# Ulam-Hyers stability, well-posedness and limit shadowing property of the fixed point problems for some contractive mappings in $M_s$-metric spaces

Volume 10, Issue 5, pp 2296--2308 Publication Date: May 22, 2017
• 474 Views

### Authors

Mi Zhou - School of Science and Technology, Sanya College, 572022, Sanya, Hainan, China.
Xiao-lan Liu - College of Science, Sichuan University of Science and Engineering, 643000, Zigong, Sichuan, China.
Yeol Je Cho - Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea.

### Abstract

In this paper, first, we introduce several types of the Ulam-Hyers stability, the well-posedness and the limit shadowing property of fixed point problems in $M_s$-metric spaces. Second, we give such results for fixed point problems of Banach and Kannan contractive mappings in $M_s$-metric spaces. Finally, we give some examples to illustrate the validity of our main results.

### Keywords

• Fixed point problem
• Ulam-Hyers stability
• well-posedness
• $M_s$-metric spaces.

### References

• [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.

• [2] M. F. Bota, E. Karapınar, O. Mleşniţe, Ulam-Hyers stability results for fixed point problems via $\alpha-\psi$-contractive mapping in (b)-metric space, Abstr. Appl. Anal., 2013 (2013), 6 pages.

• [3] M. F. Bota-Boriceanu, A. Petruşel, Ulam-Hyers stability for operatorial equations, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 57 (2011), 65–74.

• [4] F. S. De Blasi, J. Myjak, Sur la porosité de l’ensemble des contractions sans point fixe, (French) [[On the porosity of the set of contractions without fixed points]] C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 51–54.

• [5] R. Kannan , Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405–408.

• [6] B. K. Lahiri, P. Das, Well-posedness and porosity of a certain class of operators, Demonstratio Math., 38 (2005), 169–176.

• [7] N. M. Mlaiki, A contraction principle in partial S-metric spaces, Univers. J. Math. Math. Sci., 5 (2014), 109–119.

• [8] N. M. Mlaiki, N. Souayah, K. Abodayeh, T. Abdeljawad, Contraction principles inMs-metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 575–582.

• [9] A. Pansuwan, W. Sintunavarat, J. Y. Choi, Y. J. Cho, Ulam-Hyers stability, well-posedness and limit shadowing property of the fixed point problems in M-metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 4489–4499.

• [10] S. Reich, A. J. Zaslavski, Well-posedness of fixed point problems, Far East J. Math. Sci. (FJMS), Special Volume, Part III, (2001), 393–401.

• [11] S. Sedghi, N. Shobe, A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik, 64 (2012), 258–266.

• [12] P. V. Subrahmanyam, Completeness and fixed-points, Monatsh. Math., 80 (1975), 325–330.