PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS

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1 Fixed Point Theory, 15(2014), No. 2, nodeacj/sfptcj.html PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS ANDRZEJ CEGIELSKI AND RAFA L ZALAS Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, ul. Szafrana 4a, Zielona Góra, Poland a.cegielsi@wmie.uz.zgora.pl Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, ul. Szafrana 4a, Zielona Góra, Poland r.zalas@wmie.uz.zgora.pl Abstract. In this paper we present an application of a class of quasi-nonexpansive operators to iterative methods for solving the following variational inequality problem VIP(F, C): Find ū C such that F ū, z ū 0 for all z C, where C is a closed and convex subset of a Hilbert space H and F : H H is strongly monotone and Lipschitz continuous. A classical method for VIP(F, C) is the gradient projection (GP) method x +1 = P C (x µf x ) which generates sequences converging to the unique solution of VIP(F, C) if µ > 0 is sufficiently small. Unfortunately, in many optimization problems the GP method cannot be applied, because it requires an explicit computation of P C u in each iteration, where u = x µf x. To overcome this disadvantage of the GP method, one can replace the operator P C employed in the -th iteration of the method by a quasi-nonexpansive operator T and a constant µ by λ 0, 0, satisfying =0 Fix T Fix T and lim λ = 0. The new method can be presented equivalently as a so called general hybrid steepest descent (GHSD) method in the form u +1 = T u λ F T u. One should, however, suppose something more on the operators T in order to guarantee the convergence of u to the solution of VIP(F, C). In this paper we introduce a class of approximately shrining operators, prove the closedness of this class with respect to compositions and convex combinations and apply the operators from this class to a general hybrid steepest descent method for solving VIP(F, C). We give sufficient conditions for the convergence of the GHSD method as well as present several examples of methods which satisfy these conditions. In particular, we apply the results in the case, when C = m i=1 Fix U i and U i : H H are quasi-nonexpansive operators having a common fixed point, i = 1, 2,..., m. Key Words and Phrases: variational inequality problem, fixed point, quasi-nonexpansive operator, hybrid steepest descent method, string-averaging Mathematics Subject Classification: 47H05, 47H09, 47H10, 65K Introduction Let H be a real Hilbert space equipped with an inner product, and with the corresponding norm. In this paper we consider the following variational inequality problem (VIP): Given a nonlinear operator F : H H and a closed convex subset C H, find ū C such that F ū, z ū 0 for all z C. (1.1) 399

2 400 ANDRZEJ CEGIELSKI AND RAFA L ZALAS This problem, denoted by VIP(F, C), is a fundamental problem in optimization theory, because many optimization problems can be translated into VIPs. The VIP problem was intensively studied in the last decades; see, e.g., the two-volume boo by Facchinei and Pang [22], the boo of Kinderlehrer and Stampacchia [29] and papers by Yamada [38], Yamada and Ogura [39] and Hirstoaga [28]. In the whole paper we suppose that F is Lipschitz continuous and strongly monotone which guarantees the existence and the uniqueness of a solution of VIP(F, C) (see, e.g., [40, Theorem 46.C]). The standard method for VIP(F, C) is the gradient projection (GP) method x +1 = P C (x µf x ), (1.2) where P C : H H denotes the metric projection onto C (see [25]). If µ is sufficiently small, then the operator P C (Id µf ) is a contraction, consequently, the Banach fixed point theorem yields the convergence of sequences generated by the GP method to the unique solution of VIP(F, C). Denoting u = x µf x one can present the method in an equivalent form u +1 = P C u µf P C u. (1.3) The method can be efficiently applied if the metric projection P C u can be easily computed. Otherwise, the method can be essentially affected, because P C u should be computed in each iteration. Some researchers overcome this obstacle by replacing the metric projection P C by a nonexpansive or a quasi-nonexpansive operator T with Fix T = C and the constant µ > 0 by a sequence λ with lim λ = 0 (see, e.g., [27, 36, 38, 39]). This leads to the method u +1 = T u λ F T u (1.4) called by Yamada a hybrid steepest descent method (see [38]). Other researchers apply different nonexpansive or quasi-nonexpansive operators T with Fix T C in particular iterations (see [1, 10, 28, 30]). This leads to the method u +1 = T u λ F T u (1.5) called in [10] the generalized hybrid descent method. In this paper we study generalized hybrid descent method (1.5) for solving VIP(F, C), where T are quasi-nonexpansive operators satisfying =0 Fix T C. Roughly spoen, we suppose that the method generates sequences for whose the displacement T u u cannot be too small for those u which are far from C (see also [39] and [28] for related assumptions). The paper is organized as follows. Section 2 contains preliminaries on nonexpansive, firmly nonexpansive, quasi-nonexpansive as well as demi-closed operators. The properties recalled in Section 2 will be applied in the rest part of the paper. In Sections 3 and 4 we introduce the notion of approximately shrining (AS) operators, present some basic examples of these operators and give properties of AS operators, in particular, we prove that the family of AS operators is closed under composition and convex combination. These properties are applied in Section 5, where we extend the notion and the properties of AS operators to the uniformly approximately shrining (UAS) families of operators. Section 6 contains one of the main results of the paper (Theorem 6.3), where we show that

3 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 401 an application of UAS operators in (1.5), satisfying some additional conditions guarantees the convergence of {u } =0 to a unique solution of VIP(F, C). In Section 7 we present several examples of GHSD method for VIP(F, C) with C = m i=1 Fix U i, where U i : H H are quasi-nonexpansive operators having a common fixed point. We apply the results of Sections 3 6 in order to prove that the sequences generated by particular methods converge to a unique solution of VIP(F, C). The methods are combined with a a string-averaging scheme which is presented in Section 7. Moreover, we present an independent convergence result (Theorem 7.6) dedicated to this specific scheme. In the Appendix we prove the equivalence of some conditions for real valued functions (Lemma 8.1). This result is used in the proof of the equivalence of various definitions of approximately shrining operators as well as in the proof of the equivalence of various definitions of boundedly regular operators. 2. Preliminaries 2.1. Nonexpansive and firmly nonexpansive operators. We say that U is η- strongly monotone, where η > 0, if Ux Uy, x y η x y 2 for all x, y H. We say that U is κ-lipschitz continuous, where κ 0, if Ux Uy κ x y for all x, y H. If κ = 1, then U is called a nonexpansive (NE) operator. If κ < 1 then we say that U is a contraction. An example of a nonexpansive and monotone operator is a firmly nonexpansive (FNE) one, i.e., an operator U satisfying Ux Uy, x y Ux Uy 2 for all x, y H (apply the Cauchy Schwarz inequality). A relaxation U α := Id +α(u Id) of an FNE operator, where α [0, 2], is called an α-relaxed firmly nonexpansive (α-rfne or RFNE) operator. If α (0, 2) then the relaxation U α of an FNE operator is called a strictly relaxed firmly nonexpansive operator. Furthermore, U is FNE if and only if S := 2U Id is NE (see [23, Chapter 12]). Equivalently, an operator S : H H is NE if and only if U := 1 2 (S + Id) is FNE. Proposition 2.1. Let U i : H H be (strictly) relaxed firmly nonexpansive, i I := {1, 2,..., m}. Then (i) a composition U := U m... U 1 and (ii) a convex combination U := I ω iu i, where ω i 0, i I and I ω i = 1 are (strictly) relaxed firmly nonexpansive. Proof. See [33, Theorem 3(b)], [5, Lemma 2.2 and Proposition 2.1] or [7, Theorems , ]. Let C H and x H. If there is y C such that y x z x for all z C then y is called a metric projection of x onto C and is denoted by P C x. Let C be nonempty, closed and convex. Then for any x H the metric projection y := P C x is uniquely defined and characterized by the following inequality z y, x y 0 (see, e.g., [20, Theorems 3.4(2) and 4.1]). A function d(, C) : H R defined by d(x, C) := P C x x is continuous and convex, consequently, it is wealy lower semi-continuous (see [21, Chapter I, Corollary 2.2]).

