Untitled Natural Sciences issue 26 CONVERGENCE OF MANN ITERATION PROCESS TO A FIXED POINT OF ( , ) NONEXPANSIVE MAPPINGS IN p L SPACES Huynh Thi Be Trang1 and Nguyen Trung Hieu2* 1 Student, Department[.]
Natural Sciences issue CONVERGENCE OF MANN ITERATION PROCESS TO A FIXED POINT OF ( , ) -NONEXPANSIVE MAPPINGS IN Lp SPACES Huynh Thi Be Trang1 and Nguyen Trung Hieu2* Student, Department of Mathematics Teacher Education, Dong Thap University Department of Mathematics Teacher Education, Dong Thap University * Corresponding author: ngtrunghieu@dthu.edu.vn Article history Received: 10/03/2020; Received in revised form: 20/04/2020; Accepted: 15/05/2020 Abstract In this paper, we prove the convergence of Mann iteration to fixed points of ( , ) nonexpansive and strictly pseudo-contractive mappings in Lp spaces In addition, by using the obtained results, we state the convergence of Mann iteration to solutions of the nonlinear integral equations Keywords: ( , ) -nonexpansive mapping, fixed point, Lp contractive mapping spaces, strictly pseudo- - SỰ HỘI TỤ CỦA DÃY LẶP MANN ĐẾN ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ ( , ) -KHÔNG GIÃN TRONG KHÔNG GIAN Lp Huỳnh Thị Bé Trang1 Nguyễn Trung Hiếu2* Sinh viên, Khoa Sư phạm Toán học, Trường Đại học Đồng Tháp Khoa Sư phạm Toán học, Trường Đại học Đồng Tháp * Tác giả liên hệ: ngtrunghieu@dthu.edu.vn Lịch sử báo Ngày nhận: 10/03/2020; Ngày nhận chỉnh sửa: 20/04/2020; Ngày duyệt đăng: 15/05/2020 Tóm tắt Trong báo này, chúng tơi chứng minh hội tụ dãy lặp Mann đến điểm bất động ánh xạ ( , ) -không giãn giả co chặt không gian Lp Đồng thời, sử dụng kết đạt được, khảo sát hội tụ dãy lặp Mann đến nghiệm lớp phương trình tích phân phi tuyến Từ khóa: Ánh xạ ( , ) -không giãn, điểm bất động, không gian Lp , ánh xạ giả co chặt 26 Dong Thap University Journal of Science, Vol 9, No 5, 2020, 26-32 Introduction and preliminaries In fixed point theory, the nonexpansive mapping has received attention and been studied by many authors in several various ways Some authors established the sufficient conditions for the existence of fixed points of nonexpansive mappings and proved some convergence results of iteration processes to fixed points and common fixed points of nonexpansive mappings Furthermore, by constructing some inequalities which are more generalized than the inequality in the definition of a nonexpansive mapping, some authors extended a nonexpansive mapping to generalized nonexpansive mappings such as strictly pseudo-contractive mappings (Chidume, 1987), mappings satisfying 2008), mappings condition (C ) (Suzuki, satisfying condition (E ) (Garcia-Falset et al., 2011), ( ) -nonexpansive mappings (Aoyama and Kohsaka, 2011) Also, many convergence results of iteration processes to fixed points of such mappings were established In 2018, Amini-Harandi, Fakhar and Hajisharifi introduced the generalization of a nonexpansive mapping and an ( )nonexpansive mapping, and is called an ( , ) nonexpansive mapping The authors also established a sufficient condition for the existence of an approximate fixed point sequence of ( , ) -nonexpansive mappings However, the approximating fixed point of an mapping by some ( , ) -nonexpansive iteration processes has not established yet Therefore, the purpose of the current paper is to establish and prove the convergence of Mann iteration process to fixed points of ( , ) -nonexpansive and strictly pseudo-contractive mappings in Lp spaces Now, we recall some notions and lemmas found useful in what follows Definition 1.1 (Amini-Harandi et al., 2018, Definition 2.2; Chidume, 1987, p 283) Let X be a normed space, C be a nonempty subset of X and T : C C Then (1) T is called an ( , ) -nonexpansive such that for mapping if there exist , all u, v C , we have Tv ||2 || Tu v ||2 || Tu Tu ||2 || u (1 T u ||2 || Tv Tv ||2 || v v ||2 ) || u (2) is called a strictly pseudocontractive mapping