PROXIMAL POINT ALGORITHMS INVOLVING FIXED POINT OF NONSPREADING-TYPE MULTIVALUED MAPPINGS IN HILBERT SPACES

PROXIMAL POINT ALGORITHMS INVOLVING FIXED POINT OF NONSPREADING-TYPE MULTIVALUED MAPPINGS IN HILBERT SPACES Shih-sen Chang 1, Ding Ping Wu 2, Lin Wang 3,, Gang Wang 3 1 Center for General Educatin, China Medical University, Taichung, 40402, Taiwan 2 School of Applied Mathematics, Chengdu University of Information Technology Chengsu, Sichuan 610103, China 3 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China E-mail: changss2013@163.com; wdp68@163.com; wl64mail@aliyun.com; wg631208@sina.com Abstract. In this paper, a new modified proximal point algorithm involving fixed point of nonspreadingtype multivalued mappings in Hilbert spaces is proposed. Under suitable conditions, some weak convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the nonspreading-type multivalued mappings in Hilbert space are proved. The results presented in the paper are new. Keywords: Convex minimization problem; resolvent identity; proximal point algorithm; nonspreadingtype multivalued mapping; Weak and strong convergence theorem. 2000 AMS Subject Classification: 47J05, 47H09, 49J25. 1. Introduction Throughout this paper we always assume that H is a real Hilbert space and C is a nonempty closed and convex subsets of H. In the sequel we denote by CB(C) and K(C) the families of nonempty closed bounded subsets and nonempty compact subsets of C, respectively. The Hausdorff metric on CB(C) is defined by H(A, B) = max{sup x A where d(x, B) = inf b B d(x, b). d(x, B), sup d(y, A)}, A, B CB(C) y B In what follows, we denote by F ix(t ) the fixed point set of a mapping T. And write x n x to indicate that the sequence {x n } converges weakly to x and x n x implies that {x n } converges strongly to x. Recall that a singlevalued mapping T : C C is said to be nonexpansive if T x T y x y x, y C. A multivalued mapping T : C CB(C) is said to be nonexpansive if H(T x, T y) x y, x, y C, and T : C CB(C) is said to be quasi-nonexpansive if F ix(t ) and H(T x, T p) x p, x C, p F ix(t ). 0 The corresponding authors: wl64mail@aliyun.com (Lin Wang). 0 This work was supported by the Natural Science Foundation of Center for General Educatin, China Medical University, Taichung, Taiwan and The Natural Science Foundation of China (Grant No. 11361070) 1

2 PROXIMAL POINT ALGORITHMS Recall that a single-valued mapping T : C C is said to be nonspreading mappings [1] if 2 T x T y 2 x T y 2 + y T x 2, x, y C. (1.1) It is easy to prove that T : C C is nonspreading if and only if T x T y 2 x y 2 + 2 x T x, y T y x, y C. (1.2) A mapping T : C CB(C) is said to be nonspreading-type multi-valued mapping [2] if 2H(T x, T y) 2 d(x, T y) 2 + d(y, T x) 2, x, y C. (1.3) It is easy to see that, if T is a nonspreading-type multi-valued mapping, and F ix(t ), then T is a quasi-nonexpansive multi-valued mapping, i.e., H(T x, T p) x p, x C and p F ix(t ). (1.4) Indeed, for all x C and p F ix(t ), we have This implies that 2H(T x, T p) 2 d(p, T x) 2 + d(x, T p) 2 H(T x, T p) 2 + x p 2. H(T x, T p) x p. Example of nonspreading-type multi-valued mapping [2] Let C = [ 3, 0] with the usual norm. Define a multivalued mapping T : C CB(C) by { {0}, if x [ 2, 0]; T x = [ exp{x + 2}, 0], if x [ 2, 0]; It is easy to prove that T is a nonspreading-type multi-valued mapping but it does not a multi-valued nonexpansive mapping. Recall that a multivalued mapping T : C CB(C) is said to be demiclosed at 0, if {x n } C such that x n x and lim d(x n, T x n ) = 0 imply x T x. Let f : H (, ] be a proper convex and lower semi-continuous function. One of the major problems in optimization in Hilbert space H is to find x H such that f(x) = min f(y). (1.5) y H We denote by argmin y H f(y) the set of minimizers of f in H. A successful and powerful tool for solving this problem is the well-known proximal point algorithm (shortly, the PPA) which was initiated by Martinet [3] in 1970. In 1976, Rockafellar [4] generally studied, by the PPA, the convergence to a solution of the convex minimization problem in the framework of Hilbert spaces. Indeed, let f be a proper, convex, and lower semi-continuous function on a Hilbert space H which attains its minimum. The PPA is defined by x 1 H x n+1 = argmin y H (f(y) + 1 y x n 2 (1.6) ), n 1, where λ n > 0 for all n 1. It was proved that the sequence {x n } converges weakly to a minimizer of f provided Σ n=1λ n =.