4 402 ANDRZEJ CEGIELSKI AND RAFA L ZALAS 2.2. Quasi-nonexpansive operators. Let U : H H be an operator having a fixed point. We say that U is quasi-nonexpansive (QNE) if Ux z x z for all x H and all z Fix U. We say that U is γ-strongly quasi-nonexpansive (γ-sqne), where γ 0, if Ux z 2 x z 2 γ Ux x 2 for all x H and all z Fix U. If γ 1 < γ 2 and U is γ 2 -SQNE then it is γ 1 -SQNE. If γ > 0 then U is called strongly quasi-nonexpansive (SQNE). We say that U is a cutter if z Ux, x Ux 0 for all x H and all z Fix U. A NE operator having a fixed point is QNE. The fixed point set of a QNE operator is closed and convex (see [4, Proposition 2.6(ii)]). U is a cutter if and only if its α-relaxation is 2 α α -SQNE, where α (0, 2] (see [19, Proposition 2.3(ii)] and [7, Theorem ]). In particular, U is a cutter if and only if U is 1-SQNE. Furthermore, U is QNE if and only if 1 2 (U + Id) is a cutter (see [4, Proposition 2.3(v) (vi)] or [7, Corollary (ii)]) A FNE operator having a fixed point is a cutter (see [24, pp ] or [7, Theorem 2.2.5]). Proposition 2.2. Let U i : H H be ρ i -strongly quasi-nonexpansive, i I := {1, 2,..., m}, with i I Fix U i and ρ := min i I ρ i > 0. Then: (i) a composition U := U m... U 1 is ρ m- strongly quasi-nonexpansive; (ii) a convex combination U := I ω iu i, where ω i > 0, i I and I ω i = 1, is ρ-strongly quasi-nonexpansive. Moreover, in both cases, Fix U = i I Fix U i. Proof. See [7, Theorems , , and ] A comprehensive review of the properties of NE, QNE, SQNE and FNE operators can be found in [7, Chapter 2] Demi-closed operators. We say that an operator U : H H is demi-closed at 0 if for any sequence {x } =0 H converging wealy to y H and such that Ux 0 it holds Uy = 0. If we replace the wea convergence by the strong one then we obtain the definition of the closedness of U at 0. If U is nonexpansive then U Id is demi-closed at 0 (see [34, Lemma 2]). It is clear that in a finite dimensional Hilbert space the notions of a demi-closed operator and a closed one coincide. Note that a continuous operator defined on a finite dimensional Hilbert space is closed. 3. Approximately shrining operators and basic examples In this Section we introduce some subclasses of quasi-nonexpansive operators U : H H with the property that the displacement Ux x cannot be too small for those x which are far from Fix U. This property is important for the convergence analysis of methods for solving variational inequality problem over Fix U. These convergence analysis will be presented in Sections 6 and 7.

5 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 403 Definition 3.1. We say that a quasi-nonexpansive operator U : H H is approximately shrining (AS) on a subset D H if for any sequence {x } =0 D it holds lim Ux x = 0 = lim d(x, Fix U) = 0. (3.1) We say that U is approximately shrining if U is AS on any bounded subset D H. If U is AS then we say that the AS property holds for U. Proposition 3.2. Let U : H H be quasi-nonexpansive. Then the following conditions are equivalent: (i) U is approximately shrining. (ii) For any bounded sequence {x } =0 H and for any η > 0 there are γ > 0 and 0 0 such that for all 0 it holds Ux x < γ = d(x, Fix U) < η. (3.2) (iii) For any bounded subset S H and for any η > 0 there is γ > 0 such that for any x S it holds Ux x < γ = d(x, Fix U) < η. (3.3) Proof. The proposition follows from Lemma 8.1 (see Appendix) with f, g : H [0, ) defined as f(x) := x Ux, g(x) := d(x, Fix U). Definition 3.3. We say that a quasi-nonexpansive operator U : H H is linearly shrining (LS) if there is δ > 0 such that for any x H it holds Ux x δd(x, Fix U). (3.4) It is clear that if U is linearly shrining then it is approximately shrining. Conditions which play a similar role to (3.1) or (3.4) appear in many applications (see [3, 6, 8, 26, 28, 31, 32, 34, 35, 39]). Proposition 3.4. Let U : H H be quasi-nonexpansive. If U is AS (LS) then its relaxation U α is AS (LS) for all α [0, 1]. If, furthermore, U is a cutter then its relaxation U α is AS (LS) for all α [0, 2]. Proof. Let U be AS (LS). If α = 0 then U α = Id and the claim is obvious. We have U α x x = α (Ux x) for all x H and all α R, therefore, Fix U α = Fix U for any α > 0. If α (0, 1] then the convexity of the norm and the quasi nonexpansivity of U yield U α x z = (1 α)(x z) + α(ux z) (1 α) x z + α Ux z x z for all x H and z Fix U α, i.e., U α is quasi-nonexpansive. If U is a cutter and α (0, 2], then U α is quasi-nonexpansive. In both cases we have U α x x = α Ux x, consequently, U α is AS (LS).

