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Fixed point theorems for mixed monotone operators and applications to integral equations. (English) Zbl 1218.54040
The authors generalize Theorem 1 of [T. G. Bhaskar and V. Lakshmikantham, Nonlinear Anal., Theory Methods Appl. 65, No. 7, A, 1379–1393 (2006; Zbl 1106.47047)] by using altering distance functions and present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order. They also provide an application to integral equations.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E50 Complete metric spaces 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
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##### References:
 [1] Guo, D.; Lakshmikantham, V., Coupled fixed points of nonlinear operators with applications, Nonlinear anal., 11, 623-632, (1987) · Zbl 0635.47045 [2] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045 [3] Guo, D., Existence and uniqueness of positive fixed point for mixed monotone operators with applications, Appl. anal., 46, 91-100, (1992) · Zbl 0792.47053 [4] Zhang, Z., New fixed point theorems of mixed monotone operators and applications, J. math. anal. appl., 204, 307-319, (1996) · Zbl 0880.47036 [5] Zhang, S.S.; Ma, Y.H., Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solution for a class of functional equations arising in dynamic programming, J. math. anal. appl., 160, 468-479, (1991) · Zbl 0753.47029 [6] Sun, Y., A fixed point theorem for mixed monotone operator with applications, J. math. anal. appl., 156, 240-252, (1991) · Zbl 0761.47040 [7] Bhaskar, T.G.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal., 65, 1379-1393, (2006) · Zbl 1106.47047 [8] Agarwal, R.P.; El-Gebeily, M.A.; O’Regan, D., Generalized contractions in partially ordered metric spaces, Appl. anal., 87, 109-116, (2008) · Zbl 1140.47042 [9] Burgic, Dz.; Kalabusic, S.; Kulenovic, M.R.S., Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces, Fixed point theory appl., (2009), Article ID 762478 · Zbl 1168.54327 [10] Ciric, L.; Cakid, N.; Rajovic, M.; Ume, J.S., Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed point theory appl., (2008), Article ID 131294 [11] Harjani, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear anal., 71, 3403-3410, (2009) · Zbl 1221.54058 [12] Harjani, J.; Sadarangani, K., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear anal., 72, 1188-1197, (2010) · Zbl 1220.54025 [13] Harjani, J.; Sadarangani, K., Fixed point theorems for mappings satisfying a condition of integral type in partially ordered sets, J. convex anal., 17, 597-609, (2010) · Zbl 1192.54018 [14] Lakshmikantham, V.; Ciric, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal., 70, 4341-4349, (2009) · Zbl 1176.54032 [15] Nieto, J.J.; Rodríguez-López, R., Existence of extremal solutions for quadratic fuzzy equations, Fixed point theory appl., 321-342, (2005) · Zbl 1102.54004 [16] Nieto, J.J.; Rodríguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239, (2005) · Zbl 1095.47013 [17] Nieto, J.J.; Rodríguez-López, R., Applications of contractive-like mapping principles to fuzzy equations, Rev. math. comput., 19, 361-383, (2006) · Zbl 1113.26030 [18] Nieto, J.J.; Pouso, R.L.; Rodríguez-López, R., Fixed point theorems in ordered abstract spaces, Proc. amer. math. soc., 135, 2505-2517, (2007) · Zbl 1126.47045 [19] Nieto, J.J.; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. sinica, 23, 2205-2212, (2007) · Zbl 1140.47045 [20] O’Regan, D.; Petrusel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. math. anal. appl., 341, 1241-1252, (2008) · Zbl 1142.47033 [21] Petrusel, A.; Rus, I.A., Fixed point theorems in ordered $$L$$-spaces, Proc. amer. math. soc., 134, 411-418, (2006) · Zbl 1086.47026 [22] Ran, A.C.M.; Reurings, M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. math. soc., 132, 1435-1443, (2004) · Zbl 1060.47056 [23] Wu, Y., New fixed point theorems and applications of mixed monotone operator, J. math. anal. appl., 341, (2008), 883-393 · Zbl 1137.47044 [24] Cabada, A.; Nieto, J.J., Fixed points and approximate solutions for nonlinear operator equations, J. comput. appl. math., 113, 17-25, (2000) · Zbl 0954.47038 [25] Khan, M.S.; Swaleh, M.; Sessa, S., Fixed point theorems by altering distances between the points, Bull. austral. math. soc., 30, 1, 1-9, (1984) · Zbl 0553.54023
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