PROXIMAL POINT ALGORITHMS 3 However, as shown by Güler [5], the PPA does not necessarily converges strongly in general. In 2000, Kamimura-Takahashi [6] combined the PPA with Halpern s algorithm [7] so that the strong convergence is guaranteed. In the recent years, the problem of finding a common element of the set of solutions of various convex minimization problems and the set of fixed points for a singlevalued mapping in the framework of Hilbert spaces and Banach spaces have been intensively studied by many authors, for instance, (see [8-12]) and the references therein. The purpose of this paper is to introduce and study the following modified proximal point algorithm involving fixed point for nonspreading-type multivalued mappings in Hilbert spaces. u n = argminy y C [f(y) + 1 y x n 2 ], y n = (1 β n )x n + β n w n, n 1, (1.7) x n+1 = (1 α n )x n + α n v n, where T : C CB(C) is a nonspreading-type multivalued mapping, w n T u n and v n T y n for all n 1. Under suitable conditions, some weak convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the nonspreading-type multivalued mappings in Hilbert space are proved. The results presented in the paper are new. 2. Preliminaries In order to prove the main results of the paper, we need the following notations and lemmas Let f : H (, ] be a proper convex and lower semi-continuous function. For any λ > 0, define the Moreau-Yosida resolvent of f in H by J λ (x) = argmin y H [f(y) + 1 2λ y x 2 ], x H. (2.1) It was shown in [13] that the fixed point set F ix(j λ ) of the resolvent associated of f coincides with the set argmin y H f(y) of minimizers of f. Lemma 2.1 [14] Let f : H (, ] be proper convex and lower semi-continuous. For any λ > 0, the resolvent J λ of f is nonexpansive. Lemma 2.2 (Sub-differential inequality) [15] Let f : H (, ] be proper convex and lower semi-continuous. Then, for all x, y H and λ > 0, the following sub-differential inequality holds: 1 2λ J λx y 2 1 2λ x y 2 + 1 2λ x J λx 2 + f(j λ x) f(y). (2.2) Lemma 2.3 [2] Let C be a nonempty closed and convex subset of H. (I) If T : C CB(C) is a nonspreading-type multivalued mapping and F ix(t ). Then the following conclusions hold. (i) F ix(t ) is closed; (ii) If, in addition, T satisfies the condition: T p = {p}, p F ix(t ), then F ix(t ) is convex. (II) Let T : C K(C) be a nonspreading-type multivalued mapping.