6 404 ANDRZEJ CEGIELSKI AND RAFA L ZALAS Example 3.5. Let C H be a nonempty, closed and convex subset. By the characterization of the metric projection, P C is a cutter. Furthermore, Fix P C = C. Since a cutter is QNE and P C x x = d(x, C), P C is linearly shrining. By Proposition 3.4, an α-relaxation of P C, where α (0, 2], is linearly shrining. Example 3.6. Let dim H <, f : H R be a convex function and P f : H H be a subgradient projection relative to f i.e., { x f(x) + P f (x) := g f (x) g 2 f (x) if g f (x) 0, x otherwise, where g f (x) denotes a subgradient of f at x H. The existence of g f (x) and, consequently, of P f (x) follows from [3, Corollary 7.9]. Suppose that the sublevel set S(f, 0) := {x f(x) 0}. Then Fix P f = S(f, 0) (see [7, Lemma 4.2.5]) and P f is a a cutter (see [7, Corollary 4.2.6]). By [10, Lemma 24], P f is approximately shrining. By Proposition 3.4, an α-relaxation of P f, where α (0, 2], is approximately shrining. Example 3.7. Let C H be a nonempty, closed and convex subset and δ (0, 1]. Define set valued mappings T C, U C, C δ : H 2 H by and T C (x) := {y H y z x z for all z C}, U C (x) := {y H x y, z y 0 for all z C} C δ (x) := {y H y x δ P C x x }. Further, define set valued mappings T δ C, U δ C : H 2H by T δ C(x) := T C (x) C δ (x) and U δ C(x) := U C (x) C δ (x), x H. It follows from the characterization of the metric projection that P C (x) U δ C(x), x H. Furthermore, UC δ (x) T C δ (x), x H, because a cutter is a quasi-nonexpansive operator. Therefore, TC δ (x), x H. Note that an operator T with Fix T = C is quasi-nonexpansive (a cutter) if and only if T is a selection of T C (of U C ). Therefore, it follows from [7, Corollary (ii)] that U C (x) = 1 2 (T C(x) + x), x H. By definition, any selection T : H H of TC δ is linearly shrining. In particular, any selection U : H H of UC δ is linearly shrining. In Figure 1 there are presented the subsets TC δ(x), U C δ (x) for a closed convex subset C H and for some x H.

7 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 405 x U δ C (x) C T δ C (x) Subsets TC δ(x) and U C δ (x) Note that there are quasi-nonexpansive operators which are not approximately shrining, e.g., the operator U : H H defined by { PD x if x / D Ux := P C x if x D, where C := {x H x 1}, D := {x H x 2}. 4. Properties of approximately shrining operators In this section we present further properties of approximately shrining operators. Proposition 4.1. Let U : H H, with Fix U, be quasi-nonexpansive. Then the following hold: (i) If U is approximately shrining, then U Id is demi-closed at 0. (ii) If dim H < and U Id is closed at zero, then U is approximately shrining. Proof. (i) Let U be AS and {x } =0 H be bounded. Assume that x x and Ux x 0. Let {x n } =0 {x } =0 be such that lim inf d(x, Fix U) = lim d(x n, Fix U). Then, by the AS property of U and by the wea lower semi continuity of d(, Fix U), we have 0 = lim d(xn, Fix U) = lim inf d(x, Fix U) d(x, Fix U). The closedness of Fix U (see Subsection 2.2) yields now x Fix U which proves the demi closedness of U. (ii) Assume that dim H < and U Id is closed at 0. Let {x } =0 H be bounded. Hence, there is a subsequence {x n } =0 of {x } =0, which converges to x H. Without loss of generality we can assume that lim sup d(x, Fix U) = lim d(x n, Fix U). Suppose that Ux x 0. The continuity of d(, Fix U) implies that 0 lim sup d(x, Fix U) = lim d(xn, Fix U) = d(x, Fix U). By the closedness of U Id at 0 we get that x Fix U, i.e., d(x, Fix U) = 0. Therefore, lim d(x, Fix U) = 0, which completes the proof.

8 406 ANDRZEJ CEGIELSKI AND RAFA L ZALAS Because a continuous operator is closed, Proposition 4.1 yields the following result. Corollary 4.2. If a quasi-nonexpansive operator U : R n R n is continuous (e.g., a nonexpansive one), then U is approximately shrining. An important subclass of quasi-nonexpansive operators with similar properties to approximately shrining operators was introduced by Yamada and Ogura [39, Definition 1]. Below, we present an equivalent definition. A proof of equivalence can be found in [9, Proposition 2.9]. Definition 4.3. We say that a quasi-nonexpansive operator U is quasi-shrining on a subset C H if for any sequence {u } =0 C the following implication holds lim (d(u, Fix U) d(uu, Fix U)) = 0 = lim d(u, Fix U) = 0. (4.1) If U is quasi-shrining on H, then U is just called quasi-shrining. Proposition 4.4. If an operator U : H H is quasi-shrining then it is approximately shrining. Moreover, if dim H < and U is approximately shrining and strongly quasi-nonexpansive, then U is quasi-shrining. Proof. Suppose that C H is bounded, U is quasi-shrining on C and denote S := P Fix U. Let {u } =0 C and lim Uu u = 0. Since the metric projection is nonexpansive, we have i.e., 0 lim SUu Su lim Uu u = 0, lim SUu Su = 0. (4.2) By the quasi nonexpansivity of U and by the triangle inequality, consequently, 0 Su u Su Uu Uu u, lim ( Su u Su Uu ) = 0. (4.3) Applying again the triangle inequality, (4.2) and (4.3), we obtain i.e., 0 lim sup( Su u SUu Uu ) lim sup[ Su u ( Su Uu SUu Su )] = lim sup( Su u Su Uu ) = 0, lim ( Su u SUu Uu ) = 0. Since U is quasi-shrining, we have lim d(u, Fix U) = lim Su u = 0 which completes the first part of the proof. Let now dim H < and U be approximately shrining and strongly quasi-nonexpansive. By Proposition 4.1(i), U Id is closed at 0. Consequently, [9, Proposition 2.11] yields, that U is quasi-shrining.

9 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 407 Let U := {U i : H H i I} be a finite family of operators, where I := {1,..., m}. The following result is a simple extension of [7, Theorem 4.8.2], where the operators U i : H H are assumed to be cutters (or, equivalently, 1-SQNE), i I. The proof goes along lines of [7, Theorem 4.8.2] with some simple modifications. Proposition 4.5. Let U i : H H be ρ i -SQNE, ρ i > 0, i I, with i I Fix U i, U := i I ω iu i, where ω i 0, i I, and I ω i = 1, and let x H and z i I Fix U i be arbitrary. Then Consequently, Ux z 2 x z 2 i I 1 2R for any positive R x z. ω i ρ i U i x x 2. (4.4) ω i ρ i U i x x 2 Ux x (4.5) i I Proof. By [7, Corollary ], U i is an α i -relaxed cutter with α i := 2 1+ρ i (0, 2). Hence, α i U i x x, x z U i x x 2 (see [7, Remar ]). Therefore, the triangle inequality and the convexity of the function 2 yield Ux z 2 = x + ı I ω i (U i x x) z 2 x z 2 + ı I ω i (U i x x) i I ω i x z, U i x x x z 2 + ı I = x z 2 ı I = x z 2 ı I ω i U i x x 2 2ω i U i x x 2 α i ı I ( ) 2 ω i 1 U i x x 2 α i ω i ρ i U i x x 2, i.e., (4.4) is satisfied. Let R > 0 be such that x z R. Inequality (4.5) is clear for x i I Fix U i. Suppose that x / i I Fix U i. Then x / Fix U (see Proposition 2.2) and Ux x > 0. By the Cauchy Schwarz inequality, we have Now (4.4) and (4.6) yield Ux z 2 = Ux x 2 + x z Ux x, x z Ux x 2 + x z 2 2R Ux x. (4.6) 1 2R which completes the proof. ω i ρ i U i x x 2 Ux x i I

