Iterative Algorithms and Existence Theorems for Solutions of Nonlinear Equations in Banach Spaces.

Abstract

Let E be a q-uniformly smooth real Banach space and D be a nonempty, open and convex subset of E. Assume that T : D → CB(D) is a multi valued continuous (with respect to the Hausdorff metric), bounded and pseudo-contractive mapping with F(T) 6= ∅. Let {xn} be a sequence generated iteratively from arbitrary x1 D by xn+1 := (1 − λn)xn + λnun − λnθn(xn − x1), un ∈ T xn, where {λn} and {θn} are real sequences in (0, 1) satisfying the following conditions: (i) {θn} decreases to 0; (ii) λn(1 + θn) < 1, X∞ n=1λnθn = ∞, λq−1 n = o(θn); (iii) lim sup n→∞ θn−1θn − 1 λnθn ≤ 0, X∞ n=1 λq n < ∞. Then, there exists a real constant γ0 > 0 such that if λq−1 n < γ0θn for all n ≥ 1, 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. Suppose that T : K → CB(K) is a multi-valued k-strictly pseudo contractive mapping such that F(T) 6= ∅. Assume that T p = {p} for all p ∈ F(T). Let {xn} be a sequence defined by x0 ∈ K, xn+1 = (1 − λ)xn + λyn, where yn ∈ T xn and λ ∈ (0, 1 − k). Then, limn→∞ d(xn, T xn) = 0. Let q > 1 be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Suppose that T : K → CB(K) is a multi-valued k-strictly pseudo-contractive mapping such that F(T) 6= ∅ and such that T p = {p} for all p ∈ F(T). For arbitrary x1 ∈ K and λ ∈ (0, µ) with µ := min 1, qkq−1 dq 1q−1, let {xn} be a sequence defined iteratively by xn+1 = (1 − λ)xn + λyn, where yn ∈ T xn. Then, limn→ dist(xn, T xn) = 0. Let K be a nonempty, closed and convex subset of a uniformly convex real Banach space E. Suppose that T : K → CB(K) is a multi-valued quasi-nonexpansive mapping such that T p = {p} for some p ∈ F(T). Then for any fixed x0 ∈ K and arbitrary λ ∈ (0, 1), define a sequence {xn} by xn+1 = (1 − λ)xn + λyn, n ≥ 0, where yn ∈ T xn. Then, limn→∞ dist(xn, T xn) = 0. Let H be a real Hilbert space, K : D(K) ⊂ H → H, F : D(F) ⊂ H →H be two Lipschitz monotone mappings such that D(F) and D(K) are closed, convex, bounded and R(F) ⊂ D(K). Let EH := H × H and let A : D(A) ⊂ EH → EH be a mapping such that D(F) × D(K) = D(A) and A[u, v] = [F u − v, Kv + u], [u, v] ∈ D(A). Suppose that lim λ→0+ dist(w − λAw, D(A)) λ = 0 for all w ∈ D(A). Then, the Hammerstein equation u + KF u = 0 has a solution. Let K : D(K) ⊂ L p → Lp, F : D(F) ⊂ Lp → Lp be two Lipschitz mappings satisfying the following conditions: (a) there exists α > 0 such that for each u1, u2 ∈ D(F), hF u1 − F u2, j(u1 − u2)i ≥ α||u1 − u2||2 ; (b) there exists β > 0 such that for each u1, u2 ∈ D(K), hKu1 − Ku2, j(u1 − u2)i ≥ β||u1 − u2||2 . Let D(F) and D(K) be closed, convex, bounded such that R(F) ⊂ D(K). Let E := Lp×L p and let A : D(A) ⊂ E → E be a mapping such that D(F) × D(K) = D(A) and A[u, v] = [F u − v, Kv + u], ∀[u, v] ∈ D(A). Suppose that lim λ→0+ dist(w − λAw, D(A)) λ = 0 for all w ∈ D(A). Let γ := min{α, β}. If 2 ≤ p < γ + p γ 2 + 4 or 1 + √1 − γ < p ≤ 2, then the Hammerstein equation u + KF u = 0 has a solution. Let K : D(K) ⊂ L p → Lp , F : D(F) ⊂ L p → L p be two continuous accretive mappings satisfying the following conditions: (a) there exists α > 0 such that for each u1, u2 ∈ D(F), hF u1 − F u2, j(u1 − u2)i ≥ α||u1 − u2||2 ; (b) there exists β > 0 such that for each u1, u2 ∈ D(K), hKu1 − Ku2, j(u1 − u2)i ≥ β||u1 − u2||2. Let D(F) and D(K) be closed, convex, such that R(F) ⊂ D(K). Let E := L p × L p and let A : D(A) ⊂ E → E be a mapping such that D(F)×D(K) = D(A) and A[u, v] = [F u−v, Kv+u] for [u, v] ∈ D(A). Suppose that hAw, J(w)i ≥ 0 for all w ∈ D(A) with kwk ≥ R or lim kAwk = ∞ as kwk → ∞ and suppose limλ→0+ dist(w − λAw, D(A))λ= 0 for all w ∈ D(A). Let γ := min{α, β}. If 2 ≤ p < γ +p γ2 + 4 or 1 + √1 − γ < p ≤ 2, then the Hammerstein equation u + KF u = 0 has a solution. Let H be a real Hilbert space, K : H → H, F : H → H be two Lip schitz strongly monotone mappings with constants α, β, respectively. Let A : EH → EH be a mapping defined by A[u, v] = [F u−v, Kv +u]. Then, the Hammerstein equation u + KF u = 0 has a solution. Let K : Lp → Lp, F : Lp → Lp be two Lipschitz mappings satisfying mappings satisfying the following conditions: (a)there exists α > 0 such that for each u1, u2 ∈ Lp, hF u1 − F u2, j(u1 − u2)i ≥ α||u1 − u2||2;there exists β > 0 such that for each u1, u2 ∈ Lp,hKu1 − Ku2, j(u1 − u2)i ≥ β||u1 − u2||2 Let E := Lp × L p and let A : E → E be a mapping defined by A[u, v] = [F u−v, Kv+u]. Let γ := min{α, β}. If 2 ≤ p < γ+pγ2 + 4or 1 + √1 − γ < p ≤ 2, then, the Hammerstein equation u + KF u = 0 has a solution