On the steepest descent approximation method for the zeros of generalized accretive operators

Nonlinear Analysis 69 (2008) 763–769 www.elsevier.com/locate/na

On the steepest descent approximation method for the zeros of generalized accretive operators ´ c a,∗ , Siniˇsa Jesi´c b , Marina M. Milovanovi´c a , Jeong Sheok Ume c Ljubomir Ciri´ a Faculty of Mechanical Engineering, Kraljice Marije 16, 11070 Belgrade, Serbia b Faculty of Electrical Engineering, Bul. Kralja Aleksandra 73, 11 000 Belgrade, Serbia c Department of Applied Mathematics, Changwon National University Changwon 641-773, Republic of Korea

Received 1 August 2006; accepted 5 June 2007

Abstract In this paper we present certain characteristic conditions for the convergence of the generalized steepest descent approximation process to a zero of a generalized strongly accretive operator, defined on a uniformly smooth Banach space. Our study is based on an important result of Reich [S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978) 85–92] and given results extend and improve some of the earlier results which include the steepest descent approximation method. c 2007 Elsevier Ltd. All rights reserved.

MSC: primary 47H10; secondary 47H06; 54H25 Keywords: Compatible mappings; Common fixed point; Contractive type condition

1. Introduction Let X be a real Banach space, X ∗ be the duality space of X and let (x, f ) → hx, f i for (x, f ) ∈ X × X ∗ be the duality mapping. For 1 < p < +∞, let J p (x) = { f ∈ X ∗ : hx, f i = kxk p , k f k = kxk p−1 },

x ∈ X.

Then an operator T : D(T ) ⊆ X → X is said to be: (i) accretive if or each x, y ∈ D there is j2 (x − y) ∈ J2 (x − y) such that hT x − T y, j2 (x − y)i ≥ 0,

(1)

(ii) strongly accretive if there exists a constant k > 0 such that for each x, y ∈ D there is j2 (x − y) ∈ J2 (x − y) such that ∗ Corresponding address: University of Belgrade, Department of Mathematics, Kraljice Marije 16, 11070 Belgrade, Serbia. Tel.: +381 112602157. ´ c), [email protected] (S. Jesi´c), [email protected] (M.M. Milovanovi´c), E-mail addresses: [email protected] (L. Ciri´ [email protected] (J.S. Ume).

c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.06.021

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hT x − T y, j2 (x − y)i ≥ kkx − yk2 ,

(2)

(iii) φ-strongly accretive if there exists a strictly increasing function φ : [0, +∞) → [0, +∞) with φ(0) = 0 and for each x, y ∈ D there is j2 (x − y) ∈ J2 (x − y) such that hT x − T y, j2 (x − y)i ≥ φ(kx − yk)kx − yk.

(3)

Let Z (T ) = {x ∈ X : T x = 0}. If Z (T ) 6= ∅ and for each x ∈ D(T ) and y ∈ Z (T ) the inequality (1), or (2), or (3) holds, then the corresponding operator T is called quasi-accretive, strongly quasi-accretive and φ-strongly quasi-accretive, respectively. Accretive operators were introduced independently by Browder [1] and Kato [8], and the φ-strongly accretive operators were introduced by Morales [12]. Accretive, strongly accretive and monotone operators have been extensively studied by many researchers [3–6,16,17,21]. The φ-strongly accretive operators have been studied by Chidume [2], Liu and Kang [10], Osilike [14], Morales and Chidume [13] and many others. In the present paper we consider a class of operators somehow more general then the class of φ-strongly accretive operators, which are defined as follows. Definition 1. An operator T : D(T ) ⊆ X → X is said to be generalized strongly accretive if there exists a nondecreasing function ϕ : [0, +∞) → [0, +∞), with ϕ(t) = 0 if and only if t = 0 and for each x, y ∈ D there is j p (x − y) ∈ J p (x − y) such that hT x − T y, j p (x − y)i ≥ ϕ(kx − yk).

