A CONVERGENCE THEOREM FOR SOME MEAN VALUE FIXED POINT ITERATION PROCEDURES VASILE BERINDE
Abstract. A general convergence theorem for the Ishikawa fixed point iteration procedure in a large class of quasi-contractive type operators is given. As particular cases, it contains convergence theorems for Picard, Krasnoselskij and Mann iterations, theorems which extend and generalize several results in the literature.
1. Introduction In the last four decades, numerous papers were published on the iterative approximation of fixed points of self and nonself contractive type operators in metric spaces, Hilbert spaces or several classes of Banach spaces, see for example the recent monograph [1] and the references therein. While for strict contractive type operators, the Picard iteration can be used to approximate the (unique) fixed point, see e.g. [1], [15], [26], [27], for operators satisfying slightly weaker contractive type conditions, instead of Picard iteration, which does not generally converge, it was necessary to consider other fixed point iteration procedures. The Krasnoselskij iteration [16], [6], [13], [14], the Mann iteration [17], [9], [20] and the Ishikawa iteration [11] are certainly the most studied of these fixed point iteration procedures, see [1]. Let E be a normed linear space and T : E −→ E a given operator. Let x0 ∈ E be arbitrary and {αn } ⊂ [0, 1] a sequence or real numbers. The sequence {xn }∞ n=0 ⊂ E defined by (1.1)
xn+1 = (1 − αn )xn + αn T xn ,
n = 0, 1, 2, . . .
is called the Mann iteration or Mann iterative procedure, in light of [17]. The sequence {xn }∞ n=0 ⊂ E defined by ( xn+1 = (1 − αn )xn + αn T yn , n = 0, 1, 2, . . . (1.2) yn = (1 − βn )xn + βn T xn , n = 0, 1, 2, . . . , where {αn } and {βn } are sequences of positive numbers in [0, 1], and x0 ∈ E arbitrary, is called the Ishikawa iteration or Ishikawa iterative procedure, due to [11]. 02000
Mathematics Subject Classification: 47H10; 54H25 Key words and phrases: Banach space; quasi contraction; fixed point; Ishikawa iteration; convergence theorem 1
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Vasile Berinde
Remark 1. For αn = λ (constant), the iteration (1.1) reduces to the so called Krasnoselskij iteration, while for αn ≡ 1 we obtain the Picard iteration or method of successive approximations, as it is commonly known, see [1]. Obviously, for βn ≡ 0 the Ishikawa iteration (1.2) reduces to (1.1). The classical Banach’s contraction principle is one of the most useful results in fixed point theory. In a metric space setting it can be briefly stated as follows. Theorem B. Let (X, d) be a complete metric space and T : X −→ X an a-contraction, i.e. a map satisfying (1.3)
d(T x, T y) ≤ a d(x, y) ,
for all x, y ∈ X ,
where 0 < a < 1 is constant. Then T has a unique fixed point p and the Picard iteration {xn }∞ n=0 defined by (1.4)
xn+1 = T xn ,
n = 0, 1, 2, . . .
converges to p, for any x0 ∈ X. Theorem B has many applications in solving nonlinear equations, but suffers from one drawback - the contractive condition (1.3) forces T be continuous on X. In 1968 R. Kannan [12], obtained a fixed point theorem which extends Theorem B to mappings that need not be continuous, by considering instead of (1.3) the next condition: there exists b ∈ (0, 1/2 ) such that (1.5) d(T x, T y) ≤ b d(x, T x) + d(y, T y) , for all x, y ∈ X . Following Kannan’s theorem, a lot of papers were devoted to obtaining fixed point theorems for various classes of contractive type conditions that do not require the continuity of T , see for example, Rus [26],[27], and references therein. One of them, actually a sort of dual of Kannan fixed point theorem, due to Chatterjea [7], is based on a condition similar to (1.5): there exists c ∈ (0, 1/2 ) such that (1.6) d(T x, T y) ≤ c d(x, T y) + d(y, T x) , for all x, y ∈ X It is known, see Rhoades [22] that (1.3) and (1.5), (1.3) and (1.6), respectively, are independent contractive conditions. In 1972, Zamfirescu [28] obtained a very interesting fixed point theorem, by combining (1.3), (1.5) and (1.6). Theorem Z. Let (X, d) be a complete metric space and T : X −→ X a map for which there exist the real numbers a, b and c satisfying 0 < a < 1, 0 < b, c < 1/2 such that for each pair x, y in X, at least one of the following is true: (z1 ) d(T x, T y) ≤ a d(x, y); (z2 ) d(T x, T y) ≤ b d(x, T x) + d(y, T y) ; (z3 ) d(T x, T y) ≤ c d(x, T y) + d(y, T x) .
