Commuting functions with no common fixed point

Common Fixed Point Theorems With Applications to Nonlinear Integral Equations

“…Therefore, Az = Bz = Sz = Tz = z and z is a common fixed point of A, B, S and T. The uniqueness of a common fixed point can be easily verified by using (2). The other cases (a i), (02) and (03) Ty) for all x, y, 6 X, where 0 < a < 1 and any one of (ai)-(a4).…”

“…The fixed point theory of commuting mappings has been an interesting and ever growing area of enquiry since the famous problem that whether two commuting continuous mappings on an interval have a common fixed point, was settled in the negative (see [2], [12]). Various applications (see for example, [1][2][3][4][5][6][7][8][9][10][11], [13][14][15]), weakenings (resp.…”

“…Various applications (see for example, [1][2][3][4][5][6][7][8][9][10][11], [13][14][15]), weakenings (resp. generalizations) of commutativity in fixed point theory have been suggested in recent years, and one of them, known as compatibility condition, introduced by Jungck [16] has been very sensational as far as its fertility and utility are concerned (see for example, [16][17][18][19][20][21][22][23][24], [26,27], [29][30][31][32]).…”

Some problems in metric fixed point theory

“…In 1967 both W. M. Boyce [4] and H. Huneke [29] gave counterexamples to the problem. It is interesting to note that the mappings in Huneke's counterexample have Lipschitz constant 3 + √ 6.…”

Kakutani-type fixed point theorems: A survey

“…of more than one function. In 1969, Boyce [4] and Huneke [19] independently published samples of two commuting continuous maps f, g of [0, 1] into itself without a common f.p. ; i.e., for no point x ∈ [0, 1], f (x) = x = g(x) is true.…”