# Contributions to Iterative Algorithms for Nonlinear Equations in Banach Spaces

Shehu, Yekini (2012-12-13)

Thesis

• Let H be a real Hilbert space. Let K, F : H → H be bounded, continuous and monotone mappings. Let {un }∞ and {vn }∞ be n=1 n=1 sequences in E defined iteratively from arbitrary u1 , v1 ∈ H by un+1 = un − βn (F un − vn ) − βn (un − u1 ), vn+1 = vn − βn (Kvn + un ) − βn (vn − v1 ), n ≥ 1 2 where {βn }∞ is a real sequence in (0, 1) such that ∞ βn < ∞ and n=1 n=1 ∞ ∞ n=1 βn = +∞. Then, the sequence {un }n=1 converges strongly to ∗. u • Let H be a real Hilbert space. For each i = 1, 2, ...m, let Fi , Ki : H → H be bounded and monotone mappings. Let {un }∞ , {vi,n }∞ , i = n=1 n=1 1, 2, . . . , m be sequences in H defined iteratively from arbitrary u1 , vi,1 ∈ H by un+1 = un − λn αn un + m i=1 Ki vi,n − λn θn (un − u1 ), vi,n+1 = vi,n − λn αn (Fi un − vi,n ) − λn θn (vi,n − vi,1 ), i = 1, 2, . . . , m where {λn }∞ , {αn }∞ and {θn }∞ are real sequences in (0, 1) n=1 n=1 n=1 such that λn = o(θn ), αn = o(θn ) and ∞ λn θn = +∞. Suppose i=1 that u + m Ki Fi u = 0 has a solution in H. Then, there exist real i=1 constants ε0 , ε1 > 0 and a set Ω ⊂ W such that if αn ≤ ε0 θn and λn ≤ ε1 θn , ∀n ≥ n0 , for some n0 ∈ N and w∗ := (u∗ , x∗ , x∗ , . . . , x∗ ) ∈ m 1 2 Ω (where x∗ = Fi u∗ , i = 1, 2, . . . , m), the sequence {un }∞ converges n=1 i strongly to u∗ . • Let E be a reflexive real Banach space with uniformly Gˆteaux dif- a ferentiable norm. Let K be a nonempty closed convex subset of E and {Tn }∞ be a sequence of Ln -Lipschitzian mappings of K into itself with Ln ≥ 1, ∞ (Ln − 1) < ∞. Let n=1 F (Tn ) = ∅. For n=1 a fixed δ ∈ (0, 1) and each n ∈ N, define Sn : K → K by Sn x := (1 − δ)x + δTn x, ∀x ∈ K. Let {αn }∞ be a sequence in [0, 1] such n=1 ∞ that lim αn = 0 and n→∞ n=1 αn = ∞ . Let {xn }∞ be a sequence in K n=1 defined by x1 = u ∈ K and xn+1 = αn u + (1 − αn )Sn xn , ∞ for all n ∈ N. Suppose that (Kmin ) F (Tn ) = ∅ and lim ||Tn+1 xn − n=1 n→∞ Tn xn || = 0. Then, {xn }∞ converges strongly to some common fixed n=1 points of {Tn }∞ . n=1 • Let K be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bi-function from K × K satisfying (A1) − (A4), ψ a μ-inverse-strongly monotone mapping of K into H, A an α-inverse- strongly monotone mapping of K into H and M : H → 2H a maximal monotone mapping. Let T : H → H be a nonexpansive mapping such that Ω := F (T ) ∩ I(A, M ) ∩ EP = ∅ and suppose f : H → H is a contraction map with constant γ ∈ (0, 1). Suppose {xn }∞ and n=1 {un }∞ are generated by x1 ∈ H, n=1 F (un , y) + ψxn , y − un + 1 rn y − un , un − xn ≥ 0 ∀y ∈ K, xn+1 = βn xn + (1 − βn )T αn f (xn ) + (1 − αn )JM,λ (un − λAun ) , for all n ≥ 1, where {αn }∞ and {βn }∞ are sequences in [0,1] and n=1 n=1 {rn }∞ ⊂ (0, ∞) satisfying: n=1 (i) 0 < c ≤ βn ≤ d < 1, ∞ (ii) αn = ∞, lim αn = 0, n→∞ n=1 (iii) λ ∈ (0, 2α], (iv) 0 < a ≤ rn ≤ b < 2μ, {xn }∞ n=1 lim |rn+1 − rn | = 0, n→∞ then converges strongly to z, where z := PΩ f (z) and PΩ f (z) is the metric projection of f (z) onto Ω. • Let K be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bi-function from K × K to R satisfying (A1) − (A4), ψ a