A convergence theorem for some mean value fixed point iteration procedures
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Then T has a unique fixed point p and the Picard iteration {xn }∞ n=0 defined by xn+1 = T xn ,
n = 0, 1, 2, . . .
converges to p, for any x0 ∈ X. Remarks. An operator T which satisfies the contractive conditions in Theorem Z will be called a Zamfirescu operator. The class of Zamfirescu operators is one of the most studied classes of quasicontractive type operators. In this class all important fixed point iteration procedures, i.e., the Picard [28], Mann [20] and Ishikawa [21] iterations, are known to converge to the unique fixed point of T . The class of Zamfirescu operators is independent, see Rhoades [20], of the class of strictly (strongly) pseudocontractive operators, extensively studied by several authors in the last years. For the case of pseudocontractive type operators, the pioneering convergence theorems, due to Browder [5] and Browder and Petryshyn [6], established in Hilbert spaces, were successively extended to more general Banach spaces and to weaker conditions on the parameters that define the fixed point iteration procedures, as well as to several classes of weaker contractive type operators. For a recent survey and a comprehensive bibliography, we refer to the author’s monograph [1]. Harder and Hicks [10] introduced a concept of stability for fixed point iteration procedures and proved that, in a complete metric space setting, the Picard iteration is stable with respect to the class of Zamfirescu operators. The same authors proved that, in a linear normed space, certain Mann iterations are stable with respect to any Zamfirescu operator. Rhoades [25], Theorem 31, completed the previous results, showing that the Ishikawa iteration converges to the fixed point of, and is stable with respect to, Zamfirescu operators. Some of the convergence results in Rhoades [20] and [21] were very recently extended from uniformly convex to arbitrary Banach spaces, by simultaneously weakening the assumptions on the sequence {αn } which is involved in the definition of the aforementioned fixed point iterations, see Berinde [3], [4]. The author, starting from the fact that all important fixed point iterations (Picard, Mann, Ishikawa and, in particular, Krasnoselskij) can be used to approximate fixed points of Zamfirescu operators, has showed in [2] that the Picard iteration converges faster than Mann iteration, as suggested by some empirical studies reported in Berinde [1]. Motivated by such a rich literature concerning Zamfirescu operators, the present paper proves a general convergence theorem for the Ishikawa iteration in the larger class of mappings satisfying (2.1), that contains the class of Zamfirescu operators.
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Vasile Berinde
2. The main result Let E be an arbitrary Banach space and T : E −→ E an operator for which there exist 0 < δ < 1 and L ≥ 0 such that (2.1)
kT x − T yk ≤ δ · kx − yk + L · kx − T xk
holds, for all x, y ∈ E. Obviously, any a-contraction satisfies (2.1), with δ = a and L = 0. Other examples of operators satisfying (2.1) in the general context of metric spaces are given by the next lemmas. Lemma 1. Any Kannan mapping, i.e. any mapping satisfying the contractive condition (1.5), also satisfies (2.1). Proof. By condition (1.3) and triangle rule, we get d(T x, T y) ≤ b d(x, T x) + d(y, T y) ≤ n o ≤ b d(x, T x) + d(y, x) + d(x, T x) + d(T x, T y) which yields (1 − b)d(T x, T y) ≤ bd(x, y) + 2b · d(x, T x) and which implies d(T x, T y) ≤
b 2b d(x, y) + d(x, T x) , 1−b 1−b
that is, (2.1) holds with δ = .
for all x, y ∈ X ,
b 2b 1 and L = in view of 0 < b < , 1−b 1−b 2
Lemma 2. Any mapping T satisfying the contractive condition (1.6), also satisfies (2.1). Proof. Using d(x, T y) ≤ d(x, T x)+d(T x, T y), and d(y, T x) ≤ d(y, x)+ d(x, T x), by (1.6) we get after simple computations, d(T x, T y) ≤ which is (2.1), with δ =
c 2c d(x, y) + d(x, T x) , 1−c 1−c
c 2c < 1 (since c < 1/2) and L = ≥ 0. 1−c 1−c
An immediate consequence of Lemmas 1 and 2 is the following Corollary 1. Any Zamfirescu mapping, i.e., any mapping satisfying the assumptions in Theorem Z, does satisfy (2.1). It is known, see Osilike [18] that the operators satisfying (2.1) need not have a fixed point but, if F (T ) = {x ∈ E : T x = x} 6= ∅, then F (T ) is a singleton.
