Best Proximity Point Theorems for α-Rational Cyclic Contraction
Keywords:
Cyclic contraction, K-cyclic contraction, C-cyclic contraction, best proximity point and complete metric space
Abstract
By using the classical fundamental cyclic contraction and, we introduce the new notion of α-rational cyclic contraction, and then we establish some best proximity point theorems for non-self mapping in the frame work of complete metric space. Our results generalized and improve some main results in the literature. Some examples are given to support our main results.
References
M.A.Al-Thagafi and N.Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal., 70(10)(2009), 3665-3671.
J.Anuradha and P.Veeramani, Proximal pointwise contraction, Topol. Appl., 156(2009), 2942-2948.
C.Di Bari, T.Suzuki and C.Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Analysis A, 69(11)(2008), 3790-3794.
A.A.Eldred, W.A.Kirk and P.Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Mathematica, 171(3)(2005), 283-293.
A.A.Eldred and P.Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323(2006), 1001-1006.
S.Karpagam and S.Agrawal, Best proximity point theorems for p-cyclic MeirKeeler contractions, Fixed Point Theory Appl., (2009), Article ID 197308.
W.A.Kirk, P.S.Srinavasan and P.Veeramani, Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory, 4(2003), 79-89.
M.A.Petric, Bets proximity point theorems for weak cyclic Kann contractions, Faculty of Sciences and Mathematixs, University of Nis, Serbia, 25(1)(2011), 145-154.
S.Sadiq Basha, Extensions of Banachs contraction principle, Numer. Funct. Anal. Optim., 31(2010), 569-576.
S.Sadiq Basha, Best proximity points: global optimal approximate solutions, J. Glob. Optim. 49(2011), 15-21.
S.Sadiq Basha, N.Shahzad and R.Jeyaraj, Optimal approximate solutions of fixed point equations, Abstract and Applied Analysis, 2011(2011P), Article ID 174560.
S.Sadiq Basha, N.Shahzad and R.Jeyaraj, Common best proximity points: global optimal solutions, J. Optim. Theory Appl., 148(2011), 69-78.
V.Sankar Raj and P.Veeramani, Best proximity pair theorems for relatively nonexpansive mappings, Appl. General Topol., 10(2009), 21-28.
K.Wlodarczyk, R.Plebaniak and A.Banach, Best proximity points for cyclic and noncyclic set-valued relatively quasiasymptotic contractions in uniform spaces, Nonlinear Anal., 70(2009), 3332-3341.
K.Wlodarczyk, R.Plebaniak and C.Obczynski, Convergence theorems, best approximation and best proximity for setvalued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Anal., 72(2010), 794-805.
J.Anuradha and P.Veeramani, Proximal pointwise contraction, Topol. Appl., 156(2009), 2942-2948.
C.Di Bari, T.Suzuki and C.Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Analysis A, 69(11)(2008), 3790-3794.
A.A.Eldred, W.A.Kirk and P.Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Mathematica, 171(3)(2005), 283-293.
A.A.Eldred and P.Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323(2006), 1001-1006.
S.Karpagam and S.Agrawal, Best proximity point theorems for p-cyclic MeirKeeler contractions, Fixed Point Theory Appl., (2009), Article ID 197308.
W.A.Kirk, P.S.Srinavasan and P.Veeramani, Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory, 4(2003), 79-89.
M.A.Petric, Bets proximity point theorems for weak cyclic Kann contractions, Faculty of Sciences and Mathematixs, University of Nis, Serbia, 25(1)(2011), 145-154.
S.Sadiq Basha, Extensions of Banachs contraction principle, Numer. Funct. Anal. Optim., 31(2010), 569-576.
S.Sadiq Basha, Best proximity points: global optimal approximate solutions, J. Glob. Optim. 49(2011), 15-21.
S.Sadiq Basha, N.Shahzad and R.Jeyaraj, Optimal approximate solutions of fixed point equations, Abstract and Applied Analysis, 2011(2011P), Article ID 174560.
S.Sadiq Basha, N.Shahzad and R.Jeyaraj, Common best proximity points: global optimal solutions, J. Optim. Theory Appl., 148(2011), 69-78.
V.Sankar Raj and P.Veeramani, Best proximity pair theorems for relatively nonexpansive mappings, Appl. General Topol., 10(2009), 21-28.
K.Wlodarczyk, R.Plebaniak and A.Banach, Best proximity points for cyclic and noncyclic set-valued relatively quasiasymptotic contractions in uniform spaces, Nonlinear Anal., 70(2009), 3332-3341.
K.Wlodarczyk, R.Plebaniak and C.Obczynski, Convergence theorems, best approximation and best proximity for setvalued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Anal., 72(2010), 794-805.
How to Cite
Sharma, & M.R.Yadav. (2016). Best Proximity Point Theorems for α-Rational Cyclic Contraction. International Journal of Current Research in Science and Technology, 2(1), 79-84. Retrieved from https://crst.gfer.org/index.php/crst/article/view/63
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