Positive solutions of semi-positone Hammerstein integral equations and applications. (English) Zbl 1134.45005
The author considers the following Hammerstein integral equation:
\[ y(t) = \int_0^1 k(t,s) g(s) f(s,y(s))\,ds, \quad t\in[0,1],\tag{1} \]
where \( f: [0,1]\times\mathbb{R}_+ \to\mathbb{R}\) is continuous, and such that \(f>-\eta \) for some \(\eta >0; \) the kernel \(k: [0,1]\times[0, 1] \to \mathbb{R}_+ \) is continuous; \(g: [0,1]\to\mathbb{R}_+ \) is measurable. The author uses the fixed-point index theory to prove the existence of one or several positive solutions for problem (1). Furthermore, he applies the obtained results to some \(n\)th order ordinary differential equations with boundary conditions.
\[ y(t) = \int_0^1 k(t,s) g(s) f(s,y(s))\,ds, \quad t\in[0,1],\tag{1} \]
where \( f: [0,1]\times\mathbb{R}_+ \to\mathbb{R}\) is continuous, and such that \(f>-\eta \) for some \(\eta >0; \) the kernel \(k: [0,1]\times[0, 1] \to \mathbb{R}_+ \) is continuous; \(g: [0,1]\to\mathbb{R}_+ \) is measurable. The author uses the fixed-point index theory to prove the existence of one or several positive solutions for problem (1). Furthermore, he applies the obtained results to some \(n\)th order ordinary differential equations with boundary conditions.
Reviewer: Mohamed O. El-Doma (Beirut)
MSC:
45G10 | Other nonlinear integral equations |
47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
45M20 | Positive solutions of integral equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |