scholarly journals On Positive Radial Solutions for a Class of Elliptic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ying Wu ◽  
Guodong Han
Keyword(s):  
Fixed Point Index ◽  
Nonlinear Term ◽  
Elliptic Boundary ◽  
Index Theory ◽  
First Eigenvalue ◽  
Radial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Positive Radial Solutions

A class of elliptic boundary value problem in an exterior domain is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where the variables of nonlinear termfs,uneed not to be separated. Several new theorems on the existence and multiplicity of positive radial solutions are obtained by means of fixed point index theory. Our conclusions are essential improvements of the results in Lan and Webb (1998), Lee (1997), Mao and Xue (2002), Stańczy (2000), and Han and Wang (2006).

Positive Radial Solutions for Elliptic Systems with Nonlocal Conditions

2012 ◽  
Vol 204-208 ◽  
pp. 4800-4808
Author(s):  
Sheng Li Xie
Keyword(s):  
Fixed Point ◽  
Elliptic System ◽  
Fixed Point Index ◽  
Nonlocal Conditions ◽  
Elliptic Systems ◽  
Index Theory ◽  
Point Index ◽  
Radial Solutions ◽  
Fixed Point Index Theory ◽  
Positive Radial Solutions

By using fixed point index theory,we study the existence of positive radial solutions and multiple positive radial solutions for the elliptic system with nonlocal conditions. Our results extend and improve some existing ones.

scholarly journals Non-Zero Radial Solutions for Elliptic Systems with Coupled Functional BCs in Exterior Domains

2019 ◽  
Vol 62 (3) ◽  
pp. 747-769 ◽  
Author(s):  
Filomena Cianciaruso ◽  
Gennaro Infante ◽  
Paolamaria Pietramala
Keyword(s):  
Elliptic Boundary ◽  
Elliptic Systems ◽  
Index Theory ◽  
Radial Solutions ◽  
Exterior Domains ◽  
Nonlinear Functional ◽  
Elliptic Boundary Value ◽  
Fixed Point Index Theory ◽  
Functional Boundary ◽  
Open Question

AbstractWe prove new results on the existence, non-existence, localization and multiplicity of non-trivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to non-local, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do Ó, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.

scholarly journals Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Liyun Jin ◽  
Hua Luo
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Fully Nonlinear ◽  
Fixed Point Index Theory ◽  
Weaker Conditions ◽  
Lower Limits

Abstract In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

scholarly journals Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Wen Lian ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Degree Theory ◽  
Index Theory ◽  
Growth Conditions ◽  
First Eigenvalue ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0,   0<x<1, y0=y′1=y″0=0 is established, where 2<α≤3,  CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

scholarly journals Positive Solutions for a Class of Fourth-Orderp-Laplacian Boundary Value Problem Involving Integral Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yan Sun
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Integral Boundary Conditions ◽  
Index Theory ◽  
Integral Conditions ◽  
Interesting Point ◽  
Integral Boundary ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

Under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, we obtain sharp optimal criteria for the existence of positive solutions forp-Laplacian problems with integral boundary conditions. The main methods in the paper are constructing an available integral operator and combining fixed point index theory. The interesting point of the results is that the nonlinear term contains all lower-order derivatives explicitly. Finally, we give some examples to demonstrate the main results.

scholarly journals Existence and multiplicity results for some generalized Hammerstein equations with a parameter

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Lucía López-Somoza ◽  
Feliz Minhós
Keyword(s):  
Kernel Function ◽  
Fixed Point Index ◽  
Sufficient Conditions ◽  
Index Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Carathéodory Function ◽  
New Type ◽  
Derivatives Of

Abstract This paper considers the existence and multiplicity of fixed points for the integral operator $$ {\mathcal{T}}u(t)=\lambda \int _{0}^{T}k(t,s) f\bigl(s,u(s),u^{\prime }(s), \dots ,u^{(m)}(s)\bigr) \,\mathrm{d}s,\quad t\in {[} 0,T]\equiv I, $$ T u ( t ) = λ ∫ 0 T k ( t , s ) f ( s , u ( s ) , u ′ ( s ) , … , u ( m ) ( s ) ) d s , t ∈ [ 0 , T ] ≡ I , where $\lambda >0$ λ > 0 is a positive parameter, $k:I\times I\rightarrow \mathbb{R}$ k : I × I → R is a kernel function such that $k\in W^{m,1} ( I \times I ) $ k ∈ W m , 1 ( I × I ) , m is a positive integer with $m\geq 1$ m ≥ 1 , and $f:I\times \mathbb{R} ^{m+1}\rightarrow [ 0,+\infty [ $ f : I × R m + 1 → [ 0 , + ∞ [ is an $\mathrm{L}^{1}$ L 1 -Carathéodory function. The existence of solutions for these Hammerstein equations is obtained by fixed point index theory on new type of cones. Therefore some assumptions must hold only for, at least, one of the derivatives of the kernel or, even, for the kernel on a subset of the domain. Assuming some asymptotic conditions on the nonlinearity f, we get sufficient conditions for multiplicity of solutions. Two examples will illustrate the potentialities of the main results, namely the fact that the kernel function and/or some derivatives may only be positive on some subintervals, which can degenerate to a point. Moreover, an application of our method to general Lidstone problems improves the existent results in the literature in this field.

scholarly journals Existence and multiplicity of positive solutions for a singular Riemann-Liouville fractional differential problem

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 3931-3942
Author(s):  
Rodica Luca
Keyword(s):  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
Differential Problem ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We investigate the existence and multiplicity of positive solutions for a nonlinear Riemann-Liouville fractional differential equation with a nonnegative singular nonlinearity, subject to Riemann-Stieltjes boundary conditions which contain fractional derivatives. In the proofs of our main results, we use an application of the Krein-Rutman theorem and some theorems from the fixed point index theory.

scholarly journals Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongsen Fan ◽  
Zhiying Deng
Keyword(s):  
Variational Method ◽  
Boundary Value Problem ◽  
Critical Exponent ◽  
Positive Solution ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problem ◽  
Nontrivial Solutions ◽  
Elliptic Boundary Value ◽  
Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

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.

scholarly journals On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiang Li ◽  
Yongxiang Li
Keyword(s):  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Index Theory ◽  
Second Order ◽  
Multiple Delays ◽  
First Eigenvalue ◽  
Fixed Point Index Theory ◽  
Existence Conditions ◽  
Functional Differential ◽  
Periodic Boundary Problem

The existence results of positiveω-periodic solutions are obtained for the second-order functional differential equation with multiple delaysu″(t)+a(t)u(t)=f(t,u(t),u(t−τ1(t)),…,u(t−τn(t))), wherea(t)∈C(ℝ)is a positiveω-periodic function,f:ℝ×[0,+∞)n+1→[0,+∞)is a continuous function which isω-periodic int, andτ1(t),…,τn(t)∈C(ℝ,[0,+∞))areω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

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