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A careful analysis of the proofs in [3] and [4] shows that all arguments in proving the convergence theorems there work if we consider, instead of Zamfirescu operators, the operators that have at least one fixed point and satisfy (2.1). The main result of this paper is given by the next theorem. Theorem 1. Let E be a normed linear space, K a closed convex subset of E, and T : K −→ K an operator with F (T ) 6= ∅, satisfying (2.1). Let {xn }∞ n=0 be the Ishikawa iteration defined by (1.2) and x0 ∈ K, arbitrary, where {αn } and {βn } are sequences in [0, 1] with {αn } satisfying ∞ X (i) αn = ∞ . n=0
Then {xn } converges strongly to the unique fixed point of T . Proof. Let p be the unique fixed point of T and {xn }∞ n=0 be the Ishikawa iteration defined by (1.2) and x0 ∈ K arbitrary. Then
kxn+1 − pk = (1 − αn )xn + αn T yn − (1 − αn + αn )p =
= (1 − αn )(xn − p) + αn (T yn − p) ≤ (2.2) ≤ (1 − αn )kxn − pk + αn kT yn − pk . With x := p and y := yn , from (2.1) we obtain kT yn − pk ≤ δ · kyn − pk .
(2.3) Further we have
kyn − pk = (1 − βn )xn + βn T xn − (1 − βn + βn )p =
= (1 − βn )(xn − p) + βn (T xn − p) ≤ (2.4)
≤ (1 − βn )kxn − pk + βn kT xn − pk .
Again by (2.1), this time with x := p; y := xn , we find that kT xn − pk ≤ δkxn − pk
(2.5)
and hence, by (2.2) - (2.5) we obtain kxn+1 − pk ≤ 1 − (1 − δ)αn (1 + δβn ) · kxn − pk , which, by the inequality 1 − (1 − δ)αn (1 + δβn ) ≤ 1 − (1 − δ)2 αn , implies (2.6)
kxn+1 − pk ≤ 1 − (1 − δ)2 αn · kxn − pk ,
n = 0, 1, 2, . . . .
By (2.6) we inductively obtain (2.7) kxn+1 − pk ≤
n Y 1 − (1 − δ)2 αk · kx0 − pk , k=0
n = 0, 1, 2, . . . .
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Vasile Berinde
Using the fact that 0 ≤ δ < 1, αk , βn ∈ [0, 1], and
∞ P
αn = 0, by (i) it
n=0
results that lim
n→∞
n Y 1 − (1 − δ)2 αk = 0 , k=0
which by (2.7) implies lim kxn+1 − pk = 0 ,
n→∞
i.e., {xn }∞ n=0 converges strongly to p.
In view of Lemma 3, by Theorem 1 we obtain the main result in [4]. Corollary 2. Let E be an arbitrary Banach space, K a closed convex subset of E, and T : K −→ K an operator satisfying condition Z. Let {xn }∞ n=0 be the Ishikawa iteration defined by (1.2) and x0 ∈ K, where {αn } and {βn } are sequences of positive numbers in [0, 1] with {αn } satisfying (i). Then {xn }∞ n=0 converges strongly to the fixed point of T . If we take {βn } ≡ 0 in Corollary 2, we obtain the main result in [3]. Corollary 3. Let E be an arbitrary Banach space, K a closed convex subset of E, and T : K −→ K an operator satisfying condition Z. Let {xn }∞ n=0 be the Mann iteration defined by (1.1) and x0 ∈ K, with {αn } ⊂ [0, 1] satisfying (i). Then {xn }∞ n=0 converges strongly to the fixed point of T . Remarks. 1) Theorem 1 also contains Theorem 5 of Osilike [19] and Theorem 2 of Osilike [18], where more restrictive conditions on {αn } are imn n P Q posed, i.e., 0 < α ≤ αn for some α, in [19], and (1 − αk + δ αk ) j=0 k=j+1
converges, in [18], where δ is that in (2.1). 2) Since the Kannan’s and Chattejea’s contractive conditions are both included in the class of Zamfirescu operators, by Theorem 1 we obtain corresponding convergence theorems for the Ishikawa iteration in these classes of operators. 3) The contractive condition of Kannan (1.5) is a special case of that of Zamfirescu, Theorems 2 and 3 of Kannan [12] are special cases of Theorem 2, with αn = 1/2 and βn = 0. Theorem 3 of Kannan [13] is the special case of Theorem 2 with αn = λ, 0 < λ < 1 and βn = 0. Note, however, that all results of Kannan [12], [13] are obtained in uniformly Banach spaces, like Theorem 8 of Rhoades [18] and Theorem 4 of Rhoades [17], which are extended by Corollary 2 and Corollary 3, respectively.
