Global optimization and applications to a variational inequality problem

Azhar Hussain; Muhammad Adeel; Hassen Aydi; Dumitru Baleanu

doi:10.1515/math-2021-0104

Article Open Access

Global optimization and applications to a variational inequality problem

Azhar Hussain , Muhammad Adeel , Hassen Aydi and Dumitru Baleanu

Published/Copyright: December 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In the present paper, we study the existence and convergence of the best proximity point for cyclic Θ -contractions. As consequences, we extract several fixed point results for such cyclic mappings. As an application, we present some solvability theorems in the topic of variational inequalities.

Keywords: cyclic contraction; best proximity point; variational inequality

MSC 2010: 65K10; 47H10

1 Introduction and preliminaries

Let C and D be two non-empty closed subsets of a complete metric space ( X , ϖ ) . A mapping Ψ : C ∪ D → C ∪ D is called cyclic if Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C . In the case that C ∩ D ≠ ϕ , the mapping Ψ satisfying the contractive condition

(1) ϖ ( Ψ x , Ψ y ) ≤ γ ϖ ( x , y ) ,

where 0 < γ < 1 , has a fixed point in C ∩ D (note that Ψ need not to be continuous [1]). On the other hand, an element x ∗ ∈ C ∪ D is a best proximity point of Ψ if ϖ ( x ∗ , Ψ x ∗ ) = dist ( C , D ) = inf { ϖ ( ζ 1 , ζ 2 ) : ζ 1 ∈ C , ζ 2 ∈ D } .

The existence and approximation of best proximity points is an important theme in optimization theory. Elderd and Veeramani [2] extended the result of Kirk et al. [1] and proved the existence and uniqueness of best proximity points in complete metric spaces.

Definition 1.1

[2] Let C and D be non-empty subsets of ( X , ϖ ) . A map Ψ : C ∪ D → C ∪ D is a cyclic contraction map if Ψ ( C ) ⊆ D , Ψ ( D ) ⊆ C and for some 0 < γ < 1 ,

(2) ϖ ( Ψ x , Ψ y ) ≤ γ ϖ ( x , y ) + ( 1 − γ ) dist ( C , D ) .

Thagafi and Shahzad [3] provided a constructive answer to the question proposed by Elderd and Veeramani [2] on the existence of a best proximity point for a cyclic contraction map on a reflexive Banach space. Abkar and Gabeleh [4] have shown best proximity point theorems for cyclic contractions in ordered metric spaces. For more work in this direction, one can follow [5,6,7,8,9,10,11].

On the other hand, in 2014 the contractive condition of Banach was generalized by Jleli and Samet [12] in the following way:

(3) Θ ( ϖ ( Ψ x , Ψ y ) ) ≤ [ Θ ( ϖ ( x , y ) ) ] k ,

for 0 < k < 1 and x , y ∈ X , where Θ : [ 0 , ∞ ) → [ 1 , ∞ ) satisfies

Θ is non-decreasing;
for the sequence { γ n } ⊆ R + , lim n → ∞ Θ ( γ n ) = 1 if and only if lim n → ∞ ( γ n ) = 0 ;
there exist 0 < r < 1 and l ∈ ( 0 , ∞ ) such that lim γ → 0 + Θ ( γ ) − 1 γ r = l .

They [12] proved a version of Banach contraction principle for such mappings. The condition ( Θ 2 ) can be replaced by the simpler condition:

( Θ 2 ) : Θ ( γ ) > 1 for any γ > 0 .

Ahmad et al. [13] have used the following alternative condition:

Θ is continuous on ( 0 , ∞ ) .

The class of functions satisfying ( Θ 1 ) , ( Θ 2 ) and ( Θ 3 ′ ) is denoted by Λ . Many researchers have generalized (3) and have shown fixed point theorems for single and multivalued contractive mappings in many directions (see [13,14,15,16,17,18,19]).

In this article, we attempt to generalize the results of [1] and [2] for cyclic Θ -contractions. We also obtain certain fixed point results for such type of contractions. Some few examples are presented to prove the significance of our findings. Moreover, as an application, we prove solvability theorems for variational inequality problems.

2 Best proximity point results

In this section, we present the notion of cyclic Θ -contractions and discuss the existence and convergence of best proximity points for this kind of mappings in complete metric spaces.

