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Article

Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

1
Department of Mathematics and Physics, Hebei University of Architecture, Zhangjiakou 075024, China
2
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
3
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
4
Department of Mathematics, Usak University, Usak 64100, Turkey
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(1), 93; https://doi.org/10.3390/sym11010093
Submission received: 10 December 2018 / Revised: 5 January 2019 / Accepted: 7 January 2019 / Published: 15 January 2019
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

Abstract

:
In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + i = 1 m A i γ ( X ) A i and give a numerical example.
MSC:
54H25; 47H10

1. Introduction and Preliminaries

In 1922, Polish mathematician Banach [1] proved an interesting result known as “Banach contraction principle" which led to the foundation of metric fixed point theory. His contribution gave a positive answer to the existence and uniqueness of the solution of problems concerned. Later on, many authors extended and generalized Banach’s result in many directions (see [2,3,4]). Samet et al. [5] introduced the contractive condition called α - ψ -contraction by
α ( x , y ) d ( F x , F y ) ψ ( d ( x , y ) ) ,
where the functions ψ : [ 0 , ) [ 0 , ) satisfy the following conditions:
(ψ1)
ψ is nondecreasing;
(ψ2)
n = 1 + ψ n ( t ) < for all t > 0 , where ψ n is the nth iterate of ψ and ψ ( t ) < t for any t > 0 ;
and that F is α -admissible if for all x , y X
α ( x , y ) 1 α ( F x , F y ) 1 ,
where α : X × X [ 0 , ) and proved some fixed point results for such mappings in the context of complete metric spaces ( X , d ) . Subsequently, Salimi et al. [6] and Hussain et al. [2,7] modified the notions of α - ψ -contractive, α -admissible mappings and proved certain fixed point results. In 2014, Jleli et al. [4] generalized the contractive condition by considering a function Θ : ( 0 , ) ( 1 , ) satisfying,
1)
Θ is nondecreasing;
2)
for each sequence { α n } R + , lim n Θ ( α n ) = 1 if and only if lim n ( α n ) = 0 ;
3)
there exist 0 < k < 1 and l ( 0 , ) such that lim α 0 + Θ ( α ) 1 α k = l ,
in the following way,
Θ ( d ( F x , F y ) ) [ Θ ( d ( x , y ) ) ] k ,
where k ( 0 , 1 ) and x , y X and proved the following fixed point theorem.
Theorem 1.
Suppose that F: X X is a Θ-contraction, where ( X , d ) a complete metric space; hen, F possesses a unique u X such that F u = u .
Recently, Ahmad et al. [8] used the following weaker condition instead of the condition ( Θ 3 ) :
  • ( Θ 3 ) Θ is continuous on ( 0 , ) .
Many authors generalized (2) in many directions and proved fixed point theorems for single and multivalued contractive mappings (see [8,9,10]).
However, the mapping involved in all these results were self mappings. For non-empty subsets A and B of a complete metric space ( X , d ) , the contractive mapping F : A B may not have a fixed point. The case lead to the search for an element x (say) such that d ( x , F x ) is minimum, that is, the distance between the points x and F x is proximity closed. In view of the fact that d ( x , F x ) d ( A , B ) , an absolute optimal approximate solution is an element x for which the error d ( x , F x ) assumes the least possible value d ( A , B ) . Thus, a best proximity pair theorem furnishes sufficient conditions for the existence of an optimal approximate solution x, known as a best proximity point of the mapping F, satisfying the condition that d ( x , F x ) = d ( A , B ) . Many authors established the existence and convergence of fixed and best proximity points under certain contractive conditions in different metric spaces (see e.g., [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and references therein).
The purpose of this paper is to define the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove some best proximity point results in the frame work of complete metric spaces. Moreover, we prove best proximity point results in partially ordered complete metric spaces through our main results. As an application, we obtain some fixed point results for such contraction in metric and partially ordered metric spaces. Some examples to prove the validity and the existence of solution of nonlinear matrix equation with a numerical example to show the usability of our results is presented.
In the sequel, we denote Ψ the set of all functions ψ satisfying ( ψ 1 , ψ 2 ) and Ω the set of all functions Θ satisfying ( Θ 1 , Θ 2 , Θ 3 ).
