An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems

Remark 2.1

[38, 39] If E is a reflexive, strictly convex and smooth Banach space, then for x , y ∈ E , ϕ ( x , y ) = 0 if and only if x = y .

Lemma 2.2

[46] Given a number s > 0 . A real Banach space E is uniformly convex if and only if there exists a continuous strictly increasing function g : [ 0 , ∞ ) → [ 0 , ∞ ) with g ( 0 ) = 0 such that

for all x , y ∈ E , λ ∈ [ 0 , 1 ] , with ‖ x ‖ < s and ‖ y ‖ < s .

Lemma 2.3

[36] Let C be a nonempty, closed and convex subset of a reflexive, strictly convex and smooth Banach space E . If x ∈ E and q ∈ C , then

and

Lemma 2.4

[47] Let E be a smooth and uniformly convex real Banach space and let { x k } and { y k } be two sequences in E . If either { x k } or { y k } is bounded and ϕ ( x k , y k ) → 0 as k → ∞ , then ‖ x k − y k ‖ → 0 as k → ∞ .

Remark 2.5

[48] Using P5, it is easy to see that converse of Lemma 2.4 is also true whenever both { x k } and { y k } are bounded.

Lemma 2.6

[49] Let E be a 2-uniformly convex and smooth Banach space. Then for every x , y ∈ E

where ν > 0 is the 2-uniformly convexity constant of E .

Lemma 2.7

[46] Let E be a 2-uniformly smooth Banach space with the best smoothness constant d > 0 . Then, the following inequality holds:

Definition 2.8

[20,50] Let X and Y be two Hausdorff topological spaces and let D be a nonempty, convex subset of X and P be a pointed, proper, closed and convex cone of Y with int P ≠ ∅ . Let 0 be the zero point of Y , N ( 0 ) be the neighbourhood set of 0, N ( x 0 ) be the neighbourhood set of x 0 and f : D → Y be a mapping.

1. If, for any V ∈ N ( 0 ) ⊂ Y , there exists U ∈ N ( x 0 ) such that

   f ( x ) ∈ f ( x 0 ) + V + P , ∀ x U ∩ D ,

   then f is called upper P -continuous on x 0 . If f is upper P -continuous for all x ∈ D , then f is called upper P -continuous on D ;

2. If, for any V ∈ N ( 0 ) ⊂ Y , there exists ℬ ∈ N ( x 0 ) such that

   f ( x ) ∈ f ( x 0 ) + V − P , ∀ x U ∩ D ,

   then f is called lower P -continuous on x 0 . f is called lower P -continuous on D , if f is lower P -continuous at every x ∈ D ;

3. If, for any x , y ∈ D and λ ∈ [ 0 , 1 ] , the mapping f satisfies f ( x ) ∈ f ( λ x + ( 1 − λ ) y ) + P or f ( y ) ∈ f ( λ x + ( 1 − λ ) y ) + P , then f is called proper P -quasiconvex;

4. If, for x , y ∈ D and λ ∈ [ 0 , 1 ] , the mapping f satisfies

   λ f ( x ) + ( 1 − λ ) f ( y ) ∈ f ( λ x + ( 1 − λ ) y ) + P ,

   then f is called P -convex.

Lemma 2.9

[51] Let X and Y be two real Hausdorff topological spaces, D is a nonempty, compact, convex subset of X and P is a pointed, proper, closed and convex cone of Y with int P ≠ ∅ . Assume f : D × D → Y and Φ : D → Y are two vector mappings. Suppose f and Φ satisfy

1. f ( x , x ) ∈ P , for all x ∈ D ;

2. Φ is upper P -continuous on D ;

3. f ( ⋅ , y ) is lower P -continuous for all y ∈ D ;

4. f ( x , ⋅ ) + Φ ( ⋅ ) is proper P -quasiconvex for all x ∈ D .

