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Fixed point theorems for a class of nonlinear operators in Banach spaces and applications. (English) Zbl 1168.47041

Summary: We study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using cone theory and the Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions.

The results are applied to a class of abstract semilinear evolution equations with a noncompact semigroup in Banach spaces and to initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces.

47H07Monotone and positive operators on ordered topological linear spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J05Equations involving nonlinear operators (general)
47N20Applications of operator theory to differential and integral equations
34G20Nonlinear ODE in abstract spaces