10 408 ANDRZEJ CEGIELSKI AND RAFA L ZALAS Proposition 4.6. Let U i : H H be ρ i -SQNE, ρ i > 0, i I, with i I Fix U i, U := U m U m 1...U 1, and let x H and z i I Fix U i be arbitrary. Then m Ux z 2 x z 2 ρ i S i x S i 1 x 2, (4.7) where S i := U i U i 1... U 1, i I, S 0 := Id. Consequently, 1 m ρ i S i x S i 1 x 2 Ux x. (4.8) 2R i=1 for any positive R x z. Proof. By the definition of an SQNE operator and by the definition of S i, i = 0, 1,..., m, we have Ux z 2 = U m U m 1...U 1 x z 2 i=1 S m 1 x z 2 ρ m S m x S m 1 x 2 S m 2 x z 2 ρ m 1 S m 1 x S m 2 x 2 ρ m S m x S m 1 x 2 m... x z 2 ρ i S i x S i 1 x 2, i=1 i.e., (4.7) is satisfied. The rest of the proof is similar to the proof of Proposition 4.5, with an application of (4.7) instead of (4.4). Before we formulate two most important results of this section, we recall a definition of a boundedly regular family of operators. Let C i H, i I := {1, 2,..., m}, be closed convex subsets with C := i I C i. Denote C := {C i i I}. Definition 4.7. (cf. [18, Definition 5.1]) We say that the family C is boundedly regular (BR) if for any bounded sequence {x } =0 H the following implication holds lim max i I d(x, C i ) = 0 = lim d(x, C) = 0. Similarly to the definition of an approximately shrining operator, the notion of bounded regularity has also equivalent forms which are presented in the proposition below. We will apply this equivalence in the sequel. Proposition 4.8. The following conditions are equivalent: (i) C is boundedly regular. (ii) For any bounded sequence {x } =0 H and for any η > 0 there are γ > 0 and 0 0 such that for all 0 it holds max{d(x, C i ) i I} < δ = d(x, C) < η. (iii) (cf. [3, Definition 5.1]) For any bounded subset S H and for any η > 0, there is δ > 0 such that for any x S it holds max{d(x, C i ) i I} < δ = d(x, C) < η.

11 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 409 Proof. The proposition follows from Lemma 8.1 (see Appendix) with f, g : H [0, ) defined as f(x) := max{d(x, C i ) i I}, and g(x) := d(x, C). The following proposition gives sufficient conditions for a family C to be boundedly regular. The proof of the proposition can be found in [3, Proposition 5.4 (iii), Corollary 5.14 and Corollary 5.22]. Proposition 4.9. Let C i H, i I := {1, 2,..., m}, be closed convex with C := i I C i. If one of the following conditions is satisfied: (i) dim H <, (ii) C m int m 1 i=1 C i, (iii) C i are half-spaces, i I then the family C := {C i i I} is boundedly regular. An example of a family which in not boundedly regular can be found in [2, Example 5.5]. Note that a subfamily of a BR family C needs not to be BR. To see it consider two closed convex subsets A, B H having a common point such that the family {A, B} is not BR. It follows from the definition that the family {A, B, A B} is boundedly regular. For quasi-nonexpansive operators U i, i I := {1, 2,..., m}, denote F := {Fix U i i I}. Theorem Let U i : H H be ρ i -SQNE, ρ i > 0, i I, with i I Fix U i. If U i, i I, are approximately shrining and the family F is boundedly regular, then the operator U := i I ω iu i, where ω i > 0, i I, i I ω i = 1, is approximately shrining. Proof. Let {x } =0 H be bounded. Assume that Ux x 0. Then, by (4.5) with R x z for all 0 and some z Fix U, it holds The AS property of U i implies that lim U i x x = 0, for all i I. lim d(x, Fix U i ) = 0, for all i I. Since {x } =0 is bounded and the family F is boundedly regular, we have which completes the proof. lim d(x, Fix U) = 0, Theorem Let U i : H H be ρ i -SQNE, ρ i > 0, i I, with i I Fix U i. If U i, i I, are approximately shrining and the family F is boundedly regular, then the operator U := U m U m 1... U 1 is approximately shrining. Proof. Let {x } =0 H be bounded. Since S i := U i U i 1...U 1 is QNE as a composition of SQNE operators (see Proposition 2.2), the sequence {S i x } =0 is also bounded, i I. Therefore, by the AS property of U i, it holds lim U i (S i 1 x ) S i 1 x = 0 = lim d(s i 1 x, Fix U i ) = 0, (4.9)

12 410 ANDRZEJ CEGIELSKI AND RAFA L ZALAS i I. Assume that Ux x 0. By (4.8) with R x z for all 0 and some z Fix U, we have lim U i (S i 1 x ) S i 1 x = 0, (4.10) i I. Moreover, the definition of the metric projection and the triangle inequality yield d(x, Fix U i ) = x P Fix Ui x x P Fix Ui S i 1 x i 1 S j x S j 1 x + S i 1 x P Fix Ui S i 1 x. j=1 Therefore, (4.9) and (4.10) imply lim d(x, Fix U i ) = 0, for all i I. By the bounded regularity of F, we have lim d(x, Fix U) = 0 which completes the proof. 5. Uniformly approximately shrining families of operators In this section we extend the notion and the properties of an approximately shrining operator to a family of operators. The most important results of this section are Theorems 5.4 and 5.6, where we present conditions under which a uniformly approximately shrining family of operators is closed under convex combination and composition. This property is applied in Section 7, where we present several methods for solving variational inequality problem over the common fixed point set of quasi-nonexpansive operators. Let U := {U i H H, i I} be a family of QNE operators. In this section we do not suppose that I is finite. Definition 5.1. We say that the family U is uniformly approximately shrining (UAS) if for any bounded sequence {x } =0 H and for any η > 0 there are γ U > 0 and U 0, such that for all U and for all i I it holds U i x x < γ U = d(x, Fix U i ) < η. It is clear that if U is a finite family of AS operators then U is uniformly approximately shrining. Example 5.2. Let C H be a nonempty, closed and convex subset and δ (0, 1]. Let U be the family of all selections of the set valued mapping TC δ (see Example 3.7). Then U is uniformly approximately shrining. To verify this fact, it suffices to choose γ U = ηδ and U = 0. Note that γ U and U do not dependent on {x } =0. Lemma 5.3. Let U be a uniformly approximately shrining family of cutters and ε (0, 2].Then the family U α := {U α U U, α [ε, 2]} is uniformly approximately shrining. Proof. Let {x } =0 be bounded, η > 0 and let γ U > 0 and U 0 be such that for any U and for any U U, it holds d(x, Fix U) < η whenever Ux x < γ U. Let U U and α [ε, 2]. Then U α is QNE and Fix U α = Fix U. Put γ 0 := εγ U, 0 := U and let 0. Suppose that U α x x < γ 0. Then εγ U = γ 0 > U α x x = α Ux x ε Ux x

