JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 284937 10.1155/2013/284937 284937 Research Article On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces 0000-0002-5294-4447 Kang Shin Min 1 Rafiq Arif 2 Cho Sun Young 3 Muglia Luigi 1 Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea gnu.ac.kr 2 School of CS and Mathematics Hajvery University 43-52 Industrial Area Gulberg-III Lahore 54660 Pakistan hup.edu.pk 3 Department of Mathematics Gyeongsang National University Jinju 660-701 Republic of Korea gnu.ac.kr 2013 11 4 2013 2013 14 12 2012 26 03 2013 2013 Copyright © 2013 Shin Min Kang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009).

1. Introduction and Preliminaries

Let E be a real Banach space with dual E*. The symbol D(T) stands for the domain of T.

Let T:D(T)E be a mapping.

Definition 1.

The mapping T is said to be Lipschitzian if there exists a constant L>0 such that (1)Tx-TyLx-y for all x,yD(T).

Definition 2.

The mapping T is called strongly pseudocontractive if there exists t>1 such that (2)x-y(1+r)(x-y)-rt(Tx-Ty) for all x,yD(T) and r>0. If t=1 in inequality (2), then T is called pseudocontractive.

We will denote by J the normalized duality mapping from E to 2E* defined by (3)J(x)={f*E*:x,f*=x2=f*2}, where ·,· denotes the generalized duality pairing. It follows from inequality (2) that T is strongly pseudocontractive if and only if there exists j(x-y)J(x-y) such that (4)(I-T)x-(I-T)y,j(x-y)kx-y2 for all x,yD(T), where k=(t-1)/t(0,1). Consequently, from inequality (4) it follows easily that T is strongly pseudocontractive if and only if (5)x-yx-y+s[(I-T-kI)x-(I-T-kI)y] for all x,yD(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.

Definition 3.

The operator A is called accretive if (6)x-yx-y+s(Ax-Ay) for all x,yD(A) and s>0.

Also, as a consequence of Kato , this accretive condition can be expressed in terms of the duality mapping as follows.

For each x,yD(A), there exists j(x-y)J(x-y) such that (7)Ax-Ay,j(x-y)0. Consequently, inequality (2) with t=1 yields that A is accretive if and only if T:=(I-A) is pseudocontractive. Furthermore, from setting A:=(I-T), it follows from inequality (5) that T is strongly pseudocontractive if and only if (A-kI) is accretive, and, using (7), this implies that T=(I-A) is strongly pseudocontractive if and only if there exists k(0,1) such that (8)Ax-Ay,j(x-y)kx-y2 for all x,yD(A). The operator A satisfying inequality (8) is called strongly accretive. It is then clear that A is strongly accretive if and only if T=(I-A) 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  and Browder . An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem (9)dudt+Au=0,u(0)=u0 is solvable if A is locally Lipschitzian and accretive on E. If u is independent of t, then Au=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., . Two well-known iterative schemes, the Mann iterative method (see, e.g., ) and the Ishikawa iterative scheme (see, e.g., ), have successfully been employed.

The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the order O(n-1/2). It is clear that if, for an operator U, the classical iterative sequence of the form, xn+1=Uxn, x0D(U) (the so-called 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 [22, 23], Chidume proved the following results.

Theorem 4.

Let E be an arbitrary real Banach space and A:EE Lipschitz (with constant L>0) and strongly accretive with a strong accretive constant k(0,1). Let x* denote a solution of the equation Ax=0. Set ϵ:=(1/2)(k/(1+L(3+L-k))) and define Aϵ:EE by Aϵx:=x-ϵAx for each xE.

For arbitrary x0E, define the sequence {xn}n=0 in E by (10)xn+1=Aϵxn,n0. Then {xn}n=0 converges strongly to x* with (11)xn+1-x*δnx0-x*, where δ=(1-(1/2)kϵ)(0,1). Moreover, x* is unique.

Corollary 5.

