scholarly journals On Linear and Nonlinear Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Dongming Yan
Keyword(s):  
Fixed Point Index ◽  
Linear Problem ◽  
Eigenvalue Problems ◽  
Nonlocal Boundary Condition ◽  
Principal Eigenvalue ◽  
Nonlocal Boundary ◽  
Index Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

We determine the principal eigenvalue of the linear problem ,  , , where and . Moreover, we investigate the existence of positive solutions for the corresponding nonlinear problem. The proofs of our main results are based upon the Krein-Rutman theorem and fixed point index theory.

scholarly journals On a singular Riemann–Liouville fractional boundary value problem with parameters

2021 ◽  
Vol 26 (1) ◽  
pp. 151-168
Author(s):  
Alexandru Tudorache ◽  
Rodica Luca
Keyword(s):  
Fixed Point ◽  
Fixed Point Index ◽  
Nonlocal Boundary ◽  
Index Theory ◽  
Nonlocal Boundary Conditions ◽  
Principal Characteristic ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Semipositone Problem ◽  
Fractional Differential

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

scholarly journals Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Li Wu ◽  
Chuanzhi Bai
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Fractional Boundary Value Problem ◽  
Point Index ◽  
Advanced Argument ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

scholarly journals On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs

2021 ◽  
Vol 22 (2) ◽  
pp. 259
Author(s):  
Svetlin Georgiev Georgiev ◽  
Karima Mebarki
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
First Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Theoretical Results ◽  
Positive Half

The aim of this work is two fold: first  we  extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction  obtained in \cite{DjebaMeb, Svet-Meb}, to  the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction.  Secondly, as  illustration of some our theoretical results,  we study  the existence of positive solutions  for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as  a class of  partial differential equations (PDEs for short).

Existence, multiplicity and non-existence of positive solutions for two-point boundary-value problems with strong singularity

2010 ◽  
Vol 140 (6) ◽  
pp. 1187-1196
Author(s):  
Chan-Gyun Kim
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Boundary ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problemswhere $\varphi_{p}(s)=|s|^{p-2}s$, $p>1$, λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.

scholarly journals Positive solutions of singular nonlinear Sturm-Liouville boundary value problems

2004 ◽  
Vol 45 (4) ◽  
pp. 557-571
Author(s):  
Yan Sun ◽  
Lishan Liu ◽  
Yeol Je Cho
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Sturm Liouville

AbstractBy using fixed point index theory, we present the existence of positive solutions for a Sturm-Liouville singular boundary value problem with at least one positive solution. Our results significantly extend and improve many known results even for non-singular cases.

Positive Solution of Nonlinear Two-Order Three-Point Boundary Value Problem for Difference Equation with Change of Sign

2014 ◽  
Vol 687-691 ◽  
pp. 1232-1236
Author(s):  
Chun Li Wang
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Boundary ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper we investigate the existence of positive solution of the following discrete two-order three-point boundary value problemWherandis sign-changing on . By using the fixed-point index theory, the existence of positive solutions for the above boundary value problem is obtained.

scholarly journals Positive Solutions for Nonlinearq-Fractional Difference Eigenvalue Problem with Nonlocal Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Wafa Shammakh ◽  
Maryam Al-Yami
Keyword(s):  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Index Theory ◽  
Nonlocal Boundary Conditions ◽  
Fractional Difference ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

The problem of positive solutions for nonlinearq-fractional difference eigenvalue problem with nonlocal boundary conditions is investigated. Based on the fixed point index theory in cones, sufficient existence of positive solutions conditions is derived for the problem.

scholarly journals Multiplicity of Positive Solutions to Nonlocal Boundary Value Problems with Strong Singularity

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 7
Author(s):  
Chan-Gyun Kim
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlocal Boundary ◽  
Index Theory ◽  
Open Interval ◽  
Nonlocal Boundary Conditions ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions ◽  
The Given

In this paper, we consider generalized Laplacian problems with nonlocal boundary conditions and a singular weight, which may not be integrable. The existence of two positive solutions to the given problem for parameter λ belonging to some open interval is shown. Our approach is based on the fixed point index theory.

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

2020 ◽  
Vol 24 (1) ◽  
pp. 109-129
Author(s):  
Abdulkadir Dogan ◽  
John R. Graef
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Third Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results.

scholarly journals Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yongxiang Li ◽  
Qiang Li
Keyword(s):  
Periodic Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Existence Results ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
The Third ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the third-order ordinary differential equation with delaysu′′′(t)+a(t)u(t)=f(t,u(t-τ0),u′(t-τ1),u′′(t-τ2)),t∈ℝ,wherea∈C(ℝ,(0,∞))isω-periodic function andf:ℝ×[0,∞)×ℝ2→[0,∞)is a continuous function which isω-periodic int,and τ0,τ1,τ2are positive constants. The discussion is based on the fixed-point index theory in cones.

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