13 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 411 and, consequently, d(x, Fix U α ) = d(x, Fix U) < η. Suppose that i I Fix U i. For a subset J I denote F J := {Fix U j j J} and F := F I. Suppose that F is finite. Note that F can be finite even if I is infinite. Further, for a fixed m N and ε (0, 1 m ] denote and J m := {J I J = m and F J is BR} V m := V := ω j U j J J m, ω j [ε, 1], j J, ω j = 1. j J j J Note that if J m and the operators U i are SQNE, i I, then any element of V m is SQNE (see Proposition 2.2(ii)). Theorem 5.4. Let U i : H H be ρ i -SQNE, i I, ρ := inf i I ρ i > 0 and i I Fix U i. Assume that F is finite, m I and J m is nonempty. If U is uniformly approximately shrining then V m is uniformly approximately shrining. Proof. We have to show that for every bounded sequence {x } =0 H and for any η > 0 there exist γ V > 0 and V 0 such that for all V V m and all V it holds V x x < γ V = d(x, Fix V ) < η, Let η > 0, {x } =0 H be bounded and V V m, i.e., V = j J ω ju j, for some J J m, where ω j [ε, 1], j J, j J ω j = 1. By Proposition 2.2, Fix V = j J Fix U j. Let z Fix V and R > 0 be such that x z R for all 0. By Proposition 4.5, we have ερ 2R U ix x 2 1 2R ω j ρ j U j x x 2 V x x (5.1) i J for all i J and all 0. Since the sequence {x } =0 is bounded and F J is boundedly regular, there exist δ > 0 and 0 0 such that max j J d(x, Fix U j ) < δ = d(x, Fix V ) < η (5.2) for all 0. Note that δ and 0 may be chosen independently of J J m. To prove this, consider the families F J, J J m. For any F J we can find δ J > 0 and J 0 such that (5.2) holds with δ := δ J and 0 := J (note that F J is BR). It is easily seen that δ J, J J m and J 0, can be chosen such that δ := inf J Jm δ J > 0 and 0 := sup J Jm J < +, because F is finite. The family U is uniformly AS, hence, we can find γ U > 0 and U 0 such that for all j I and all U it holds U j x x < γ U = d(x, Fix U j ) < δ. (5.3)

14 412 ANDRZEJ CEGIELSKI AND RAFA L ZALAS Let V := max{ U, 0 }, γ V := εγu 2 ρ/2r and let V. Assume that V x x < γ V. Then, by (5.1), U j x x < γ U for all j J. Therefore, (5.3) yields max j J d(x, Fix U j ) < δ, which, by (5.2), completes the proof. Let U be AS and η > 0. Consider a family of bounded sequences X := {{x n} =0 n I}, where I is finite. By the AS property of U, it is clear that for any sequence {x n} =0, n I, there are γ n > 0 and n 0, such that for all n it holds Ux n x n < γ n = d(x n, Fix U) < η. If we put γ := min n I γ n and 0 := max n I n then for all 0 and for all n I it holds Ux n x n < γ = d(x n, Fix U) < η. Such an approach cannot be applied in the case when I is infinite. Nevertheless, if I is countable we can omit such inconveniences by presenting another approach for choosing γ and 0. Moreover, such an approach can be presented not only for one AS operator U, but for a family of UAS operators. Lemma 5.5. Let U be uniformly approximately shrining and let X := {{x n} =0 n N} H be bounded. Then, for any η > 0 there are γ > 0 and 0 0 such that for all U U, n N and 0 it holds Ux n x n < γ = d(x n, Fix U) < η. Proof. Let η > 0 be fixed. The family X is countable, hence there is a sequence {u l } l=0 which consists of all x n,, n 0, and maintains the order of indices, i.e., if 1 2, n 1 n 2 and x 1 n 1, x 2 n 2 have the positions l 1, l 2 in {u l } l=0, respectively, then l 1 l 2 (one can apply, e.g., a well nown construction which shows that N 2 is countable). By the boundedness of X, the sequence {u l } l=0 is also bounded. By the UAS property of U, there are γ U > 0 and l U 0, such that for all U U and l l U it holds Uu l u l < γ U = d(u l, Fix U) < η. Let γ := γ U and 0 be the smallest integer such that x 0 1 has a position l 0 l U in {u l } l=0. Suppose that 0, n 0, Ux n x n < γ and x n is situated at the position l,n in the sequence {u l } =0, i.e., x n = u l,n. Then it follows from the properties of the sequence {u l } l=0 that x n has a position l,n l U in {u l } l=0. Hence, which completes the proof. and For a fixed m N denote d(x n, Fix U) = d(u l,n, Fix U) < η J m := {J = (j 1,..., j m ) j i I, i = 1, 2,..., m, and F J is BR} (5.4) S m := {S = U jm... U j1 J J m }. We identify J := (j 1,..., j m ) in (5.4) wit a subset {j 1,..., j m } I and, for simplicity, we write J = {j 1,..., j m }. Note that if J m and the operators U i are SQNE then any element of S m is SQNE (see Proposition 2.2(i)).

15 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 413 Theorem 5.6. Let U i : H H be ρ i -SQNE for all i I, ρ := inf i I ρ i > 0 and i I Fix U i. Assume that U is countable, F is finite, m N is fixed and J m is nonempty. If U is uniformly approximately shrining then S m is uniformly approximately shrining. Proof. Let η > 0 and {x } =0 be bounded. We have to show that there exist γ S > 0 and S 0 such that for all S S m and all S it holds Denote and Sx x < γ S = d(x, Fix S) < η. (5.5) J := {J = (j 1,..., j t ) j i I, i = 1, 2,..., t, 1 t m}, S := {S = U jt... U j1 J = (j 1,..., j t ) J }, Y := {{y } =0 := {Sx } =0 S S} {{x } =0}. We divide the proof in 5 steps. Step 1: Let z i I Fix U i and R > 0 be such that x z R, for all 0 and let S S. The operator S is QNE, hence Sx z x z R, for all 0 and z Fix S = i I Fix U i (see Proposition 2.2). Therefore, Y is bounded. Step 2: Let S S m, i.e., S = U jm... U j1 for some J = (j 1,..., j m ) J m. Assume, for simplicity, that J = (1,..., m). Let z Fix S and R > 0 be such that x z R. By Proposition 4.6, we have Sx x ρ 2R m S j x S j 1 x 2, (5.6) j=1 where S j := U j... U 1, j = 1, 2,..., m and S 0 := Id. Step 3 : Since Y is bounded and F J is boundedly regular, there exist δ > 0 and 0 0 such that it holds max j J d(x, Fix U j ) < δ = d(x, Fix S) < η (5.7) for all 0. Note that δ and 0 may be chosen independently of J J m, by similar arguments to those made in the proof of Theorem 5.4. Step 4: The family U is countable, hence, by definition, Y consists of a countable number of sequences. By Step 1, Y is bounded and, by assumption, U is uniformly approximately shrining. Therefore, it follows from Lemma 5.5 that there are γ U > 0 and U 0 such that for all U U, {y } =0 Y and U it holds Uy y < γ U = d(y, Fix U) < δ/2. (5.8) By (5.8) with U = U j and y = S j 1 x, we have U j (S j 1 x ) S j 1 x < γ U = d(s j 1 x, Fix U j ) < δ/2, (5.9) for all j J. Step 5: Now we go to the main part of the proof. Let 0 < γ U δ/2m, S := max{ U, 0 }, γ S := ργ 2 U/2R (5.10)

