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Fixed points of cyclic weakly (\(\psi, \phi ,L, A, B\))-contractive mappings in ordered \(b\)-metric spaces with applications. (English) Zbl 06334850
Summary: We introduce the notion of ordered cyclic weakly (\(\psi, \phi ,L, A, B\))-contractive mappings, and we establish some fixed and common fixed point results for this class of mappings in complete ordered \(b\)-metric spaces. Our results extend several known results from the context of ordered metric spaces to the setting of ordered \(b\)-metric spaces. They are also cyclic variants of some very recent results in ordered \(b\)-metric spaces with even weaker contractive conditions. Some examples support our results and show that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI
[1] Kirk, WA; Srinivasan, PS; Veeramani, P, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 79-89, (2003) · Zbl 1052.54032
[2] Berinde, V, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum, 9, 43-53, (2004) · Zbl 1078.47042
[3] Berinde, V, Some remarks on a fixed point theorem for ćirić-type almost contractions, Carpath. J. Math, 25, 157-162, (2009) · Zbl 1249.54078
[4] Berinde, V, Common fixed points of noncommuting almost contractions in cone metric spaces, Math. Commun, 15, 229-241, (2010) · Zbl 1195.54070
[5] Berinde, V, Approximating common fixed points of noncommuting almost contractions in metric spaces, Fixed Point Theory, 11, 179-188, (2010) · Zbl 1218.54031
[6] Pacurar, M, Fixed point theory for cyclic berinde operators, Fixed Point Theory, 11, 419-428, (2012) · Zbl 1237.54057
[7] Suzuki, T, Fixed point theorems for berinde mappings, Bull. Kyushu Inst. Technol., Pure Appl. Math, 58, 13-19, (2011) · Zbl 1244.54100
[8] Babu, GVR; Sandhya, ML; Kameswari, MVR, A note on a fixed point theorem of berinde on weak contractions, Carpath. J. Math, 24, 8-12, (2008) · Zbl 1199.54205
[9] Ćirić, L; Abbas, M; Saadati, R; Hussain, N, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput, 217, 5784-5789, (2011) · Zbl 1206.54040
[10] Aghajani, A; Radenović, S; Roshan, JR, Common fixed point results for four mappings satisfying almost generalized \(####\)-contractive condition in partially ordered metric spaces, Appl. Math. Comput, 218, 5665-5670, (2012) · Zbl 1245.54035
[11] Pacurar, M, Remark regarding two classes of almost contractions with unique fixed point, Creative Math, 19, 178-183, (2010) · Zbl 1210.47078
[12] Khan, MS; Swaleh, M; Sessa, S, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc, 30, 1-9, (1984) · Zbl 0553.54023
[13] Bakhtin, IA, The contraction principle in quasimetric spaces, 26-37, (1989), Ul’yanovsk
[14] Czerwik, S, Contraction mappings in \(b\)-metric spaces, Acta Math. Inform. Univ. Ostrav, 1, 5-11, (1993) · Zbl 0849.54036
[15] Czerwik, S, Nonlinear set-valued contraction mappings in \(b\)-metric spaces, Atti Semin. Mat. Fis. Univ. Modena, 46, 263-276, (1998) · Zbl 0920.47050
[16] Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered \(b\)-metric spaces. Math. Slovaca (2012, in press) · Zbl 1349.54078
[17] Aydi, H; Bota, M-F; Karapinar, E; Moradi, S, A common fixed point for weak \(ϕ\)-contractions on \(b\)-metric spaces, Fixed Point Theory, 13, 337-346, (2012) · Zbl 1297.54080
[18] Boriceanu, M, Fixed point theory for multivalued contraction on a set with two \(b\)-metrics, Creative Math, 17, 326-332, (2008) · Zbl 1265.54155
[19] Boriceanu, M, Strict fixed point theorems for multivalued operators in \(b\)-metric spaces, Int. J. Mod. Math, 4, 285-301, (2009) · Zbl 1221.54051
[20] Bota, M, Multivalued fractals in \(b\)-metric spaces, Cent. Eur. J. Math, 8, 367-377, (2010) · Zbl 1235.54011
[21] Bota, M; Molnar, A; Varga, C, On ekeland’s variational principle in \(b\)-metric spaces, Fixed Point Theory, 12, 21-28, (2011) · Zbl 1278.54022
[22] Hussain, N; Ðorić, D; Kadelburg, Z; Radenović, S, Suzuki-type fixed point results in metric type spaces, No. 2012, (2012) · Zbl 1274.54128
[23] Khamsi, MA, Remarks on cone metric spaces and fixed point theorems of contractive mappings, No. 2010, (2010) · Zbl 1194.54065
[24] Khamsi, MA; Hussain, N, KKM mappings in metric type spaces, Nonlinear Anal, 73, 3123-3129, (2010) · Zbl 1321.54085
[25] Pacurar, M, Sequences of almost contractions and fixed points in \(b\)-metric spaces, An. Univ. Vest. Timiş., Ser. Mat.-Inform, 3, 125-137, (2010) · Zbl 1249.54086
[26] Roshan, JR, Shobkolaei, N, Sedghi, S, Abbas, M: Common fixed point of four maps in \(b\)-metric spaces. Hacet. J. Math. Stat. (2013, in press) · Zbl 1321.54099
[27] Roshan, JR; Parvaneh, V; Sedghi, S; Shobkolaei, N; Shatanawi, W, Common fixed points of almost generalized \(####\)-contractive mappings in ordered \(b\)-metric spaces, No. 2013, (2013) · Zbl 1295.54080
[28] Singh, SL; Prasad, B, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl, 55, 2512-2520, (2008) · Zbl 1142.65360
[29] Hussain, N; Shah, MH, KKM mappings in cone \(b\)-metric spaces, Comput. Math. Appl, 62, 1677-1684, (2011) · Zbl 1231.54022
[30] Ran, ACM; Reurings, MCB, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Am. Math. Soc, 132, 1435-1443, (2004) · Zbl 1060.47056
[31] Nieto, JJ; Lopez, RR, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239, (2005) · Zbl 1095.47013
[32] Shatanawi, W; Postolache, M, Common fixed point results of mappings for nonlinear contraction of cyclic form in ordered metric spaces, No. 2013, (2013) · Zbl 1286.54053
[33] Agarwal, RP; Hussain, N; Taoudi, M-A, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, No. 2012, (2012)
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