Let ( X , d ) be a metric space, A and B two nonempty subsets of X . Define
d ( A , B ) = inf { d ( a , b ) : a A , b B } , A 0 = { a A : there exists some b B such that d ( a , b ) = d ( A , B ) } , B 0 = { b B : there exists some a A such that d ( a , b ) = d ( A , B ) } .
Definition 1.
Let ( X , d ) be a metric space and A 0 ϕ , we say that the pair ( A , B ) has the weak P-property if
d ( x 1 , y 1 ) = d ( A , B ) d ( x 2 , y 2 ) = d ( A , B ) d ( x 1 , x 2 ) d ( y 1 , y 2 )
for all x 1 , x 2 A and y 1 , y 2 B . [31]
Definition 2.
Let ( X , d ) be a metric space and A , B two subsets of X, a non-self mapping T : A B is called α-proximal admissible if
α ( x 1 , x 2 ) 1 , d ( u 1 , T x 1 ) = d ( A , B ) , α ( u 1 , u 2 ) 1 , d ( u 2 , T x 2 ) = d ( A , B )
for all x 1 , x 2 , u 1 , u 2 A , where α : A × A [ 0 , ) [4].

2. Best Proximity Point Results for C ´ iri c ´ Type Contraction

We begin this section with the following definition:
Definition 3.
Let A , B be two subsets of a metric space ( X , d ) and and α: A × A [ 0 , ) be a function. A mapping F: A B is said to be C ´ iri c ´ type α - ψ - Θ -contraction if for ψ Ψ , Θ Ω , there exists k ( 0 , 1 ) and for x , y A with α ( x , y ) 1 and d ( F x , F y ) > 0 , we have
α ( x , y ) Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 d ( A , B ) , d ( x , F y ) + d ( y , F x ) 2 d ( A , B ) .
Theorem 2.
Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a C ´ iri c ´ type α - ψ - Θ -contraction satisfying
(i) 
F is α-proximal admissible;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iii) 
F is continuous;
(iv) 
there exist x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) such that α ( x 0 , x 1 ) 1 .
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Consider x 0 in A 0 , since F ( A 0 ) B 0 , there exists an element x 1 in A 0 such that d ( x 1 , F x 0 ) = d ( A , B ) , by assumption (iv), α ( x 0 , x 1 ) 1 . Since x 1 A 0 and F ( A 0 ) B 0 , there exists x 2 A 0 such that d ( x 2 , F x 1 ) = d ( A , B ) . By α-proximal admissibility of F, we have that α ( x 1 , x 2 ) 1 . Continuing in this way, we get
d ( x n + 1 , F x n ) = d ( A , B ) and α ( x n , x n + 1 ) 1 n N .
Now if there exists n 0 N such that x n 0 = x n 0 + 1 , we have
d ( x n 0 , F x n 0 ) = d ( x n 0 + 1 , F x n 0 ) = d ( A , B ) .
Then, x n 0 is the point of best proximity. Therefore, we assume that x n x n + 1 , i.e., d ( x n , x n + 1 ) > 0 for all n N { 0 } .
By weak P-property of the pair ( A , B ) and from (3), (4), we have for all n N
1 < Θ [ d ( x n + 1 , x n ) ] Θ [ d ( F x n , F x n 1 ) ] α ( x n , x n 1 ) Θ [ d ( F x n , F x n 1 ) ] [ ψ ( Θ ( M ( x n , x n 1 ) ) ) ] k ,
where
M ( x n , x n 1 ) = m a x d ( x n , x n 1 ) , d ( x n , F x n ) + d ( x n 1 , F x n 1 ) 2 d ( A , B ) , d ( x n , F x n 1 ) + d ( x n 1 , F x n ) 2 d ( A , B ) m a x d ( x n , x n 1 ) , d ( x n , x n + 1 ) + d ( x n + 1 , F x n ) + d ( x n 1 , x n ) + d ( x n , F x n 1 ) 2 d ( A , B ) , d ( x n , F x n 1 ) + d ( x n 1 , x n + 1 ) + d ( x n + 1 , F x n ) 2 d ( A , B ) = m a x d ( x n , x n 1 ) , d ( x n , x n + 1 ) + d ( x n 1 , x n ) 2 , d ( x n 1 , x n + 1 ) 2 m a x d ( x n , x n 1 ) , d ( x n , x n + 1 ) + d ( x n 1 , x n ) 2 m a x d ( x n , x n 1 ) , d ( x n , x n + 1 ) .
