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Fixed point theorems for $$\alpha$$-$$\psi$$-contractive type mappings. (English) Zbl 1242.54027
Let $$\Psi$$ be the family of nondecreasing functions $$\psi:[0,+\infty)\to[0,+\infty)$$ such that $$\sum_{n=1}^{\infty}\psi^n(t)<+\infty$$ for each $$t\geq0$$, where $$\psi^n$$ is the $$n$$-th iterate of $$\psi$$. Let $$(X,d)$$ be a metric space and $$T:X\to X$$. The authors call the mapping $$T$$ $$\alpha$$-$$\psi$$-contractive if there exist functions $$\alpha:X\times X\to[0,+\infty)$$ and $$\psi\in\Psi$$ such that $$\alpha(x,y)\,d(Tx,Ty)\leq\psi(d(x,y))$$ for all $$x,y\in X$$. Furthermore, $$T$$ is called $$\alpha$$-admissible if $$\alpha(x,y)\geq1$$ implies that $$\alpha(Tx,Ty)\geq1$$.
The authors prove that, if the space $$(X,d)$$ is complete and the mapping $$T$$ is continuous, $$\alpha$$-admissible and $$\alpha$$-$$\psi$$-contractive, then $$T$$ has a fixed point in $$X$$, provided that $$\alpha(x_0,Tx_0)\geq1$$ for some $$x_0\in X$$. Continuity of $$T$$ can be omitted if, for each sequence $$\{x_n\}$$ in $$X$$, $$\alpha(x_n,x_{n+1})\geq1$$ for all $$n$$ and $$x_n\to x\in X$$ as $$n\to\infty$$ imply that $$\alpha(x_n,x)\geq1$$ for all $$n$$. The uniqueness of the fixed point holds if, for all $$x,y\in X$$, there is some $$z\in X$$ such that $$\alpha(x,z)\geq1$$ and $$\alpha(y,z)\geq1$$.
Several known (and some new) fixed and coupled fixed point theorems in metric, as well as in ordered metric spaces can be obtained from these results. Some examples are presented, showing how the obtained theorems can be used. Also, some applications to ordinary differential equations are given.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E40 Special maps on metric spaces 54E50 Complete metric spaces
##### Keywords:
fixed point; coupled fixed point; contractive mapping
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##### References:
  Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales, Fund. math., 3, 133-181, (1922) · JFM 48.0201.01  Abbas, M.; Ali Khan, M.; Radenović, S., Common coupled fixed point theorems in cone metric spaces for $$w$$-compatible mappings, Appl. math. comput., 217, 195-202, (2010) · Zbl 1197.54049  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  Altun, I.; Durmaz, G., Some fixed point theorems on ordered cone metric spaces, Rend. circ. mat. Palermo, 58, 319-325, (2009) · Zbl 1184.54038  Altun, I.; Simsek, H., Some fixed point theorems on ordered metric spaces and application, Fixed point theory appl., 2010, (2010), Article ID 621492, 17 pages · Zbl 1197.54053  Beg, I.; Butt, A.R., Fixed point for set-valued mappings satisfying an implicit relation in ordered metric spaces, Nonlinear anal., 71, 3699-3704, (2009) · Zbl 1176.54028  Bhaskar, T.G.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal., 65, 1379-1393, (2006) · Zbl 1106.47047  Choudhury, Binayak S.; Kundu, A., A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear anal., 73, 2524-2531, (2010) · Zbl 1229.54051  Caballero, J.; Harjani, J.; Sadarangani, K., Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed point theory appl., 2010, (2010), Article ID 916064, 14 pages · Zbl 1194.54057  Chifu, C.; Petruşel, G., Fixed-point results for generalized contractions on ordered gauge spaces with applications, Fixed point theory appl., 2011, (2011), Article ID 979586, 10 pages · Zbl 1214.54032  Ćirić, Lj., A generalization of Banach principle, Proc. amer. math. soc., 45, 727-730, (1974)  Ćirić, Lj., Non-self mappings satisfying nonlinear contractive condition with applications, Nonlinear anal., 71, 2927-2935, (2009) · Zbl 1166.47052  Ćirić, Lj.; Cakić, N.; Rajović, M.; Ume, J.S., Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed point theory appl., 2008, (2008), Article ID 131294, 11 pages · Zbl 1158.54019  Ćirić, Lj.; Lakshmikantham, V., Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. anal. appl., 27, 1246-1259, (2009) · Zbl 1176.54030  Du, W.-S., Coupled fixed point theorems for nonlinear contractions satisfied mizoguchi – takahashi’s condition in quasiordered metric spaces, Fixed point theory appl., 2010, (2010), Article ID 876372, 9 pages · Zbl 1194.54061  Harjani, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear anal., 71, 3403-3410, (2008) · Zbl 1221.54058  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  Harjani, J.; López, B.; Sadarangani, K., Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear anal., 74, 1749-1760, (2011) · Zbl 1218.54040  Hussain, N.; Shah, M.H.; Kutbi, M.A., Coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a Q-function, Fixed point theory appl., 2011, (2011), Article ID 703938, 21 pages · Zbl 1215.54020  Jachymski, J., Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear anal., 74, 768-774, (2011) · Zbl 1201.54034  Jungck, G.; Rhoades, B.E., Fixed points for set valued functions without continuity, Indian J. pure appl. math., 29, 227-238, (1998) · Zbl 0904.54034  Kannan, R., Some results on fixed points, Bull. Calcutta math. soc., 60, 71-76, (1968) · Zbl 0209.27104  Karapinar, E., Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. math. appl., 59, 3656-3668, (2010) · Zbl 1198.65097  Lakshmikantham, V.; Ćirić, Lj., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal., 70, 4341-4349, (2009) · Zbl 1176.54032  Nashine, H.K.; Samet, B., Fixed point results for mappings satisfying $$(\psi, \varphi)$$-weakly contractive condition in partially ordered metric spaces, Nonlinear anal., 74, 2201-2209, (2011) · Zbl 1208.41014  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  Nieto, J.J.; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. sin. (engl. ser.), 23, 2205-2212, (2007) · Zbl 1140.47045  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, (2003) · Zbl 1060.47056  Nieto, J.J.; Pouso, R.L.; Rodríguez-López, R., Fixed point theorems in ordered abstract spaces, Proc. amer. math. soc., 132, 2505-2517, (2007) · Zbl 1126.47045  O’Regan, D.; Petruşel, A., Fixed point theorems for generalized contractions in ordered metric spaces, J. math. anal. appl., 341, 1241-1252, (2008) · Zbl 1142.47033  Turkoglu, D.; Binbasioglu, D., Some fixed point theorems for multivalued monotone mappings in ordered uniform space, Fixed point theory appl., 2011, (2011), Article ID 186237, 12 pages · Zbl 1213.54086  Samet, B., Coupled fixed point theorems for a generalized meir – keeler contraction in partially ordered metric spaces, Nonlinear anal., 72, 4508-4517, (2010) · Zbl 1264.54068  Samet, B.; Vetro, C., Coupled fixed point, $$F$$-invariant set and fixed point of $$N$$-order, Ann. funct. anal., 1, 46-56, (2010) · Zbl 1214.54041  Samet, B.; Vetro, C., Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces, Nonlinear anal., 74, 4260-4268, (2011) · Zbl 1216.54021
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