Lemma 2.10

[31] Let f and ψ satisfy restrictions ( R 1 ) – ( R 7 ) . Define a mapping K r : E 2 → X as follows:

Then,

1. K r ( x ) ≠ ∅ for all x ∈ E 2 ;

2. K r is single valued;

3. K r is firmly nonexpansive-type mapping, that is for all x , y ∈ E 2 ,

   ⟨ K r ( x ) − K r ( y ) , J K r ( x ) − J K r ( y ) ⟩ ≤ ⟨ K r x − K r y , J x − J y ⟩ ;

4. Fix ( K r ) = Sol ( GV EP ) ;

5. Sol ( GV EP ) is closed and convex.

Definition 2.11

[25] A mapping F : E → E ∗ is said to be

1. weakly hemicontinuous if F is upper semicontinuous from line segments to the weak topology of E ;

2. sequentially weakly semicontinuous if for each sequence { x k } in E with x k ⇀ q , then F x k ⇀ F q .

Lemma 2.12

[21] A solution of D V I P is always a solution of V I P , if the operator F is weakly hemicontinuous.

Remark 2.13

It is well known that p ∈ C is a solution of (5) if and only if p is a fixed point of the operator P C ( I − λ F ) for all λ > 0 .

Definition 2.14

Let C be a nonempty closed and subset of a real Banach space E with dual E ∗ . The mapping F : C → E ∗ is said to be:

1. m -inverse strongly monotone on C with constant m > 0 if for each u , v ∈ C , there holds

   ⟨ u − v , F ( u ) − F ( v ) ⟩ ≥ m ‖ F ( u ) − F ( v ) ‖ 2 ;

2. strongly monotone on C with constant τ > 0 if for each u , v ∈ C , there holds

   ⟨ u − v , F ( u ) − F ( v ) ⟩ ≥ τ ‖ u − v ‖ 2 ;

3. strictly monotone on C if for all distinct u , v ∈ C , there holds

   ⟨ u − v , F ( u ) − F ( v ) ⟩ > 0 ;

4. monotone on C if for each u , v ∈ C , there holds

   ⟨ u − v , F ( u ) − F ( v ) ⟩ ≥ 0 ;

5. pseudo-monotone on C if for each u , v ∈ C , there holds

   ⟨ u − v , F ( v ) ⟩ ≥ 0 ⇒ ⟨ u − v , F ( u ) ⟩ ≥ 0 ;

6. quasi-monotone on C if for each u , v ∈ C , there holds

   ⟨ u − v , F ( v ) ⟩ > 0 ⇒ ⟨ u − v , F ( u ) ⟩ ≥ 0 ;

7. (see [52]) semistrictly quasi-monotone on C if F is quasi-monotone on C and for all distinct of points u , v ∈ C , we have that

   ⟨ u − v , F ( v ) ⟩ > 0 ⇒ ⟨ u − v , F ( w ) ⟩ > 0 ,

   for some w ∈ 1 2 ( u + v ) , u .

Lemma 2.15

[53] Let C be a nonempty, closed and convex subset of E and F : C → E be a weakly hemicontinuous and semistrictly quasi-monotone mapping. Then D V I ( C , F ) at least has one solution.

Lemma 2.16

[54] Let { a k } be a sequence of nonnegative real numbers satisfying the following relation:

where

1. { α k } ⊂ [ 0 , 1 ] , lim k → ∞ α k = 0 and ∑ k = 1 ∞ α k = ∞ ;

2. limsup k → ∞ σ k ≤ 0 ;

3. γ k ≥ 0 , ( k ≥ 1 ) and ∑ k = 1 ∞ γ k < ∞ .

Lemma 2.17

[9,42] Let { a k } be a sequence of real numbers such that there exists a subsequence { k j } of { k } such that a k j < a k j + 1 for all j ∈ N . Then, there exists a nondecreasing subsequence { m k } ⊂ N such that m k → ∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ N : a m k < a m k + 1 and a k < a m k + 1 . In fact, m k = max { i ≤ k : a i < a i + 1 } .

