Some common fixed point theorems for Banach operator pairs with applications in best approximation

Abstract

The ordered pair ( T , I ) of two self-maps of a metric space ( X , d ) is called a Banach operator pair if the set F ( I ) of fixed points of I is T -invariant i.e. T ( F ( I ) ) ⊆ F ( I ) . Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318–323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149–155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466–1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241–244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249–252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349–351], Shahzad [N. Shahzad, Invariant approximations and R -subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39–45] and of few others.