16 414 ANDRZEJ CEGIELSKI AND RAFA L ZALAS and suppose that Sx x < γ S for some S. Hence, by (5.6), U j (S j 1 x ) S j 1 x < γ U, for all j J. By (5.9), the definition of the metric projection and the triangle inequality, we get d(x, Fix U j ) = x P Fix Uj x x P Fix Uj S j 1 x < j 1 S l x S l 1 x + S j 1 x P Fix Uj S j 1 x l=1 m γ U + δ/2 δ. l=1 By the bounded regularity (see (5.7)) d(x, Fix S) < η, which completes the proof. The following two Corollaries follow immediately from Theorems 5.4 and 5.6. Corollary 5.7. Let U i : H H, i I := {1,..., m}, be approximately shrining cutters with a common fixed point. Let ε (0, 1 m ] be arbitrary. If F is boundedly regular, then the family { } V := V = i I is uniformly approximately shrining. ω i U i,αi α i [ε, 2 ε], ω i [ε, 1], i I, ω i = 1 i I Corollary 5.8. Let U i : H H, i I := {1,..., m}, be approximately shrining cutters with a common fixed point. If F is boundedly regular and A [ε, 2 ε], where ε (0, 1), is a countable subset, then the family is uniformly approximately shrining. S = {S = U m,αm... U 1,α1 α 1,..., α m A} 6. General method for VIP over the common fixed point set of quasi-nonexpansive operators Let U i : H H, i I := {1, 2,..., m}, be quasi-nonexpansive with C := i I Fix U i and F : H H be κ-lipschitz continuous and η-strongly monotone, where 0 < η κ. It is clear that C is a closed convex subset, because Fix U i, i I, are closed convex. Consider the following variational inequality problem, called VIP(F, C): Find u C such that F u, u u 0 for all u C. (6.1) The existence and uniqueness of a solution of VIP(F, C) follows from [40, Theorem 46.C]. In this Section we will analyze the convergence properties of the following method, which we call a generalized hybrid steepest descent (GHSD) method: u 0 H u +1 := T u λ F T u, arbitrary (6.2)

17 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 415 where {T } =0 is a sequence of quasi-nonexpansive operators T : H H satisfying C 0 Fix T and {λ } =0 [0, 2η/κ2 ] (cf. [10, Section 3]). Before we present our main result of this Section, we recall a definition of an s-approximately shrining method (cf. [10, Definition 11]). Definition 6.1. Let T : H H be ρ -strongly quasi-nonexpansive, where ρ 0, 0. We say that method (6.2) is s-approximately shrining with respect to a subset D H if for any η > 0 there are γ > 0 and 0 0, such that for all 0 it holds s 1 ρ l T l u l u l 2 < γ = d(u, D) < η. (6.3) l=0 We say that the method is approximately shrining with respect to D if it is 1- approximately shrining with respect to D. The following result follows from [10, Lemma 9 and Theorem 12]. Theorem 6.2. Let {u } =0 be generated by the GHSD method (6.2). If the following conditions are satisfied: (i) the operators T are ρ -SQNE with ρ 0, 0, (ii) lim λ = 0 and =0 λ = +, then {u } =0 and {F T u } =0 are bounded. Moreover, if (iii) the method is s-approximately shrining with respect to C for some s 1, then {u } =0 converges in norm to a unique solution of VIP(F, C). The most difficult problem in applications of Theorem 6.2 is to guarantee that the method is s-as with respect to C. In the theorem below, which is the main theorem of this section, we present sufficient conditions for the method to be s-as. These conditions correspond directly to the quasi-nonexpansive operators T, 0, and U i, i I, and not to the method. Such conditions are in most applications easier to verify than the condition that the method is s-as (see Section 7). One can prove that the iteration in (6.2) is equivalent to the following one u +1 = (1 α )T u + α QT u, (6.4) where Q : H H is a contraction. Recently, Hirstoaga in [28] proposed a method for solving VIP(F, C), where C := 0 Fix T, which is slightly more general than (6.4). Hirstoaga supposed that the method has the following property: if u +1 T u 0 then any wea cluster point of {u } =0 belongs to C. Furthermore, Hirstoaga gave several examples of methods satisfying this condition, where T, 0, are assumed to be nonexpansive (see [28, Section 3]). In Section 7 we will present, however, examples of method (6.2), where T, 0, are quasi-nonexpansive. In many applications these operators are simpler to evaluate than the nonexpansive ones with corresponding properties. As an example we give here the subgradient projection P f relative to a convex function f : H R defined on a finite dimensional Hilbert space H with S(f, 0) := {x H f(x) 0} (see Example 3.6). In general, P f x is simpler to evaluate than P S(f,0) x for x H with f(x) > 0.

18 416 ANDRZEJ CEGIELSKI AND RAFA L ZALAS Theorem 6.3. Let {u } =0 be generated by method (6.2) and the following conditions be satisfied: (i) the operators T are ρ -SQNE, 0, with ρ := inf ρ > 0, (ii) lim λ = 0 and =0 λ = +, (iii) there is s m, such that for any i I and s 1, there is l,i {0,..., s 1} with Fix T l,i Fix U i, (iv) the family T := {T 0} is uniformly approximately shrining, (v) the family F := {Fix U i i I} is boundedly regular. Then the method is s-approximately shrining with respect to C, consequently {u } =0 converges in norm to a unique solution of VIP(F, C). Proof. Let η > 0 be fixed and {u } =0 be a sequence generated by method (6.2). By Theorem 6.2, the sequences {u } =0 and {F T u } =0 are bounded. By the bounded regularity of F, there are δ > 0 and 1 0, such that for any 1 it holds max i I d(u, Fix U i ) < δ = d(u, C) < η. (6.5) By assumption, T is uniformly AS, consequently, there are γ T > 0 and T 0 such that for all T it holds T u u < γ T = d(u, Fix T ) < δ/2. (6.6) Put γ := ρ min{γ 2 T, (δ/4s)2 } and 0 := max{ T, 1, 2 } + s, where 2 is such that for all 2 s 2 λ l F T l u l < δ/4. l=0 The existence of 2 follows from the assumption lim λ = 0 and from the boundedness of {F T u } =0. Assume that 0 and s 1 ρ l T l u l u l 2 < γ. (6.7) l=0 Hence, T l u l u l < γ T, for all l {0,..., s 1}. In particular, this inequality holds for l = l,i where l,i is the smallest integer from {0,..., s 1} such that Fix T l,i Fix U i, i I. By (6.6), we have Furthermore, (6.7) yields d(u l,i, Fix T l,i ) < δ/2. (6.8) T l u l u l < δ 4s

