Stability of Jungck-type iterative procedures

Let (X ,d) be a metric space and T : X → X . Several physical problems, expressed as a fixed point equation Tx = x, are solved by approximating a sequence {xn} ⊂ X generated by an iterative procedure f (T ,xn). Let the sequence {xn} converge to a fixed point of T . The iterative procedure f (T ,xn) is considered numerically stable if and only if a sequence {yn} ⊂ X approximatively close to {xn} converges to the desired solution of the equation Tx = x. This kind of study in R1 was initiated by Urabe [32] during the middle of the last century (see also, Collatz [3] and Ortega and Rheinboldt [17]). However, a formal definition of the stability of general iterative procedures is due to Harder and Hicks [7, 8] (cf. definition below). Ostrowski [20] appears to be the first to discuss the stability of iterative procedures on metric spaces. Due to its increasing importance in computational mathematics, especially due to the revolution in computer programming, the stability theory has extensively been studied by various authors (see, e.g., Berinde [1], Czerwik et al. [4], Istrăţescu [9], Jachymski [11], Matkowski and Singh [15], Osilike [18, 19], Rus et al. [27], Rhoades [24, 25], and several references of [1, 27]). However, in all these references cited above, special cases of the following fundamental iterative procedure