4 PROXIMAL POINT ALGORITHMS (iii) If x, y C and u T x, then there exists v T y such that H(T x, T y) 2 x y 2 + 2 x u, y v. (2.3) (iv) (Demiclosed principle) If {x n } is a bounded sequence in C such that x n p and lim x n y n = 0 for some y n T x n, then p T p. Lemma 2.4 [20] (The resolvent identity) Let f : H (, ] be a proper convex and lower semi-continuous function. Then the following identity holds: J λ x = J µ ( λ µ λ J λx + µ x), x H and λ > µ > 0. (2.4) λ Definition 2.5 A Banach space X is said to satisfy Opial condition, if x n z (as n ) and z y imply that lim sup x n z < lim sup x n y. It is well known that each Hilbert space H satisfies the Opial condition. 3. Weak convergence theorems We are now in a position to give the following main result. Theorem 3.1 Let f : C (, ] be a proper convex and lower semi-continuous function, and T : C K(C) be a nonspreading-type multivalued mapping. Let {α n }, {β n } be sequences in [0, 1] with 0 < a α n, β n < b < 1, n 1. Let {λ n } be a sequence such that λ n λ > 0 for all n 1 and some λ. For any given x 0 C, let {x n } be the sequence generated in the following manner: u n = argminy y C [f(y) + 1 y x n 2 ], y n = (1 β n )x n + β n w n, x n+1 = (1 α n )x n + α n v n, n 1, (3.1) where w n T u n and v n T y n for all n 1. If Ω := F ix(t ) argmin y C f(y), then {x n } converges weakly to a point x Ω which is a minimizer of f in C as well as it is also a fixed point of T in C. Proof Let q Ω. Then q = T q and f(q) f(y), y C. This implies that f(q) + 1 q q 2 f(y) + 1 y q 2, y C, and hence q = J λn q, n 1, where J λn is the Moreau-Yosida resolvent of f in H defined by (2.1). (I) First we prove that the limit lim x n q exists for all q Ω. Indeed, since u n = J λn x n, by Lemma 2.1, J λn is nonexpansive. Hence we have u n q = J λn x n J λn q x n q. (3.2)

PROXIMAL POINT ALGORITHMS 5 It follows from (3.1), (3.2) and (1.4) that y n q (1 β n )x n + β n w n q (1 β n ) x n q + β n w n q (1 β n ) x n q + β n H(T u n, T q) (1 β n ) x n q + β n u n q (by (1.4) x n q. From (3.1), (3.3) and (1.4) we have x n+1 q = (1 α n )x n + α n v n q (1 α n ) x n q + α n v n q (1 α n ) x n q + α n H(T (y n ), T q) (1 α n ) x n q + α n y n q x n q, n 1. (3.3) (3.4) This shows that { x n q } is decreasing and bounded below. Hence the limit lim x n q exists. Without loss of generality, we can assume that lim x n q = c. (3.5) Therefore the sequences {x n }, {y n }, {u n }, {w n }, {T z n } and {T y n } all are bounded. (II) Next we prove that Indeed, by the sub-differential inequality (2.2) we have lim x n u n = 0 (3.6) 1 { u n q 2 x n q 2 + x n u n 2 } f(q) f(u n ). Since f(q) f(u n ), n 1, it follows that x n u n 2 x n q 2 u n q 2. (3.7) Therefore in order to prove lim x n u n = 0, it suffices to prove u n q c. In fact, it follows from (3.4) that Simplifying we have x n+1 q (1 α n ) x n q + α n y n q. x n q 1 α n [ x n q x n+1 q ] + y n q This together with (3.5) shows that 1 a [ x n q x n+1 q ] + y n q. c = lim inf On the other hand, it follows from (3.3) and (3.5) that lim sup This together with (3.8) implies that x n q lim inf y n q. (3.8) y n q lim sup x n q = c. lim y n q = c. (3.9)

6 PROXIMAL POINT ALGORITHMS Also, by (3.3), y n q (1 β n ) x n q + β n u n q, which can be rewritten as This together with (3.9) shows that From (3.2), it follows that x n q 1 β n [ x n q y n q ] + u n q 1 a [ x n q y n q ] + u n q c = lim inf lim sup x n q lim inf u n q. (3.10) u n q lim sup x n q = c. This together with (3.10) shows that lim u n q = c. Therefore it follows from (3.7) that lim x n u n = 0. (III) Now we prove that lim x n w n = 0 lim x n y n = 0 and lim u n w n = 0. (3.11) Indeed, it follows from (3.1), (3.2) and (1.4) that y n q 2 = (1 β n )x n + β n w n q 2 (1 β n ) x n q 2 + β n w n q 2 β n (1 β n ) x n w n 2 (1 β n ) x n q 2 + β n H(T u n, T q 2 β n (1 β n ) x n w n 2 (1 β n ) x n q 2 + β n u n q 2 β n (1 β n ) x n w n 2 x n q 2 β n (1 β n ) x n w n 2 (3.12) After simplifying and by using the condition that 0 < a α n, β n < b < 1, it follows from (3.12) that a(1 b) x n w n 2 β n (1 β n ) x n w n 2 x n q 2 y n q 2 0 (as n ). This implies that lim x n w n = 0. (3.13) Hence from (3.6) and (3.13) we have So is y n x n = (1 β n )x n + β n w n, x n = β n ) x n w n 0 (as n ) u n w n u n x n + x n w n 0(as n ) (3.14) (IV) Now we prove that In fact, by Lemma 2.3(iii), for each u n, such that lim d(x n, T x n ) = 0, (3.15) x n and w n T u n there exists an k n T x n H(T u n, T x n ) 2 u n x n 2 + 2 u n w n, x n k n, for each n 1. (3.16)

