scholarly journals Existence and Nonexistence of Solutions to p-Laplacian Problems on Unbounded Domains

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 438 ◽  
Author(s):  
Jeongmi Jeong ◽  
Chan-Gyun Kim ◽  
Eun Kyoung Lee
Keyword(s):  
Fixed Point Index ◽  
Index Theorem ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Multiplicity Results ◽  
One Dimensional ◽  
Existence And Multiplicity ◽  
Nonexistence Of Solutions ◽  
Fixed Point Index Theorem ◽  
Infinite Intervals

In this article, using a fixed point index theorem on a cone, we prove the existence and multiplicity results of positive solutions to a one-dimensional p-Laplacian problem defined on infinite intervals. We also establish the nonexistence results of nontrivial solutions to the problem.

scholarly journals Positive solutions of three-point boundary value problems for higher-orderp-Laplacian with infinitely many singularities

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Fuyi Xu ◽  
Yonghong Wu ◽  
Lishan Liu ◽  
Yunming Zhou
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Index Theorem ◽  
Point Boundary ◽  
Point Index ◽  
Fixed Point Index Theorem ◽  
Infinitely Many Singularities

We study a three-point nonlinear boundary value problem with higher-orderp-Laplacian. We show that there exist countable many positive solutions by using the fixed point index theorem for operators in a cone.

Positive Solutions for Nonlinear Lidstone Boundary Value Problems

2013 ◽  
Vol 313-314 ◽  
pp. 1201-1204 ◽  
Author(s):  
Lei Wang ◽  
Li Li
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Existence Result ◽  
Boundary Value ◽  
Index Theorem ◽  
Point Index ◽  
Existence Of Positive Solutions ◽  
Fixed Point Index Theorem

In this paper, we consider the existence of positive solutions for nonlinear Lidstone boundary value problems. An new existence result is obtained by applying the fixed point index theorem.

scholarly journals Positive Solutions for Second-Order Three-Point Eigenvalue Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Chuanzhi Bai
Keyword(s):  
Fixed Point ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Existence Theorem ◽  
Fixed Point Index ◽  
Eigenvalue Problems ◽  
Index Theorem ◽  
Second Order ◽  
Point Index ◽  
Fixed Point Index Theorem

With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problemx′′(t)+λf(t,x(t))=0,  0≤t≤1,  x(0)=0,  x(1)=x(η), whereλis a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

scholarly journals Positive Fixed Points for Semipositone Operators in Ordered Banach Spaces and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Zengqin Zhao ◽  
Xinsheng Du
Keyword(s):  
Fixed Points ◽  
Integral Equations ◽  
Fixed Point Index ◽  
Index Theorem ◽  
Point Index ◽  
Hammerstein Integral Equations ◽  
Polynomial Type ◽  
Fixed Point Index Theorem ◽  
Abstract Spaces ◽  
Ordered Banach Spaces

The theory of semipositone integral equations and semipositone ordinary differential equations has been emerging as an important area of investigation in recent years, but the research on semipositone operators in abstract spaces is yet rare. By employing a well-known fixed point index theorem and combining it with a translation substitution, we study the existence of positive fixed points for a semipositone operator in ordered Banach space. Lastly, we apply the results to Hammerstein integral equations of polynomial type.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 330
Author(s):  
Gennaro Infante
Keyword(s):  
Fixed Point ◽  
Boundary Conditions ◽  
Integral Equations ◽  
General Class ◽  
Fixed Point Index ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Hammerstein Integral Equations ◽  
Functional Boundary ◽  
Functional Boundary Conditions

We discuss the solvability of a fairly general class of systems of perturbed Hammerstein integral equations with functional terms that depend on several parameters. The nonlinearities and the functionals are allowed to depend on the components of the system and their derivatives. The results are applicable to systems of nonlocal second order ordinary differential equations subject to functional boundary conditions, this is illustrated in an example. Our approach is based on the classical fixed point index.

Schrödinger systems with quadratic interactions

2019 ◽  
Vol 21 (08) ◽  
pp. 1850077
Author(s):  
Rushun Tian ◽  
Zhi-Qiang Wang ◽  
Leiga Zhao
Keyword(s):  
Variational Formulation ◽  
Variational Problems ◽  
Nontrivial Solutions ◽  
New Techniques ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Coupled Schrödinger System ◽  
Schrödinger Systems ◽  
Bose Einstein ◽  
Schrödinger System

In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are constants and [Formula: see text], [Formula: see text]. Such type of systems stem from applications in nonlinear optics, Bose–Einstein condensates and plasma physics. The existence (and nonexistence), multiplicity and asymptotic behavior of vector solutions of the system are established via variational methods. In particular, for multiplicity results we develop new techniques for treating variational problems with only partial symmetry for which the classical minimax machinery does not apply directly. For the above system, the variational formulation is only of even symmetry with respect to the first component [Formula: see text] but not with respect to [Formula: see text], and we prove that the number of vector solutions tends to infinity as [Formula: see text] tends to infinity.

Solutions of second order multi-point boundary value problems

2008 ◽  
Vol 145 (2) ◽  
pp. 489-510 ◽  
Author(s):  
JOHN R. GRAEF ◽  
LINGJU KONG
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problems ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Linear Integral ◽  
The Relationship

AbstractWe consider classes of second order boundary value problems with a nonlinearity f(t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞ and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.

scholarly journals The Existence and Multiplicity of Positive Solutions for Second-Order Periodic Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Feng Wang ◽  
Fang Zhang ◽  
Fuli Wang
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Periodic Boundary Value Problem ◽  
Point Index ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

The existence and multiplicity of positive solutions are established for second-order periodic boundary value problem. Our results are based on the theory of a fixed point index for A-proper semilinear operators defined on cones due to Cremins. Our approach is different in essence from other papers and the main results of this paper are also new.

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.

Sign in / Sign up

Export Citation Format

Share Document