μ-inverse-strongly monotone mapping of K into H, A an α-inverse- strongly monotone mapping of K into H and M : H → 2H a maximal monotone mapping. Let := {T (u) : 0 ≤ u < ∞} be a one-parameter nonexpansive semigroup on H such that Ω := F ( )∩I(A, M )∩EP = ∅ and suppose f : H → H is a contraction mapping with a constant γ ∈ (0, 1). Let {tn } ⊂ (0, ∞) be a real sequence such that limn→∞ tn = ∞. Suppose {xn }∞ and {un }∞ are generated by x1 ∈ H, n=1 n=1   F (un , y) + ψxn , y − un + r1 y − un , un − xn ≥ 0 ∀y ∈ K  n wn = JM,λ (un − λAun )   xn+1 = βn xn + (1 − βn ) 1 tn T (u)[αn f (xn ) + (1 − αn )wn ]du , tn 0 for all n ≥ 1, where {αn }∞ and {βn }∞ are sequences in (0,1) and n=1 n=1 {rn }∞ ⊂ (0, ∞) satisfying: n=1 ∞ (i) lim βn = 0, n=1 |βn+1 − βn | < ∞, n→∞ (ii) ∞ αn = ∞, lim αn = 0, n→∞ n=1 (iii) λ ∈ (0, 2α], (iv) 0 < a ≤ rn ≤ b < 2μ, 1 (v) lim |tn −tn−1 | αn (1−βn ) = 0, tn ∞ n=1 |αn+1 ∞ n=1 |rn+1 − αn | < ∞, − rn | < ∞, n→∞ then {xn }∞ converges strongly to z, where z := PΩ f (z) and PΩ f (z) n=1 is the metric projection of f (z) onto Ω. • Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty, closed and convex subset of E. Let F be a bifunction from C × C satisfying (A1) − (A4). Suppose {Tn }∞ n=0 is a countable family of relatively nonexpansive mappings of C into E such that Ω := (∩∞ F (Tn )) ∩ EP (F ) = ∅. Let {αn }, {βn } and {γn } n=0 be sequences in (0, 1) such that αn + βn + γn = 1. Suppose {xn }∞ n=0 is iteratively generated by u, u0 ∈ E, xn = Trn un , un+1 = J −1 (αn Ju + βn Jxn + γn JTn xn ), n ≥ 0, with the conditions ∞ (i) lim αn = 0, αn = ∞; n→∞ n=0 (ii) 0 < b ≤ βn γn ≤ 1; (iii) {rn }∞ ⊂ (0, ∞) satisfying lim inf rn > 0. n=0 n→∞ Then, {xn }∞ converges strongly to ΠΩ u, where ΠΩ u is the general- n=0 ized projection of u onto Ω. • Let E be a 2-uniformly convex real Banach space which is also uni- formly smooth. Let C be a nonempty, closed and convex subset of E. Suppose B : C → E ∗ is an operator satisfying (B1) − (B3) and {Tn }∞ is an infinite family of relatively-quasi nonexpansive mappings n=1 of C into itself such that F := V I(C, B) ∩ ∩∞ F (Tn ) n=1 ∩m GM EP (Fk , φk ) ∩ k=1 = ∅. Let {xn }∞ be iteratively generated by x0 ∈ n=0 C, C1 = C, x1 = ΠC1 x0 ,   υn = ΠC J −1 (Jxn − λn Bxn )    y = J −1 (α Jυ + (1 − α )JT υ )  n n n n n n    z = J −1 (β Jx + (1 − β )Jy ) n n 0 n n G2 G1 Gm Gm−1  un = Trm,n Trm−1,n ...Tr2,n Tr1,n zn    C 2  n+1 = {w ∈ Cn : φ(w, un ) ≤ φ(w, xn ) + βn (||x0 || + 2 w, Jxn − Jx0 )}    x n+1 = ΠCn+1 x0 , n ≥ 1, where J is the duality mapping on E. Suppose {αn }∞ , {βn }∞ n=1 n=1 and {γn }∞ are sequences in (0, 1) such that lim inf αn (1 − αn ) > n=1 n→∞ 0, lim βn = 0 and {λn }∞ ⊂ [a, b] for some a, b with 0 < a < n=1 n→∞ 2 b < c 2α , where 1 is the 2-uniformly convexity constant of E and c {rk,n }∞ ⊂ (0, ∞), (k = 1, 2, ..., m) satisfying lim inf rk,n > 0, (k = n=1 n→∞ 1, 2, ..., m). Suppose that for each bounded subset D of C, the ordered pair ({Tn }, D) satisfies either condition AKTT or condition ∗ AKTT. Let T be the mapping from C into E defined by T x := lim Tn x for all n→∞ x ∈ C and suppose that T is closed and F (T ) = ∩∞ F (Tn ). Then, n=1 {xn }∞ converges strongly to ΠF x0 .