Abstract
In this research work, we generalize classical results on the existence of common fixed points of generalized expanding mappings by removing the constraint on the signs of the underlining coefficients in the inequality considered. The hybrid class obtained surprisingly regroups both classes of generalized contractive maps and classes of generalized expansive maps. The research work also provides a clear understanding of the boundary between contractive and expansive mappings in literature and is applied to product metric-type spaces.
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Aage, C.T., Salunke, J.N.: Some fixed point theorems for expansion onto mappings on cone metric spaces. Acta Math. Sin. Engl. Ser. 27(6), 1101–1106 (2011)
Abbas, M., Rhoades, B.E., Nazir, T.: Common fixed points for four maps in cone metric spaces. Appl. Math. Comput. 216, 80–86 (2010)
Bakhtin, I.A.: The contraction mapping principle in almost metric spaces. Funct. Anal. Gos. Ped. Inst. Unian. 30, 26–37 (1989)
Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund. Math. 3, 133–181 (1922)
Bhashkar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal. Theory Methods Appl. 65(7), 1379–1393 (2006)
Ciric, L.B.: Generalized contractions and fixed-point theorems. Publ. l’Inst. Math. (Beograd) (NS) 12(26), 19–26 (1971)
Daffer, P.Z., Kaneko, H.: On expansive mappings. Math. Jpn. 37(1992), 733–735 (1992)
Han, Y., Xu, S.: Some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 3, 1–9 (2013)
Huang, H., Xu, S.: Correction: Fixed point theorems of contractive mappings in cone \(b\)-metric spaces and applications. Fixed Point Theory Appl. 2014, 55 (2014)
Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332(2), 1468–1476 (2007)
Hussain, N., Shah, M.H.: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, 1677–1684 (2011)
Jungck, G., Rhoades, B.E.: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 29(3), 227–238 (1998)
Kumar, S., Garg, S.K.: Expansion mapping theorems in metric spaces. Int. J. Contemp. Math. Sci. 4(36), 1749–1758 (2009)
Nashine, H.K., Kadelburg, Z., Radenovic, S.: Coupled common fixed point theorems for w*-compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 218, 5422–5432 (2012)
Olaleru, J.O.: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 1–10 (2009) (Article ID 657914)
Olaleru, J.: Common fixed points of three self-mappings in cone metric spaces. Appl. Math. E-notes 11, 41–49 (2011)
Olaoluwa, H., Olaleru, J.O., Chang, S.S.: Coupled fixed point theorems for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2013, 68 (2013)
Olaoluwa, H., Olaleru, J.: Multipled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2014, 43 (2014)
Olaoluwa, H., Olaleru, J.O.: An approach to the approximation of common coupled fixed points of contractive maps in metric-type spaces. Adv. Fixed Point Theory 5(1), 13–31 (2015)
Olaoluwa, H., Olaleru, J.O.: On common fixed points and multipled fixed points of contractive mappings in metric-type spaces (2015). doi:10.1016/j.jnnms.2015.06.001
Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)
Rhoades, B.E.: Contractive definitions. In: Rassias, T.M. (ed.) Nonlinear Analysis, pp. 513–526. Word Scientific Publishing Company, NJ (1988)
Sahin, I., Telci, M.: A theorem on common fixed points of expansion type mappings in cone metric spaces. An. St. Univ. Ovid. Const. 18, 329–336 (2010)
Samet, B., Vetro, C.: Coupled fixed point, \(F\)-invariant set and fixed point of \(N\)-order. Ann. Funct. Anal. 1(2), 46–56 (2011)
Shatanawi, W., Awawdeh, F.: Some fixed and coincidence point theorems for expansive maps in cone metric spaces. Fixed Point Theory Appl. 2012, 19 (2012)
Wang, S.-Z., Li, B.-Y., Gao, Z.-M., Lseki, K.: Some fixed points of expansion mappings. Math. Jpn. 29, 631–636 (1984)
Xu, S., Cheng, S., Han, Y.: Correction: some new theorems of expanding mappings without continuity in cone metric spaces. Fixed Point Theory Appl. 2014, 178 (2014)
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The authors are grateful to the referees for their helpful suggestions contributing to the improvement of the paper.
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Olaoluwa, H., Olaleru, J.O. A hybrid class of expansive-contractive mappings in cone b-metric spaces. Afr. Mat. 27, 825–840 (2016). https://doi.org/10.1007/s13370-015-0381-0
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DOI: https://doi.org/10.1007/s13370-015-0381-0