một số phương pháp tìm điểm bất động chung của một họ hữu hạn các ánh xạ không giãn trong không gian banach tóm tắt tiếng anh

MINISTRY OF EDUCATION AND TRAINING THAI NGUYEN UNIVERSITY TRUONG MINH TUYEN SOME METHODS OF FINDING A COMMON FIXED POINT FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES Major: Mathematics Analysis Code: 62 46 01 02 SUMMARY PHD THESIS OF MATHEMATICS THAI NGUYEN-2013 This thesis is completed at: College of Education-Thai Nguyen University, Thai Nguyen, Viet Nam Scientific supervisors: Prof Dr Nguyen Buong Prof Dr Jong Kyu Kim Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended in front of the PhD thesis committee at university level at: The thesis can be found at: - National Library - Learning Resource Center of Thai Nguyen University mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh Introduction The problem of finding a common fixed point of a finite family of nonexpansive mappings in Hilbert spaces or Banach spaces is a particular case of the convex feasibility problem: ”Finding an element belongs to nonempty intersection of a finite or infinite family of closed and convex subsets {Ci}i∈I of Hilbert spaces or Banach spaces” This problem has many important applications in different fields of science, for instance: Image recovery, signal recovery, physis, medicine When Ci = F ix(Ti), with F ix(Ti) is the set of fixed point of nonexpansive mapping Ti, i = 1, 2, , N , then there have been many proposed methods based on iterative methods known classic Such as the iterative methods Kranoselskii, Mann, Ishikawa, Halpern and viscosity approximation method The results of the research directions are presented in more detail in Chapter of the thesis We have known that, if T is a nonexpansive mapping in Banach space E, then A = I − T is a accretive operator, with I is the identity mapping of E So, the problem of finding a common fixed point of a finite family of nonexpansive mappings Ti in Banach space E is equivalent to the problem of finding a common zero of a finite family of accretive operator Ai = I − Ti with i = 1, 2, , N For the problem of finding zero of accretive operator, in this thesis we only mention of well-known methods for problem classes as: Tikhonov regularization, proximal point algorithm, some modification of proximal point algorithm, including inertial proximal point algorithm, regularization proximal point algorithm, regularization inertial proximal point algorithm For the problem of finding common solution of a finite family of equations with maximal monotone operator in Hilbert spaces, we are very interested in Browder-Tikhonov regularization method of professor Buong Nguyen (Buong N (2006), ”Regularization for unconstrained vector optimization of convex functionals in Banach spaces”, Compt Math and Math Phys., 46 (3), pp 372-378) when he mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh solved the problem of finding a common zero of a finite family of single valued monotone operators, hemi-continuous from Banach space E to its dual space E ∗ He transformed the problem of solving a system of equations with the maximal monotone operators to the problem of solving an equation and he obtained the strong convergence of his algorithm when regularization parameters are suitable chosen Next, in 2008 the author Buong Nguyen (Buong N (2008), ”Regularization proximal point algorithm for unconstrained vector convex optimization problems”, Ukrainian Mathematical Journal, 60 (9), pp 1483-1491) combined the inertial proximal point algorithm with the regularization method and its called the regularization inertial proximal point algorithm, to find a common zero of a finite family of maximal monotone operators Ai = ∂fi, with ∂fi are subdifferential of proper lower semicontinuous convex functions fi, i = 1, 2, , N on a Hilbert space H The main propose of this thesis is study Tikhonov regularization method and some modification of proximal point algorithm including proximal point algorithm of Xu H K (Xu H.