Algorithm 3.1

Pick x 0 , x 1 , u ∈ C and choose the parameters ρ ∈ ( 0 , 1 ) , η ∈ ( 0 , ν ) , where ν is the 2-uniform convexity constant of E 1 and suppose that { α k } ⊂ [ a , e ] for some 0 < a < e < 1 , { β k } ⊆ [ d , b ] for some 0 < d < b < 1 , { r k } be a sequence of positive real numbers and γ ∈ ( 0 , 1 d 2 ‖ L ‖ 2 ) and d is the best smoothness constant of E 2 . For j = 1 , 2 , … , m , let h j : E → R be a family of convex, weakly lower semicontinuous and Gâteaux differentiable functions such that h j ′ ( ⋅ ) is K j -Lipschitz continuous with K = max 1 ≤ j ≤ m K j . The feasible set C is defined by

where

We also assume the following conditions are satisfied:

1. lim k → ∞ β k = 0 , ∑ k = 1 ∞ β k = ∞ ;

2. liminf k → ∞ α k ( 1 − α k ) > 0 ;

3. ∑ k = 1 ∞ ϕ ( x k , x k − 1 ) < ∞ ;

4. lim k → ∞ θ k β k = 0 .

For each k > 0 , having the k -iterate { x k } , compute the following steps:

1. For j = 1 , 2 , … , m and given the current iterate, construct the family of half spaces

   C k j = { w ∈ E 1 : h j ( w k ) + ⟨ h j ′ ( w k ) , w − w k ⟩ ≤ 0 }

   and set

   C k = ⋂ j = 1 m C k j .

   Compute

   (14) w k = J 1 − 1 ( J 1 x k + θ k ( J 1 x k − 1 − J 1 x k ) ) , z k = Π C k J 1 − 1 ( J 1 w k − λ k F w k ) ,

   where λ k = ρ l k and l k is the smallest nonnegative integer such that

   (15) λ k ‖ F w k − F z k ‖ ≤ η ‖ w k − z k ‖ .

2. If w k = z k , ( w k ∈ VIP ( C , F ) ) , then w k = y k and go to step III. Otherwise, compute the next iterate by

   (16) y k = Π Q k J 1 − 1 ( J 1 w k − λ k F z k ) ,

   where Q k = { w ∈ E 1 : ⟨ w − z k , J 1 w k − λ k F w k − J 1 z k ⟩ ≤ 0 } .

3. Compute

   (17) u k = J 1 − 1 ( α k J 1 y k + ( 1 − α k ) J 1 T y k ) , v k = J 1 − 1 ( J 1 u k + γ L ∗ J 2 ( ( K r k ( I − r A ) − I ) L u k ) ) , x k + 1 = J 1 − 1 ( β k J 1 u + ( 1 − β k ) J 1 v k ) .

4. Set k ≔ k + 1 and go to step I.

Lemma 3.2

If w k = z k , then w k ∈ VIP ( C , F ) .

Proof

Suppose w k = z k , then by the characteristics of Π C k and (15), we have

or equivalently

Since λ k > 0 , we obtain

Hence, w k ∈ VIP ( C k , F ) . We conclude therefore from this, w k ∈ C and C ∈ C k that w k ∈ VIP ( C , F ) .□

Lemma 3.3

There exists a nonnegative integer l k satisfying (15) of Algorithm 3.1.

Proof

If w k ∈ VIP ( C , F ) , then w k = Π C k J 1 − 1 ( J 1 w k − λ k F w k ) and l k = 0 . Hence, we consider the case where w k ∉ VIP ( C , F ) and assume the contrary, that is for l > 0

This implies

Next, we consider the following possibilities, w k ∈ C k and w k ∉ C k . First, let w k ∈ C , then w k ∈ C k and w k = Π C k ( w k ) . Using the continuity of F and Π C k , then lim k → ∞ ‖ w k − z k ‖ = 0 . Consequently,