scholarly journals Application of Fixed-Point Theory for a Nonlinear Fractional Three-Point Boundary-Value Problem

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 526
Author(s):  
Ehsan Pourhadi ◽  
Reza Saadati ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Point Theorem ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Point Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions

Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term of the first order derivative ( a C D α x ) ( t ) = f ( t , x ( t ) , x ′ ( t ) ) , a < t < b , 1 < α < 2 , x ( a ) = 0 , x ( b ) = μ x ( η ) , a < η < b , μ > λ , where λ = b − a η − a and a C D α denotes the Caputo’s fractional derivative, and f : [ a , b ] × R × R → R is a continuous function satisfying the certain conditions.

scholarly journals Existence of Positive Solutions form-Point Boundary Value Problems on Time Scales

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Zhang ◽  
ShiDong Qiao
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Time Scales ◽  
Point Theorem ◽  
Boundary Value ◽  
Point Boundary ◽  
One Dimensional ◽  
Existence Of Positive Solutions

We study the one-dimensionalp-Laplacianm-point boundary value problem(φp(uΔ(t)))Δ+a(t)f(t,u(t))=0,t∈[0,1]T,u(0)=0,u(1)=∑i=1m−2aiu(ξi), whereTis a time scale,φp(s)=|s|p−2s,p>1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by usingKrasnosel′skll′sfixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensionalp-Laplacianm-point boundary value problem on time scales has been studied.

scholarly journals On a Nonlocal Multipoint and Integral Boundary Value Problem of Nonlinear Fractional Integrodifferential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Lahcen Ibnelazyz ◽  
Karim Guida ◽  
Said Melliani ◽  
Khalid Hilal
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Integrodifferential Equations ◽  
Integral Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Integral Boundary Value Problem ◽  
Uniqueness Results

The aim of this paper is to give the existence as well as the uniqueness results for a multipoint nonlocal integral boundary value problem of nonlinear sequential fractional integrodifferential equations. First of all, we give some preliminaries and notations that are necessary for the understanding of the manuscript; second of all, we show the existence and uniqueness of the solution by means of the fixed point theory, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Last, but not least, we give two examples to illustrate the results.

scholarly journals EXISTENCE OF POSITIVE SOLUTIONS FOR SUPERLINEAR SEMIPOSITONE $m$-POINT BOUNDARY-VALUE PROBLEMS

2003 ◽  
Vol 46 (2) ◽  
pp. 279-292 ◽  
Author(s):  
Ruyun Ma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Positive Parameter ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
The Fixed Point Theorem

AbstractIn this paper we consider the existence of positive solutions to the boundary-value problems\begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*}where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones.AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15

scholarly journals Solvability of a Class of Higher-Order Fractional Two-Point Boundary Value Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Xiangshan Kong ◽  
Haitao Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Green's Function ◽  
Point Theorem ◽  
Caputo Derivative ◽  
Boundary Value ◽  
Higher Order ◽  
Sufficient Condition ◽  
Point Boundary

This paper investigates the solvability of a class of higher-order fractional two-point boundary value problem (BVP), and presents several new results. First, Green’s function of the considered BVP is obtained by using the property of Caputo derivative. Second, based on Schaefer’s fixed point theorem, the solvability of the considered BVP is studied, and a sufficient condition is presented for the existence of at least one solution. Finally, an illustrative example is given to support the obtained new results.

scholarly journals Fixed-Point Theory on a Frechet Topological Vector Space

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Afif Ben Amar ◽  
Mohamed Amine Cherif ◽  
Maher Mnif
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Vector Space ◽  
Topological Vector Space ◽  
Point Theorem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Weakly Compact ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Continuous Map

We establish some versions of fixed-point theorem in a Frechet topological vector spaceE. The main result is that every mapA=BC(whereBis a continuous map andCis a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem forU-contractions and weakly compact mappings, while the second one, by assuming that the family{T(⋅,y):y∈C(M)whereM⊂EandC:M→Ea compactoperator}is nonlinearφequicontractive, we give a fixed-point theorem for the operator of the formEx:=T(x,C(x)).

scholarly journals Existence of Positive Solutions for Third-Order -Point Boundary Value Problems with Sign-Changing Nonlinearity on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Fatma Tokmak ◽  
Ilkay Yaslan Karaca
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solutions ◽  
Time Scales ◽  
Point Theorem ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Third Order ◽  
Existence Of Positive Solutions

A four-functional fixed point theorem and a generalization of Leggett-Williams fixed point theorem are used, respectively, to investigate the existence of at least one positive solution and at least three positive solutions for third-order -point boundary value problem on time scales with an increasing homeomorphism and homomorphism, which generalizes the usual -Laplacian operator. In particular, the nonlinear term is allowed to change sign. As an application, we also give some examples to demonstrate our results.

Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem

2017 ◽  
Vol 67 (2) ◽  
Author(s):  
Ilkay Yaslan Karaca ◽  
Fatma Tokmak Fen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Point Theorem ◽  
Boundary Value ◽  
Existence Of Positive Solutions ◽  
Impulsive Boundary Value Problem

AbstractIn this paper, six functionals fixed point theorem is used to investigate the existence of at least three positive solutions for a nonlinear

scholarly journals Three-Point Boundary Value Problems of Nonlinear Second-Orderq-Difference Equations Involving Different Numbers ofq

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Thanin Sitthiwirattham ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Difference Equations ◽  
Boundary Value ◽  
Point Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Uniqueness Results ◽  
Nonlinear Alternative

We study a new class of three-point boundary value problems of nonlinear second-orderq-difference equations. Our problems contain different numbers ofqin derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

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