if there exist t such that for all u, v C and r 0, we have || u v || || (1 r )(u v) rt(Tu Tv) || Let X be a Banach space and X * be a dual space of X The normalized duality mapping J : X 2X defined by u* J (u) For p X * :|| u * ||2 || u ||2 u, u * Lp ( ) the set of 2, we denote E such that | f |p is measurable functions on Lebesgue integrable on In E Lp ( ), the normalized duality mapping J is single-valued and is denoted by j (Chidume, 1987, p 284) We shall need the following lemmas Lemma 1.2 (Chidume, 1987, Lemma 1) For E Lp ( ) and for all u, v E, we have || u v ||2 (p 1) || u ||2 || v ||2 u, j(v) (1.1) Lemma 1.3 (Chidume, 1987, Lemma 3) For E Lp ( ) and let T : C C be a strictly pseudo-contractive mapping with constant t Then, for all u, v E, we have (I T )u (I T )u, j(u Main results v) t t || u v ||2 (1.2) We denote F(T ) {p C : Tp p} the set of fixed points of the mapping T : C C , 27 Natural Sciences issue I1 {( , ) : 1, I2 {( , ) : and 0} 1, First, 0} we prove that for ( , ) I or ( , ) I , an ( , ) -nonexpansive mapping is a quasi Lipschitz mapping, that is, there exists L such that || Tu p || L || u p || for all u C and p F (T ) Proposition 2.1 Let X be a normed space, C be a nonempty subset of X and T : C C be an ( , ) -nonexpansive mapping Then (1) If ( , ) I 1, then for all u C and p F (T ), we have p ||2 || Tu p ||2 || u (2) If ( , ) I , then for all u p F (T ), we have p ||2 || Tu Proof (1) For p F (T ), we have Tp p Since T is an ( , ) -nonexpansive mapping, for u C , we have p ||2 || Tu Tp ||2 || Tu p ||2 || Tp u ||2 || u Tu ||2 || p Tp ||2 (1 It follows from || Tu || u p || ) || Tu || Tu (2.3) (2) Since p ||2 ) p ||2 ) || u p ||2 (1 p ||2 ) || u ) || Tu p ||2 (1 ) || u 1, we have (1 Since Then, by (2.4), we get || Tu p ||2 || u p ||2 (2.4) p ||2 Remark 2.2 Put max , 1 Then, for ( , ) I or ( , ) I 2, inequalities (2.1) and (2.2) can be rewritten in the following form: for all u C and p F (T ), || Tu p || || u p || (2.5) Next, we prove that the set of fixed points mappings with ( , ) -nonexpansive ( , ) I or ( , ) I is closed p || (1 2 ) || u p ||2 (1 ) || u p ||2 || u p || p ||2 0, from (2.3), we get p ||2 ( , ) -nonexpansive I or ( , ) mapping with I Then F (T ) is closed Proof Let {pn } be a sequence in F (T ) such 1, || Tp pn || || p pn || Taking the limit in (2.6) as n lim || p pn || 0, we have n lim || Tp n 28 || Tu that {pn } converges to p C We prove that p F (T ) Since T is an ( , ) -nonexpansive mapping with ( , ) I or ( , ) I , by using inequality (2.5), we have and (2.3) that By combining the above inequality with we get || Tu Tu ||)2 This gives ( , ) Tu ||2 This implies that (1 p ||2 (2 || u ) || Tu an p ||2 ) || u (1 || p Proposition 2.3 Let X be a normed space, C be a nonempty subset of X and T : C C be p ||2 ) || u p ||2 || Tu p ||2 p || p ||2 ) || u || Tu ( (|| u of || Tu (1 (1 p ||2 and p ||2 (2.2) || u || Tu (2.1) C p ||2 || Tu pn || 0, (2.6) and using Dong Thap University Journal of Science, Vol 9, No 5, 2020, 26-32 that is, the sequence {pn } converges to Tp By combining this with the convergence of the sequence {pn } to p, we obtain Tp p, that is, p F (T ) This implies that F (T ) is closed Next, we establish and prove the convergence of Mann iteration process to fixed points of ( , ) -nonexpansive and strictly pseudo-contractive mappings with ( , ) I or ( , ) I in E Lp spaces (1) C is a nonempty convex subset of E (2) T : C C is an ( , ) -nonexpansive mapping with ( , ) I or ( , ) I and strictly pseudo-contractive mapping with constant t such that F (T ) (3) {un } is the sequence generated by C , un (1 an )un anTun with n 1, where the sequence {an } satisfies t with [(p an (0,1), t constant 1) 1] and the an n || (1 defined by (2.5) (p un u, j(un u) un u, j(un u) (I T )un || un u ||2 || un (1 )|| un u ||2 u, j(un un u) Tun , j(un un (I u) T )u, j(un u) u ||2 || un (1 u ||2 an )2 || un 2(1 [1 u ||2 )an (1 2an2 )(1 (2an 2 n (p u || n 2an (1 an ) || un ) 1) a ] || un {1 a [(p || un 1) 2 u ||2 e u ||2 1]} || un u ||2 u ||2 u ||2 an ) || un an2 2an 1) 2an2 || un (p (2.7) an || un t e u ||2 for all (2.8) Then, let n get some values 1, ,1 in (2.8), we find that for all n || uN u ||2 e ( aN ) e ( aN ) e ( aN ) u ||2 || uN e ( aN e ( aN 1 ) ) || uN e u ||2 ( a1 ) || u1 u ||2 N an u) 1)an2 || Tun 1) 2an2 || un (1 e u || anTun 2an (1 (p un u ||2 an )un ||an (Tun Tun N, N Proof Let u be a fixed point of T Since T is an ( , ) -nonexpansive mapping, by using the inequality (1.1), we have u) By using the inequality t t 0, from (2.7), we obtain Then the sequence {un } converges to fixed points of T || un u, j(un Therefore, Theorem 2.4 Suppose that u1 Tun an )(un u ||2 (1 an )2 || un u, j(un u ||2 (1 an )2 || un u, j(un || u1 u ||2 Taking the limit in (2.9) as N u) ||2 an ) Tun 2an (1 an ) Tun n u ||2 the assumption an (2.9) and using , we conclude that n u) u ||2 u) Moreover, by using the inequality (1.2), we find that the sequence {un } converges to u Finally, we apply Theorem 2.4 in order to study the convergence of Mann iteration process to solutions of a nonlinear integral equation Example 2.5 E L2 ([0,1]) denotes a Banach space with normed 29 Natural Sciences issue 1 Tu(x ) || u || | u(x ) | dx g(x ) K (x, s, u(s ))ds 0 Consider the following nonlinear integral equation u(x ) g(x ) (2.10) K (x, s, u(s ))ds where g : E E and K : [0,1] [0,1] E E are given mappings Put C {u E : u(s) for all s [0,1]} for all [0,1], x Then C is a nonempty convex subset of E For u C , x [0,1], put Tu(x ) g(x ) K (x , s, u(s ))ds (1) For all x [0,1] and u, v inequality Holder, we find that Assume that (H1) For all u C , we have Tu for all u C , x [0,1] Then, by assumption (H1), we conclude that T is well-defined Note that u C is a solution of the equation (2.10) if and only if u C is a fixed point of T Therefore, in order to prove the sequence {un } converges to solution u C of the equation (2.10), we shall prove that the sequence {un } converges to u F (T ) Now, we prove that all assumptions in Theorem 2.4 are satisfied Indeed, | Tu(x ) C C , using the Tv(x ) | (H2) There exists ( , ) I or ( , ) I such that for all x, s [0,1] and u, v C , we have | K (x, s, u(s )) v(s) |2 | Tu(s) u(s) |2 2 ) | u(s) v(s) | (H3) There exists t such that for all x, s [0,1] and u, v C , we have | u(s ) v(s ) | t Consider the sequence {un } defined by C , un u1 with n an n (1 an )un an t t [ (0,1), 2 1] (2.11) Then, from (2.11) and using the assumption (H2), we obtain | Tu(x ) Tv(x ) |2 K (x, s, v(s )) |2ds | K (x, s, u(s )) [ | Tu(s ) v(s ) |2 and Then, if the equation (2.10) has a | Tv(s ) u(s ) |2 | u(s) Tu(s) |2 (1 2 ) | u(s) | v(s) Tv(s) |2 v(s) |2 ]ds | Tu(s ) v(s ) |2 ds solution u C , the sequence {un } converges to u C Proof Consider the mapping T : C C defined by 30 K (x, s, v(s )) |2ds | K (x, s, u(s )) 1 0 anTun 1, where the sequence {an } satisfies with K (x, s, v(s )) | K (x, s, v(s )) |2ds | v(s) Tv(s) |2 | K (x, s, u(s )) | K (x, s, u(s )) ds | Tv(s) | u(s) Tu(s) |2 (1 K (x, s, v(s)) |2 | K (x, s, u(s)) K (x, s, v(s )) |ds | Tv(s ) 1 | u(s ) Tu(s ) |2ds | v(s ) Tv(s ) |2ds (1 u(s ) |2ds 2 ) | u(s ) v(s) |2 ds Dong Thap University Journal of Science, Vol 9, No 5, 2020, 26-32 || Tu v ||2 || v Tv ||2 || Tv (1 u ||2 || u Tu ||2 ) || u v ||2 (2.12) By taking the integral both sides of (2.12) with respect to the variable x on [0,1], we have Tv(x ) |2dx | Tu(x ) [ || Tu v || || Tv u || Tv ||2 (1 v ||2 [ || Tu || u ) || u u ||2 || Tv Tu || v ||2 ]dx Tu ||2 || u || v Tv || (1 2 ) || u v || ] dx This gives that || Tu Tv ||2 || Tv u ||2 || u Tu ||2 || v Tv ||2 ) || u v ||2 This implies that T is an ( , ) -nonexpansive (1 mapping (2) For all x we have [0,1] and u, v C , from (2.11), | Tu(x ) Tv(x ) |2 | K (x, s, u(s )) K (x, s, v(s )) |2ds By combining this with the assumption (H3), there