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On fixed points of quasi-contraction type multifunctions. (English) Zbl 1243.54062
Summary: D. Ilić and V. Rakočević [Appl. Math. Lett. 22, No. 5, 728–731 (2009; Zbl 1179.54060)] proved that quasi-contraction maps on normal cone metric spaces have a unique fixed point. Then, Kadelburg, Radenović and Rakočević generalized their results by considering an additional assumption [Z. Kadelburg, S. Radenović and V. Rakočević, Appl. Math. Lett. 22, No. 11, 1674–1679 (2009; Zbl 1180.54056)]. Also, they proved that quasi-contraction maps on cone metric spaces have the property (P) whenever $\lambda \in \left(0,\frac{1}{2}\right)$. Later, Haghi, Rezapour and Shahzad proved same results without the additional assumption and for $\lambda \in \left(0,1\right)$ by providing a new technical proof [Sh. Rezapour, R.H. Haghi and N. Shahzad, Appl. Math. Lett. 23, No. 4, 498–502 (2010; Zbl 1206.54061)]. In 2011, D. Wardowski published a paper [Appl. Math. Lett. 24, No. 3, 275–278 (2011; Zbl 1206.54067)] and tried to test fixed point results for multifunctions on normal cone metric spaces. Of course, he used a special view in his results. Recently, Amini-Harandi proved a result on the existence of fixed points of set-valued quasi-contraction maps in metric spaces by using the technique of Rezapour et al. [Zbl 1206.54061]. But, like Kadelburg et al. [Zbl 1180.54056], he could prove it only for $\lambda \in \left(0,\frac{1}{2}\right)$ A. Amini-Harandi [Appl. Math. Lett. 24, No. 11, 1791–1794 (2011; Zbl 1230.54034)]. In this work, we prove again the main result of Amini-Harandi [Zbl 1230.54034] by using a simple method. Also, we introduce quasi-contraction type multifunctions and show that the main result of Amini-Harandi [Zbl 1230.54034] holds for quasi-contraction type multifunctions.
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54C60 Set-valued maps (general topology)
##### References:
 [1] Ćirić, Lj.: Fixed point theory: contraction mapping principle, (2003) [2] Pathak, H. K.; Shahzad, N.: Fixed point results for generalized quasicontraction mappings in abstract metric spaces, Nonlinear anal. 71, 6068-6076 (2009) · Zbl 1189.54036 · doi:10.1016/j.na.2009.05.052 [3] Amini-Harandi, A.: Fixed point theory for set-valued quasi-contraction maps in metric spaces, Appl. math. Lett. 24, 1791-1794 (2011) · Zbl 1230.54034 · doi:10.1016/j.aml.2011.04.033 [4] Rezapour, Sh.; Haghi, R. H.; Shahzad, N.: Some notes on fixed points of quasi-contraction maps, Appl. math. Lett. 23, 498-502 (2010) · Zbl 1206.54061 · doi:10.1016/j.aml.2010.01.003 [5] Haghi, R. H.; Rezapour, Sh.; Shahzad, N.: Some fixed point generalizations are not real generalizations, Nonlinear anal. 74, 1799-1803 (2011) [6] Ilić, D.; Rakocević, V.: Quasi-contraction on a cone metric space, Appl. math. Lett. 22, 728-731 (2009) · Zbl 1179.54060 · doi:10.1016/j.aml.2008.08.011 [7] Kadelburg, Z.; Radenović, S.; Rakocević, V.: Remarks on ”quasi-contraction on a cone metric space”, Appl. math. Lett. 22, 1674-1679 (2009) · Zbl 1180.54056 · doi:10.1016/j.aml.2009.06.003 [8] Wardowski, D.: On set-valued contractions of nadler type in cone metric spaces, Appl. math. Lett. 24, 275-278 (2011) · Zbl 1206.54067 · doi:10.1016/j.aml.2010.10.003