In this paper, a cyclic algorithm for approximating a class of split variational inequality probl... more In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. e results presented here improve, unify, and generalize certain recent results in the literature.
Let E be a real q−uniformly smooth Banach space with constant d q , q ≥ 2. Let T : E → E and G : ... more Let E be a real q−uniformly smooth Banach space with constant d q , q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η−strongly accretive map which is also κ− Lipschitzian, respectively. Let {λ n } be a real sequence in [0, 1] satisfying some appropriate conditions. For δ ∈ (0, (qη dq κ q) q−1), define a sequence {x n } iteratively in E by x 0 ∈ E, x n+1 = T λ n+1 x n = T x n − δλ n+1 G(T x n), n ≥ 0. Then, {x n } converges strongly to the unique solution x * of the variational inequality problem V I(G, K) (search for x * ∈ K such that Gx * , j q (y − x *) ≥ 0 ∀ y ∈ K), where K := F ix(T) = {x ∈ E : T x = x} = ∅. A convergence theorem related to finite family of nonexpansive maps is also proved.
Let H be a real Hilbert space and T1,T2,…,TN be a family of asymptotically nonexpansive self-mapp... more Let H be a real Hilbert space and T1,T2,…,TN be a family of asymptotically nonexpansive self-mappings of H with sequences {1+kp(n)i(n)}, such that kp(n)i(n)→0 as n→∞ where p(n)=j+1 if jNn≤(j+1)N,j=0,1,2,… and n=jN+i(n),i(n)∈{1,2,…,N}. Let F≔⋂i=1NFix(Ti)≠0̸ and let f:H→H be a contraction mapping with coefficient α∈(0,1), also let A be a strongly positive bounded linear operator with coefficient γ¯>0, and 0γγ¯α. Let {αn},{βn} be
Nonlinear Analysis-theory Methods & Applications, Jun 1, 2008
... uniformly smooth. It was observed that both Halpern's and Lion's conditions on the re... more ... uniformly smooth. It was observed that both Halpern's and Lion's conditions on the real sequence {λ n } excluded the canonical choice . This ... p<∞). In 2002, Xu [29] (see also [30]) improved the result of Lion twofold. First, he ...
Abstract In this paper, we consider the generalized variational inequality GVI (F, g, C), where F... more Abstract In this paper, we consider the generalized variational inequality GVI (F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions ...
We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banac... more We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of Bregman-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the Bregman-mapping is the set of common fixed points of { } =1. Using the Bregman-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.
Boletin De La Sociedad Matematica Mexicana, Mar 1, 2021
In this paper, we study a modified inertial subgradient extragradient algorithm in reflexive Bana... more In this paper, we study a modified inertial subgradient extragradient algorithm in reflexive Banach spaces and prove a strong convergence theorem for approximating common solutions of a fixed point equation of a demigeneralized mapping and a variational inequality problem of a monotone and Lipschitz mapping. Our result extends and improves important recent results announced by many authors.
In this paper, a cyclic algorithm for approximating a class of split variational inequality probl... more In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. e results presented here improve, unify, and generalize certain recent results in the literature.
Let E be a real q−uniformly smooth Banach space with constant d q , q ≥ 2. Let T : E → E and G : ... more Let E be a real q−uniformly smooth Banach space with constant d q , q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η−strongly accretive map which is also κ− Lipschitzian, respectively. Let {λ n } be a real sequence in [0, 1] satisfying some appropriate conditions. For δ ∈ (0, (qη dq κ q) q−1), define a sequence {x n } iteratively in E by x 0 ∈ E, x n+1 = T λ n+1 x n = T x n − δλ n+1 G(T x n), n ≥ 0. Then, {x n } converges strongly to the unique solution x * of the variational inequality problem V I(G, K) (search for x * ∈ K such that Gx * , j q (y − x *) ≥ 0 ∀ y ∈ K), where K := F ix(T) = {x ∈ E : T x = x} = ∅. A convergence theorem related to finite family of nonexpansive maps is also proved.
Let H be a real Hilbert space and T1,T2,…,TN be a family of asymptotically nonexpansive self-mapp... more Let H be a real Hilbert space and T1,T2,…,TN be a family of asymptotically nonexpansive self-mappings of H with sequences {1+kp(n)i(n)}, such that kp(n)i(n)→0 as n→∞ where p(n)=j+1 if jNn≤(j+1)N,j=0,1,2,… and n=jN+i(n),i(n)∈{1,2,…,N}. Let F≔⋂i=1NFix(Ti)≠0̸ and let f:H→H be a contraction mapping with coefficient α∈(0,1), also let A be a strongly positive bounded linear operator with coefficient γ¯&amp;gt;0, and 0γγ¯α. Let {αn},{βn} be
Nonlinear Analysis-theory Methods & Applications, Jun 1, 2008
... uniformly smooth. It was observed that both Halpern's and Lion's conditions on the re... more ... uniformly smooth. It was observed that both Halpern's and Lion's conditions on the real sequence {λ n } excluded the canonical choice . This ... p<∞). In 2002, Xu [29] (see also [30]) improved the result of Lion twofold. First, he ...
Abstract In this paper, we consider the generalized variational inequality GVI (F, g, C), where F... more Abstract In this paper, we consider the generalized variational inequality GVI (F, g, C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions ...
We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banac... more We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of Bregman-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the Bregman-mapping is the set of common fixed points of { } =1. Using the Bregman-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.
Boletin De La Sociedad Matematica Mexicana, Mar 1, 2021
In this paper, we study a modified inertial subgradient extragradient algorithm in reflexive Bana... more In this paper, we study a modified inertial subgradient extragradient algorithm in reflexive Banach spaces and prove a strong convergence theorem for approximating common solutions of a fixed point equation of a demigeneralized mapping and a variational inequality problem of a monotone and Lipschitz mapping. Our result extends and improves important recent results announced by many authors.
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