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A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces. (English) Zbl 1213.65082

The article deals with the equilibrium problem

f(x,y)0forallyC(1)

where C is a nonempty closed convex subset of a real Banach space with the dual E * , f:C×C. The special case of this problem is the known problem of solving a variational inequality; in this problem f(x,y)=Tx,y-x, T:C(E)E * . It is assumed that E is a strictly convex reflexive Banach space with the Kadec–Klee property and Fréchet differentiable norm, {S λ :λΛ} a family of closed hemi-relatively nonexpansive mappings of C into itself with a common fixed point, {α n } a sequence in [0,1] converging to zero. The following iterative scheme is studied:

x 1 C,C 1 =C,y n,λ =J -1 (α n Jx 1 +(1-α n )JS λ x n ),C n+1 ={zC n :sup λΛ φ(z,y n,λ )α n φ(z,x 1 )+(1-α n )φ(z,x n )],x n+1 =Π C n+1 x;

here J is the duality mapping, φ(x,y)=x 2 -2x,Jy+y 2 , Π C n is the generalized projection from C onto C n . It is proved that {x n } converges strongly to Π F x, where Π F is the generalized projection from C onto F, F= λΛ F(S λ ) is the set of common fixed points of {S λ }. The following modified iterative scheme

x 1 C,C 1 =C,f λ (u n ,y)+1 ry-u n ,Ju n -Jx n 0,forallyC,r>0,y n,λ =J -1 (α n Jx 1 +(1-α n )Ju n,λ ),C n+1 ={zC n :sup λΛ φ(z,y n,λ )α n φ(z,x 1 )+(1-α n )φ(z,x n )},x n+1 =Π C n+1 x

is considered too; in this case it is assumed that f(x,y) satisfies the following properties: (A1) f(x,x)=0; (2) f(x,y)+f(y,x)0; (3) lim t0 f(tz+(1-t)x,y)f(x,y); (4) the functions yf(x,y), xC, are convex and lower semicontinuous.


MSC:
65J15Equations with nonlinear operators (numerical methods)
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
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