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

• Published in: Proceedings of the Edinburgh Mathematical Society 2006-10, Vol.49 (3), p.637-656
• Main Authors: Infante, Gennaro, Webb, J. R. L.
• Format: Article
• Language: English
• Subjects: 47H10; 47H30; Boundary value problems; cone; fixed-point index; positive solution; Primary 34B18; Secondary 34B10
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091505000532

Motivated 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.