In this paper two new fixed point results are studied. The first result is a theorem that involves (α −β) type rational singlevalued contractions, in the sense of Geraghty type operators. The second result consists of multivalued modified Hardy Rogers operators, namely the existence of the fixed point, data dependence, local version involving two metrics and homotopy theorems involving two metrics are studied.
Cristian Daniel Alecsa
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
fixed point theorems; complete metric space; rational contractions; multivalued; homotopy; Geraghty;
C.-D. Alecsa, Some fixed point results linked to α –β rational contractions and modified multivalued Hardy-Rogers operators, J. Fixed Point Theory, 2018, 2018:3.
About this paper
J. Fixed Point Theory
google scholar link
 C.D. Agarwal, D. O’Regan, Fixed point theory for generalized contractions on spaces with two metrics, J. Math. Anal. Appl. 248 (2000), no.2, 402-414.
 C.D. Agarwal, J.H. Dshalalow, D. O’Regan, Fixed point and homotopy results for generalized contractive maps of Reich type, Appl. Anal. 82 (2003), no.4, 329-350.
 M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.
 P.S. Kumari, D. Panthi, Connecting various type of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings, Fixed Point Theory Appl. 2016 (2016), Article ID 15.22 C.D. ALECSA
 T. Lazar, D. O’Regan, A. Petrusel, Fixed points and homotopy results for Ciric-type multivalued operators on a set with two metrics, Bull. Korean Math. Soc. 45 (2008), no.1, 67-73.
 A. Oprea, Fixed point theorems for multivalued generalized contractions of rational type in complete metric spaces, Creat. Math. Inform. 23 (2014), 99-106.
 N.S. Papageorgiou, S. Hu, Handbook of Multivalued Analysis (vol. I and II), Kluwer Acad. Publ., Dordrecht (1997 and 1999).
 L. Paunovic, P. Kaushik, S. Kumar, Some applications with new admissibility contractions in b-metric spaces, J. Nonlinear Sci. Appl. 10 (2017), 4162-4174.
 A. Petrusel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn. 59, 169-202.
 I.A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.
 I.A. Rus, A. Petrusel, A. Sîntmarian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003), no.8, 1947-1959.
 R.J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 373.
 W. Sintunavarat, Generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for α −β−contraction mappings in metric spaces, Sci. World J. 2014(2014), article ID 569174, 7 pages.
 F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, J. Appl. Math., 2014(2014), Article ID 929821