(4)

If Z (T ) 6= ∅ and for each x ∈ D(T ) and y ∈ Z (T ) the inequality (4) holds, then T is called the generalized strongly quasi-accretive operator. Obviously, every φ-strongly accretive operator is a generalized strongly accretive operator and every φ-strongly quasi-accretive operator is a generalized strongly quasi-accretive operator, where ϕ(t) is defined by ϕ(t) = φ(t)t. The example below shows that the converse is not true. Throughout the paper we assume that X is a real uniformly smooth Banach space. Recall that an operator T : X → X is said to be bounded if it maps bounded sets into bounded sets. The following lemma, due to Reich, is useful in a study of uniformly smooth Banach spaces. Lemma 1 (Reich [15]). Let X be a uniformly smooth Banach space. Then there exists a non-decreasing function b : [0, +∞) → [0, +∞) which satisfies the following conditions: (i) b(ct) ≤ cb(t) for all c ≥ 1; (ii) limt→0+ b(t) = 0; and (iii) kx + yk2 ≤ kxk2 + 2Rehy, J (x)i + max{kxk, 1}kykb(kyk). The steepest descent approximation process for monotone operators was introduced independently by Vainberg [18] and Zarantonello [20]. Mann [11] introduced an iteration process (now generally referred to as the Mann iteration process) which, under suitable conditions, converges to a zero in Hilbert spaces. The Mann iteration scheme was further developed by Ishikawa [7]. Recently Zhou and Guo [22], Morales and Chidume [13], Chidume [2], Xu and Roach [19] and many others have studied the characteristic conditions for the convergence of the steepest descent approximations. Morales and Chidume proved the following theorem: Theorem 2 (Morales and Chidume [13]). Let X be a uniformly smooth Banach space and let T : X → X be a φ-strongly accretive operator, which is bounded and demicontinuous. Let z ∈ X and let xo be an arbitrary initial value in X for which limt→∞ inf φ(t) > kT x0 k. Then the steepest descent approximation scheme xn+1 = xn − αn (T xn − z),

n = 0, 1, 2, . . . ,

(5)

converges strongly to the unique solution of the equation T x = z provided that the sequence {αn } of positive real numbers satisfies the following: (i) {α Pn } is bounded above by the constant r0 , defined below; (ii) P∞ n=0 αn = +∞; and (iii) ∞ n=0 αn b(αn ) < +∞.

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We may observe that in the above theorem the condition (iii) seems to be strong and difficult to test. The purpose of this paper is to continue a study of sufficient conditions for the convergence of the steepest descent approximation process to the zero of a bounded generalized strongly accretive operator. The condition (iii) in the above theorem is weakened in our main theorem with αn → 0 as n → ∞. We give a new general lemma on the convergence of the real sequences, which appears useful for the proof of our main results. The presented results extend and improve some of the earlier results which include the steepest descent method considered by Morales and Chidume [13], Chidume [2] and Xu and Roach [19] for a bounded φ-strongly quasi-accretive operator and also the generalized steepest descent method considered by Zhou and Guo [22] for a bounded φ-strongly quasi-accretive operator. 2. Main results We prove a new general lemma on the convergence of the real sequences, which appears useful for the proof of our main result. Lemma 3. Let {ρn } be a sequence of non-negative real numbers and let {λn } and {µn } be two bounded sequences of non-negative real numbers such that p

p

ρn+1 ≤ ρn − λn ϕ(ρn ) + λn µn ,

(6)

where p > 0 and ϕ : [0, +∞) → [0, +∞) is a non-decreasing real function such that ϕ(t) = 0 if and only if t = 0. If P (i) ∞ n=1 λn = +∞ and (ii) limn→∞ µn = 0, then lim ρn = 0.

n→∞

Proof. First we show that 2δ = lim inf ρn = 0. n→∞

(7)

Suppose, to the contrary, that δ > 0. Then there exists some positive integer n 0 such that ρn ≥ δ > 0 for all n ≥ n 0 . Since ϕ(t) is non-decreasing and δ > 0, then ϕ(ρn ) ≥ ϕ(δ) > 0 for all n ≥ n 0 . As limn→∞ µn = 0, there exists a positive integer n 1 ≥ n 0 such that ϕ(δ) 2 for all n ≥ n 1 . Then from (6) and using the fact that ϕ(t) is non-decreasing, for all n ≥ n 1 we get µn ≤

1 p p ρn+1 ≤ ρn − λn ϕ(δ) + λn ϕ(δ), 2 or, rewritten, 1 p p ϕ(δ)λn ≤ ρn − ρn+1 . 2 Hence, for any n > n 1 , n X 1 p p p ϕ(δ) λ j ≤ ρn 1 − ρn ≤ ρn 1 . 2 j=n 1

This implies ϕ(δ)

∞ X j=n 1

p

λ j ≤ 2ρn 1 ,

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which is in contradiction with (i). Thus δ = 0, i.e., limn→∞ inf ρn = 0. Now we show that limn→∞ ρn = 0. Assume, to the contrary, that limn→∞ sup ρn > 0. Then there is an  > 0 such that lim sup ρn ≥ 31/ p ε.

