scholarly journals Two Positive Solutions of Third-Order BVP with Integral Boundary Condition and Sign-Changing Green's Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Bing-Wei Niu ◽  
Jian-Ping Sun ◽  
Qiu-Yan Ren
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Integral Boundary Condition ◽  
Third Order ◽  
Integral Boundary

We are concerned with the following third-order boundary value problem with integral boundary condition:  u′′′(t)=f(t,u(t)),  t∈[0,1],  u′(0)=u(1)=0,  u′′(η)+∫αβ‍u(t)dt=0,where1/2<α≤β≤1,  α+β≤4/3, andη∈(1/2,α]. Although the corresponding Green's function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions onfby using the two-fixed-point theorem due to Avery and Henderson. An example is also included to illustrate the main results obtained.

scholarly journals Positive Solutions of a Third-Order Three-Point BVP with Sign-Changing Green’s Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Li-Juan Gao ◽  
Jian-Ping Sun
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Point Theorem ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order

We are concerned with the following third-order three-point boundary value problem:u′′′t=ft, ut,   t∈0, 1,   u′0=u1=0and u′′η-αu′1=0,whereα∈0, 1andη∈(14+α)/(24-3α),1. Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions onfby using the two-fixed-point theorem due to Avery and Henderson.

scholarly journals Multiple Positive Solutions of Third-Order BVP with Advanced Arguments and Stieltjes Integral Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Jian Chang ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Integral Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Stieltjes Integral ◽  
Integral Conditions ◽  
Third Order ◽  
Integral Boundary

We consider the following third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions:u′′′t+ft,uαt=0,  t∈0,1,  u0=γuη1+λ1uandu′′0=0,  u1=βuη2+λ2u, where0<η1<η2<1,0≤γ,β≤1,α:[0,1]→[0,1]is continuous,α(t)≥tfort∈[0,1], andα(t)≤η2fort∈[η1,η2]. Under some suitable conditions, by applying a fixed point theorem due to Avery and Peterson, we obtain the existence of multiple positive solutions to the above problem. An example is also included to illustrate the main results obtained.

scholarly journals POSITIVE SOLUTIONS OF NTH-ORDER BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS

2015 ◽  
Vol 20 (2) ◽  
pp. 188-204 ◽  
Author(s):  
Ilkay Yaslan Karaca ◽  
Fatma Tokmak Fen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary

In this paper, by using double fixed point theorem and a new fixed point theorem, some sufficient conditions for the existence of at least two and at least three positive solutions of an nth-order boundary value problem with integral boundary conditions are established, respectively. We also give two examples to illustrate our main results.

scholarly journals Existence of Three Positive Solutions for a Class of Boundary Value Problems of Caputo Fractional q-Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Huiqin Chen ◽  
Shugui Kang ◽  
Lili Kong ◽  
Ying Gao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Green's Function ◽  
Point Theorem ◽  
Boundary Value ◽  
Existence Criteria

A class of boundary value problems of Caputo fractional q-difference equation is introduced. Green’s function and its properties for this problem are deduced. By applying these properties and the Leggett-Williams fixed-point theorem, existence criteria of three positive solutions are obtained. At last, some examples are given to illustrate the validity of our main results.

scholarly journals Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Luchao Zhang ◽  
Weiguo Zhang ◽  
Xiping Liu ◽  
Mei Jia
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Existence Of Positive Solutions ◽  
Two Parameters ◽  
Laplacian Equations

AbstractWe consider a class of Caputo fractional p-Laplacian differential equations with integral boundary conditions which involve two parameters. By using the Avery–Peterson fixed point theorem, we obtain the existence of positive solutions for the boundary value problem. As an application, we present an example to illustrate our main result.

Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions

2018 ◽  
Vol 21 (3) ◽  
pp. 716-745 ◽  
Author(s):  
Seshadev Padhi ◽  
John R. Graef ◽  
Smita Pati
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Integral Boundary ◽  
Existence Of Positive Solutions

Abstract In this paper, we study the existence of positive solutions to the fractional boundary value problem $$\begin{array}{} \displaystyle D^{\alpha }_{0+}x(t)+q(t)f(t,x(t))=0, \,\, 0\lt t \lt1, \end{array}$$ together with the boundary conditions $$\begin{array}{} \displaystyle x(0)=x^{\prime}(0)= \cdots = x^{(n-2)}(0)=0, D_{0+}^{\beta }x(1)= \int^{1}_{0}h(s,x(s))\,dA(s), \end{array}$$ where n > 2, n – 1 < α ≤ n, β ∈ [1,α – 1], and $\begin{array}{} \displaystyle D^{\alpha }_{0+} \end{array}$ and $\begin{array}{} \displaystyle D^{\beta }_{0+} \end{array}$ are the standard Riemann-Liouville fractional derivatives of order α and β, respectively. We consider two different cases: f, h : [0, 1] × R → R, and f, h : [0, 1] × [0, ∞) → [0, ∞). In the first case, we prove the existence and uniqueness of the solutions of the above problem, and in the second case, we obtain sufficient conditions for the existence of positive solutions of the above problem. We apply a number of different techniques to obtain our results including Schauder’s fixed point theorem, the Leray-Schauder alternative, Krasnosel’skii’s cone expansion and compression theorem, and the Avery-Peterson fixed point theorem. The generality of the Riemann-Stieltjes boundary condition includes many problems studied in the literature. Examples are included to illustrate our findings.

scholarly journals New results for higher-order Hadamard-type fractional differential equations on the half-line

2021 ◽  
Author(s):  
Tugba Senlik Cerdik ◽  
Fulya Yoruk Deren
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Fractional Integral ◽  
Boundary Value ◽  
Integral Boundary Condition ◽  
Fractional Boundary Value Problem ◽  
Integral Boundary ◽  
Multiplicity Results ◽  
Fractional Differential

The purpose of this paper is to analyze a new kind of Hadamard fractional boundary value problem combining integral boundary condition and multipoint fractional integral boundary condition on an infinite interval. By the help of the Bai-Ge’s fixed point theorem, multiplicity results of positive solutions are derived for the Hadamard fractional boundary value problem. In the end, to illustrative the main result, an example is also presented.

Positive solutions for fractional differential equations with non-separated type nonlocal multi-point and multi-term integral boundary conditions

2021 ◽  
Vol 66 (4) ◽  
pp. 691-708
Author(s):  
Habib Djourdem ◽  
◽  
Slimane Benaicha ◽  
Keyword(s):  
Boundary Condition ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Positive Solutions ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Fractional Differential

In this paper, we investigate a class of nonlinear fractional differential equations that contain both the multi-term fractional integral boundary condition and the multi-point boundary condition. By the Krasnoselskii fixed point theorem we obtain the existence of at least one positive solution. Then, we obtain the existence of at least three positive solutions by the Legget-Williams fixed point theorem. Two examples are given to illustrate our main results.

scholarly journals Existence of Positive Solutions for Nonlinear Third-Order Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Tiaoxia Dun ◽  
Pengyu Chen
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Boundary Value ◽  
Index Theorem ◽  
Third Order ◽  
Existence Of Positive Solutions ◽  
Fixed Point Index Theorem ◽  
Linear Boundary Value Problem

We are concerned with the existence of positive solutions for the nonlinear third-order three-point boundary value problemu′′′(t)+λg(t)f(u(t))=0,0<t<1,u(0)=αu(η),u′(0)=u′′(1)=0, where0<η<1,0<α<1,λis a positive parameter,g:(0,1)→[0,∞),  and  f:[0,∞)→[0,∞)is continuous. We construct Green’s function for the associated linear boundary value problem and obtain some useful properties of Green’s function. Finally, by using fixed-point index theorem in cones, we establish the existence results of positive solutions for the boundary value problem an example illustrates the application of the results obtained.

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