Definition 2.1

Let C , D be two non-empty subsets of ( X , ϖ ) . Let Θ ∈ Λ be a continuous function. A mapping Ψ : C ∪ D → C ∪ D is called a cyclic Θ -contraction if Ψ ( C ) ⊆ D , Ψ ( D ) ⊆ C and for all x ∈ C and y ∈ D , we have

(4) Θ ( ϖ ∗ ( Ψ x , Ψ y ) ) ≤ [ Θ ( ϖ ∗ ( x , y ) ) ] k ,

where k ∈ ( 0 , 1 ) and ϖ ∗ ( x , y ) = ϖ ( x , y ) − dist ( C , D ) .

Remark 2.1

Every cyclic contraction in a classical sense is a cyclic Θ -contraction for Θ ( t ) = e t with k ∈ ( 0 , 1 ) . Indeed, let Ψ be a cyclic contraction on C ∪ D , so for all x ∈ C and y ∈ D , we have

ϖ ( Ψ x , Ψ y ) ≤ k ϖ ( x , y ) + ( 1 − k ) dist ( C , D ) .

This implies that

(5) ϖ ( Ψ x , Ψ y ) − dist ( C , D ) ≤ k ( ϖ ( x , y ) − dist ( C , D ) ) ,

which further yields by taking exponential on both sides

e ϖ ∗ ( Ψ x , Ψ y ) ≤ e k ϖ ∗ ( x , y ) .

Hence,

Θ ( ϖ ∗ ( Ψ x , Ψ y ) ) ≤ [ Θ ( ϖ ∗ ( x , y ) ) ] k .

Remark 2.2

From (5), one has ϖ ( Ψ x , Ψ y ) ≤ ϖ ( x , y ) for all x ∈ C and y ∈ D .

Example 2.1

Let X = { 1 , 2 , 5 , 6 , 9 } be endowed with the metric ϖ given as

ϖ ( x , y ) = 5 if x , y ∈ { 1 , 5 } , 6 if x , y ∈ { 2 , 6 } , 0 if x = y , 1 otherwise .

Then ( X , ϖ ) is a complete metric space. Let C = { 1 , 2 } and D = { 5 , 6 } be subsets of X . We have dist ( C , D ) = 1 . Define Ψ : C ∪ D → C ∪ D by Ψ ( 1 ) = 5 , Ψ ( 2 ) = 6 , Ψ ( 5 ) = 2 and Ψ ( 6 ) = 1 . Then Ψ ( C ) = D , Ψ ( D ) = C . If we take Θ ( t ) = 5 t , then clearly, Ψ is a cyclic Θ -contraction for all k ∈ ( 0 , 1 ) .

First, we give an approximation result.

Lemma 2.1

Let C and D be non-empty subsets of a metric space ( X , ϖ ) and Ψ : C ∪ D → C ∪ D be a cyclic Θ -contraction. For x 0 ∈ C ∪ D , the sequence { x n } defined by x n + 1 = Ψ x n , n = 0 , 1 , 2 , … , verifies

(6) lim n → ∞ ϖ ( x n , x n + 1 ) = dist ( C , D ) .

Proof

If there is an integer n 0 so that ϖ ( x n 0 , x n 0 + 1 ) = dist ( C , D ) , the proof is done. Assume that for each n , ϖ ( x n , x n + 1 ) > dist ( C , D ) , so ϖ ∗ ( x n , x n + 1 ) > 0 for each n ∈ N . One writes

1 < Θ ( ϖ ∗ ( x n , x n + 1 ) ) ≤ [ ( Θ ( ϖ ∗ ( x n − 1 , x n ) ) ) ] k ≤ [ ( Θ ( ϖ ∗ ( x n − 2 , x n − 1 ) ) ) ] k 2 . . . ≤ [ ( Θ ( ϖ ∗ ( x 0 , x 1 ) ) ) ] k n .

Taking the limit as n → ∞ , we have

Θ ( ϖ ∗ ( x n , x n + 1 ) ) → 1 .

By ( Θ 2 ) , we obtain

(7) lim n → ∞ ϖ ∗ ( x n , x n + 1 ) = 0 .

This gives that

lim n → ∞ ϖ ( x n , x n + 1 ) = dist ( C , D ) . □

Next, the existence of a best proximity point for a cyclic Θ -contraction mapping is as follows.