This together with inequality (5) gives
1 < Θ [ d ( x n , x n + 1 ) ] [ ψ ( Θ ( m a x { d ( x n , x n 1 ) , d ( x n , x n + 1 ) } ) ) ] k .
If
m a x { d ( x n , x n 1 ) , d ( x n , x n + 1 ) } = d ( x n , x n + 1 ) ,
we have
1 < Θ [ d ( x n , x n + 1 ) ] [ ψ ( Θ ( d ( x n , x n + 1 ) ) ) ] k < Θ ( d ( x n , x n + 1 ) ) ,
a contradiction, so we have
1 < Θ [ d ( x n , x n + 1 ) ] [ ψ ( Θ ( d ( x n , x n 1 ) ) ) ] k .
By induction, we get
1 < Θ [ d ( x n , x n + 1 ) ] [ ψ ( Θ ( d ( x n 1 , x n ) ) ) ] k [ ψ ( Θ ( d ( x n 2 , x n 1 ) ) ) ] k 2 . . . [ ψ ( Θ ( d ( x 0 , x 1 ) ) ) ] k n .
Taking limit as n in above inequality, we have
Θ [ d ( x n , x n + 1 ) ] 1
and by Θ 2 , we obtain
lim n d ( x n , x n + 1 ) = 0 .
Now, we show that { x n } is a Cauchy sequence in A. Suppose on the contrary that it is not, that is, ∃ ϵ > 0 , we can find the sequences { p n } and { q n } of natural numbers such that for p n > q n > n , we have
d ( x p n , x q n ) ϵ .
Then,
d ( x p n 1 , x q n ) < ϵ
for all n N . Thus, by triangle inequality and (7), we get
ϵ d ( x p n , x q n ) d ( x p n , x p n 1 ) + d ( x p n 1 , x q n ) < d ( x p n , x p n 1 ) + ϵ .
Taking limit and using inequality (6), we get
lim n d ( x p n , x q n ) = ϵ .
Again by triangle inequality, we have
d ( x p n , x q n ) d ( x p n , x p n + 1 ) + d ( x p n + 1 , x q n + 1 ) + d ( x q n + 1 , x q n )
and
d ( x p n + 1 , x q n + 1 ) d ( x p n + 1 , x p n ) + d ( x p n , x q n ) + d ( x q n , x q n + 1 ) .
Taking limit as n , from Equations (6) and (8), we have that
lim n d ( x p n + 1 , x q n + 1 ) = ϵ .
Thus, Equation (8) holds. Then by assumption, α ( x p n , x q n ) 1 , we get
1 Θ ( d ( x p n + 1 , x q n + 1 ) ) Θ ( d ( F x p n , F x q n ) ) α ( x p n , x q n ) Θ ( d ( F x p n , F x q n ) ) [ ψ ( Θ ( M ( x p n , x q n ) ) ) ] k < Θ ( M ( x p n , x q n ) ) .
By taking limit as n in above inequality, using ( Θ 3 ) and Equation (6), we get
lim n d ( x p n , x q n ) = 0 < ϵ ,
which is a contradiction. Thus, { x n } is a Cauchy sequence. Since { x n } A and A is closed in a complete metric space ( X , d ) , we can find u A such that x n u . Since F is continuous, we have F x n F u . This implies that d ( x n + 1 , F x n ) d ( u , F u ) .
Since the sequence { d ( x n + 1 , F x n ) } is a constant sequence with value d ( A , B ) , we deduce
d ( u , F u ) = d ( A , B ) .
This completes the proof. □
Example 1.