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4) I view of our paper [2], where a comparison of the Picard and Mann iterations is provided, and based on similar arguments to those used in the papers [3]-[4], we can compare the Picard, Krasnoselskij, Mann and Ishikawa iterations, in the more general context of the operators satisfying (2.1). The conclusion is that in this class of operators and, consequently, in the class of Zamfirescu operators, too, the best one amongst Picard, Krasnoselskij, Mann and Ishikawa fixed point iteration procedures, is the Picard iteration. 5) Since the class of Zamfirescu operators coincides, see Rhoades [22], with the class of mappings satisfying the following contraction condition: there exists a constant 0 < h < 1 such that d(T x, T y) ≤ h · max d(x, y), [d(x, T x) + d(y, T y)]/2, [d(x, T y) + d(y, T x)]/2 , for all x, y ∈ X . many other results in literature regarding the approximation of fixed points by Picard iteration or by other fixed point iteration procedures, are also contained in our Theorem 1. References [1] Berinde, V., Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002 [2] Berinde, V., Picard iteration converges faster than the Mann iteration for the class of quasi-contractive operators Fixed Point Theory and Applications 1 (2004) (to appear) [3] Berinde, V., On the convergence of Mann iteration for a class of quasi contractive operators (submitted) [4] Berinde, V., On the convergence of Ishikawa iteration for a class of quasi contractive operators (submitted) [5] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banach spaces Proc. Sympos. Pure Math. 18, Pt. 2, Amer. Math. Soc., Providence, R. I., 1976 [6] Browder, F.E. and Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert spaces J. Math. Anal. Appl. 20 (1967), 197-228 [7] Chatterjea, S.K., Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727-730 [8] Ciric, Lj.B., A generalization of Banach’s contraction principle, Proc. Am. Math. Soc. 45 (1974) 267-273 [9] Groetsch, C.W., A note on segmenting Mann iterates J. Math. Anal. Appl. 40 (1972), 369-372 [10] Harder, A.M. and Hicks, T.L., Stability results for fixed point iteration procedures Math. Japonica 33 (1988), No. 5, 693-706 [11] Ishikawa, S., Fixed points by a new iteration method Proc. Amer. Math. Soc. 44(1) (1974), 147-150 [12] Kannan, R. Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968) 71-76 [13] Kannan, R., Some results on fixed points. III Fund. Math. 70 (1971), 169-177 [14] Kannan, R., Construction of fixed points of a class of nonlinear mappings J. Math. Anal. Appl. 41 (1973), 430-438
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[15] Kirk, W.A. and Sims, B., Handbook of Metric Fixed Point Theory Kluwer Academic Publishers, 2001 [16] Krasnoselskij, M.A., Two remarks on the method of successive approximations (Russian) Uspehi Mat. Nauk. 10 (1955), no. 1 (63), 123-127 [17] Mann, W.R., Mean value methods in iteration Proc. Amer. Math. Soc. 44 (1953), 506-510 [18] Osilike, M.O., Stability results for fixed point iteration procedures J. Nigerian Math. Soc. 14/15 (1995/96) 17-29 [19] Osilike, M.O., Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings, Indian J. Pure Appl. Math. 30 (12) (1999) 1229-1234 [20] Rhoades, B.E., Fixed point iterations using infinite matrices Trans. Amer. Math. Soc. 196 (1974), 161-176 [21] Rhoades, B.E., Comments on two fixed point iteration methods J. Math. Anal. Appl. 56 (1976), No. 2, 741-750 [22] Rhoades, B.E., A comparison of various definitions of contractive mappings Trans. Amer. Math. Soc. 226 (1977), 257-290 [23] Rhoades, B.E., Contractive definitions revisited Contemporary Math. 21 (1983), 189-205 [24] Rhoades, B.E., Contractive definitions and continuity, Contemporay Math., 72 (1988), 233-245 [25] Rhoades, B.E., Some fixed point iteration procedures Int. J. Math. Math. Sci. 14 (1991), No. 1, 1-16 [26] Rus, I.A., Principles and Applications of the Fixed Point Theory (Romanian) Editura Dacia, Cluj-Napoca, 1979 [27] Rus, I.A., Generalized Contractions and Applications Cluj University Press, Cluj-Napoca, 2001 [28] Zamfirescu, T., Fix point theorems in metric spaces Arch. Math. (Basel), 23 (1972), 292-298
Department of Mathematics and Computer Science North University of Baia Mare Victorie1 76, 430072 Baia Mare ROMANIA E-mail: vasile
[email protected];
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