Lemma 2.2

Let C and D be non-empty subsets of a metric space ( X , ϖ ) and Ψ : C ∪ D → C ∪ D be a cyclic Θ -contraction. If { x 2 n } and { x 2 n + 1 } have convergent subsequences in C and D , respectively, then there exists ( x , y ) ∈ C × D such that

ϖ ( x , Ψ x ) = dist ( C , D ) = ϖ ( y , Ψ y ) = ϖ ( x , y ) .

Proof

Since Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C , likewise in Lemma 2.1, we have a sequence { x n } in C ∪ D such that { x 2 n } is contained in C and { x 2 n + 1 } is contained in D . Let { x 2 n k } be a subsequence of { x 2 n } such that

(8) lim k → ∞ x 2 n k = x

for some x ∈ C . One writes

(9) dist ( C , D ) ≤ ϖ ( x , x 2 n k − 1 ) ≤ ϖ ( x , x 2 n k ) + ϖ ( x 2 n k , x 2 n k − 1 ) .

Taking the limit as k → ∞ in (9) and using Lemma 2.1 with (8), we get

(10) lim k → ∞ ϖ ( x , x 2 n k − 1 ) = dist ( C , D ) .

Since Ψ is a cyclic Θ -contraction, we have

(11) Θ [ ϖ ∗ ( Ψ x 2 n k − 1 , Ψ x ) ] ≤ [ Θ ( ϖ ∗ ( x 2 n k − 1 , x ) ) ] k < Θ ( ϖ ∗ ( x 2 n k − 1 , x ) ) .

That is,

(12) ϖ ( Ψ x 2 n k − 1 , Ψ x ) < ϖ ( x 2 n k − 1 , x ) .

Now,

(13) dist ( C , D ) ≤ ϖ ( Ψ x 2 n k − 1 , Ψ x ) < ϖ ( x 2 n k − 1 , x ) ,

so, by letting k → ∞ in (13) and using (8) and (10), we get ϖ ( x , Ψ x ) = dist ( C , D ) . Similarly, if { x 2 n k + 1 } is a subsequence of { x 2 n + 1 } such that x 2 n k + 1 → y ∈ D as k → ∞ . It can be proven that ϖ ( y , Ψ y ) = dist ( C , D ) . Furthermore,

ϖ ( x , y ) = lim n → ∞ ϖ ( x 2 n k , x 2 n k + 1 ) = dist ( C , D ) . □

Example 2.2

Let X = R be equipped with the usual metric

ϖ ( x , y ) = ∣ y − x ∣ .

Consider C = 0 , 3 10 and D = 7 10 , 1 , then dist ( C , D ) = 2 5 . Define Ψ : C ∪ D → C ∪ D by

Ψ ( x ) = 7 10 if x ∈ 0 , 3 20 ⊂ C ; 1 − x if x ∈ 3 20 , 3 10 ⊂ C ; 3 10 if x ∈ 7 10 , 41 50 ⊂ D ; x 2 − 1 5 if x ∈ 41 50 , 1 ⊂ D ;

and Θ : ( 0 , ∞ ) → ( 1 , ∞ ) by Θ ( t ) = exp [ t ] . Note that Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C . Now, for x = 3 20 ∈ C and y = 41 50 ∈ D , we have

Θ [ ϖ ∗ ( Ψ x , Ψ y ) ] = Θ ϖ ∗ 1 − x , y 2 − 1 5 = Θ ϖ 1 − x , y 2 − 1 5 − ϖ ( C , D ) = Θ y 2 + x − 6 5 − 2 5 = exp 6 25

and

[ Θ ( ϖ ∗ ( x , y ) ) ] k = [ Θ ( ϖ ( x , y ) − ϖ ( C , D ) ) ] k = [ Θ ∣ y − x ∣ − ϖ ( C , D ) ] k = Θ 27 100 k < Θ 27 100 = exp 27 100 .

This implies that

Θ [ ϖ ∗ ( Ψ x , Ψ y ) ] = exp 6 25 < exp 27 100 = [ Θ ( ϖ ∗ ( x , y ) ) ] k .

