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

The ordered pair (T,I)(T,I) of two self-maps of a metric space (X,d)(X,d) is called a Banach operator pair if the set F(I)F(I) of fixed points of II is TT-invariant i.e. T(F(I))⊆F(I)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 RR-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39–45] and of few others.