# Iterative Algorithms for Single-Valued and Multi-Valued Nonexpansive-Type Mappings in Real Lebesgue Spaces

Thesis

Algorithms for single-valued and multi-valued nonexpansive-type mappings have continued to attract a lot of attentions because of their remarkable utility and wide applicability in modern mathematics and other reasearch areas,(most notably medical image reconstruction, game theory and market economy). The first part of this thesis presents contributions to some crucial new concepts and techniques for a systematic discussion of questions on algorithms for single-valued and multi-valued mappings in real Hilbert spaces. Novel contributions are made on iterative algorithms for fixed points and solutions of the split equality fixed point problems of some single-valued pseudocontractive-type mappings in real Hilbert spaces. Interesting contributions are also made on iterative algorithms for fixed points of a general class of multivalued strictly pseu-docontractive mappings in real Hilbert spaces using a new and novel approach and the thorems were gradually extended to a countable family of multi-valued mappings in real Hilbert spaces.It also contains contains original research and important results on iterative approximations of fixed points of multi-valued tempered Lipschitz pseudocontractive mappings in Hilbert spaces. Apart from using some well known iteration methods and identities, some very new and innovative iteration schemes and identities are constructed. The thesis serves as a basis for unifying existing ideas in this area while also generalizing many existing concepts. In order to demonstrate the wide applicability of the theorems, there are given some nontrivial examples and the technique is demonstrated to be more valuable than other methods currently in the literature. The second part of the thesis focuses on some related optimization problems in some Banach spaces. Some iterative algorithms are proposed for common solutions of zeroes of a monotone mapping and a finite family of nonexpansive mappings in Lebesgue spaces. The thesis presents in a unified manner, most of the recent works of this author in this direction, namely: • Let H1 , H2 , H3 be real Hilbert spaces, S : H1 → H1 and T : H2 → H2 two Lipschitz hemicontractive mappings, and A : H1 → H3 and B : H2 → H3 are two bounded linear mappings. Then the coupled sequence (xn , yn ) generated by the algorithm (x1 , y1 ) ∈ H1 × H2 , chosen arbitarily, (xn+1 , yn+1 ) = (1 − α)[(xn − λA∗ (Axn − Byn ), yn + λB ∗ (Axn − Byn )] +αG(un , vn ), (un , vn ) = (1 − α)[(xn − λA∗ (Axn − Byn ), yn + λB ∗ (Axn − Byn )] +αG(x , y ), n n α ∈ (0, L−2 (√L2 + 1 − 1)) λ ∈ (0, 2α ), ̄ λ(A,B) converges weakly to a solution (x∗ , y ∗ ) of the Split Equality Problem. • Let K be a nonempty, closed, convex subset of a real Hilbert space H. Let T : K → CB(K) be a mapping satisfying D(T x, T y) ≤ x − y 2+ kD(Ax, Ay), k ∈ (0, 1), A := I − T. Assume that F (T ) = ∅ and T p = {p} ∀p ∈ F (T ). Then, the sequence {xn } generated by a certain Krasnolselskii type algorithm is an approx- imate fixed point sequence of T and under