Let X = R 2 with metric d defined as d ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) = | x 1 y 1 | + | x 2 y 2 | . Suppose A = { ( 4 , 4 ) , ( 7 , 8 ) , ( 20 , 0 ) , ( 25 , 30 ) } and B = { ( 4 , 0 ) , ( 0 , 4 ) , ( 9 , 10 ) , ( 11 , 8 ) } . Then, d ( A , B ) = 4 , A 0 = { ( 4 , 4 ) , ( 7 , 8 ) } and B 0 = { ( 4 , 0 ) , ( 0 , 4 ) , ( 9 , 10 ) , ( 11 , 8 ) } . Define F : A B by F ( 4 , 4 ) = ( 9 , 10 ) , F ( 7 , 8 ) = ( 11 , 8 ) , F ( 20 , 0 ) = ( 4 , 0 ) , F ( 25 , 30 ) = ( 0 , 4 ) and α : A × A [ 0 , ) by α ( ( x , y ) , ( u , v ) ) = 11 10 . Clearly, F ( A 0 ) B 0 . Now, let ( 4 , 4 ) , ( 7 , 8 ) A and ( 4 , 0 ) , ( 9 , 10 ) B such that
d ( ( 4 , 4 ) , ( 4 , 0 ) ) = d ( A , B ) = 4 , d ( ( 7 , 8 ) , ( 9 , 10 ) ) = d ( A , B ) = 4 . d ( ( 4 , 4 ) , ( 7 , 8 ) ) < d ( ( 4 , 0 ) , ( 9 , 10 ) ) .
Similarly, for all ( x 1 , y 1 ) , ( x 2 , y 2 ) A and ( u 1 , v 1 ) , ( u 2 , v 2 ) B , we have
d ( ( x 1 , y 1 ) , ( u 1 , v 1 ) ) = d ( A , B ) d ( ( x 2 , y 2 ) , ( u 2 , v 2 ) ) = d ( A , B ) d ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) < d ( ( u 1 , v 1 ) , ( u 2 , v 2 ) ) ,
that is, the pair ( A , B ) has weak P-property. Suppose
α ( ( 7 , 8 ) , ( 20 , 0 ) ) 1 , d ( ( 9 , 10 ) , ( 11 , 8 ) ) = d ( A , B ) = 4 , d ( ( 4 , 4 ) , ( 4 , 0 ) ) = d ( A , B ) = 4 ,
then α ( ( 9 , 10 ) , ( 4 , 4 ) ) = 11 10 > 1 . Hence, α ( ( x , y ) , ( u , v ) ) 1 for all x , y , u , v A . Thus, F is α-proximal admissible mapping. Now, we show that F is C ´ iri c ´ type α - ψ - Θ contraction. For ((−4, −4), (20, 0)), define ψ: [ 0 , ) [ 0 , ) by ψ ( t ) = 999 1000 t and Θ: ( 0 , ) ( 1 , ) by Θ ( t ) = t + 1 .
Now,
α ( ( 4 , 4 ) , ( 20 , 0 ) ) Θ [ d ( F ( 4 , 4 ) , F ( 20 , 0 ) ) ] = 88 5
and for
M ( ( 4 , 4 ) , ( 20 , 0 ) ) = m a x { d ( ( 4 , 4 ) , ( 20 , 0 ) ) , d ( ( 4 , 4 ) , F ( 4 , 4 ) ) + d ( ( 20 , 0 ) , F ( 20 , 0 ) ) 2 d ( A , B ) , d ( ( 4 , 4 ) , F ( 20 , 0 ) ) + d ( ( 20 , 0 ) , F ( 4 , 4 ) ) 2 d ( A , B ) } = m a x { 28 , d ( ( 4 , 4 ) , ( 9 , 10 ) ) + d ( ( 20 , 0 ) , ( 4 , 0 ) ) 2 4 , d ( ( 4 , 4 ) , ( 4 , 0 ) ) + d ( ( 20 , 0 ) , ( 9 , 10 ) ) 2 4 } = 28 ,
we have
[ ψ ( Θ ( M ( ( 4 , 1 ) , ( 7 , 4 ) ) ) ) ] k = 999 1000 ( 29 ) k .
Hence, from Equation (12), (13) and for k = 0.83 , we have
88 5 < 999 1000 ( 29 ) k .
Similarly, inequality holds for the remaining cases. Hence, all the assertions of Theorem 2 are satisfied and F has a best proximity point ( 7 , 8 ) .
Example 2.
Let X = R 2 with metric d defined as d ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) = | x 1 y 1 | + | x 2 y 2 | . Suppose A = ( , 1 ] × { 1 } and B = { 0 } × [ 5 4 , + ) . Then, d ( A , B ) = d ( ( 1 , 1 ) , ( 0 , 5 4 ) ) = 5 4 and A 0 = { ( 1 , 1 ) } , B 0 = { ( 0 , 5 4 ) } . Define F : A B by
F ( x , 1 ) = ( 0 , x + | x + 3 | | x + 4 | e x ) i f x ( , 2 ) , ( 0 , x 4 + 1 ) i f x [ 2 , 1 ] ,
and α : A × A [ 0 , ) by
α ( ( x , y ) , ( u , v ) ) = 1 , i f ( x , y ) , ( u , v ) [ 2 , 1 ] × [ 2 , 1 ] , 0 , o t h e r w i s e .