Let E be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:KK be Lipschitz (with constant L>0) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,yK). Assume that T has a fixed point x*K. Set ϵ0:=(1/2)(k/(1+L(3+L-k))) and define Tϵ0:KK by Tϵ0x=(1-ϵ0)x+ϵ0Tx for each xK. For arbitrary x0K, define the sequence {xn}n=0 in K by (12)xn+1=Tϵ0xn,n0.

Then {xn}n=0 converges strongly to x* with (13)xn+1-x*δnx0-x*, where δ=(1-(1/2)kϵ0)(0,1). Moreover, x* is unique.

Recently, Ćirić et al.  improved the results of Chidume [22, 23], Liu , and Sastry and Babu  as in the following results.

Theorem 6.

Let E be an arbitrary real Banach space and A:EE a Lipschitz (with constant L>0) and strongly accretive with a strong accretive constant k(0,1). Let x* denote a solution of the equation Ax=0. Set ε:=(k-η)/L(2+L), η(0,k) and define Aε:EE by Aεx:=x-εAx for each xE. For arbitrary x0E, define the sequence {xn}n=0 in E by (14)xn+1=Aεxn,n0. Then {xn}n=0 converges strongly to x* with (15)xn+1-x*θnx0-x*, where θ=(1-((k-η)/(k(k-η)+L(2+L)))η)(0,1). Thus the choice η=k/2 yields θ=1-k2/2[k+2L(2+L)]. Moreover, x* is unique.

Corollary 7.

Let E be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:KK be Lipschitz (with constant L>0) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,yK). Assume that T has a fixed point x*K. Set ε0:=(k-η)/L(2+L), η(0,k) and define Tε0:KK by Tε0x=(1-ε0)x+ε0Tx for each xK. For arbitrary x0K, define the sequence {xn}n=0 in K by (16)xn+1=Tε0xn,n0. Then {xn}n=0 converges strongly to x* with (17)xn+1-x*θnx0-x*, where θ=(1-((k-η)/(k(k-η)+L(2+L)))η)(0,1). Moreover, x* is unique.

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 Ćirić et al. .

2. Main Results

In the following theorems, L>1 will denote the Lipschitz constant of the operator A and k will denote the strong accretive constant of the operator A as in inequality (8). Furthermore, ϵ>0 is defined by (18)ϵ:=k-ηL+(1+L)(k-η),η(0,k). With these notations, we prove the following theorem.

Theorem 8.

Let E be an arbitrary real Banach space and A:EE Lipschitz and strongly accretive with a strong accretive constant k(0,1). Let x* denote a solution of the equation Ax=0. Define Aϵ:EE by Aϵxn=(1-ϵ)xn-1+ϵxn-ϵAxn for each xnE. For arbitrary x0E, define the sequence {xn}n=0 in E by (19)xn=Aϵxn,n1. Then {xn}n=0 converges strongly to x* with (20)xn+1-x*ρnx0-x*, where ρ=(1-((k-η)/(L+((k-η)(1+L+k)))η)(0,1). Thus the choice η=k/2 yields ρ=1-k2/2[2L+k(1+L+k)]. Moreover, x* is unique.

Proof.

