A generalization on weak contractions in partially ordered \(b\)-metric spaces and its application to quadratic integral equations. (English) Zbl 1347.54064
Summary: We introduce the notion of almost generalized \((\psi, \varphi, L)\)-contractive mappings, and establish the coincidence and common fixed point results for this class of mappings in partially ordered complete \(b\)-metric spaces. Our results extend and improve several known results from the context of ordered metric spaces to the setting of ordered \(b\)-metric spaces. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
Keywords:
fixed point; common fixed point; coincidence point; integral equations; \(b\)-metric space; partially ordered setReferences:
| [1] | Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations.Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4 · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 |
| [2] | Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.Order 2005, 22: 223-239. 10.1007/s11083-005-9018-5 · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |
| [3] | Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces.Appl. Anal. 2008,87(1):109-116. 10.1080/00036810701556151 · Zbl 1140.47042 · doi:10.1080/00036810701556151 |
| [4] | Altun I, Durmaz G: Some fixed point theorems on ordered cone metric spaces.Rend. Circ. Mat. Palermo 2009, 58: 319-325. 10.1007/s12215-009-0026-y · Zbl 1184.54038 · doi:10.1007/s12215-009-0026-y |
| [5] | Beg I, Rashid Butt A: Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces.Nonlinear Anal. 2009, 71: 3699-3704. 10.1016/j.na.2009.02.027 · Zbl 1176.54028 · doi:10.1016/j.na.2009.02.027 |
| [6] | Ćirić, L.; Cakić, N.; Rajović, M.; Ume, JS, Monotone generalized nonlinear contractions in partially ordered metric spaces, No. 2008 (2008) · Zbl 1158.54019 |
| [7] | Grana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications.Nonlinear Anal. 2006, 65: 1379-1393. 10.1016/j.na.2005.10.017 · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 |
| [8] | Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets.Nonlinear Anal. 2009, 71: 3402-3410. · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240 |
| [9] | Berinde V: Approximating fixed points of weak contractions using the Picard iteration.Nonlinear Anal. Forum 2004, 9: 43-53. · Zbl 1078.47042 |
| [10] | Berinde V: Some remarks on a fixed point theorem for Ćirić-type almost contractions.Carpath. J. Math. 2009, 25: 157-162. · Zbl 1249.54078 |
| [11] | Berinde V: Common fixed points of noncommuting almost contractions in cone metric spaces.Math. Commun. 2010, 15: 229-241. · Zbl 1195.54070 |
| [12] | Berinde V: Approximating common fixed points of noncommuting almost contractions in metric spaces.Fixed Point Theory 2010, 11: 179-188. · Zbl 1218.54031 |
| [13] | Hussain, N.; Ðorić, D.; Kadelburg, Z.; Radenović, S., Suzuki-type fixed point results in metric type spaces, No. 2012 (2012) · Zbl 1274.54128 |
| [14] | Jovanović, M.; Kadelburg, Z.; Radenović, S., Common fixed point results in metric-type spaces, No. 2010 (2010) · Zbl 1207.54058 |
| [15] | Pacurar M: Fixed point theory for cyclic Berinde operators.Fixed Point Theory 2012, 11: 419-428. · Zbl 1237.54057 |
| [16] | Radenović S, Kadelburg Z: Quasi-contractions on symmetric and cone symmetric spaces.Banach J. Math. Anal. 2011,5(1):38-50. 10.15352/bjma/1313362978 · Zbl 1297.54058 · doi:10.15352/bjma/1313362978 |
| [17] | Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A: Some results on weakly contractive maps.Bull. Iran. Math. Soc. 2012,38(3):625-645. · Zbl 1391.54036 |
| [18] | Shah MH, Simić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces.J. Comput. Anal. Appl. 2012,14(2):290-297. · Zbl 1256.54081 |
| [19] | Shukla, S., Partial rectangular metric spaces and fixed point theorems, No. 2014 (2014) |
| [20] | Suzuki T: Fixed point theorems for Berinde mappings.Bull. Kyushu Inst. Technol., Pure Appl. Math. 2011, 58: 13-19. · Zbl 1244.54100 |
| [21] | Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions.Carpath. J. Math. 2008, 24: 8-12. · Zbl 1199.54205 |
| [22] | 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. 10.1016/j.amc.2010.12.060 · Zbl 1206.54040 · doi:10.1016/j.amc.2010.12.060 |
| [23] | Aghajani A, Radenović S, Roshan JR: Common fixed point results for four mappings satisfying almost generalized (S,T)-contractive condition in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5665-5670. 10.1016/j.amc.2011.11.061 · Zbl 1245.54035 · doi:10.1016/j.amc.2011.11.061 |
| [24] | Czerwik S: Contraction mappings inb-metric spaces.Acta Math. Inform. Univ. Ostrav. 1993, 1: 5-11. · Zbl 0849.54036 |
| [25] | Aghajani, A.; Arab, R., Fixed points of (ψ,ϕ,θ)-contractive mappings in partially ordered b-metric spaces and application to quadratic integral equation, No. 2013 (2013) · Zbl 1469.54037 |
