Fixed point results for single and set-valued \(\alpha\)-\(\eta\)-\(\psi\)-contractive mappings. (English) Zbl 1293.54025
Summary: B. Samet et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 4, 2154–2165 (2012; Zbl 1242.54027)] introduced \(\alpha\)-\(\psi\)-contractive mappings and proved some fixed point results for these mappings. More recently, P. Salimi et al. [Fixed Point Theory Appl. 2013, Article ID 151, 19 p. (2013; Zbl 1293.54036)] modified the notion of \(\alpha\)-\(\psi\)-contractive mappings and established certain fixed point theorems. Here, we continue to utilize these modified notions for single-valued Geraghty and Meir-Keeler-type contractions, as well as multi-valued contractive mappings. The presented theorems provide main results of N. Hussain et al. [J. Inequal. Appl. 2013, Article ID 114, 11 p. (2013; Zbl 1293.54023)], N. Hussain et al. [Fixed Point Theory Appl. 2013, Article ID 34, 14 p. (2013; Zbl 1293.54024)] and [J. Hasanzade Asl et al. [Fixed Point Theory Appl. 2012, Article ID 212, 6 p. (2012; Zbl 1293.54017)]] as corollaries. Moreover, some examples are given here to illustrate the usability of the obtained results.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
| 54C60 | Set-valued maps in general topology |
Keywords:
modified Meir-Keeler-type contractions; triangular \(\alpha\)-admissible map; \(\alpha\)-\(\eta\)-\(\psi\)-contractive map; metric spaceReferences:
| [1] | Akbar, F.; Khan, AR, Common fixed point and approximation results for noncommuting maps on locally convex spaces, No. 2009 (2009) · Zbl 1168.47044 |
| [2] | Ćirić L, Hussain N, Cakic N: Common fixed points for Ćirić typef-weak contraction with applications.Publ. Math. (Debr.) 2010,76(1-2):31-49. · Zbl 1261.47071 |
| [3] | Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces.Appl. Math. Comput. 2011, 217:5784-5789. · Zbl 1206.54040 · doi:10.1016/j.amc.2010.12.060 |
| [4] | Geraghty MA: On contractive mappings.Proc. Am. Math. Soc. 1973, 40:604-608. · Zbl 0245.54027 · doi:10.1090/S0002-9939-1973-0334176-5 |
| [5] | Hussain N, Berinde V, Shafqat N: Common fixed point and approximation results for generalizedϕ-contractions.Fixed Point Theory 2009, 10:111-124. · Zbl 1196.47040 |
| [6] | Hussain N, Khamsi MA, Latif A: Common fixed points forJH-operators and occasionally weakly biased pairs under relaxed conditions.Nonlinear Anal. 2011, 74:2133-2140. · Zbl 1270.47042 · doi:10.1016/j.na.2010.11.019 |
| [7] | Hussain, N.; Kadelburg, Z.; Radenovic, S.; Al-Solamy, FR, Comparison functions and fixed point results in partial metric spaces, No. 2012 (2012) · Zbl 1242.54023 |
| [8] | Hussain, N.; Djorić, D.; Kadelburg, Z.; Radenović, S., Suzuki-type fixed point results in metric type spaces, No. 2012 (2012) · Zbl 1274.54128 |
| [9] | Hussain, N.; Karapinar, E.; Salimi, P.; Akbar, F., α-admissible mappings and related fixed point theorems, No. 2013 (2013) · Zbl 1293.54023 |
| [10] | Hussain, N.; Karapinar, E.; Salimi, P.; Vetro, P., Fixed point results for [InlineEquation not available: see fulltext.]-Meir-Keeler contractive and G-[InlineEquation not available: see fulltext.]-Meir-Keeler contractive mappings, No. 2013 (2013) |
| [11] | Karapinar, E.; Kumam, P.; Salimi, P., On α-ψ-Meir-Keeler contractive mappings, No. 2013 (2013) · Zbl 1423.54083 |
| [12] | Asl, JH; Rezapour, S.; Shahzad, N., On fixed point of α-contractive multifunctions, No. 2012 (2012) · Zbl 1293.54017 |
| [13] | Karapinar, E.; Samet, B., Generalized α-ψ contractive type mappings and related fixed point theorems with applications, No. 2012 (2012) · Zbl 1252.54037 |
| [14] | Salimi, P.; Latif, A.; Hussain, N., Modified α-ψ-contractive mappings with applications, No. 2013 (2013) · Zbl 1293.54036 |
| [15] | Samet B, Vetro C, Vetro P: Fixed point theorem forα-ψcontractive type mappings.Nonlinear Anal. 2012, 75:2154-2165. · Zbl 1242.54027 · doi:10.1016/j.na.2011.10.014 |
| [16] | Kadelburg Z, Radenović S: Meir-Keeler-type conditions in abstract metric spaces.Appl. Math. Lett. 2011,24(8):1411-1414. · Zbl 1220.54027 · doi:10.1016/j.aml.2011.03.021 |
| [17] | Suzuki T: A generalized Banach contraction principle that characterizes metric completeness.Proc. Am. Math. Soc. 2008, 136:1861-1869. · Zbl 1145.54026 · doi:10.1090/S0002-9939-07-09055-7 |
| [18] | Takahashi W, Itoh S: The common fixed point theory of singlevalued mappings and multivalued mappings.Pac. J. Math. 1978,79(2):493-508. · Zbl 0371.47042 · doi:10.2140/pjm.1978.79.493 |
| [19] | Takahashi W, Shimizu T: Fixed points of multivalued mappings in certain convex metric spaces.Topol. Methods Nonlinear Anal. 1996,8(1):197-203. · Zbl 0902.47049 |
| [20] | Takahashi W, Lin L-J, Wang SY: Fixed point theorems for contractively generalized hybrid mappings in complete metric spaces.J. Nonlinear Convex Anal. 2012,13(2):195-206. · Zbl 1244.54094 |
| [21] | Azam A, Arshad M: Fixed points of a sequence of locally contractive multivalued maps.Comput. Math. Appl. 2009, 57:96-100. · Zbl 1165.47305 · doi:10.1016/j.camwa.2008.09.039 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