Clearly, F ( A 0 ) B 0 . Now, let ( x 1 , 1 ) , ( x 2 , 1 ) A and ( 0 , u 1 ) , ( 0 , u 2 ) B such that
d ( ( x 1 , 1 ) , ( 0 , u 1 ) ) = d ( A , B ) = 5 4 , d ( ( x 2 , 1 ) , ( 0 , u 2 ) ) = d ( A , B ) = 5 4 .
Necessarily, ( x 1 = u 1 [ 2 , 1 ] ) and ( x 2 = u 2 [ 2 , 1 ] ) . In this case,
d ( ( x 1 , 1 ) , ( x 2 , 1 ) ) = d ( ( 0 , u 1 ) , ( 0 , u 2 ) ) ,
that is, the pair ( A , B ) has weak P-property.
Suppose
α ( ( x 1 , 1 ) , ( x 2 , 1 ) ) 1 , d ( ( u 1 , 1 ) , F ( x 1 , 1 ) ) = d ( A , B ) = 5 4 , d ( ( u 2 , 1 ) , F ( x 2 , 1 ) ) = d ( A , B ) = 5 4 ,
then
( x 1 , 1 ) , ( x 2 , 1 ) [ 2 , 1 ] , d ( ( u 1 , 1 ) , F ( x 1 , 1 ) ) = 5 4 , d ( ( u 2 , 1 ) , F ( x 2 , 1 ) ) = 5 4 .
Thus, ( x 1 , x 2 ) [ 2 , 1 ] × [ 2 , 1 ] . We also have u 1 = x 4 + 1 a n d u 2 = x 4 + 1 , that is ( u 1 = x 4 + 1 , 1 ) , ( u 2 = x 4 + 1 , 1 ) [ 2 , 1 ] × [ 2 , 1 ] . Thus, α ( ( u 1 , 1 ) , ( u 2 , 1 ) ) 1 . That is, F is an α-proximal admissible mapping. Now, we show that F is C ´ iri c ´ type α - ψ - Θ contraction. For this, define ψ: [ 0 , ) [ 0 , ) by ψ ( t ) = 999 1000 t and Θ: ( 0 , ) ( 1 , ) by Θ ( t ) = t + 1 . We will verify the following inequality
α ( ( x , 1 ) , ( y , 1 ) ) Θ [ d ( F ( x , 1 ) , F ( y , 1 ) ) ] [ ψ ( Θ ( M ( x , 1 ) , ( y , 1 ) ) ) ] k ,
where k ( 0 , 1 ) . The left-hand side of inequality (14) gives
α ( ( x , 1 ) , ( y , 1 ) ) Θ [ d ( F ( x , 1 ) , F ( y , 1 ) ) ] = | x y | 4 + 1
and the right side of inequality (14) is
[ ψ ( Θ ( M ( ( x , 1 ) , ( y , 1 ) ) ) ) ] k ,
where
M ( ( x , 1 ) , ( y , 1 ) ) = m a x d ( ( x , 1 ) , ( y , 1 ) ) , d ( ( x , 1 ) , F ( x , 1 ) ) + d ( ( y , 1 ) , F ( y , 1 ) ) 2 d ( A , B ) , d ( ( x , 1 ) , F ( y , 1 ) ) + d ( ( y , 1 ) , F ( x , 1 ) ) 2 d ( A , B ) = m a x d ( ( x , 1 ) , ( y , 1 ) ) , d ( ( x , 1 ) , ( 0 , x 4 + 1 ) ) + d ( ( y , 1 ) , ( 0 , y 4 + 1 ) ) 2 5 4 , d ( ( x , 1 ) , ( 0 , y 4 + 1 ) ) + d ( ( y , 1 ) , ( 0 , x 4 + 1 ) ) 2 5 4 = m a x | x y | , | x | + | 1 + x 4 1 | + | y | + | 1 + y 4 1 | 2 5 4 , | x | + | 1 + y 4 1 | + | y | + | 1 + x 4 1 | 2 5 4 = m a x | x y | , | x | + | 1 + x 4 1 | + | y | + | 1 + y 4 1 | 2 5 4 = m a x | x y | , | x | + | x 4 | + | y | + | y 4 | 2 5 4 .
