| In this paper, the main purpose is to study the convergence theorem of fixed point for several nonlinear operators.Firstly, we discuss the iteration for nonexpansive mapping :in a reflexive banach space, the sequence of generated by (1) strongly converges to a fixed point by the following theorems.Theorem 1 Let E be a reflexive Banach space having a weakly continuous duality mapping J(?) with a gauge function (?). Suppose that K is a nonempty closed convex subset of E, and T is a non-expansive mapping from K into itself with F(T)≠(?). Let {xn} be the sequence generated by (1). Suppose that {αn} and {βn} satisfied the following condition:Then {xn} converges strongly to Pu, where P is the unique sunny non-expansive retractionfrom K onto F(T).Colollary 1 Let E be a reflexive Banach space having a weakly continuous duality mapping J(?) with a gauge function (?). Suppose that K is a nonempty closed convex subset of E, and T is a non-expansive mapping from K into itself with F(T)≠(?). Let {xn} be the sequence generated by:where {αn}∈[0,1] satisfied the condition (C1), (C2) and (C3). Then {xn} converges strongly to Pu, where P is the unique sunny non-expansive retraction from K onto F(T). We extended (1) to the viscosity iteration.Theorem 2 Let E be a reflexive Banach space having a weakly continuous duality mapping J(?) with a gauge function (?). Suppose that K is a nonempty closed convex subset of E, and T is a non-expansive mapping from K into itself with F(T)≠(?). For arbitrary initial value x0∈K and a fixed weak contractive mapping f : K→K with a function (?), let {xn} be the sequence generated by:where {αn}, {βn}, {γn} (?) [0,1], andαn +βn +γn= 1. Suppose that {αn}, {βn} satisfied the conditions of theorem 1. Then {xn} converges to a z = Pf(z), where P is the unique sunny non-expansive retraction from K onto F(T).Colollary 2 Let E be a reflexive Banach space having a weakly continuous duality mapping J(?) with a gauge function (?) Suppose that K is a nonempty closed convex subset of E, and T is a non-expansive mapping from K into itself with F(T)≠(?). For arbitrary initial value x0∈K and a fixed weak contractive mapping f : K→K with a function (?), let {xn} be the sequence generated by:where {αn},{βn},{γn}(?)[0,1],andαn+βn+γn=1.Suppose that {αn},{βn} satisfiedthe conditions (C1), (C2) and (C3). Then {xn} converges strongly to z = Pf(z), where P is the unique sunny non-expansive retraction from K onto F(T).Secondly, we study the implicit and explicit viscosity iteration scheme for nonexpansivesemigroup.Theorem 3 Let E be a real reflexive strictly convex Banach space with a uniformlyGateaux differentiable norm. and K a nonempty closed convex subset of E, and {T(t)} a u.a.r. nonexpansive semigroup from K into itself such that F := Fix(F) =∩t>0Fix(T(t))≠(?),and f:K→K a fixed contractive mapping with contractive coefficientficientβ∈(0,1).Suppose (?)tn=∞andαn∈(0,1) such that (?)αn=0. If {xn} isdefined byThen as n→∞,{xn} converges strongly to some common fixed point p of F such that p is the unique solution in F to the following co-variational inequality: Corollary 3 Let E be an uniformly convex Banach space with a uniformly Gateaux differentiable norm, and K a nonempty closed convex subset of E, and {T(t)} a nonexpansivesemigroup from K into itself such that F := Fix(F) =∩t>0 Fix(T(t))≠(?)0, and f: K→K a fixed contractive mapping with contractive coefficientβ∈(0,1). Suppose (?)tn =∞andαn∈(0,1) such that (?)αn= 0 . If {xn} is given by the following equation:Then as n→∞, {xn} converges strongly to some common fixed point p of F such that p is the unique solution in F to the following co-variational inequality:Theorem 4 Let E be a real reflexive strictly convex Banach space with a uniformlyGateaux differentiable norm, and K a nonempty closed convex subset of E, and {T(t)} a u.a.r. nonexpansive semigroup from K into itself such that F := Fix(F) =∩t>0 Fix(T(t))≠(?), and f : K→K a fixed contractive mapping with contractive coefficientβ∈(0,1). Suppose (?)tn =∞andαn∈(0,1) such that (?)αn = 0 and∑(?)αn=∞. If {xn} is defined byThen as n→∞, {xn} converges strongly to some common fixed point p of F such that p is the unique solution in F to the following co-variational inequality:Corollary 4 Let E be a real reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, and K a nonempty closed convex subset of E, and {T(t)} a nonexpansive semigroup from K into itself such that F := Fix(F) =∩t>0 Fix(T(t))≠(?), and f : K→K a fixed contractive mapping with contractive coefficientβ∈(0,1). Suppose (?) tn =∞andαn∈(0,1) such that (?)αn = 0 and∑(?)αn =∞. If {xn} given byThen as n→∞, {xn} converges strongly to some common fixed point p of F such that p is the unique solution in F to the following co-variational inequality:Lastly, this paper shows a convergence theorem of approximating fixed point for stricly asympotically pseudo-contractive mapping in a Hilbert space. Theorem 5 Let C be a closed convex subset of a Hilbert, space H, Ti: C→C be asymptoticallyλ-strict pseudo-contraction mapping, for each Ti there exists a uin and uin≥0, (?)<∞, 0 <λn<1,i∈{0,1. 2,…, N - 1}, have the following inequality:Where F(Ti)is the fixed point set of {Ti}, suppose F:=(?)F(Ti)≠(?). Assume that there exists T∈{Ti} is semicompact. x0∈K is any given, letλ= max{λi: 0≤i≤N - 1}, {xn} is given by.Suppose {αn} satisfiedλ+ε≤αn≤1 -ε, for any n∈N and someε∈(0,1). Then {xn}converges strongly to a commn fixed point of {Ti}. |
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