In this paper, two open questions on strong $b$-metric spaces posed by Kirk and Shahzad [11, Chapter 12] are investigated. A counterexample is constructed to give a negative answer to the first question, and a theorem on the completion of a strong $b$-metric space is proved to give a positive answer to the second question.
References
T. Van An, N. Van Dung, Z. Kadelburg and S. Radenović, Various generalizations of metric spaces and fixed point theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109 (2015), no. 1, 175–198. 10.1007/s13398-014-0173-7 MR3315711 1318.54038 T. Van An, N. Van Dung, Z. Kadelburg and S. Radenović, Various generalizations of metric spaces and fixed point theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109 (2015), no. 1, 175–198. 10.1007/s13398-014-0173-7 MR3315711 1318.54038
T. V. An, L. Q. Tuyen and N. V. Dung, Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015), 50–64. 10.1016/j.topol.2015.02.005 MR3319463 1322.54014 T. V. An, L. Q. Tuyen and N. V. Dung, Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015), 50–64. 10.1016/j.topol.2015.02.005 MR3319463 1322.54014
R. M. Connell, R. Kwok, J. Curlander, W. Kober and S. Pang, $\Psi$-$S$ correlation and dynamic time warping: Two methods for tracking ice floes in SAR images, IEEE Trans. Geosci. Remote Sens. 29 (1991), no. 6, 1004–1012. 10.1109/36.101377 R. M. Connell, R. Kwok, J. Curlander, W. Kober and S. Pang, $\Psi$-$S$ correlation and dynamic time warping: Two methods for tracking ice floes in SAR images, IEEE Trans. Geosci. Remote Sens. 29 (1991), no. 6, 1004–1012. 10.1109/36.101377
S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), no. 1, 5–11. MR1250922 0849.54036 S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), no. 1, 5–11. MR1250922 0849.54036
A. L. Dontchev and W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), no. 2, 481–489. 10.1090/s0002-9939-1994-1215027-7 0804.49021 A. L. Dontchev and W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), no. 2, 481–489. 10.1090/s0002-9939-1994-1215027-7 0804.49021
N. V. Dung, N. T. T. Ly, V. D. Thinh and N. T. Hieu, Suzuki-type fixed point theorems for two maps on metric-type spaces, J. Nonlinear Anal. Optim. 4 (2013), no. 2, 17–29. MR3145675 N. V. Dung, N. T. T. Ly, V. D. Thinh and N. T. Hieu, Suzuki-type fixed point theorems for two maps on metric-type spaces, J. Nonlinear Anal. Optim. 4 (2013), no. 2, 17–29. MR3145675
R. Fagin, R. Kumar and D. Sivakumar, Comparing top $k$ lists, SIAM J. Discrete Math. 17 (2003), no. 1, 134–160. 10.1137/s0895480102412856 R. Fagin, R. Kumar and D. Sivakumar, Comparing top $k$ lists, SIAM J. Discrete Math. 17 (2003), no. 1, 134–160. 10.1137/s0895480102412856
N. T. Hieu and N. V. Dung, Some fixed point results for generalized rational type contraction mappings in partially ordered $b$-metric spaces, Facta Univ. Ser. Math. Inform. 30 (2015), no. 1, 49–66. MR3317562 06749334 N. T. Hieu and N. V. Dung, Some fixed point results for generalized rational type contraction mappings in partially ordered $b$-metric spaces, Facta Univ. Ser. Math. Inform. 30 (2015), no. 1, 49–66. MR3317562 06749334
W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. 10.1007/978-3-319-10927-5 1308.58001 W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. 10.1007/978-3-319-10927-5 1308.58001
P. Kumam, N. V. Dung and V. T. L. Hang, Some equivalences between cone $b$-metric spaces and $b$-metric spaces, Abstr. Appl. Anal. 2013 (2013), Art. ID 573740, 8 pp. 10.1155/2013/573740 P. Kumam, N. V. Dung and V. T. L. Hang, Some equivalences between cone $b$-metric spaces and $b$-metric spaces, Abstr. Appl. Anal. 2013 (2013), Art. ID 573740, 8 pp. 10.1155/2013/573740
Q. Xia, The geodesic problem in quasimetric spaces, J. Geom. Anal. 19 (2009), no. 2, 452–479. 10.1007/s12220-008-9065-4 MR2481970 1200.54013 Q. Xia, The geodesic problem in quasimetric spaces, J. Geom. Anal. 19 (2009), no. 2, 452–479. 10.1007/s12220-008-9065-4 MR2481970 1200.54013