If m a x { | x y | , | x | + | x 4 | + | y | + | y 4 | 2 5 4 } = | x y | , then inequality (14) becomes
| x y | 4 + 1 [ ψ ( Θ ( | x y | ) ) ] k = [ ψ ( | x y | + 1 ) ] k < ψ ( | x y | + 1 ) = 999 1000 ( | x y | + 1 ) .
Thus, | x y | 4 + 1 < 999 1000 ( | x y | + 1 ) , which is true.
Now, if
m a x | x y | , | x | + | x 4 | + | y | + | y 4 | 2 5 4 = | x | + | x 4 | + | y | + | y 4 | 2 5 4 ,
then
| x y | 4 + 1 ψ ( Θ ( | x | + | x 4 | + | y | + | y 4 | 2 5 4 ) ) k = ψ ( 2 ( | x | + | x 4 | + | y | + | y 4 | ) 4 ) k < ψ 2 ( | x | + | x 4 | + | y | + | y 4 | ) 4 = 999 1000 2 | x | + | x 4 | + | y | + | y 4 | 2
implies
| x y | 4 + 1 < 999 1000 [ 2 | x | + | x 4 | + | y | + | y 4 | 2 ] ,
which is also true. Thus, F is C ´ iri c ´ type α - ψ - Θ contraction. Similar argument holds for the rest of the interval. Hence, all the hypotheses of Theorem 2 are verified. Thus F has best proximity point ( 1 , 1 ) .
Condition of continuity of the mapping in Theorem 2 can be replaced with the following condition to prove the existence of best proximity point of F: H : If { x n } is a sequence in A such that α ( x n , x n + 1 ) 1 for all n and x n x A as n , then there exists a subsequence { x n ( p ) } of { x n } such that α ( x n ( p ) , x ) 1 for all p.
Theorem 3.
Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a C ´ iri c ´ type α - ψ - Θ -contraction satisfying
(i) 
F is α-proximal admissible;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iii) 
there exists x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) such that α ( x 0 , x 1 ) 1 ;
(iv) 
condition H holds.
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Following the proof of Theorem 2, there is a Cauchy sequence { x n } in A such that x n u A . Then, by condition (iv), there exists a subsequence { x n ( p ) } of { x n } such that α ( x n ( p ) , u ) 1 for all p. Since F is C ´ iri c ´ type α - ψ - Θ -contraction, we have by weak P-property and for all p
1 Θ ( d ( x n ( p ) + 1 , u ) ) Θ ( d ( F x n ( p ) , F u ) ) α ( x n ( p ) , u ) Θ ( d ( F x n ( p ) , F u ) ) [ ψ ( Θ ( M ( x n ( p ) , u ) ) ) ] k ,
where
M ( x n ( p ) , u ) = m a x { d ( x n ( p ) , u ) , d ( x n ( p ) , F x n ( p ) ) + d ( u , F u ) 2 d ( A , B ) , d ( x n ( p ) , F u ) + d ( u , F x n ( p ) ) 2 d ( A , B ) } m a x { d ( x n ( p ) , u ) , d ( x n ( p ) , x n ( p ) + 1 ) + d ( x n ( p ) + 1 , F x n ( p ) ) + d ( u , F u ) 2 d ( A , B ) , d ( x n ( p ) , u ) + d ( u , F u ) + d ( u , x n ( p ) + 1 ) + d ( x n ( p ) + 1 , F x n ( p ) ) 2 d ( A , B ) } = m a x { d ( x n ( p ) , u ) , d ( x n ( p ) , x n ( p ) + 1 ) + d ( A , B ) + d ( u , F u ) 2 d ( A , B ) , d ( x n ( p ) , u ) + d ( u , F u ) + d ( u , x n ( p ) + 1 ) + d ( A , B ) 2 d ( A , B ) } .
Letting p in the above inequality, we get that
lim p M ( x n ( p ) , u ) d ( u , F u ) d ( A , B ) 2 .