19 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 417 for any l = 0, 1,..., s 1. Now, for any i I we have u u l,i l,i 1 l=0 l=0 s 2 u l u l 1 u l u l 1 l=0 s 2 s 2 T l 1 u l 1 u l 1 + λ l 1 F T l 1 u l 1 < δ/4 + δ/4 = δ/2. l=0 Consequently, (iii), the definition of the metric projection, the triangle inequality and (6.8) yield for any i I u P Fix Ui u u P Fix T l,i u u P Fix T l,i u l,i u u l,i + u l,i P Fix T l,i u l,i < δ/2 + δ/2 = δ. By implication (6.5), we obtain d(u, C) < η, consequently, method (6.2) is s- approximately shrining with respect to C. Now, Theorem 6.2 yields the convergence of u to a unique solution of VIP(F, C). 7. Examples In this Section we present several examples of GHSD method with various constructions of operators T basing on a finite family of operators and analyze their convergence properties by applying Theorem 6.3. Let U i : H H, i I := {1, 2,..., m}, be quasi-nonexpansive with C := i I Fix U i and F : H H be κ-lipschitz continuous and η-strongly monotone, where 0 < η κ. We assume without loss of generality that all U i, i I, are cutters, because any QNE operator U : H H can be treated as a relaxation of a cutter S : H H and Fix U = Fix S (see Subsection 2.2). We consider the GHSD method (6.2) for solving VIP(F, C), where {T } =0 is a sequence of quasi-nonexpansive operators T : H H satisfying C 0 Fix T and {λ } =0 [0, 2η/κ2 ] is a sequence satisfying lim λ = 0 and =1 λ =. Denote U := {U i i I} and F := {Fix U i i I}. At the beginning we present a general procedure of a construction of operators T basing on the family U. This procedures is related to the so called string-averaging scheme introduced by Censor et al. in [12] and studied in [13, 14, 15, 16]. The scheme has the following form. Procedure 7.1 (String-Averaging Scheme). Let 0 and r N. Step 1. For p = 1, 2,..., t, where t m, choose nonempty, ordered subsets I p elements taen from I, called strings, of the form of I p = (ip 1, ip 2,..., ip,m p ),

20 418 ANDRZEJ CEGIELSKI AND RAFA L ZALAS where i p j I, j = 1, 2,..., mp and mp r. Further, choose vectors ap of corresponding relaxation parameters of the form a p = (αp 1, αp 2,..., αp,m p ), where α p j [ε, 2 ε], j = 1, 2,..., mp, and ε (0, 1] is a constant. Step 2. For p = 1, 2,..., t, and j = 1, 2,..., m p p, denote by Vj the relaxation of U i p j with a relaxation parameter α p j, i.e., and create operators and V p V p j := Id +αp j (U i p j Id), S p j := V p j V p,j 1...V p 1, := Sp,m p Step 3. For p = 1, 2,..., t choose weights ω p ω (0, 1 m ] is a constant, and define T := = V p V p,m p,m p 1...V p 1 (7.1) t p=1 ω satisfying t p=1 ω p = 1, where ω p V p. (7.2) In the original string-averaging scheme there is supposed that the strings do not depend of, i.e., t = t and m p = mp for all, and that α p j = 1, j = 1, 2,..., mp, p = 1, 2,..., t, consequently, V p = V p and T = T (cf. [13]). First, we present some properties of the operators defined above, basing on the facts given in Subsection 2.2. It follows from Proposition 2.2 that for all 0, p = 1, 2,..., t and j = 1, 2,..., m p the operators defined in Procedure 7.1 have the following properties: (1) V p j is ε p 2-SQNE because Vj is an αp j -relaxation of a cutter U i p and ε j α p j 2; (2) S p j is ε 2r -SQNE because Sp j which are ε 2 -SQNE and j mp r; (3) V p (4) T is ε is ε 2r -SQNE because V p = Sp,m p ; p is a composition of j operators Vi, i = 1, 2,..., j, 2r -SQNE because T is a convex combination of operators V p which are ε 2r -SQNE. Consider the GHSD method combined with the string-averaging scheme of the construction of operators T. This approach gives a very flexible framewor related to a lot of results nown in literature. Because the operators T are ρ -SQNE, with ρ := inf ρ > 0, Theorem 6.3 implies that only conditions (iii), (iv) and (v) have to be checed in order to guarantee the convergence of sequences generated by the GHSD method. In the examples below we present several constructions of T satisfying these conditions, which are special cases of the string-averaging scheme.

21 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 419 Example 7.2. (Hybrid steepest descent method) Let m = 1. Then the family U consists of only one operator U : H H which is a cutter. Trivially F is BR. Assume that U is AS. GHSD method (6.2) combined with Procedure 7.1 has the form: u +1 = U α u λ F U α u, where U α is an α -relaxation of a cutter U with α [ε, 2 ε] and ε (0, 1]. Condition (iii) of Theorem 6.3 is trivial and condition (iv) follows from Lemma 5.3. Consequently, u converges in norm to a unique solution of the VIP(F, Fix U). If we tae α = α [0, 2], 0, then T := U α is quasi-nonexpansive and we obtain the hybrid steepest descent method u +1 = T u λ F T u studied by Yamada in [38] and developed by Yamada and Ogura in [39]. In [38] T is supposed to be nonexpansive and the convergence was proved under an additional assumption lim (λ λ +1 )/λ 2 +1 = 0 (see [38, Theorem 3.2 ]). Recall that if H is finite dimensional, then a nonexpansive operator is approximately shrining (see Corollary 4.2). In [39, Theorem 4] the convergence was proved under the assumption that T is quasi-shrining. This assumption implies that T is approximately shrining (see Proposition 4.4). Therefore, for α (0, 2) the convergence follows from Theorem 6.3. In the next examples we assume that m > 1. Furthermore, we assume that U i, i I, are approximately shrining and the family F is boundedly regular. Example 7.3. (GHSD with successive projections) Consider GHSD method (6.2) combined with Procedure 7.1, where t = 1, m := m 1 = 1, I := I 1 = i and α := α1 1, 0. The method has the form u +1 = U i,α u λ F U i,α u. Condition (iv) of Theorem 6.3 follows from Lemma 5.3, because α [ε, 2 ε], 0, for a constant ε (0, 1]. The sequence {i } =0 is called a control. We consider two types of control: (a) (almost cyclic control) Suppose that there is s m such that I {i,..., i +s 1 } for all 0. Such a control {i } =0 is called s-almost cyclic (cf. [11, Definition 3.4]). Then condition (iii) of Theorem 6.3 is satisfied. Consequently, u converges in norm to a unique solution of VIP(F, C). The convergence of the method was proved in [10, Theorem 21]. A special case of the above iteration with a cyclic control and T being nonexpansive was studied by Lions [30, Theorem 4], Bausche [1, Theorem 3.1], Yamada in [38, Theorem 3.3], Xu and Kim in [37, Theorem 3.2] and Hirstoaga in [28, Corollary 3.1], where the convergence was proved under various assumptions on the sequence {λ } =0 and the family F. (b) (maximal displacement control) Let i := argmax{ U i u u i I}. We show that the method is approximately shrining with respect to C. Let η > 0. Because F is boundedly regular and {u } =0 is bounded (see Theorem

