On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

⟨(I − T) x − (I − T) y, j (x − y)⟩ ≥ k 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 (4) for all x, y ∈ D(T), where k = (t − 1)/t ∈ (0, 1). Consequently, from inequality (4) it follows easily that T is strongly pseudocontractive if and only if 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩x − y + s [(I − T − kI) x − (I − T − kI) y] 󵄩󵄩󵄩󵄩 (5) for all x, y ∈ D(T) and s > 0. Closely related to the class of pseudocontractive maps is the class of accretive operators. Let A : D(A) → E be an operator.


Introduction and Preliminaries
Let  be a real Banach space with dual  * .The symbol () stands for the domain of .
Also, as a consequence of Kato [1], this accretive condition can be expressed in terms of the duality mapping as follows.
For each ,  ∈ (), there exists (−) ∈ (−) such that Consequently, inequality (2) for all ,  ∈ ().The operator  satisfying inequality (8) is called strongly accretive.It is then clear that  is strongly accretive if and only if  = ( − ) is strongly pseudocontractive.Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive mappings.We will exploit this connection in the sequel.The notion of accretive operators was introduced independently in 1967 by Kato [1] and Browder [2].An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem is solvable if  is locally Lipschitzian and accretive on .
If  is independent of , then  = 0 and the solution of this equation corresponds to the equilibrium points of the system (9).Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].Two well-known iterative schemes, the Mann iterative method (see, e.g., [20]) and the Ishikawa iterative scheme (see, e.g., [21]), have successfully been employed.The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the order ( −1/2 ).It is clear that if, for an operator , the classical iterative sequence of the form,  +1 =   ,  0 ∈ () (the socalled Picard iterative sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and, moreover, its rate of convergence is always at least as fast as that of a geometric progression.
In this paper, we study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings.We have also improved the results of Ćiri ć et al. [24].

Main Results
In the following theorems,  > 1 will denote the Lipschitz constant of the operator  and  will denote the strong accretive constant of the operator  as in inequality (8).Furthermore,  > 0 is defined by ,  ∈ (0, ) .
With these notations, we prove the following theorem.
Then {  } ∞ =0 converges strongly to  * with Observe that  * = 0 if and only if  * is a fixed point of .Moreover,  is strongly pseudocontractive since  is strongly accretive, and so  also satisfies inequality ( 5) for all ,  ∈  and  > 0. Furthermore, the recursion formula   =     becomes Observe that and from the recursion formula ( 21) which implies that This implies using inequality (5) with  = /(1 + ) and where From ( 29) and (30), we get       − which is similar to (21).Following the method of computations as in the proof of the Theorem 8, we obtain (39) Thus the relation between Ćirić et al. [24] and our parameter of convergence, that is, between  and , respectively, is the following: Our convergence parameter  shows the overall improvement for , and consequently the results of Ćirić et al. [24] are improved.
(7)h  = 1 yields that  is accretive if and only if  := ( − ) is pseudocontractive.Furthermore, from setting  := ( − ), it follows from inequality (5) that  is strongly pseudocontractive if and only if ( − ) is accretive, and, using(7), this implies that  = ( − ) is strongly pseudocontractive if and only if there exists