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...he operator under consideration and some other iterative procedures should be considered. This results in the generalization of the function iteration in various ways in different settings. The Mann iteration scheme [6], for ,...2,1,0=n and ],1,0[⊂nα is defined as ,)1(1 nnnnn Txxx αα +−=+ (1) 1238 B. Prasad and R. Sahni This is further generalized by Ishikawa [4] in the following way: ,)1( )1(1 nnnnn nnnnn Txxy Tyxx ββ αα +−= +−=+ (2) where ]1,0[, ⊂nn βα and ,...2,1,0=n . Further, these iterative schemes are developed by taking two mappings ,:, XYTS → where )()( YSYT ⊆ and Yx ∈0 . Singh et al [11] discuss the following iterative procedure: ,...1,0),,(1 ==+ nxTfSx nn (3) It is called Jungck-iterative procedures (also see [5]). If ),( nxTf in (3) is replaced by nTx and nnnn TxSx αα +− )1( , it becomes Jungck-Picard and Jungck-Mann iteration respectively. Jungck–Ishikawa iteration (see [7]) is defined in the following manner: ⎭ ⎬ ⎫ +−= +−=+ nnnnn nnnnn TxSxSz TzSxSx ββ αα )1( )1(1 (4) For a detailed discussion and stability of various iterative schemes, see Berinde [2], Rhoades [8], Rhoades and Saliga [9], Singh and Prasad [10] and Singh et al. [11] and references thereof. 2. Preliminarie...

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...mappings was due to Ostrowski [16] where he established the stability of the Picard iteration by using Banach contraction condition . Harder and Hicks [6], Rhoades [19], Osilike [14] and Singh et al. =-=[20]-=- used the method of the summability theory of infinite matrices to prove various stability results for certain contractive definitions. Harder [6] established applications of stability results to firs...

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...} n = 0, 1, · · · , where {αn}∞n=0 and {βn}∞n=0 are sequences in [0, 1], is called the Ishikawa iteration process (see Ishikawa [14]). The following is the iteration process introduced by Singh et al =-=[39]-=- to establish some stability results: Let S and T be operators on an arbitrary set Y with values in E such that T (Y ) ⊆ S(Y ). S(Y ) is a complete subspace of E. Then, for x0 ∈ Y, the sequence {Sxn}∞...

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...s, one may refer to Agarwal et al [1], Berinde [2], Czerwik et al. [4], Harder and Hicks [5]-[6], Istrǎţescu [8], Osilike [10], Rhoades [13]-[15], Rus et al. [16], Prasad and Sahni [12], Singh et al. =-=[17]-=- Singh and Prasad [18] and references thereof. Recently Timis and Berinde [20] observed that the concept of stability is not very precise because of the choice of an arbitrary sequence in place of an ...

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...approximated over the years by various authors and we shall discuss some of these iterative processes in the present paper. Convergence results for Jungck-type iterative processes. . . 81 Singh et al =-=[28]-=- defined the following general iterative process to prove some stability results: Let S, T : Y → E and T (Y ) ⊆ S(Y ). For any x0 ∈ Y , define {Sxn}∞n=0 ⊂ E iteratively by Sxn+1 = f(T, xn), n = 0, 1, ...

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...1]. Normed spaces and their subsets are hyperbolic spaces as well as convex metric spaces. The class of hyperbolic spaces is properly contained in the class of convex metric spaces (see [18], [20]). It is worth to mention that an important and interesting example of convex metric spaces is a CAT (0) space (see [4], [6], [13], [16]). Search for new iterative scheme to solve nonlinear problems is one of the main themes of fixed point theory. For various iterative schemes, convergency, stability and data dependency have been investigated in ([1], [2], [5], [7], [9], [11], [12], [14], [15], [17], [23], [24]). Let C be a nonempty convex subset of a convex metric space M and T : C → C be an operator. Let {α(1)n }, {α(2)n }, {α(3)n }, {α(4)n } and {α(5)n } be sequences in [0, 1] satisfying certain control condition (s). Recently, Fukhar-ud-din and Berinde [8] have proposed the following scheme for a given x (1) 1 ∈ C: (1.6) x (1) n+1 = W ( Tx (2) n ,W ( Tx (3) n , x (1) n , α(2)n 1−α(1)n ) , α (1) n ) , x (2) n = W ( Tx (3) n ,W ( Tx (1) n , x (1) n , α(5)n 1−α(4)n ) , α (4) n ) , x (3) n = W ( Tx (1) n , x (1) n , α (3) n ) , n ≥ 1. It has been proved by Fukhar-ud-din and Berinde t...