| [26] | Azam, A.; Mehmood, N.; Ahmad, J.; Radenović, S., Multivalued fixed point theorems in cone b-metric spaces, No. 2013 (2013) · Zbl 1291.54051 |
| [27] | Boriceanu M: Fixed point theory for multivalued generalized contraction on a set with twob-metrics.Stud. Univ. Babeş-Bolyai, Math. 2009,LIV(3):3-14. · Zbl 1240.54118 |
| [28] | Boriceanu M: Strict fixed point theorems for multivalued operators inb-metric spaces.Int. J. Mod. Math. 2009,4(3):285-301. · Zbl 1221.54051 |
| [29] | George, R, Radenović, S, Reshma, KP, Shukla, S: Rectangular b-metric spaces and contraction principle. J. Nonlinear Sci. Appl. (2014, in press) · Zbl 1249.54078 |
| [30] | Hussain, N.; Parvaneh, V.; Roshan, JR; Kadelburg, Z., Fixed points of cyclic (ψ,φ,L,A,B)-contractive mappings in ordered b-metric spaces with applications, No. 2013 (2013) · Zbl 1469.54114 |
| [31] | Mustafa, Z.; Roshan, JR; Parvaneh, V., Coupled coincidence point results for (ψ,φ)-weakly contractive mappings in partially ordered Gb-metric spaces, No. 2013 (2013) · Zbl 1470.54087 |
| [32] | Mustafa, Z.; Roshan, JR; Parvaneh, V.; Kadelburg, Z., Some common fixed point results in ordered partial b-metric spaces, No. 2013 (2013) · Zbl 1489.54180 |
| [33] | Popović B, Radenović S, Shukla S: Fixed point results to TVS-coneb-metric spaces.Gulf J. Math. 2013, 1: 51-64. · Zbl 1389.54109 |
| [34] | Roshan JR, Shobkolaei N, Sedhi S, Parvaneh V, Radenović S:Common fixed point theorems for three maps in discontinuousGb-metric spaces.Acta Math. Sci. Ser. B 2014,34(5):1-12. |
| [35] | Shukla S: Partialb-metric spaces and fixed point theorems.Mediterr. J. Math. 2014, 11: 703-711. 10.1007/s00009-013-0327-4 · Zbl 1291.54072 · doi:10.1007/s00009-013-0327-4 |
| [36] | Zabihi, F.; Razani, A., Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, No. 2014 (2014) · Zbl 1338.54221 |
| [37] | Roshan JR, Parvaneh V, Altun I: Some coincidence point results in orderedb-metric spaces and applications in a system of integral equations.Appl. Math. Comput. 2014, 226: 725-737. · Zbl 1354.45010 · doi:10.1016/j.amc.2013.10.043 |
| [38] | Roshan, JR, Parvaneh, V, Kadelburg, Z: Common fixed point theorems for weakly isotone increasing mappings in ordered b-metric spaces. J. Nonlinear Sci. Appl. (to appear). www.tjnsa.com · Zbl 1392.54035 |
| [39] | Roshan, JR; Parvaneh, V.; Sedghi, S.; Shobkolaei, N.; Shatanawi, W., Common fixed points of almost generalized (ψ,φ)s-contractive mappings in ordered b-metric spaces, No. 2013 (2013) · Zbl 1295.54080 |
| [40] | Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures.Comput. Math. Appl. 2008, 55: 2512-2520. 10.1016/j.camwa.2007.10.026 · Zbl 1142.65360 · doi:10.1016/j.camwa.2007.10.026 |
| [41] | Pacurar M: Sequences of almost contractions and fixed points inb-metric spaces.An. Univ. Vest. Timiş., Ser. Mat.-Inform. 2010, 48: 125-137. · Zbl 1249.54086 |
| [42] | Hussain N, Shah MH: KKM mappings in coneb-metric spaces.Comput. Math. Appl. 2011, 62: 1677-1684. 10.1016/j.camwa.2011.06.004 · Zbl 1231.54022 · doi:10.1016/j.camwa.2011.06.004 |
| [43] | Czerwik S: Nonlinear set-valued contraction mappings inb-metric spaces.Atti Semin. Mat. Fis. Univ. Modena 1998,46(2):263-276. · Zbl 0920.47050 |
| [44] | Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces. Math. Slovaca (in press) · Zbl 1349.54078 |
| [45] | Khamsi MA, Hussain N: KKM mappings in metric type spaces.Nonlinear Anal. 2010,73(9):3123-3129. 10.1016/j.na.2010.06.084 · Zbl 1321.54085 · doi:10.1016/j.na.2010.06.084 |
| [46] | Ðukić, D.; Kadelburg, Z.; Radenović, S., Fixed points of Geraghty-type mappings in various generalized metric spaces, No. 2011 (2011) · Zbl 1231.54030 |
| [47] | Parvaneh, V.; Roshan, JR; Radenović, S., Existence of tripled coincidence point in ordered b-metric spaces and application to a system of integral equations, No. 2013 (2013) · Zbl 1425.54030 |
| [48] | Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces.Comput. Math. Appl. 2010, 60: 1776-1783. 10.1016/j.camwa.2010.07.008 · Zbl 1202.54039 · doi:10.1016/j.camwa.2010.07.008 |
| [49] | An, TV; Dung, NV; Kadelburg, Z.; Radenović, S., Various generalizations of metric spaces and fixed point theorems (2014) |
| [50] | Khamsi, MA, Remarks on cone metric spaces and fixed point theorems of contractive mappings, No. 2010 (2010) · Zbl 1194.54065 |
| [51] | Altman M: A fixed point theorem in compact metric spaces.Am. Math. Mon. 1975, 82: 827-829. 10.2307/2319801 · Zbl 0326.40001 · doi:10.2307/2319801 |
| [52] | Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points.Bull. Aust. Math. Soc. 1984, 30: 1-9. 10.1017/S0004972700001659 · Zbl 0553.54023 · doi:10.1017/S0004972700001659 |
| [53] | Ray BK: On Ciric’s fixed point theorem.Fundam. Math. 1977,94(3):221-229. · Zbl 0345.54044 |
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