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Common fixed points in best approximation for Banach operator pairs with Ćirić type $$I$$-contractions. (English) Zbl 1134.47039
A 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., $$T(F(I))\subseteq F(I)$$. A mapping $$T:(X,\|\cdot\|)\to (X,\|\cdot\|\}$$ is said to satisfy Ćirić type contractive condition if
$\|Tx-Ty\|\leq a\max\{\|x-y\|, c(\|x-Ty\|+ \|y-Tx\|)\}+ b\max\{\|x-Tx\|, \|y-Ty\|\},$ where $$a\in (0,1)$$, $$a+b=1$$, $$0\leq c<n$$, and $$n= \min\{\frac{2+q}{5+q}, \frac{2-q}{4}, \frac {4}{q+a}\}$$. In this paper, some common fixed point theorems, similar to those of Lj. B. Ćirić [Publ. Inst. Math., Nouv. Sér. 49(63), 174–178 (1991; Zbl 0753.54023); Arch. Math., Brno 29, No. 3–4, 145–152 (1993; Zbl 0810.47051); Czech. Math. J. 50, No. 3, 449–458 (2000; Zbl 1079.47509)], B. Fisher and S. Sessa [Int. J. Math. Math. Sci. 9, 23–28 (1986; Zbl 0597.47036)], G. Jungck [Int. J. Math. Math. Sci. 13, No. 3, 497–500 (1990); Zbl 0705.54034)], and R. N. Mukherjee and V. Verma [Math. Jap. 33, No. 5, 745–749 (1988; Zbl 0655.47047)], are proved for a Banach operator pair. As applications, common fixed point and approximation results for Banach operator pair satisfying Ćirić type contractive conditions are obtained without the assumption of linearity or affinity of either $$T$$ or $$I$$. The results proved in the present paper unify and generalize various known results to a more general class of noncommuting mappings.

##### MSC:
 47H10 Fixed-point theorems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A50 Best approximation, Chebyshev systems
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