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Positive solutions for nonlinear operator equations and several classes of applications. (English) Zbl 1198.47078
The authors study a class of nonlinear operator equations $x=Ax+{x}_{0}$ on ordered Banach spaces, where $A$ is a monotone generalized concave operator. Employing the properties of cones and monotone iterative technique, they prove existence and uniqueness of solutions for such equations without requiring existence of upper-lower solutions and compactness and continuity conditions. Applications are proposed to first-order initial value problems and two-point boundary value problems with nonlinear term which is monotone with respect to its second argument, as well as to nonlinear systems of equations and to nonlinear matrix equations.
##### MSC:
 47J05 Equations involving nonlinear operators (general) 47N20 Applications of operator theory to differential and integral equations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H07 Monotone and positive operators on ordered topological linear spaces 34B18 Positive solutions of nonlinear boundary value problems for ODE 35F25 Initial value problems for first order nonlinear PDE 15A30 Algebraic systems of matrices
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