Hence, Ψ is a cyclic Θ -contraction. Similarly, the inequality (4) holds for the remaining cases. Also, the sequences { x 2 n = Ψ 2 n ( 0 ) } and { x 2 n + 1 = Ψ 2 n + 1 ( 0 ) } have convergent subsequences in C and D , respectively. Thus, all the conditions of Lemma 2.2 are satisfied and so there exists 3 10 ∈ C such that

ϖ ( x , Ψ x ) = ϖ 3 10 , 1 − 3 10 = 7 10 − 3 10 = dist ( C , D ) .

Lemma 2.3

Let C and D be non-empty subsets of a metric space ( X , ϖ ) and Ψ : C ∪ D → C ∪ D be a cyclic Θ -contraction. Then for every x 0 ∈ C , likewise in Lemma 2.1, there exists a sequence { x n } in C ∪ D , which is bounded.

Proof

Suppose that x 0 ∈ C , by Lemma 2.1, there exists a sequence { x n } in C ∪ D such that { ϖ ( x 2 n , x 2 n + 1 ) } converges to dist ( C , D ) . We claim that { x 2 n } is bounded. On contrary, assume that { x 2 n } is not bounded, then there exists q ∈ N satisfying M < ϖ ∗ ( x 2 n , x 2 ( n + q ) + 1 ) and ϖ ∗ ( x 2 n , x 2 ( n + q ) − 1 ) ≤ M , where M = 2 q dist ( C , D ) . Thus, we have

(14) M < ϖ ∗ ( x 2 n , x 2 ( n + q ) + 1 ) = ϖ ( x 2 n , x 2 ( n + q ) + 1 ) − dist ( C , D ) ≤ ϖ ( x 2 n , x 2 n + 1 ) + ϖ ( x 2 n + 1 , x 2 n + 2 ) + ⋯ + ϖ ( x 2 ( n + q ) , x 2 ( n + q ) + 1 ) − dist ( C , D ) = ( ϖ ( x 2 n , x 2 n + 1 ) − dist ( C , D ) ) + ( ϖ ( x 2 n + 1 , x 2 n + 2 ) − dist ( C , D ) ) + ⋯ + ( ϖ ( x 2 ( n + q ) , x 2 ( n + q ) + 1 ) − dist ( C , D ) ) + ( 2 q dist ( C , D ) − dist ( C , D ) ) ≤ ϖ ∗ ( x 2 n , x 2 n + 1 ) + ϖ ∗ ( x 2 n + 1 , x 2 n + 2 ) + ⋯ + ϖ ∗ ( x 2 ( n + q ) , x 2 ( n + q ) + 1 ) + ( 2 q − 1 ) dist ( C , D ) .

Letting n → ∞ in (14) and using Lemma 2.1, we get

M < M − dist ( C , D ) ,

which leads to a contradiction. We can also prove that { x 2 n + 1 } is bounded and thus { x n } is bounded.□

Theorem 2.1

Let C and D be non-empty subsets of a metric space ( X , ϖ ) and Ψ : C ∪ D → C ∪ D be a cyclic Θ -contraction. If either C or D is boundedly compact, then there exists x ∈ C ∪ D such that ϖ ( x , Ψ x ) = dist ( C , D ) .

Proof

It follows from Lemmas 2.2 and 2.3.□

3 Particular case: fixed point results

In this section, we deduce some of the fixed point results for cyclic Θ -contractions.

Theorem 3.1

Let C and D be non-empty subsets of a metric space ( X , ϖ ) with dist ( C , D ) = 0 and Ψ : C ∪ D → C ∪ D be a mapping satisfying Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C . Assume that there exists Θ ∈ Λ such that

(15) Θ ( ϖ ( Ψ x , Ψ y ) ) ≤ [ Θ ( ϖ ( x , y ) ) ] k .

Then Ψ has a unique fixed point in C ∩ D provided that { x 2 n } and { x 2 n + 1 } have convergent subsequences in C and D , respectively, where { x n } is the sequence of Lemma 2.1, for some x 0 ∈ C .

Proof

Since dist ( C , D ) = 0 , equation (15) implies that Ψ is a cyclic Θ -contraction. Hence, from Lemma 2.2, we have x ∈ C and y ∈ D with x = y such that ϖ ( x , Ψ x ) = 0 . Consequently, C ∩ D ≠ ∅ and Ψ has a fixed point in C ∩ D . For uniqueness, suppose that u ∈ C ∪ D such that Ψ u = u with x ≠ u , then by (15), we get

Θ ( ϖ ( x , u ) ) = Θ ( ϖ ( Ψ x , Ψ u ) ) ≤ ( Θ ( ϖ ( x , u ) ) ) k < Θ ( ϖ ( x , u ) ) .