PROXIMAL POINT ALGORITHMS 7 Therefore by (3.16), (3.13), (3.6), (3.14) and noting that the sequences {x n } and {x n } are bounded, we have d(x n, T x n ) x n w n + d(w n, T x n ) (IV) Now we prove that x n w n + H(T u n, T x n ) x n w n + u n x n 2 + 2 u n w n, x n k n 0. lim x n J λ x n = 0, where λ n λ > 0. (3.17) In fact, it follows from (3.6) and Lemma 2.4 that J λ x n x n J λ x n u n + u n x n = J λ x n J λn x n + u n x n = J λ x n J λ ( λ n λ λ n J λn x n + λ λ n x n )) + u n x n x n (1 λ λ n )J λn x n λ λ n x n + u n x n (1 λ λ n ) x n J λn x n + u n x n = (1 λ λ n ) x n u n + u n x n 0. (3.18) (IV) Finally we prove that {x n } converges weakly to p (some point in Ω). In fact, since {u n } is bounded, there exists a subsequence u ni {u n } such that u ni p C (some point in C). By (3.11), u ni w ni 0. It follows from Lemma 2.3 (iv) that p F ix(t ). Again by (3.6) x ni u ni 0, hence x ni p. Therefore from (3.17) we have x ni J λ x ni 0. Since J λ is a single-valued nonexpansive mapping, it is demi-closed at 0. Hence p F ix(j λ ) = argmin y C f(y). This shows that p Ω. If there exists another subsequence {x nj } {x n } such that x nj q C and p q. By the same method as given above we can also prove that q Ω. Since H has the Opial property, we have lim sup n i x ni p < lim sup = lim sup n j n i x nj q < lim sup n j x ni q = lim x n q x nj p = lim x n p = lim sup x ni p. n i This is a contradiction. Therefore p = q and x n p Ω. This completes the proof of Theorem 3.1. If T : C C is a single-valued nonspreading mapping, then the following theorem can be obtained from Theorem 3.1 immediately. Theorem 3.2 Let H, C, f be the same as in Theorem 3.1. Let T : C C be a single-valued nonspreading mapping and {α n }, {β n } be sequences in [0, 1] with 0 < a α n, β n < b < 1, n 1. Let {λ n } be a sequence such that λ n λ > 0 for all n 1 and some λ. For any given x 0 C, let {x n } be the sequence generated in the

