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Fixed point theorems for mixed monotone operators with PPF dependence. (English) Zbl 1162.47042
First, the authors prove an existence result for coupled fixed points of mixed monotone operators. The uniqueness of such fixed points is also proved, but just in a region of the domain of the involved operator. In the last section, the authors use this result as well as one of their previous results to prove the existence and uniqueness for the solution of a PVBP with delay, in the minimal class.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B15Nonlinear boundary value problems for ODE
47H05Monotone operators (with respect to duality) and generalizations
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] Bernfeld, R. S.; Lakshmikantham, V.; Reddy, Y. M.: Fixed point theorems of operators with PPF dependence in a Banach space, Applicable analysis 6, 271-280 (1977) · Zbl 0375.47027 · doi:10.1080/00036817708839165
[2] Drici, Z.; Mcrae, F. A.; Devi, J. Vasundhara: Fixed point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear analysis 67, No. 2, 641-647 (2007) · Zbl 1127.47049 · doi:10.1016/j.na.2006.06.022
[3] Bhaskar, T. Gnana; Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear analysis 65, No. 7, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017
[4] Lakshmikantham, V.; Koksal, S.: Monotone flows and rapid convergence for nonlinear partial differential equations, (2003)
[5] Nieto, J. J.; Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, No. 3, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[6] Ran, A. C. M.; Reurings, M. C. R.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American mathematical society 132, 1435-1443 (2003) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4