It is a contradiction. Thus, x = u .□

Theorem 3.2

Let C and D be non-empty subsets of a metric space ( X , ϖ ) with dist ( C , D ) = 0 and Ψ : C ∪ D → C ∪ D be a mapping such that Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C satisfying (15). If C and D are boundedly compact, then Ψ has a unique fixed point in C ∩ D .

Proof

It follows from Lemma 2.3 and Theorem 3.1.□

Remark 3.1

Recently, Radenović [9] showed that fixed point results for mappings satisfying cyclical contractive conditions appearing in [1] are equivalent to fixed point results for classical contractive mappings. It can be noted in the proof of Theorem 2.5 of [9] that to obtain cyclical-type theorems from classical results, completeness of metric space and closedness of subsets C i for all i are necessary, otherwise ( ∩ i = 1 m C i , ϖ ) is not a complete metric space. Therefore, Theorem 3.2 cannot be obtained from classical fixed point theorems by using the approach of Radenović [9].

Lemma 3.1

Let C and D be non-empty subsets of a metric space ( X , ϖ ) such that dist ( C , D ) = 0 . If Ψ : C ∪ D → C ∪ D is a cyclic Θ -contraction. Then likewise in Lemma 2.1, there exists a Cauchy sequence { x 2 n } in C .

Proof

Let x 0 ∈ C . Given that dist ( C , D ) = 0 , so for all n ∈ N , from Lemma 2.1 we get

(16) lim n → ∞ ϖ ( x n , x n + 1 ) = 0 .

We claim that for ε > 0 , there exists N ∈ N such that

(17) ϖ ( x 2 n , x 2 m + 1 ) < ε ,

with n , m ≥ N . If not, then there exist two divergent sequences { 2 n k } , { 2 m k + 1 } ⊆ N such that

(18) ϖ ( x 2 n k , x 2 m k + 1 ) ≥ ε ,

for all k ∈ N . We assume that 2 m k + 1 is a minimal index for which (18) holds. Then for all k ∈ N ,

(19) ϖ ( x 2 n k , x 2 m k − 1 ) < ε .

From (18) and (19), we have

(20) ε ≤ ϖ ( x 2 n k , x 2 m k + 1 ) ≤ ϖ ( x 2 n k , x 2 m k − 1 ) + ϖ ( x 2 m k − 1 , x 2 m k ) + ϖ ( x 2 m k , x 2 m k + 1 ) < ε + ϖ ( x 2 m k − 1 , x 2 m k ) + ϖ ( x 2 m k , x 2 m k + 1 ) .

Letting k → ∞ in (20) and using (16), we get

(21) lim k → ∞ ϖ ( x 2 n k , x 2 m k + 1 ) = ε .

Now, by a triangular inequality, we have

(22) ϖ ( x 2 n k , x 2 m k + 1 ) ≤ ϖ ( x 2 n k , x 2 n k − 1 ) + ϖ ( x 2 n k − 1 , x 2 m k ) + ϖ ( x 2 m k , x 2 m k + 1 )

and

(23) ϖ ( x 2 n k − 1 , x 2 m k ) ≤ ϖ ( x 2 n k − 1 , x 2 n k ) + ϖ ( x 2 n k , x 2 m k + 1 ) + ϖ ( x 2 m k + 1 , x 2 m k ) .

Letting k → ∞ in (22) and (23) and using (16) and (21), we get

(24) lim k → ∞ ϖ ( x 2 n k − 1 , x 2 m k ) = ε .

In view of (21) and (24), there is an integer k 0 so that ϖ ( x 2 n k , x 2 m k + 1 ) > 0 and ϖ ( x 2 n k − 1 , x 2 m k ) > 0 for each k ≥ k 0 . Thus, since Θ ∈ Λ , we have

Θ ( ϖ ( x 2 n k , x 2 m k + 1 ) ) ≤ ( Θ ( ϖ ( x 2 n k − 1 , x 2 m k ) ) ) k .

Letting k tend to ∞ and using again (21) and (24) lead to a contradiction. Hence, { x 2 n } is a Cauchy sequence in C .□

Theorem 3.3

Let C and D be non-empty closed subsets of a complete metric space ( X , ϖ ) such that dist ( C , D ) = 0 . If Ψ : C ∪ D → C ∪ D is a cyclic Θ -contraction, then Ψ has a unique fixed point in C ∩ D .

