Document Type : Research Paper
1 Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran.
3 Department of Mathematics, Faculty of Science, Arak Branch, Islamic Azad University, Arak, Iran.
In this paper, a new stratification of mappings, which is called $\Psi$-simulation functions, is introduced to enhance the study of the Suzuki type weak-contractions. Some well-known results in weak-contractions fixed point theory are generalized by our researches. The methods have been appeared in proving the main results are new and different from the usual methods. Some suitable examples are furnished to demonstrate the validity of the hypothesis of our results and reality of our generalizations.
 H. Argoubi, B. Samet, and C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order, J. Nonlinear Sci. Appl., 8 (2015), pp. 1082-1094.
 A.D. Arvanitakis, A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc., 131 (2003), pp. 3647–-3656.
 S. Banach, Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math., 3 (1922), pp. 133-181.
 V. Berinde, Approximating fixed points of weak $varphi$-contractions, Fixed Point Theory, 4 (2003), pp. 131-142.
 D. Doric, Common fixed point for generalized $(psi - varphi)$-weak contraction, Appl. Math. Lett., 22 (2009), pp. 1896-1900.
 D. Doric, Z. Kadelburg, and S. Radenovic, Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces, Nonlinear Anal., 75 (2012), pp. 1927-1932.
 D. Doric and R. Lazovic, Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications, Fixed Point Theory Appl., 2011 (2011), 13 pages.
 P.N. Dutta and B.S. Choudhary, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., 2008 (2008), 8 pages.
 F. Khojasteh, S. Shukla, and S. Radenovic, A new approach to the study of fixed point theorems via simulation functions, Filomat, 29 (2015), pp. 1189-1194.
 M. Kikkawa and T. Suzuki, Some notes on fixed point theorems with constants, Bull. Kyushu Inst. Technol. Pure Appl. Math., 56 (2009), pp. 11-18.
 M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69 (2008), pp. 2942-2949.
 W. Kirk and B. Sims, Handbook of metric fixed point theory, Springer Science & Business Media., 2001.
 A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), pp. 326-329.
 A. Nastasi and P. Vetro, Fixed point results on metric and partial metrric spaces via simulations functions, J. Nonlinear Sci. Appl.,
8 (2015), pp. 1059-1069.
 M. Olgun, O. Bicer, and T. Alyildiz, A new aspect to Picard operators with simulation functions, Turk. J. Math., 40 (2016), pp. 832-837.
 K.P.R. Rao, K.P.K. Rao, and H. Aydi, A Suzuki type unique common fixed point theorem for hybrid pairs of maps under a new condition in partial metric spaces, Mathematical Sciences, 7 (2013), 8 pages.
 B.E. Rhodes, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), pp. 2683-2693.
 A. Roldan, E. Karapinar, C. Roldan, and J. Martinez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), pp. 345–-355.
 S.L. Singh, S.N. Mishra, Renu Chugh, and Raj Kamal, General common fixed point theorems and applications, J. Appl. Math., 2012 (2012), 14 pages.
 S.L. Singh, R. Kamal, M. De La Sen, and Renu Chugh, A Fixed Point Theorem for Generalized Weak Contractions, Filomat, 29 (2015), pp. 1481-1490.
 S.L. Singh, Renu Chugh, and Raj Kamal, Suzuki type common fixed point theorems and applications, Fixed Point Theory, 14 (2) (2013), pp. 497-506.
 T. Suzuki, A generalized Banach contraction principle that Characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), pp. 1861-1869.
 T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (11) (2009), pp. 5313-5317.
 Q. Zhang and Y. Song, Fixed point theory for generalized $(psi - varphi)$-weak contractions, Appl. Math. Lett., 22 (2009), pp. 75-78.