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Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. (English) Zbl 1253.47037
The authors study the existence and uniqueness of positive fixed points for mixed monotone operators with perturbation. They consider the existence and uniqueness of positive solutions for the operator equation $A\left(x,x\right)+B\left(x\right)=x$ in ordered Banach spaces, where $A$ is a mixed monotone operator and $B$ is an increasing sub-homogeneous or $\alpha$-concave operator. In fact, using the properties of cones and a fixed point theorem for mixed monotone operators, the authors obtain existence and uniqueness results for the above equation without assuming the operators to be continuous or compact. They also give some applications to boundary value problems for nonlinear fractional differential equations.
MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H07 Monotone and positive operators on ordered topological linear spaces 34A08 Fractional differential equations 47N20 Applications of operator theory to differential and integral equations