exists t such that | Tu(x ) 1 t2 Tv(x ) |2 || u t2 v || rt || Tu Tv || By adding || u v || to both sides of the above inequality, we find that || u v || (r 1) || u v || rt || Tu || (r v) Tv || 1)(u rt(Tu Tv) || This implies that T a strictly pseudocontractive mapping with constant t Therefore, all assumptions in Theorem 2.4 are satisfied Thus, by Theorem 2.4, we conclude that the sequence {un } converges to u F (T ) and hence the sequence {un } converges to solution u C of the nonlinear integral equation (2.10) The following example guarantees the existence of two mappings g, K satisfying all the assumptions in Example 2.5 Also, this example illustrates the existence of the sequence {an } in Theorem 2.4 Example 2.6 E L2 ([0,1]) denotes a Banach space with normed | u(s ) r || u v || Then, || Tu v ||2 dx || v v ||2 || u v ||2 t This gives that t || Tu Tv || || u for all r 0, we get 1 || u t2 v(s ) | ds v ||2 By taking the integral both sides of the above inequality with respect to the variable x on [0,1], we obtain | Tu(x ) 1 t | u(x ) |2 dx || u || || u Tv(x ) | dx v ||2 dx and C {u E : u(s) for all s [0,1]} Consider the following nonlinear integral equation u(x ) 11 x 12 (1 s )x 2u(s ) ds 4(1 u(s )) (2.13) for all x [0,1], where u C is a function which we must find out For all x, s [0,1] and u C , put 31 Natural Sciences issue (1 s )x 2u(s ) 4(1 u(s )) K (x , s, u(s )) and 11 x 12 Tu(x ) (1 s )x 2u(s ) ds 4(1 u(s )) We will prove the assumptions (H1), (H2) and (H3) in Example 2.5 are satisfied Indeed, (1) For u C , we have u(s) for all s [0,1] Therefore, Tu(x ) for x [0,1] Moreover, for all x [0,1], we have 11 x 12 Tu(x ) x2 s2 ) (1 ds v(s ) v(s ) (2.14) v(s ) , (3) From (2.14), we conclude that the assumption (H3) is satisfied with t Therefore, the assumptions (H1), (H2) and (H3) in Example 2.5 are satisfied Moreover, it is easy to check that u(x ) x for all x [0,1] is a solution to the nonlinear integral equation t 1 (2.13) Note that from and t 2 32 becomes for all n 2n an 1, we have Then, by Example 2.5, the sequence {un } defined by: u1 C, un (x ) 2n un (x ) 2n 11 x 2n 12 [0,1] and n (1 4(1 s )x 2un (s ) un (s )) ds converges to solution x Acknowledgments: This research supported by the project SPD2019.02.14./ This proves that the assumption (H2) is satisfied with 1] for all x [0,1] of the nonlinear integral equation (2.13) K (x, s, v(s)) | s )x u(s ) u(s ) u(s ) 2 is References | K (x, s, u(s)) (1 an n u(x ) This implies that Tu E Thus, Tu C (2) For all x, s [0,1] and u, v C , we have 2 By choosing an for all n (1 s )x 2u(s ) ds 4(1 u(s )) 11 x 12 x [ an 1, the condition Amini-Harandi, A., Fakhar, M., and Hajisharifi, H R (2018) Approximate fixed points of -nonexpansive mappings J Math Anal Appl., 467(2), 1168-1173 Aoyama, K and Kohsaka, F (2011) Fixed -nonexpansive point theorem for mappings in Banach spaces Nonlinear Anal., 74, 4387-4391 Chidume, C E (1987) Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings Proc Amer Soc, 99(2), 283-288 Garcia-Falset, J., Llorens-Fuster, E., Suzuki, T (2011) Fixed point theory for a class of generalized nonexpansive mappings J Math Anal Appl., 375(1), 185-195 Suzuki, T (2008) Fixed point theorems and convergence for some generalized nonexpansive mappings J Math Anal Appl., 340(2), 1088-1095 ... satisfying condition (E ) (Garcia-Falset et al., 2011), ( ) -nonexpansive mappings (Aoyama and Kohsaka, 2011) Also, many convergence results of iteration processes to fixed points of such mappings. .. fixed points of -nonexpansive mappings J Math Anal Appl., 467(2), 1168-1173 Aoyama, K and Kohsaka, F (2011) Fixed -nonexpansive point theorem for mappings in Banach spaces Nonlinear Anal., 74, 4387-4391... mappings were established In 2018, Amini-Harandi, Fakhar and Hajisharifi introduced the generalization of a nonexpansive mapping and an ( )nonexpansive mapping, and is called an ( , ) nonexpansive