scholarly journals Eigenvalues of a Class of Singular Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Daliang Zhao ◽  
Yansheng Liu
Keyword(s):  
Banach Spaces ◽  
Fixed Point Index ◽  
Measure Of Noncompactness ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Zero Element ◽  
Singular Boundary ◽  
Index Measure ◽  
Special Cone

This paper is devoted to investigating the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces:μx′′+f(t,x)=0,t∈(0,1),t≠ti;Δx|t=ti=αix(ti-0),i=1,2,…,k;ax(0)-bx′(0)=θ;cx(1)+dx′(1)=θ,whereθdenotes the zero element of Banach space,Δx|t=ti=x(ti+0)-x(ti-0),αi>-1,a,b,c,d∈R+,γ=ac+ad+bc>0,μis a parameter, andf(t,x)may be singular att=0,1andx=θ. The arguments are mainly based upon the theory of fixed point index, measure of noncompactness, and the special cone, which is constructed to overcome the singularity.

scholarly journals Positive Solutions for (k,n−k) Conjugate Multipoint Boundary Value Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Yulin Zhao
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Real Banach Space ◽  
Boundary Value ◽  
Index Theory ◽  
Zero Element ◽  
Singular Boundary ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Value Problems

By means of the fixed point index theory of strict-set contraction operator, we study the existence of positive solutions for the multipoint singular boundary value problem(-1)n-kun(t)=f(t,ut),0<t<1,n≥2,1≤k≤n-1,u(0)=∑i=1m-2‍aiu(ξi),u(i)(0)=u(j)(1)=θ,1≤i≤k−1,0≤j≤n−k−1in a real Banach spaceE, whereθis the zero element ofE,0<ξ1<ξ2<⋯<ξm-2<1,ai∈[0,+∞),i=1,2,…,m-2.As an application, we give two examples to demonstrate our results.

scholarly journals Existence, Nonexistence and Multiplicity of Positive Solutions for Singular Boundary Value Problems Involving φ-Laplacian

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 953 ◽  
Author(s):  
Chan-Gyun Kim
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problems ◽  
Interesting Point ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

In this paper, we establish the results on the existence, nonexistence and multiplicity of positive solutions to singular boundary value problems involving φ -Laplacian. Our approach is based on the fixed point index theory. The interesting point is that a result for the existence of three positive solutions is given.

scholarly journals Positive Solutions for a Class of Fourth-Order Boundary Value Problems in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Jingjing Cai ◽  
Guilong Liu
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Positive Solutions ◽  
Fourth Order ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence And Nonexistence

Using a specially constructed cone and the fixed point index theory, this work shows existence and nonexistence results of positive solutions for fourth-order boundary value problem with two different parameters in Banach spaces.

scholarly journals Multiple Positive Solutions for Singular Periodic Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Hongxia Fan ◽  
Yongxiang Li
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Periodic Boundary ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Point Theory ◽  
Singular Boundary ◽  
Multiple Positive Solutions ◽  
Periodic Boundary Value Problems

By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second-order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.

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.

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