22 420 ANDRZEJ CEGIELSKI AND RAFA L ZALAS 6.2), there are δ > 0 and 0 0 such that max i I d(u, Fix U i ) < δ = d(u, C) < η (7.3) for all 0. Because U i is approximately shrining, there is γ i > 0 and i 0 such that for all i it holds U i u u < γ i = d(u, Fix U i ) < δ, (7.4) i I. Let γ = min i i γ i and = max i {0,1,...,m} i. Suppose that and Ui u u < γ. Then, Ui u u < γi and (7.3) and (7.4) imply d(u, C) < η. Therefore, the method is approximately shrining. Theorem 6.2 yields that u converges to a unique solution of VIP(F, C). Example 7.4. (GHSD with simultaneous projections) Consider GHSD method (6.2) combined with Procedure 7.1, where t = m, m p = 1, Ip = p and ap = α, p = 1, 2,..., m, 0. The method has the form m T := Id +α ( ω p U p Id). (7.5) p=1 By Proposition 2.2, Fix T = C, 0, i.e., condition (iii) of Theorem 6.3 is satisfied. Condition (iv) follows from Corollary 5.7. Consequently, u converges in norm to a unique solution of VIP(F, C). The above construction of the operator T arises very often in the literature, not only related to the variational inequalities. For VIP(F, C) the method with U i, i I, being firmly nonexpansive and with constant weight vectors w = w and relaxation parameters α = α was studied by Yamada in [38, Corollary 3.6], where α = 1 and by Hirstoaga in [28, Remar 3.5], where α [0, 2]. Example 7.5. (GHSD with sequential projections) Consider GHSD method (6.2) combined with Procedure 7.1, where t = 1, m := m 1 = m and I := I 1 consists of all elements of I, 0. Denote by I := (i 1, i 2,..., i m ) the string in -th iteration and by a := (α 1, α 2,..., α m ) the corresponding vector of relaxation parameters. Then, T = U im,α m U i,m 1,α,m 1...U i1,α 1 Similarly as in Example 7.4, condition (iii) of Theorem 6.3 is satisfied and condition (iv) follows from Corollary 5.8. Therefore, u converges in norm to a unique solution of VIP(F, C). Such sequential approach for VIP was studied by Yamada in [38, Theorem 3.5] and by Hirstoaga in [28, Corollary 3.4 and Remar 3.5], where U i were supposed to be nonexpansive and T = U m U m 1...U 1, where the convergence was proved under some additional assumptions on the sequence {λ } =0 and the family F. Although, I p is an ordered subset in Procedure 7.1, by the same symbol we denote the corresponding subset of I, i.e., I p = {ip 1, ip 2,..., ip 0, p = 1, 2,..., t,m}, p. Further, denote I := I 1 I2... It. Let {T } =0 be a sequence of operators defined by Procedure 7.1 and suppose that the procedure is s-intermittent, where s m, i.e., I I I +s 1 = I for all 0 (cf. [3, Definition 3.18]). If we assume that all subfamilies of F are

23 APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS 421 boundedly regular, then, by Theorems 5.4 and 5.6, the family T := {T 0} is uniformly approximately shrining. In this case the sequence {T } =0 fits into the frame of Theorem 6.3, which allows a more general construction of T than the string-averaging scheme. Consequently, the sequence {u } =0 generated by the GHSD method converges in norm to a unique solution of VIP(F, C). Unfortunately, the assumption that all subfamilies of F are boundedly regular is very restrictive. This assumption can be weaened, if we restrict the construction of T to the stringaveraging scheme. Theorem 7.6. Let U i : H H, i I := {1, 2,..., m}, be approximately shrining cutters with C := i I Fix U i, and u be generated by GHSD method (6.2), where T, 0, is constructed by the string-averaging scheme (Procedure 7.1). If Procedure 7.1 is s-intermittent for some s m and the family F is boundedly regular, then the GHSD method is s-approximately shrining with respect to C. Consequently u converges in norm to a unique solution of VIP(F, C). Proof. Let Procedure 7.1 be s-intermittent for some s m and let η > 0. We have to show that there exist γ > 0 and 0 0 such that for all 0 it holds d(u, C) < η whenever l= s+1 ρ l T l u l u l 2 < γ. By Theorem 6.2, the sequences {u } =0 and {F T u } =0 are bounded. We divide the proof into three steps. Step 1. In this step we present three auxiliary properties of Procedure 7.1. (a) Let R > 0 be such that u z R for some z i I Fix U i and for all 0. Let δ 0 > 0 and 1 0 be such that for any 1 it holds max i I d(u, Fix U i ) < δ 0 = d(u, C) < η. (7.6) The existence of δ 0 and 1 follows from the bounded regularity of F. (b) Let i I be arbitrary. Let s 1 and l { s + 1, s + 2,..., }, p {1, 2,..., t l } and j {1, 2,..., m p l } be such that U i = U p i. The existence l,j of l, p and j follows from the assumption that Procedure 7.1 is s-intermittent. Because S p l,j 1 is QNE as a composition of strongly quasi-nonexpansive operators having a common fixed point (see Proposition 2.2) and u is bounded, we have that v := S p l,j 1 ul is bounded. Note that y := S p l,j u l = V p l,j (v ) = V p l,j (S p l,j 1 ul ) = U i,α p l,j (S p l,j 1 ul ) = U i,α p l,j (v ), (7.7) where α p l,j ε. Because U i are approximately shrining, the family {U i,α α [ε, 1], i I} is UAS (see Lemma 5.3). Consequently, there are γ U > 0 and U 0 such that for all U it holds U i,α v v < γ U = d(v, Fix U i ) < δ 0 /4. (7.8) Let δ := min{γ U, δ 0 }. Recall that V p l,j = U p i,α l,j and α p l,j ε. Therefore, U i,α p l,j v v < δ = d(v, Fix U i ) < δ 0 /4. (7.9)