Proof

Consider x 0 ∈ C , then by Lemma 3.1, there exists a Cauchy sequence { x 2 n } in C . Since X is complete and C is a closed subset of X , there exists x ∗ ∈ C such that x 2 n → x ∗ as n → ∞ . Now, we have

0 ≤ ϖ ( x 2 n − 1 , x ∗ ) ≤ ϖ ( x 2 n − 1 , x 2 n ) + ϖ ( x 2 n , x ∗ ) ,

so taking limit as n → ∞ , we get that

ϖ ( x 2 n − 1 , x ∗ ) → 0 .

Similarly, { x 2 n − 1 } is a Cauchy sequence in D , which converges to x ∗ ∈ D as D is closed in the complete metric space X . Hence, { x 2 n } and { x 2 n + 1 } have convergent subsequences in C and D , respectively. From Theorem 3.2, Ψ has a unique fixed point in C ∩ D .□

If we take Θ ( t ) = e t in Theorem 3.3, we obtain the following main result of [1].

Corollary 3.1

Let C and D be non-empty closed subsets of a complete metric space ( X , ϖ ) such that dist ( C , D ) = 0 . If Ψ : C ∪ D → C ∪ D is a cyclic contraction, then Ψ has a unique fixed point in C ∩ D .

Example 3.1

Let X = R be endowed with the usual metric ϖ . Consider C = [ − 1 , 0 ] and D = [ 0 , 1 ] . Define Ψ : C ∪ D → C ∪ D such that

Ψ ( x ) = − x 3

for all x ∈ C ∪ D and Θ : ( 0 , ∞ ) → ( 1 , ∞ ) by Θ ( t ) = 1 + t . Note that Ψ ( C ) ⊆ D and Ψ ( D ) ⊆ C . Now, for x ∈ C ∪ D , we have

Θ [ ϖ ( Ψ x , Ψ y ) ] = Θ ϖ − x 3 , − y 3 = Θ ∣ x − y ∣ 3

and

[ Θ ( ϖ ( x , y ) ) ] k = [ Θ ( ∣ x − y ∣ ) ] k .

This implies

Θ ∣ x − y ∣ 3 ≤ [ Θ ( ∣ x − y ∣ ) ] k .

Hence, Ψ is a cyclic Θ -contraction for all k ∈ ( 0 , 1 ) . Thus, all the conditions of Theorem 3.3 are satisfied and so there is a unique 0 ∈ C ∪ D such that Ψ x = x .

4 Application to a variational inequality problem

Let H be a real Hilbert space with an inner product ⟨ . , . ⟩ and an induced norm ‖ . ‖ . An element μ ∗ ∈ Ω (where ϕ ≠ Ω ⊆ H ) is known as a best approximation if ∣ ∣ μ − μ ∗ ∣ ∣ = D ( μ , Ω ) , where D ( μ , Ω ) = inf ν ∈ Ω ∣ ∣ μ − ν ∣ ∣ is called the error in approximating μ by Ω . A metric projection map P Ω : H → Ω such that for all μ ∈ H , P Ω ( μ ) = { ν ∈ Ω : ‖ μ − ν ‖ = D ( μ , Ω ) } is the set of all best approximations from μ to Ω . This metric projection plays an important role for solving the variational inequality problem.

For a non-empty, closed and convex subset Ω ⊆ H , f : Ω → H is a given operator. The Hartman-Stampacchia variational inequality (HSVI) problem is defined as

P-I: HSVI ( f , Ω ) ≔ find μ ∗ ∈ Ω such that ⟨ f μ ∗ , μ − μ ∗ ⟩ ≥ 0 for all μ ∈ Ω

and the Minty variational inequality (MVI) problem is defined by

P-II: MVI ( f , Ω ) ≔ find μ ∗ ∈ Ω such that ⟨ f μ , μ − μ ∗ ⟩ ≥ 0 for all μ ∈ Ω .

The HSVI has various applications in the field of engineering, physics and industry, while the MVI facilitates in the study of solvability of HSVI ( f , Ω ), see [20].

