Abstract existence theorems of positive solutions for nonlinear boundary value problems Original Research Article

Let View the MathML source be a bounded domain, H=L2(Ω), L:D(L)⊂H→H be a unbounded linear operator, and View the MathML source be a continuous function. This paper deals with the existence of positive solutions of the operator equation Lu=f(x,u),u∈D(L), which is the abstract form of differential equations boundary value problems. The essential conditions on L and f guaranteeing the existence of positive solution are characterized. The discussion is based on the Krasnoselskiiʹs fixed point theorem of cone mapping and the fixed point index theory in cones. The abstract results are applied to some differential equations boundary value problems, and some known results are extended and improved.