Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

SIAM Review ◽  
1976 ◽  
Vol 18 (4) ◽  
pp. 620-709 ◽  
Author(s):  
Herbert Amann
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Ordered Banach Spaces ◽  
Nonlinear Eigenvalue

scholarly journals SOLUTION BRANCHES OF NONLINEAR EIGENVALUE PROBLEMS ON RESTRICTED DOMAINS

2020 ◽  
Vol 102 (3) ◽  
pp. 490-497
Author(s):  
SHANE ARORA
Keyword(s):  
Banach Spaces ◽  
Bifurcation Point ◽  
Eigenvalue Problems ◽  
Accumulation Point ◽  
Nonlinear Eigenvalue Problems ◽  
Restricted Domains ◽  
Nonlinear Eigenvalue

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd-multiplicity points. In the simple-multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on such restricted domains.

scholarly journals Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Xiaoling Han ◽  
Hongliang Gao
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Positive Parameter ◽  
Nonlinear Eigenvalue Problems ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Nonlinear Eigenvalue

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equationD0+αu(t)+λa(t)f(t,u(t))=0,  0<t<1,  u(0)=0,  u(1)=Σi=1∞αiu(ξi)are considered, where1<α≤2is a real number,λis a positive parameter,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),αi∈[0,∞)withΣi=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,∞),[0,∞)).

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