

the existence of positive solutions for a fourth-order boundary value problem with a sign-changing nonlinear term is investigated. By using Krasnoselskii’s fixed point theorem, sufficient conditions that guarantee the existence of at least one positive solution are obtained. An example is presented to illustrate the application of our main results.

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 Multiple Positive Solutions of Nonlinear Boundary Value Problem for Finite Fractional Difference

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Yansheng He ◽  
Mingzhe Sun ◽  
Chengmin Hou
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Value Problem ◽  
Multiple Positive Solutions ◽  
Order Difference ◽  
Nonlinear Boundary Value ◽  
Order Difference Equation

We consider a discrete fractional nonlinear boundary value problem in which nonlinear termfis involved with the fractional order difference. And we transform the fractional boundary value problem into boundary value problem of integer order difference equation. By using a generalization of Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions.

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