Results in strongly minihedral cone and scalar weighted cone metric spaces and applications
Abstract
The convergence of sequences and non-unique fixed points are established in M-orbitally complete cone metric spaces over the strongly minihedral cone, and scalar weighted cone assuming the cone to be strongly minihedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions.
Keywords
Cone metric space; strongly minihedral; normal cone; nonunique fixed point
References
2. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
3. L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), no. 2, 1468–1476.
4. Z. Kadelburg, S. Radenović, and V. Rakočević, Topological vector space-valued cone metric spaces and fixed point theorems, Fixed Point Theory Appl. 2010, Art. ID 170253, 17 pp.
5. L.V. Kantorovich, On some further applications of the Newton approximation method, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 12 (1957), no. 7, 68–103 (in Russian).
6. E. Karapınar, Some nonunique fixed point theorems of Ćirić type on cone metric spaces, Abstr. Appl. Anal. 2010, Art. ID 123094, 14 pp.
7. M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, Art. ID 315398, 7 pp.
8. S.D. Lin, A common fixed point theorem in abstract spaces, Indian J. Pure Appl. Math. 18 (1987), no. 8, 685–690.
9. Sh. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mapping”, J. Math. Anal. Appl. 345 (2008), no. 2, 719–724.
10. B.E. Rhoades, Contractive definitions and continuity, in: R.F. Brown (ed.), Fixed Point Theory and Its Applications, Contemp. Math., 72, Amer. Math. Soc., Providence, 1988, pp. 233–245.
11. B. Rzepecki, On fixed point theorems of Maia type, Publ. Inst. Math. (Beograd) (N.S.) 28(42) (1980), 179–186.
12. A. Sonmez and H. Cakalli, Cone normed spaces and weighted means, Math. Comput. Modelling 52 (2010), no. 9-10, 1660–1666.
13. J.S. Vandergraft, Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432.
Government Degree College Thatyur (Tehri Garhwal), India India
https://orcid.org/0000-0001-8033-856X
S.G.R.R. (P.G.) College, India India
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
