Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

2009 ◽  
Vol 71 (12) ◽  
pp. 5979-5993 ◽  
Author(s):  
K.Q. Lan
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Value ◽  
Hammerstein Integral Equations

scholarly journals NONLINEAR NON-LOCAL BOUNDARY-VALUE PROBLEMS AND PERTURBED HAMMERSTEIN INTEGRAL EQUATIONS

2006 ◽  
Vol 49 (3) ◽  
pp. 637-656 ◽  
Author(s):  
Gennaro Infante ◽  
J. R. L. Webb
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Positive Solutions ◽  
Boundary Value ◽  
Uniqueness Result ◽  
Compact Set ◽  
Local Boundary ◽  
Flow Problems ◽  
Hammerstein Integral Equations ◽  
Non Local

AbstractMotivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

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