Definition 4.1

[21] A mapping f : Ω → H is called pseudomonotone if for all μ , ν ∈ Ω , ⟨ μ − ν , f ( ν ) ⟩ ≥ 0 implies ⟨ μ − ν , f ( μ ) ⟩ ≥ 0 .

Definition 4.2

[21] Let Ω ⊆ H be a convex set. A mapping f : Ω → H is hemicontinuous from the line segments of Ω to the weak topology of H .

The following theorem establishes a relation between the problems: P-I and P-II.

Theorem 4.1

[21] Let H ( τ ) be a locally convex space, Ω ⊆ H be a closed convex set and f : Ω → H be a pseudomonotone and hemicontinuous mapping. Then an element u 0 ∈ Ω is a solution to the problem HSVI ( f , Ω ) iff it is a solution to the problem MVI ( f , Ω ).

It is known that, for each u ∈ H , there exists a unique nearest point P Ω ( u ) ∈ Ω such that ‖ u − P Ω ( u ) ‖ ≤ ‖ u − v ‖ for all v ∈ Ω [20].

Lemma 4.1

[20] Let Ω be a convex subset of the Hilbert space H , z ∈ H and u ∈ Ω . Then u = P Ω ( z ) if and only if ⟨ z − u , v − u ⟩ ≤ 0 for all v ∈ Ω .

Lemma 4.2

[20] Let Ω be a convex subset of Hilbert space H and f : Ω → Ω . Then u ∈ Ω is a solution of ⟨ f u , v − u ⟩ ≥ 0 for all v ∈ Ω if and only if u = P Ω ( u − λ f u ) , λ > 0 .

Every set is clearly an approximately compact set in terms of itself. So the following findings are proven.

Theorem 4.2

Let Ω be a non-empty, closed and convex subset of a real Hilbert space H . Assume that for μ ∈ Ω , there exists Θ ∈ Λ such that for f : Ω → Ω , P Ω ( I Ω − λ f ) : Ω → Ω satisfies

Θ ( ∣ ∣ P Ω ( I Ω − λ f ) μ − P Ω ( I Ω − λ f ) ν ∣ ∣ ) ≤ ( Θ ∣ ∣ μ − ν ∣ ∣ ) k ,

where I Ω is the identity operator on Ω . Then for all v ∈ Ω there exists a unique element u ∈ Ω such that ⟨ f u , v − u ⟩ ≥ 0 . Moreover, for a fixed element u 0 ∈ Ω , λ > 0 and n ∈ N ∪ { 0 } , the sequence { u n } with u n + 1 = P Ω ( u n − λ g u n ) converges to u .

Proof

Define g : Ω → Ω by g x = P Ω ( x − λ f x ) for all x ∈ Ω , then g satisfies all the conditions of Theorem 3.3 by setting C = D = Ω , and so g has a unique fixed point u . Hence, by Lemma 4.2, ⟨ f u , v − u ⟩ ≥ 0 has a solution u ∈ Ω for all v ∈ Ω if and only if u is a fixed point of g . This completes the proof.□

Theorem 4.3

Suppose that all the conditions of Theorem 4.2 hold. Then for all v ∈ Ω , there exists a unique element u ∈ Ω such that ⟨ f v , v − u ⟩ ≥ 0 . Moreover, for a fixed element u 0 ∈ Ω , λ > 0 and n ∈ N ∪ { 0 } , the sequence { u n } with u n + 1 = P Ω ( u n − λ g u n ) converges to u .

Proof

From Theorems 4.1 and 4.2, we have the required results.□

5 Conclusion

In this paper, we presented the existence and convergence of the best proximity point for cyclic Θ -contractions. We derived several fixed point results for such mappings. As an application of the obtained results, we presented some solvability theorems in the topic of variational inequalities.

Funding information: This paper received no external funding.
Author contributions: All authors contributed equally and significantly in writing this article.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: No data were used to support this study.

