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New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems. (English) Zbl 1225.47074

The authors first present a fixed point theorem for mixed monotone operators defined on cones of Banach spaces; by mixed monotone operator, it is meant an operator \(A\) increasing in one variable and decreasing in the other one. Then they study the eigenvalue problem \(A(x,x)=\lambda x\). As application, the authors develop several existence-uniqueness theorems for some BVPs with various boundary conditions. These theorems are illustrated by means of concrete examples.

MSC:

47H10 Fixed-point theorems
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47N20 Applications of operator theory to differential and integral equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
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