# On multi-valued weak quasi-contractions in b-metric spaces

Volume 10, Issue 7, pp 3815--3823
Publication Date: July 27, 2017 Submission Date: May 30, 2017
• 1801 Views

### Authors

Nawab Hussain - Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Zoran D. Mitrović - Faculty of Electrical Engineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina.

### Abstract

We introduce some generalizations of the contractions for multi-valued mappings and establish some fixed point theorems for multi-valued mappings in b-metric spaces. Our results generalize and extend several known results in b-metric and metric spaces. Some examples are included which illustrate the cases when the new results can be applied while the old ones cannot.

### Share and Cite

##### ISRP Style

Nawab Hussain, Zoran D. Mitrović, On multi-valued weak quasi-contractions in b-metric spaces, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3815--3823

##### AMA Style

Hussain Nawab, Mitrović Zoran D., On multi-valued weak quasi-contractions in b-metric spaces. J. Nonlinear Sci. Appl. (2017); 10(7):3815--3823

##### Chicago/Turabian Style

Hussain, Nawab, Mitrović, Zoran D.. "On multi-valued weak quasi-contractions in b-metric spaces." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3815--3823

### Keywords

• Fixed points
• b-metric space
• set-valued mapping.

•  47H10

### References

• [1] M. Abbas, N. Hussain, B. E. Rhoades, Coincidence point theorems for multivalued f-weak contraction mappings and applications, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 105 (2011), 261–272.

• [2] A. Amini-Harandi, Fixed point theory for set-valued quasi-contraction maps in metric spaces, Appl. Math. Lett., 24 (2011), 1791–1794.

• [3] H. Aydi, M. F. Bota, E. Karapınar, S. Mitrović, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., 2012 (2012), 8 pages.

• [4] I. A. Bakhtin, The contraction mapping principle in almost metric space, (Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 30 (1989), 26–37.

• [5] M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl., 326 (2007), 772–782.

• [6] S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), 727–730.

• [7] L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) (N.S.), 12 (1971), 19–26.

• [8] L. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267–273.

• [9] M. Cosentino, P. Salimi, P. Vetro, Fixed point results on metric-type spaces, Acta Math. Sci. Ser. B Engl. Ed., 34 (2014), 1237–1253.

• [10] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.

• [11] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263–276.

• [12] N. Hussain, A. Amini-Harandi, J. Y. Cho, Approximate endpoints for set-valued contractions in metric spaces, Fixed Point Theory Appl., 2010 (2010), 13 pages.

• [13] N. Hussain, D. Dorić, Z. Kadelburg, S. Radenović, Suzuki-type fixed point results in metric type spaces, Fixed Point Theory Appl., 2012 (2012), 12 pages.

• [14] N. Hussain, V. Parvaneh, J. R. Roshan, Z. Kadelburg, Fixed points of cyclic weakly $( \psi,\varphi, L,A, B)$-contractive mappings in ordered b-metric spaces with applications, Fixed Point Theory Appl., 2013 (2013), 18 pages.

• [15] N. Hussain, R. Saadati, R. P. Agrawal, On the topology and wt-distance on metric type spaces, Fixed Point Theory Appl., 2014 (2014), 14 pages.

• [16] N. Hussain, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl., 62 (2011), 1677–1684.

• [17] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.

• [18] M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., 2010 (2010), 7 pages.

• [19] M. A. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), 3123–3129.

• [20] R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19 (2017), 2153–2163

• [21] S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488.

• [22] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121–124.

• [23] J. R. Roshan, V. Parvaneh, I. Altun, Some coincidence point results in ordered b-metric spaces and applications in a system of integral equations, Appl. Math. Comput., 226 (2014), 725–737.

• [24] J. R. Roshan, V. Parvaneh, S. Radenović, M Rajović, Some coincidence point results for generalized ($\psi,\varphi$)-weakly contractions in ordered b-metric spaces, Fixed Point Theory Appl., 2015 (2015), 21 pages.

• [25] J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei,W. Shatanawi, Common fixed points of almost generalized $(\psi,\varphi)_s$- contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl., 2013 (2013), 23 pages.

• [26] S. Shukla, S. Radenović, C. Vetro, Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces, Int. J. Math. Math. Sci., 2014 (2014), 9 pages.