(8)

n→∞

Define h = sup{λn : n ≥ 1}. Then by (i), h > 0. Since ϕ() > 0 and µn → 0 as n → ∞, there exists a positive integer n 2 such that  p  ε , ϕ() (9) µn ≤ min h for all n ≥ n 2 . From (7) there is a positive integer r ≥ n 2 such that ρr < ε.

(10)

From (8) there exists a positive integer s > r such that ρs ≥ 21/ p ε.

(11)

We show that (11) with s > r implies ρs−1 ≥ ρs .

(12)

From (11), (6) and (9) we get p

p

p

p

2ε p ≤ ρs ≤ ρs−1 − λs−1 ϕ(ρs−1 ) + λs−1 µs−1 ≤ ρs−1 + hµs−1 ≤ ρs−1 + ε p . Hence ρs−1 ≥ ε. Then from (6) and (9), p

p

p

ρs ≤ ρs−1 − λs−1 ϕ () + λs−1 ϕ () = ρs−1 . Hence ρs−1 ≥ ρs . Thus we showed (12). Since (11) and (12) imply that ρs−1 satisfies (11), similarly to above, we get ρs−2 ≥ ρs−1 . Continuing this process we obtain ρs ≤ ρs−1 ≤ ρs−2 ≤ · · · ≤ ρr . p

p

Now, by (10) and (11) we have 2ε p ≤ ρs ≤ ρr ≤ ε p , a contradiction. Thus limn→∞ ρn = 0.



Now we are in position to prove our main result. Theorem 4. Let X be a uniformly smooth Banach space and let T : X → X be a bounded and demicontinuous generalized strongly accretive operator with p = 2. Let z ∈ X and let x0 be an arbitrary initial value in X for which kT x0 k < sup{ϕ(t)/t : t > 0}. Then a steepest descent approximation scheme defined by xn+1 = xn − αn (T yn − z), yn = xn − βn (T xn − z),

n = 0, 1, 2, . . . ,

(13)

n = 0, 1, 2, . . . ,

where the sequence {αn } of positive real numbers satisfies the following conditions: (i) αP n ≤ λ, where the constant λ is defined in (16) below, (ii) ∞ n=0 αn = +∞, and (iii) αn → 0 and βn → 0 as n → ∞, converges strongly to the unique solution of the equation T x = z. Proof. From Theorem 1 of Kartsatos [9], T (X ) is open in X . Since T (X ) is also closed, T is surjective. Thus, the equation T x = z has a unique solution for an arbitrary z ∈ X . Without loss of generality we may assume that z = 0. Define the positive constants r, m and λ as follows:   ϕ(t) r = sup t > 0 : ≤ kT x0 k , (14) t m = sup{kT xk : kx − x0 k ≤ 3r }

(15)

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and    1 2ϕ(r ) λ = min r, sup t : b(t) ≤ , m m · max{2r, 1}

(16)

where b(t) is a non-negative Reich’s function, mentioned in Lemma 1. We shall show that a sequence {xn }, defined by xn+1 = xn − αn T xn , yn = xn − βn T xn ,

n = 0, 1, 2, . . . ,

(17)

n = 0, 1, 2, . . . ,

with αn , βn ≤ λ, is bounded. Define by x ∗ the unique zero of T . Then, as T is a generalized strongly accretive operator, for t = kx0 − x ∗ k we have hT x0 , j2 (x0 − x ∗ )i ϕ(kx0 − x ∗ k) ≤ kx0 − x ∗ k kx0 − x ∗ k kT x0 kkx0 − x ∗ k . ≤ kx0 − x ∗ k Hence ϕ(kx0 − x ∗ k) ≤ kT x0 k. kx0 − x ∗ k Thus from (14), kx0 − x ∗ k ≤ r.

(18)

Now we show that for any x in X , kx − x ∗ k ≤ 2r Let kx −

x ∗k

implies kT xk ≤ m.

(19)

≤ 2r . Then by the triangle inequality and (18),

kx − x0 k ≤ kx − x ∗ k + kx0 − x ∗ k ≤ 2r + r = 3r. Thus, by (15), kT xk ≤ m. Now we shall show that if kxn − x ∗ k ≤ 2r for some fixed n ≥ 0, then kyn − x ∗ k ≤ 2r,

n = 0, 1, 2, . . . .

