##### Title: On Some New Contractive Conditions for Asymptotically Regular Set-Valued Mappings

##### Pages: 586-593

##### Cite as:

Pradip Debnath, Manuel de La Sen, On Some New Contractive Conditions for Asymptotically Regular Set-Valued Mappings, Int. J. Anal. Appl., 18 (4) (2020), 586-593.#### Abstract

In this paper, we introduce two new contractive conditions of Proinov-type for asymptotically regular set-valued mappings and prove the existence of their fixed points. Our results extend some results due to Nadler and Boyd and Wong.

##### Full Text: PDF

#### References

- J. P. Aubin and J. Siegel. Fixed points and stationary points of dissipative multivalued maps. Proc. Amer. Math. Soc., 78 (1980), 391–398.
- I. Beg and A. Azam. Fixed points of asymptotically regular multivalued mappings. J. Austral. Math. Soc. (Series A), 53 (1992), 284–289.
- M. Berinde and V. Berinde. On a general class of multi-valued weakly picard mappings. J. Math. Anal. Appl., 326 (2007), 772–782.
- M. Boriceanu, A. Petrusel, and I.A. Rus. Fixed point theorems for some multivalued generalized contraction in b-metric spaces. Int. J. Math. Stat., 6 (2010), 65–76.
- D. W. Boyd and J. S. Wong. On nonlinear contractions. Proc. Amer. Math. Soc., 20 (1969), 458–464.
- F. E. Browder and W. V. Petryshyn. The solution by iteration of nonlinear functional equation in Banach spaces. Bull Am. Math. Soc., 72 (1966), 571–576.
- S. Czerwik. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Univ. Modena, 46 (1998), 263–276.
- P. Z. Daffer and H. Kaneko. Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl., 192 (1995), 655–666.
- P. Debnath and M. de La Sen. Contractive inequalities for some asymptotically regular set-valued mappings and their fixed points. Symmetry, 12 (2020), 411.
- P. Debnath and M. de La Sen. Fixed points of eventually δ-restrictive and δ( )-restrictive set-valued maps in metric spaces. Symmetry, 12 (2020), 127.
- H. W. Engl. Weak convergence of asymptotically regular sequences for nonexpansive mappings and connections with certain Chebyshef–centers. Nonlinear Anal., Theory Methods Appl., 1 (5) (1977), 495–501.
- M. D. Guay and K. L. Singh. Fixed points of asymptotically regular mappings. Math. Vesnik, 35 (1983), 101–106.
- S. Itoh and W. Takahashi. Single valued mappings, multivaluued mappings and fixed point theorems. J. Math. Anal. Appl., 59 (1977), 514–521.
- S. B. Nadler. Multi-valued contraction mappings. Pac. J. Math., 30 (2) (1969), 475–488.
- P. D. Proinov. Fixed point theorems for generalized contractive mappings in metric spaces. J. Fixed Point Theory Appl., 22 (2020), 21.
- B. E. Rhoades. A comparison of various definitions of contractive mappings. Trans. Amer. Math. Soc., 226 (1977), 257–290.
- B. E. Rhoades, S. Sessa, M. S. Khan, and M. Swaleh. On fixed points of asymptotically regular mappings. J. Austral. Math. Soc. (Ser. A), 43 (1987), 328–346.
- B. E. Rhoades, S. L. Singh, and C. Kulshrestha. Coincidence theorem for some multivalued mappings. Int. J. Math. and Math. Sci., 7 (3) (1984), 429–434.
- S. L. Singh, S. N. Mishra, and R. Pant. New fixed point theorems for asymptotically regular multi-valued maps. Nonlinear Anal., Theory Methods Appl., 71 (2009), 3299–3304.