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.

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 Existence of Three Positive Solutions form-Point Discrete Boundary Value Problems withp-Laplacian

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Yanping Guo ◽  
Wenying Wei ◽  
Yuerong Chen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Laplacian Operator ◽  
One Dimensional

We consider the multi-point discrete boundary value problem with one-dimensionalp-Laplacian operatorΔ(ϕp(Δu(t−1))+q(t)f(t,u(t),Δu(t))=0,t∈{1,…,n−1}subject to the boundary conditions:u(0)=0,u(n)=∑i=1m−2aiu(ξi), whereϕp(s)=|s|p−2s,p>1,ξi∈{2,…,n−2}with1<ξ1<⋯<ξm−2<n−1andai∈(0,1),0<∑i=1m−2ai<1. Using a new fixed point theorem due to Avery and Peterson, we study the existence of at least three positive solutions to the above boundary value problem.

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.

scholarly journals Existence of Triple Positive Solutions for Second-Order Discrete Boundary Value Problems

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Yanping Guo ◽  
Jiehua Zhang ◽  
Yude Ji
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Triple Positive Solutions

By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equationΔ2x(k−1)+q(k)f(k,x(k),Δx(k))=0, fork∈{1,2,…,n−1}, subject to the following two boundary conditions:x(0)=x(n)=0orx(0)=Δx(n−1)=0, wheren≥3.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

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 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 Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Nichaphat Patanarapeelert ◽  
Thanin Sitthiwirattham
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Integral Boundary ◽  
Uniqueness Results ◽  
Fractional Hahn Integral

The existence and uniqueness results of two fractional Hahn difference boundary value problems are studied. The first problem is a Riemann-Liouville fractional Hahn difference boundary value problem for fractional Hahn integrodifference equations. The second is a fractional Hahn integral boundary value problem for Caputo fractional Hahn difference equations. The Banach fixed-point theorem and the Schauder fixed-point theorem are used as tools to prove the existence and uniqueness of solution of the problems.

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