(20)

We will consider two cases: kxn − ≤ r and kxn − > r. Case 1. Let kxn − x ∗ k ≤ r . Then from (17), (19), (16) and (i), we have x ∗k

x ∗k

kyn − x ∗ k = kxn − x ∗ − βn T xn k ≤ kxn − x ∗ k + βn kT xn k ≤ r + λm ≤ r + r = 2r. Thus, if kxn − x ∗ k ≤ r , then kyn − x ∗ k ≤ 2r . Case 2. Let r < kxn − x ∗ k. In this case we shall use Lemma 1. Thus, by (iii) in this lemma, kyn − x ∗ k2 = kxn − βn T xn − x ∗ k2 ≤ kxn − x ∗ k2 − 2βn hT xn , J2 (xn − x ∗ )i + max{kxn − x ∗ k, 1}βn kT xn kb(βn kT xn k). Hence, by (19) and as T is generalized strongly accretive and b(t) is non-decreasing, kyn − x ∗ k2 ≤ kxn − x ∗ k2 − 2βn ϕ(kxn − x ∗ k) + βn max{kxn − x ∗ k, 1}m b(βn m). Since r < kxn − x ∗ k ≤ 2r and as ϕ(t) and b(t) are non-decreasing, then −ϕ(kxn − x ∗ k) ≤ −ϕ(r ) and max{kxn − x ∗ k, 1} ≤ max{2r, 1}. So we have kyn − x ∗ k2 ≤ kxn − x ∗ k2 − 2βn ϕ(r ) + βn max{2r, 1}mb(βn m).

(21)

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Now by (16), kyn − x ∗ k2 ≤ kxn − x ∗ k2 − 2βn ϕ(r ) + βn max{2r, 1}m

2ϕ(r ) . m · max{2r, 1}

Hence kyn − x ∗ k ≤ kxn − x ∗ k. Since by the hypothesis kxn − x ∗ k ≤ 2r , then kyn − x ∗ k ≤ 2r . Thus we have proved that in the both cases, kyn − x ∗ k ≤ 2r. We now prove that kxn − x ∗ k ≤ 2r ;

n = 0, 1, 2, . . . .

(22)

We shall use the mathematical induction. From (18) we see that for n = 0, (22) holds. Suppose now that kxn − x ∗ k ≤ 2r for some fixed n ≥ 0. Then by (20) we know that kyn − x ∗ k ≤ 2r . So by (19), kT xn k ≤ m. To show that kxn+1 − x ∗ k ≤ 2r , we will consider, as above, two cases: kxn − x ∗ k ≤ r and kxn − x ∗ k > r . At first, assume that kxn − x ∗ k ≤ r . Then kxn+1 − x ∗ k ≤ kxn+1 − xn k + kxn − x ∗ k = αn kT yn k + kxn − x ∗ k ≤ λm + r ≤ r + r = 2r. Thus, if kxn − x ∗ k ≤ r , then (22) holds. Let now r < kxn − x ∗ k ≤ 2r . Again by (iii) in Lemma 1, kxn+1 − x ∗ k2 = kxn − αn T yn − x ∗ k2 ≤ kxn − x ∗ k2 − 2αn hT yn , J2 (xn − x ∗ )i + max{kxn − x ∗ k, 1}αn kT yn kb(αn kT yn k). Then, by using the same procedure as in Case 2 above, we obtain the following inequality, analogous to (21): kxn+1 − x ∗ k2 ≤ kxn − x ∗ k2 − 2αn ϕ(r ) + αn max{2r, 1}mb(αn m).

(23)

Now by (16) we get kxn+1 − x ∗ k ≤ kxn − x ∗ k. Since by the induction hypothesis kxn − x ∗ k ≤ 2r , then kxn+1 − x ∗ k ≤ 2r . Thus, by induction, we conclude that (23) holds for all n ≥ 0. Since by the hypothesis (iii), αn → 0 as n → ∞, then αn m → 0 as n → ∞. Thus, by (ii) in Lemma 1, b(αn m) → 0 as n → ∞. Therefore, max{2r, 1}mb(αn m) → 0

as n → ∞.

Taking ρ = kxn − x ∗ k,

p = 2,

λn = 2αn

and

µn =

1 max{2r, 1}mb(αn m), 2

from (23) and Lemma 3 we get

lim xn − x ∗ = 0. n→∞

Since by (21), kyn − x ∗ k2 ≤ kxn − x ∗ k2 + 2βn µn , then also kyn − x ∗ k → 0 as n → ∞. Thus, we have proved that {xn } and {yn } converges strongly to the unique zero of the generalized accretive operator T − z.  Observe that the presented proof for the generalized strongly accretive operator in Theorem 4 is also correct for a generalized strongly quasi-accretive operator. Thus we have the following result: Theorem 5. Let X be a uniformly smooth Banach space and let T : X → X be a bounded and demicontinuous generalized strongly quasi-accretive operator with p = 2, such that Z (T ) 6= ∅. Let z ∈ X and let x0 be an arbitrary initial value in X for which kT x0 k < limt→∞ inf ϕ(t)/t. Then a generalized steepest descent approximation scheme defined by xn+1 = xn − αn (T yn − z), n = 0, 1, 2, . . . , yn = xn − βn (T xn − z), n = 0, 1, 2, . . . , where the sequences {αn } and {βn } of positive real numbers satisfy the following conditions: (i) {αn } and {βn } are bounded above by the constant λ defined in (16) above,