8 PROXIMAL POINT ALGORITHMS following manner: u n = argminy y C [f(y) + 1 y x n 2 ], y n = (1 β n )x n + β n T u n, x n+1 = (1 α n )x n + α n T y n, n 1, (3.19) If Ω := F ix(t ) argmin y C f(y), then {x n } converges weakly to a point x Ω which is a minimizer of f in C as well as it is also a fixed point of T in C. Remark 3.3 1. Theorem 3.1 is a generalization of Agarwal et al [16], Bačák [8], and the corresponding results in Ariza-Ruiz et al [13]. In fact, we present a new modified proximal point algorithm for solving the convex minimization problem as well as the fixed point problem of nonspreading-type multivalued mappings. 2. Theorem 3.2 is an improvement and generalization of the main result in Rockafellar [4] and Güler [5]. 4. Some strong convergence theorems Let (X, d) be a metric space, and C be a nonempty closed and convex subset of X. Recall that a mapping T : C CB(C) is said to be demi-compact, if for any bounded sequence {x n } in C such that d(x n, T x n ) 0 (as n ), then there exists a subsequence {x ni } {x n } such that {x ni } converges strongly (i.e., in metric topology) to some point p C. Theorem 4.1 Under the assumptions of Theorem 3.1, if, in addition, T or J λ is demi-compact, then the sequence {x n } defined by (3.1) converges strongly to a point p Ω. and Proof In fact, it follows from (3.15), (3.17), (3.6) and (3.11) that lim d(x n, T x n ) = 0, (4.1) lim x n J λ (x n ) = 0, (4.2) lim x n u n = 0, lim u n w n = 0. (4.3) By the assumption that T or J λ is demi-compact. Without loss of generality, we can assume T is demi-compact, it follows from (4.1) that there exists a subsequence {x ni } {x n } such that {x nj } converges strongly to some point p C. Since J λ is nonexpansive, it is demi-closed at 0. Hence it follows from (4.2) that p F ix(j λ ). Also it follows from (4.3) that u ni p and T is also demi-closed at 0. This implies that p F ix(t ). Hence p Ω. Again by (3.5) the limit lim x n p exists. Hence we have lim x n p = 0. This completes the proof of Theorem 4.1. Theorem 4.2 Under the assumptions of Theorem 3.1, if, in addition, there exists a nondecreasing function g : [0, ) [0, ) with g(0) = 0, g(r) > 0, r > 0, such that g(d(x, Ω)) d(x, J λ x) + d(x, T x), x C. (4.4) Then the sequence {x n } defined by (3.1) converges strongly to a point p Ω.

PROXIMAL POINT ALGORITHMS 9 Proof It follows from (4.1) and (4.2) that lim g(d(x n, Ω)) = 0. Since g is nondecreasing with g(0) = 0 and g(r) > 0, r > 0, we have lim d(x n, Ω) = 0. (4.5) Next we prove that {x n } is a Cauchy sequence in C. In fact, it follows from (3.4) that for any q Ω x n+1 q x n q, n 1, Hence for any positive integers n, m we have This shows that x n+m x n x n+m q + x n q 2d(x n, q), q Ω. x n+m x n 2d(x n, Ω) 0 (as n, m ). Hence {x n } is a Cauchy sequence in C. Since C is a closed subset in H, it is complete. Without loss of generality, we can assume that {x n } converges strongly to some point p C. Since F ix(j λ ) and F ix(t ) both are closed subsets in C, so is Ω. Hence p Ω. This completes the proof of Theorem 4.2. Competing interests The authors declare that they have no competing interests. Author s contributions The authors contributed equally to this work. The authors read and approved the final manuscript. Acknowledgements The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advices. This study was supported by the Natural Science Foundation of China Medical University, Taichung, Taiwan and The Natural Science Foundation of China (Grant No. 11361070). References [1] Kohsaka, F, Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansivetype mappings in Banach spaces, SIAM J. Optim. 19 (2008) 824-835. [2] Cholamjiak, W.: Shrinking projection methods for a split equilibrium problem and a nonspreading-type multivalued mapping, J. Nonlinear Sci. Appl. 9 (2016) (in print). [3] Martinet, B: Réularisation d inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154-158 (1970). [4] Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877-898 (1976). [5] Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403-419 (1991). [6] Kamimura, S, Takahashi, W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226-240 (2000). [7] Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967). [8] Bačák, M: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689-701 (2013). [9] Boikanyo, OA, Morosanu, G: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635-641 (2010). [10] Marino, G, Xu, HK: Convergence of generalized proximal point algorithm. Commun. Pure Appl. Anal. 3, 791-808 (2004). [11] Cholamjiak, P., Abdou, AA and Cho, YJ, Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Applications (2015) 2015:227.

10 PROXIMAL POINT ALGORITHMS [12] Chang, SS, Wen, CF, Yao, JC : Proximal point algorithms involving Cesàro type mean of asymptotically nonexpansive mappings in CAT(0) spaces, J. Nonlinear Sci. Appl. 9 (2016), 4317-4328. [13] Ariza-Ruiz, D, Leustean, L, Lóez, G: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 366, 4299-4322 (2014) [14] Jost, J: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659-673 (1995). [15] Ambrosio, L, Gigli, N, Savare, G: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn. Lectures in Mathematics ETH Zurich. Birkhauser, Basel (2008). [16] Agarwal, RP, ORegan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61-79 (2007).