-K (2006), A regularization method for the proximal point algorithm, J Glob Optim 36 (1) (2006), pp 115-125) and regularization inertial proximal point algorithm for common fixed point of a finite family of nonexpansive mappings in Banach spaces, along with the related problems based on the algorithm of professor Buong N We also study the stability of obtained methods following the research direction of Alber Y (Alber Y (2007), ”On the stability of iterative approximations to fixed points of nonexpansive mappings”, J Math Anal Appl., 328, pp 958-971) Specifically, the thesis focuses on the following issues: Study proximal point algorithm following the research direction of Xu H K for the problem of finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces, we also study the stability of regularization obtained methods following the research direction of Alber Y Study of extend the results of Xu H K for the problem of finding zero of m-accretive from Hilbert spaces to uniformly smooth Banach spaces mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 3 Study Tikhonov regularization and regularization inertial proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces and some modifications of it, we also study the stability of iterative obtained methods The content of the thesis is presented in three chapters Chapter is the prilaminaries chapter, this chapter presents on the geometry of Banach spaces, ill-posed problems of monotone type, the problem of finding a common fixed point of a finite family of nonexpasive mappings, overview of known methods for solving the above problem class and the finally section we introduce some lemma which need to use for the proof of the obtained results in the later chapters of the thesis Chapter presents some strong convergence theorems of proximal point algorithm following the research direction of the authors Buong N and Xu H K for the problem of finding a common fixed point of a finite family of nonexpansive mappings and the problem of finding zero of accretive operator in Banach spaces, here the stability of the iterative methods are established and studied An application of obtained results for common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces, the convex feasibility problem in Banach spaces and some examples are presented at the final of this chapter Chapter presents strong convergence theorems of Tikhonov regularization and regularization inertial proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces, and the stability of methods obtaining The fininal section in this chapter, we refers to some applications of the iterative methods to solve the problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces, the convex feasibility problem, and some examples to illustrate further the research results achieved mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh Chapter Some prepared problems Chapter of the thesis is the prepared chapter and in order to present the most basic knowledge for the presentation of research results in the later chapters of the thesis: The Section 1.1 presents on the geometry of Banach spaces as uniformly convex Banach spaces, uniformly smooth Banach spaces In this section we also present the concept of monotone operator, normalized duality mapping and nonexpansive mapping with some basis properties of them The Section 1.2 introduces the concept of ill-posed problem, Tikhonov regularization for this problems type The Section 1.3 and Section 1.4 of this chapter present the proximal point algorithm, inertial proximal point algorithm algorithm and regularization inertial proximal point algorithm algorithm for equations with monotone type operators The Section 1.5 states the problem of finding a common fixed point of a finite family of nonexpansive mappings and specially in this section we introduce some clasical iterative methods to approximate fixed point of a nonexpansive mapping or common fixed point of a finite family of nonexpansive mappings The Section 1.6 is the final section of this chapter, presentation some important lemmas which is frequently used in proving the obtained results of the thesis mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh Chapter Proximal point algorithm This chapter, we present the obtained results on the strong convergence of the proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings and the problem of finding zero of m−accretive operators in Banach spaces, and some applications of the obtained results for the problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces, the convex feasibility problem in Banach spaces 2.1 Proximal point algorithm for the problem of finding common fixed point of a finite family of nonexpansive mappings The first, we consider the following problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(Ti ) 6= ∅, (2.1) where F ix(Ti) is the set of fixed point of nonexpansive mapping Ti : E −→ E, i = 1, 2, , N Theorem 2.1 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ti : E −→ E, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ If the sequence {tn } ⊂ (0, 1) satisfies one of conditions i) limn→∞ tn = 0, P∞ ii) limn→∞ tn = 0, P∞ n=1 tn = ∞, limn→∞ n=1 tn = ∞, P∞ n=1 tn tn+1 = or |tn − tn+1| < +∞, mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh then the sequence {xn} defined by N X Ai(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ (2.2) i=1 converges strongly to QS u, where Ai = I − Ti, i = 1, 2, , N and QS : E −→ S is a sunny nonexpansive retraction from E onto S Theorem 2.2 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ti : E −→ E, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ If the sequences {rn } ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy the following conditions i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞, then the sequence {xn} defined by rn N X Ai(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ (2.3) i=1 converges strongly to QS u, where Ai = I − Ti, i = 1, 2, , N and QS : E −→ S is a sunny nonexpansive retraction from E onto S Next in this section, we give some iterative methods similar to solve the following more general problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(Ti ), (2.4) where Ti : Ci −→ Ci, i = 1, 2, , N is nonexpansive mapping and Ci is a closed, convex and nonexpansive retract of E Theorem 2.3 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ci be a closed, convex and nonexpansive retract of E and let mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh Ti : Ci −→ Ci, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ Let {xn } be sequence generated by N X Bi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ 0, (2.5) i=1 where Bi = I − TiQCi , i = 1, 2, , N and QCi : E −→ Ci is a nonexpansive retraction from E onto Ci, i = 1, 2, , N If the sequence {tn} ⊂ (0, 1) satisfies one of conditions P tn = or i) limn→∞ tn = 0, ∞ t = ∞, lim n→∞ n=1 n tn+1 ii) limn→∞ tn = 0, P∞ n=1 tn = ∞, P∞ n=1 |tn − tn+1| < +∞, then the sequence {xn} converges strongly to QS u, where QS : E −→ S is a sunny nonexpansive retraction from E onto S Theorem 2.4 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ci be a closed, convex and nonexpansive retract of E and let Ti : Ci −→ Ci, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ Let {xn } be sequence generated by rn N X Bi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ 0, (2.6) i=1 where Bi = I − TiQCi , i = 1, 2, , N and QCi : E −→ Ci is a nonexpansive retraction from E onto Ci, i = 1, 2, , N If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy the following conditions i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞, then the sequence {xn} converges strongly to QS u, where QS : E −→ S is a sunny nonexpansive retraction from E onto S Finally, in this section we give a method to solve the following problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(Ti ), mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh (2.7) mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh where Ti : Ci −→ E, i = 1, 2, , N is nonexpansive nonself mapping and Ci is a closed, convex and sunny nonexpansive retract of E Theorem 2.5 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ci be a closed, convex and sunny nonexpansive retract of E and let Ti : Ci −→ E, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ Let {xn } be sequence generated by N X fi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ 0, (2.8) i=1 where fi = I − QCi TiQCi , i = 1, 2, , N and QCi : E −→ Ci are sunny nonexpansive retractions from E onto Ci, i = 1, 2, , N If the sequence {tn} ⊂ (0, 1) satisfies one of conditions P tn i) limn→∞ tn = 0, ∞ t = ∞, lim = or n→∞ n=1 n tn+1 ii) limn→∞ tn = 0, P∞ n=1 tn = ∞, P∞ n=1 |tn − tn+1| < +∞, then the sequence {xn} converges strongly to QS u, where QS : E −→ S is a sunny nonexpansive retraction from E onto S Theorem 2.6 Suppose that E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let Ci be a closed, convex and sunny nonexpansive retract of E and let Ti : Ci −→ E, i = 1, 2, , N be nonexpansive mappings with S = ∩N i=1 F ix(Ti ) 6= ∅ Let {xn } be sequence generated by rn N X fi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ 0, (2.9) i=1 where fi = I − QCi TiQCi , i = 1, 2, , N and QCi : E −→ Ci are sunny nonexpansive retractions from E onto Ci, i = 1, 2, , N If the sequence {tn} ⊂ (0, 1) satisfies one of conditions If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy the following conditions mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 11 2.3 Proximal point algorithm and the problem of finding zero of m−accretive operator Let E be a uniformly smooth Banach cpase and let A : D(A) ⊆ E −→ 2E be a m−accretive operator with S = A−1(0) 6= ∅ We study the strong convergence of implicit iterative {xn} defined by the following algorithm: u, x0 ∈ E, rnA(xn+1) + xn+1 tnu + (1 − tn)xn, n ≥ 0, (2.13) where {tn} ⊂ (0, 1) v {rn} ⊂ (0, +∞) The first we have the following theorem: Theorem 2.9 Let E be an uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let A : D(A) ⊆ E −→ 2E be an m− accretive operator with S = A−1(0) 6= ∅ If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞, then the sequence {xn} generated by (2.13) converges strongly to QS u, where QS is a sunny nonexpansive retraction of E onto S With other assumptions posed on the sequences {rn} and {tn}, we also obtain the strong convergence of the sequence {xn} This is confirmed in the following theorem: Theorem 2.10 Let E be an uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let A : D(A) ⊆ E −→ 2E be an m− accretive operator with S = A−1(0) 6= ∅ If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy ∞ n=0 tn = +∞, n=0 |tn+1 − tn | < +∞; P − rn < +∞, ii) inf rn = r > 0, ∞ n=0 n rn+1 i) limn→∞ tn = 0; P∞ P mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 12 then the sequence {xn} generated by (2.13) converges strongly to QS u, where QS is a sunny nonexpansive retraction of E onto S Next, we study stability of algorithm (2.13) in the form rnAn(zn+1) + zn+1 tnu + (1 − tn)zn, u, z0 ∈ E, n ≥ 0, (2.14) where An : D(An) ⊆ E −→ 2E are m−accretive operators with D(An) = D(A) such that H(An(x), A(x)) ≤ g(kxk)hn, (2.15) where g is real bounded (image of a bounded set is bounded) function for t ≥ with g(0) = and {hn} is positive sequence We have the following results: Theorem 2.11 Let E be an uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let A : D(A) ⊆ E −→ 2E and An : D(An) ⊆ E −→ 2E be m− accretive operators with S = A−1(0) 6= ∅ and D(A) = D(An) for all n If the condition (2.15) is fulfilled and the sequences {rn} ⊂ (0, +∞), and {tn} ⊂ (0, 1) satisfy i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞; iii) P∞ n=1 rnhn < +∞, then the sequence {zn} generated by (2.14) converges strongly to QS u, where QS is a sunny nonexpansive retraction of E onto S Theorem 2.12 Let E be an uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let A : D(A) ⊆ E −→ 2E and An : D(An) ⊆ E −→ 2E be m− accretive operators with S = A−1(0) 6= ∅ and D(A) = D(An) for all n If the condition (2.15) is fulfilled and the sequences {rn} ⊂ (0, +∞), and {tn} ⊂ (0, 1) satisfy ∞ n=0 tn = +∞, n=0 |tn+1 − tn | < +∞; P − rn < +∞; ii) inf rn = r > 0, ∞ n=0 n rn+1 i) limn→∞ tn = 0; P∞ P mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh iii) P∞ n=1 13 rnhn < +∞, then the sequence {zn} generated by (2.14) converges strongly to QS u, where QS is a sunny nonexpansive retraction of E onto S 2.4 Applications 2.4.1 The problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings First of all, in this section we refer to the application of the iterative methods which are presented in the section 2.1 to solve the problem of finding a common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces We consider the following problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(fi ) 6= ∅, (2.16) where fi : Ci −→ Ci is ki−strictly pseudocontractive mapping from closed and convex subset Ci of Hilbert space H to Ci for all i = 1, 2, , N For each i, let Ti = (1 − t)I + tfi and Fi = I − TiPCi , where < t ≤ mini=1,2, ,N {1 − ki} and PCi is metric projection from H onto Ci Then, Ti are nonexpansive mappings from Ci to itself an the problem (2.16) is equivalent to the following problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(Ti ) 6= ∅ (2.17) From Theorem 2.1 and Theorem 2.2, we have the following results: Theorem 2.13 If the sequence {tn} ⊂ (0, 1) satisfies the following conditions P tn i) limn→∞ tn = 0, ∞ t = ∞, lim = hoc n n→∞ n=1 tn+1 ii) limn→∞ tn = 0, P∞ n=1 tn = ∞, P∞ n=1 |tn − tn+1| < +∞, then the sequence {xn} defined by N X i=1 Fi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ H, n ≥ (2.18) mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 14 converges strongly to PS u, where PS is metric projection from H onto S Theorem 2.14 If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy the following conditions i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞, then the sequence {xn} defined by rn N X Fi(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ H, n ≥ (2.19) i=1 converges strongly to PS u, where PS is metric projection from H onto S 2.4.2 The convex feasibility problem Consider the following convex feasibility problem: Finding an element x∗ ∈ S = ∩N i=1 Si 6= ∅, (2.20) where Si, i = 1, 2, , N are closed, convex and nonexpansive retracts of a uniformly convex and uniformly smooth Banach space E, and S is a sunny nonexpansive retract of E Let QSi denote the nonexpansive retraction from E onto Si, i = 1, 2, , N It is clear that F (QSi ) = Si, i = 1, 2, , N Thus, the problem (2.20) is equivalent to the problem of finding a common fixed point of finite family of nonexpansive mappings Ti = QSi , i = 1, 2, , N From Theorem 2.1 and Theorem 2.2, we have the followingb results: Theorem 2.15 If the sequence {tn} ⊂ (0, 1) satifies one of the following conditions P tn i) limn→∞ tn = 0, ∞ t = ∞, lim = or n n→∞ n=1 tn+1 ii) limn→∞ tn = 0, P∞ n=1 tn = ∞, P∞ n=1 |tn − tn+1| < +∞, mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 15 then the sequence {xn} defined by N X Ai(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ (2.21) i=1 converges strongly to QS u, where Ai = I − QSi , i = 1, 2, , N and QS : E −→ S is a sunny nonexpansive retraction from E onto S Theorem 2.16 If the sequences {rn} ⊂ (0, +∞) and {tn} ⊂ (0, 1) satisfy the conditions i) limn→∞ tn = 0; P∞ n=0 tn = +∞; ii) limn→∞ rn = +∞, then the sequence {xn} defined rn N X Ai(xn+1) + xn+1 = tnu + (1 − tn)xn, u, x0 ∈ E, n ≥ (2.22) i=1 converges strongly to QS u, where Ai = I − QSi , i = 1, 2, , N and QS : E −→ S is a sunny nonexpansive retraction from E onto S mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh Chapter Tikhonov regularization method and regularization inertial proxiaml point algorithm In this chapter, we present some obtained reseach results on the strong convergence of Tikhonov regularization and regularization inertial proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings, along with some applications of regularization methods for common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces and the convex feasibility problem in Banach spaces 3.1 Tikhonov regularization and regularization inertial proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings Firstly, in this section we consider the following problem: Finding an element x∗ ∈ S = ∩N i=1 F ix(Ti ) 6= ∅, (3.1) twhere F (Ti) is the set of fixed points of nonexpansive mappings Ti : C −→ C and C is a closed, convex and sunny nonexpansive retract subset of a uniformly convex and uniformly smooth Banach space E We introduce Tikhonov regularization and regularization inertial proximal point algorithm in the forms N X Ai(xn) + αn(xn − y) = 0, (3.2) i=1 cn( N X Ai(un+1) + αn(un+1 − y)) i=1 + un+1 = QC (un + γn(un − un−1)), mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 16 (3.3) mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh 17 respectively, where y, u0, u1 ∈ C, Ai = I − Ti, i = 1, 2, , N and QC : E −→ C is a sunny nonexpansive retraction from E onto C to solve the problem (3.1) We have the following theorem: Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let C be a nonempty closed convex sunny nonexpansive retract of E and let Ti : C −→ C, i = 1, 2, , N be nonexpansive mappings such that S = ∩N i=1 F (Ti ) 6= ∅ Then i) For each αn > the equation (3.2) has unique solution xn; ii) If the sequence of positive numbers {αn} satisfies limn→∞ αn = 0, then {xn} converges strongly to QS y, where QS : E −→ S is a sunny nonexpansive retraction from E onto S Moreover, we have the following estimate kxn+1 − xnk ≤ |αn+1 − αn| R0 ∀n ≥ 0, αn (3.4) where R0 = 2ky − QS yk Theorem 3.2 Let E be a uniformly convex and uniformly smooth Banach space which admits a weakly sequentially continuous normalized duality mapping j from E to E ∗ Let C be a nonempty closed convex sunny nonexpansive retract of E and let Ti : C −→ C, i = 1, 2, , N be nonexpansive mappings such that S = ∩N i=1 F (Ti ) 6= ∅ If the sequences {cn }, {αn } and {γn } satisfy i) < c0 < cn, αn > 0, αn → 0, P∞ |αn+1 − αn| → 0, n=0 αn = +∞; αn2 ii) γn ≥ 0, γnαn−1kun − un−1k −→ 0, then the sequence {un} defined by (3.3) converges strongly to QS y, where QS : E −→ S is a sunny nonexpansive retraction from E onto S mot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anhmot.so.phuong.phap.tim.diem.bat.dong.chung.cua.mot.ho.huu.han.cac.anh.xa.khong.gian.trong.khong.gian.banach.tom.tat.tieng.anh