appropriate mild conditions, the sequence {xn } converges strongly to a fixed point of T . • Let K be a nonempty, closed and convex subset of a real Hilbert space H. For i = 1, 2, ..., m, let Ti : K → CB(K) be a family of mappings satisfying D(Ti x, Ti y) ≤ x − y 2+ ki D(Ai x, Ai y), ki ∈ (0, 1), Ai := I − Ti , for each i. Suppose that ∩m F (Ti ) = ∅ and assume that for p ∈ i=1 ∩m F (Ti ), Ti p = {p}. Then, the sequence {xn } generated by the al-gorithm: x0 ∈ K chosen arbitarily, m x i n+1 = (λ0 )xn + λi yn , i=1 i y n i Sn ∈ := λ0 ∈ (k, 1), i zn m i ∈ Ti xn : D2 ({xn }, Ti xn ) ≤ xn − zn 2 + 1 n2 λi = 1, and k := max{ki , i = 1, 2, ..., m, }. i=0 is an approximate fixed point sequence for the finite family of mappings. • Let Ti : K → CB(K) be a countably infinite family of mappings satisfying D(Ti x, Ti y) ≤ x − y 2+ ki D(Ai x, Ai y), ki ∈ (0, 1), Ai := I − Ti . Assume that κ := sup ki ∈ (0, 1), ∩∞ F (Ti ) = ∅ and for p ∈ ∩∞ F (Ti ), Ti p = i=1 i=1 i {p}. Then, the Krasnoselskii type sequence {xn } generated by the algorithm: x0 ∈ K, arbitrary, i ζ ∈ Γi := z i ∈ Ti xn : D2 ({xn }, Ti xn ) ≤ xn − z i n n n n 2 + 1 n2 ∞ i xn+1 = δ0 xn + δi ζn , i=1 ∞ δ0 ∈ (κ, 1), i=0 δi = 1, is an approximate fixed point sequence of the family Ti . • Let H be a real Hilbert space, K ⊆ H be a nonempty, closed and convex. Let T : K → CB(K) be a multivalued mapping satisfying F (T ) = ∅, diam(T x ∪ T y) ≤ L x − y for some L > 0, and D2 (T x, T p) ≤ x − p 2+ D2 (x, T x), ∀x ∈ H, p ∈ F (T ). Let {xn } be a sequence defined by the algorithm: x 1 ∈ K √ x n+1 = (1 − λ)xn + λzn , λ ∈ (0, L−2 [ 1 + L2 − 1]) zn ∈ Γn := {un ∈ T yn : D(xn , T yn ) ≤ xn − un 2 + θn } y = (1 − λ)xn + λwn , n wn ∈ Πn := {vn ∈ T xn : D(xn , T xn ) ≤ xn − vn 2 + θn } ∞ θn < ∞ θn ≥ 0, n=1 Then p ∈ F (T ), lim xn − p exists and {xn } is an approximate fixed n→∞ point sequence of T . 1 • Let E = Lp , 1 < p ≤ 2, and E ∗ = Lq , p + 1 = 1. For k = 1, 2, ..., N, let q Tk : E → E be a finite family of nonextensive mappings and A : E → E ∗ be an η−strongly monotone mapping which is also L−Lipschitzian. Assume that S := A−1 (0) ∩ ∩N F ix(Tk ) = ∅. Then for arbitrary x1 ∈ E, k=1 the sequence {xn } defined by (0.0.2) xn+1 = j −1 j(T[n] xn ) − λA(T[n] xn ) , n ≥ 1 converges to the common solution of the problem VIP∗ (A, F ix(T[n] )), where T[n] := Tn mod N , and λ ∈ (0, 2LηL2 ), L1 , L2 the Lipschitz constants 2 1 −1 for the mappings A and j , respectively. 1 • Let E = Lp , 2 ≤ p < ∞ and A : Lp → Lq , p + 1 = 1, be an η-strongly q monotone mapping which is also Lipschitzian. For k = 1, 2, ..., N , let Tk : Lp → Lp be a finite family of nonextensive mappings. Assume that S := A−1 (0) ∩ ∩N F ix(Tk ) = ∅. Then for arbitrary x1 ∈ E, the sequence k=1 {xn } defined by (0.0.3) xn+1 = j −1 j(T[n] xn ) − λn A(T[n] xn ) , n ≥ 1 converges strongly to the unique common solution of the problem VIP∗ (A, F ix(Tk )), where T[n] := Tn mod N , and λn ∈ ∞ p 0, η p p−1 2L1 L2 satisfies ∞ λn = ∞, n=1 p−1 λn < ∞ , L1 , L2 are the Lipschitz constants for the mappings A n=1 and j −1 , respectively.

http://library.aust.edu.ng:8080/xmlui/handle/123456789/367