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P (ii) ∞ n=0 αn = +∞, and (iii) αn → 0 and βn → 0 as n → ∞, converges strongly to the unique solution of the equation T x = z. The following example shows that a generalized strongly quasi-accretive operator is not necessarily a φ-strongly quasi-accretive operator. Example. Let X = R (the set of the real numbers with the usual norm), D = [0, +∞) and let T : D ⊆ X → X be defined by x Tx = . 1 + x2 Then Z (T ) = {0} 6= ∅ and it is easy to verify that for each x ∈ D(T ) and y ∈ Z (T ), the mapping T satisfies (4) with ϕ(t) = t 2 /(1 + t 2 ). However, T is not φ-strongly quasi-accretive. Indeed, if there exists a strictly increasing function φ : [0, +∞) → [0, +∞) with φ(0) = 0 satisfying (3), then we get φ(t) ≤ t/(1 + t 2 ) for all t ∈ (0, ∞). Thus limt→∞ φ(t) = 0. This is in contradiction with the hypotheses that φ(t) is strictly increasing and φ(0) = 0. Acknowledgements This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund; KRF-2006-312-C00026). The second and third authors thank for the support of Ministry of Science and Environmental Protection of Serbia. References [1] F.E. Browder, Nonlinear mappings of non-expansive and accretive type in Banach spaces, Bull. Amer. Math. Soc. 73 (1967) 875–882. [2] C.E. Chidume, Steepest descent approximations for accretive operator equations, Nonlinear Anal. 26 (1996) 299–311. ´ c, J.S. Ume, Ishikawa process with errors for nonlinear equations of generalized monotone type in Banach spaces, Math. Nachr. 10 [3] Lj.B. Ciri´ (2005) 1137–1146. ´ c, J.S. Ume, Ishikawa iterative process for strongly pseudo-contractive operators in arbitrary Banach spaces, Math. Commun. 8 (1) [4] Lj.B. Ciri´ (2003) 43–48. ´ c, J.S. Ume, Iterative process with errors for approximation solutions to nonlinear equations of generalized monotone type in [5] Lj.B. Ciri´ arbitrary Banach spaces, Bull. Austral. Math. Soc. 69 (2004) 177–189. [6] K. Deimling, Zeros of accretive operators, Manuscripta Math. 13 (1974) 283–288. [7] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 73 (1976) 65–71. [8] T. Kato, Nonlinear semi-groups and evolution equations, J. Math. Soc. Japan 18–19 (1967) 508–520. [9] A.G. Kartsatos, Zeros of demicontinuous accretive operators in Banach spaces, J. Integral Equations 8 (1985) 175–184. [10] Z.Q. Liu, S.M. Kang, Convergence theorems for φ-strongly accretive and φ-hemi-contractive operators, J. Math. Anal. Appl. 253 (2001) 35–49. [11] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510. [12] C.H. Morales, Surjectivity theorems for multi-valued mappings of accretive types, Comment. Math. Univ. Carol. 26 (2) (1985) 397–413. [13] C.H. Morales, C.E. Chidume, Convergence of the steepest descent method for accretive operators, Proc. Amer. Math. Soc. 127 (12) (1999) 3677–3683. [14] M.O. Osilike, Iterative solution of nonlinear equations of the φ-strongly accretive type, J. Math. Anal. Appl. 200 (2) (1996) 259–271. [15] S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978) 85–92. [16] B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741–750. [17] K.K. Tan, H.K. Hu, Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces, J. Math. Anal. Appl. 178 (1993) 9–21. [18] M.M. Vainberg, On the convergence of the method of steepest descent for nonlinear equations, Sibirsk Mat. Zb. (1961) 201–220. [19] Z.B. Xu, G.F. Roach, Characteristic inequalities in uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157 (1991) 189–210. [20] E.H. Zarantonello, The closure of the numerical range contains the spectrum, Bull. Amer. Math. Soc. 70 (1964) 781–783. [21] E. Zeidler, Nonlinear Functional Analysis and its Applications. Part II. Monotone Operators, Springer-Verlag, New York, Berlin, 1985. [22] H.Y. Zhou, J. Guo, A characteristic condition for convergence of generalized steepest descent approximation to accretive equations, Indian J. Pure Appl. Math. 32 (2) (2001) 277–284.