References

[1] W. A. Kirk, P. S. Srinivasan, and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), 79–89. Search in Google Scholar

[2] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001–1006. 10.1016/j.jmaa.2005.10.081Search in Google Scholar

[3] M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), 3665–3671. 10.1016/j.na.2008.07.022Search in Google Scholar

[4] A. Abkar and M. Gabeleh, Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl. 150 (2011), 188–193. 10.1007/s10957-011-9810-xSearch in Google Scholar

[5] N. V. Dung and S. Radenović, Remarks on theorems for cyclic quasi-contractions in uniformly convex Banach spaces, Kragujevac J. Math. 40 (2016), no. 2, 272–279. 10.5937/KgJMath1602272DSearch in Google Scholar

[6] V. Parvaneh, M. R. Haddadi, and H. Aydi, On best proximity point results for some type of mappings, J. Funct. Spaces 2020 (2020), 6298138. 10.1155/2020/6298138Search in Google Scholar

[7] H. Aydi, H. Lakzian, Z. D. Mitrović, and S. Radenović, Best proximity points of MF-cyclic contractions with property UC, Numer. Funct. Anal. Optim. 41 (2020), no. 7, 871–882. 10.1080/01630563.2019.1708390Search in Google Scholar

[8] Z. Kadelburg and S. Radenović, A note on Pata-type cyclic contractions, Sarajevo J. Math. 11 (2015), no. 24, 235–245. Search in Google Scholar

[9] S. Radenović, Some remarks on mappings satisfying cyclical contractive conditions, Afr. Mat. 27 (2016), 291–295, 10.1007/s13370-015-0339-2Search in Google Scholar

[10] S. Radenović, S. Chandok, and W. Shatanawi, Some cyclic fixed point results for contractive mappings, Univ. Though Publ. Nature Sci. 6 (2016), no. 2, 38–40. 10.5937/univtho6-11813Search in Google Scholar

[11] H. Aydi, E. Karapinar, and A. F. Roldan Lopez de Hierro, w-interpolative Ciric-Reich-Rus type contractions, Mathematics 7 (2019), 57. 10.3390/math7010057Search in Google Scholar

[12] M. Jleli and B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014 (2014), 38. 10.1186/1029-242X-2014-38Search in Google Scholar

[13] J. Ahmad, A. E. Al-Mazrooei, Y. J. Cho, and Y. O. Yang, Fixed point results for generalized Θ-contractions, J. Nonlinear Sci. Appl. 10 (2017), no. 5, 2350–2358. 10.22436/jnsa.010.05.07Search in Google Scholar

[14] H. A. Hammad, H. Aydi, and N. Mlaiki, Contributions of the fixed point technique to solve the 2D Volterra integral equations, Riemann? Liouville fractional integrals, and Atangana-Baleanu integral operators, Adv. Differ. Equ. 2021 (2021), 97. 10.1186/s13662-021-03255-6Search in Google Scholar

[15] H. A. Hammad, H. Aydi, and Y. U. Gaba, Exciting fixed point results on a novel space with supportive applications, J. Funct. Spaces 2021 (2021), 6613774. 10.1155/2021/6613774Search in Google Scholar

[16] X. D. Liu, S. Chang, Y. Xiao, and L. C. Zhao, Existence of fixed points for Θ-type contraction and Θ-type Suzuki contraction in complete metric spaces, Fixed Point Theory Appl. 2016 (2016), 8. 10.1186/s13663-016-0496-5Search in Google Scholar

[17] Z. Ma, A. Asif, H. Aydi, S. Ullah Khan, and M. Arshad, Analysis of F-contractions in function weighted metric spaces with an application, Open Math. 18 (2020), 582–594. 10.1515/math-2020-0035Search in Google Scholar

[18] A. Ali, H. Isik, H. Aydi, E. Ameer, J. R. Lee, and M. Arshad, On multivalued Suzuki-type θ-contractions and related applications, Open Math. 18 (2020), 386–399. 10.1515/math-2020-0139Search in Google Scholar

[19] V. Parvaneh, F. Golkarmanesh, N. Hussain, and P. Salimi, New fixed point theorems for α-HΘ-contractions in ordered metric spaces, J. Fixed Point Theory Appl. 18 (2016), 905–925. 10.1007/s11784-016-0330-zSearch in Google Scholar

[20] F. Deutsch, Best Approximation in Inner Product Spaces, Springer-Verlag, New York, 2001. 10.1007/978-1-4684-9298-9Search in Google Scholar

[21] G. Isac, Leray-Schauder Type Alternatives, Complementarity Problems and Variational Inequalities, Springer-Verlag, New York, 2006. Search in Google Scholar

Received: 2020-11-27

Revised: 2021-06-25

Accepted: 2021-06-26

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0104

Keywords for this article

cyclic contraction; best proximity point; variational inequality

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