Existence of x* follows from [5, Theorem 13.1]. Define T=(I-A) where I denotes the identity mapping on E. Observe that Ax*=0 if and only if x* is a fixed point of T. Moreover, T is strongly pseudocontractive since A is strongly accretive, and so T also satisfies inequality (5) for all x,yE and s>0. Furthermore, the recursion formula xn=Aϵxn becomes (21)xn=(1-ϵ)xn-1+ϵTxn,n1. Observe that (22)x*=(1+ϵ)x*+ϵ(I-T-kI)x*-(1-k)ϵx*, and from the recursion formula (21) (23)xn-1=(1+ϵ)xn+ϵ(I-T-kI)xn-(1-k)ϵxn+ϵ2(xn-1-Txn), which implies that (24)xn-1-x*=(1+ϵ)(xn-x*)+ϵ[(I-T-kI)xn-(I-T-kI)x*]-(1-k)ϵ(xn-x*)+ϵ2(xn-1-Txn). This implies using inequality (5) with s=ϵ/(1+ϵ) and y=x* that (25)xn-1-x*(1+ϵ)((xn-x*)+ϵ1+ϵ×[(I-T-kI)xn-(I-T-kI)x*]ϵ1+ϵ)-(1-k)ϵxn-x*-ϵ2xn-1-Txn(1+ϵ  )xn-x*-(1-k)ϵxn-x*-ϵ2xn-1-Txn=(1+kϵ)xn-x*-ϵ2xn-1-Txn. Observe that (26)xn-1-Txnxn-1-Txn-1+Txn-1-TxnAxn-1+xn-1-xn+Axn-1-AxnLxn-1-x*+(1+L)xn-1-xn=Lxn-1-x*+(1+L)ϵxn-1-Txn, and so (27)xn-1-TxnL1-(1+L)ϵxn-1-x*, so that from (25) we obtain (28)xn-1-x*(1+kϵ)xn-x*xn-1-x*-Lϵ21-(1+L)ϵxn-1-x*. Therefore (29)xn-x*1+Lϵ2/(1-(1+L)ϵ)1+kϵxn-1-x*, where (30)ρ=1+Lϵ2/(1-(1+L)ϵ)1+kϵ=1-ϵ1+kϵ(k-Lϵ1-(1+L)ϵ)=1-ϵ1+kϵη=1-k-ηL+(k-η)(1+L+k)η. From (29) and (30), we get (31)xn-x*ρxn-1-x*ρnx0-x*0 as n. Hence xnx* as n. Uniqueness follows from the strong accretivity property of A.

The following is an immediate corollary of Theorem 8.

Corollary 9.

Let E be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:KK be Lipschitz (with constant L>1) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,yK). Assume that T has a fixed point x*K. Set ϵ0:=(k-η)/(L+(1+L)(k-η)), η(0,k) and define Aϵ0:KK by Aϵ0xn=(1-ϵ0)xn-1+ϵ0xn-ϵ0Axn for each xnK. For arbitrary x0K, define the sequence {xn}n=0 in K by (32)xn=Aϵ0xn,n1. Then {xn}n=0 converges strongly to x* with (33)xn+1-x*ρ0nx0-x*, where ρ0=(1-((k-η)/(L+(k-η)(1+L+k)))η)(0,1). Thus the choice η=k/2 yields ρ0=1-k2/2[2L+k(1+L+k)]. Moreover, x* is unique.

Proof.

Observe that x* is a fixed point of T if and only if it is a fixed point of Tϵ0. Furthermore, the recursion formula (32) is simplified to the formula (34)xn=(1-ϵ0)xn-1+ϵ0Txn, which is similar to (21). Following the method of computations as in the proof of the Theorem 8, we obtain (35)xn-x*1+Lϵ02/(1-(1+L)ϵ0)1+kϵ0xn-1-x*=(1-k-ηL+(k-η)(1+L+k)η)xn-1-x*. Set ρ0=1-((k-η)/(L+(k-η)(1+L+k)))η. Then from (35) we obtain (36)xn-x*ρ0xn-1-x*ρ0nx0-x*0 as n. This completes the proof.

Remark 10.

Since L>1 and k<L, we have (37)L>k-η. So we can easily obtain (38)1L+(k-η)(1+L+k)>1k(k-η)+L(2+L).

Now (39)ρ=1-(k-η)L+(k-η)(1+L+k)η<1-(k-η)k(k-η)+L(2+L)η=θ. Thus the relation between Ćirić et al.  and our parameter of convergence, that is, between θ and ρ, respectively, is the following: (40)ρ<θ.

Our convergence parameter ρ shows the overall improvement for θ, and consequently the results of Ćirić et al.  are improved.

Acknowledgment

The authors would like to thank the referees for useful comments and suggestions.

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