Furthermore,
d ( u , F u ) d ( u , x n ( p ) + 1 ) + d ( x n ( p ) + 1 , F x n ( p ) ) + d ( F x n ( p ) , F u ) d ( u , x n ( p ) + 1 ) + d ( A , B ) + d ( F x n ( p ) , F u ) ,
which gives
d ( u , F u ) d ( A , B ) d ( u , x n ( p ) + 1 ) d ( F x n ( p ) , F u ) .
Taking p in inequality (18), we get
d ( u , F u ) d ( A , B ) lim p d ( F x n ( p ) , F u ) .
By (15), we have
Θ ( d ( F x n ( p ) , F u ) ) [ ψ ( Θ ( M ( x n ( p ) , u ) ) ) ] k < Θ ( M ( x n ( p ) , u ) ) ,
which implies
d ( F x n ( p ) , F u ) M ( x n ( p ) , u ) .
Taking limit as p in inequality (21), we obtain
d ( u , F u ) d ( A , B ) d ( u , F u ) d ( A , B ) 2 ,
which is a contradiction. Hence, d ( u , F u ) = d ( A , B ) . □
For the uniqueness of best proximity point, we use the following condition:
U : For all x , y B P P ( F ) , α ( x , y ) 1 , where BPP(F) denote the set of best proximity points of F.
Theorem 4.
Adding condition U to the hypotheses of Theorem 2 (resp., Theorem 3), one obtains a unique u in A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Suppose that u and v are two best proximity points of F with u v , that is, d ( u , F u ) = d ( A , B ) = d ( v , F v ) . Then, by U ,
α ( u , v ) 1 .
Since the pair ( A , B ) has the weak P-property, from inequality (3), we have
Θ ( d ( u , v ) ) Θ ( d ( F u , F v ) ) α ( u , v ) Θ ( d ( F u , F v ) ) [ ψ ( Θ ( M ( u , v ) ) ) ] k = [ ψ ( Θ ( d ( u , v ) ) ) ] k < Θ ( d ( u , v ) ) ,
which is a contradiction, so u = v . □
If we take M ( x , y ) = d ( x , y ) in Theorem 2, we have the following corollary:
Corollary 1.
Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a mapping satisfying
(i) 
α ( x , y ) Θ [ d ( F x , F y ) ] [ ψ ( Θ ( d ( x , y ) ) ) ] k ;
(ii) 
F is continuous α-proximal admissible;
(iii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iv) 
there exist x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) such that α ( x 0 , x 1 ) 1 .
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
If α ( x , y ) = 1 for all x , y A in Theorem 2, we have
Corollary 2.
Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a mapping satisfying
(i) 
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iii) 
F is continuous;
(iv) 
there exist x 0 , x 1 A 0 such that d ( x 1 , F x 0 ) = d ( A , B ) ;
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
If M ( x , y ) = d ( x , y ) in Corollary 2, we have the following corollary:
Corollary 3.
Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a mapping satisfying
(i) 
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( d ( x , y ) ) ) ] k ;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iii) 
F is continuous;
(iv) 
there exist x 0 , x 1 A 0 such that d ( x 1 , F x 0 ) = d ( A , B ) ;
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
If we take ψ ( t ) = k t for k ( 0 , 1 ) and Θ ( t ) = e t in Corollary 3, we obtain the following main results of Jleli et al. [32] and Suzuki [33]:
Corollary 4
([32], Theorem 4.2). Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a mapping satisfying
(i) 
d ( F x , F y ) k ( d ( x , y ) ) ;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the P-property;
(iii) 
F is continuous;
(iv) 
there exist x 0 , x 1 A 0 such that d ( x 1 , F x 0 ) = d ( A , B ) ;
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
Corollary 5
([33], Theorem 8). Let A and B be two closed subsets of a complete metric space ( X , d ) with A 0 ϕ and let F: A B be a mapping satisfying
(i) 
d ( F x , F y ) k ( d ( x , y ) ) ;
(ii) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(iii) 
F is continuous;
(iv) 
there exist x 0 , x 1 A 0 such that d ( x 1 , F x 0 ) = d ( A , B ) ;
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .

3. Best Proximity Point Results on Metric Space Endowed with Partial Order

Let ( X , d , ) be a partially ordered metric space, A and B be two nonempty subsets of X. Many authors have proved the existence of best proximity point results in the framework of partially ordered metric spaces (see, for example, [12,17,34,35,36,37,38]). In this section, we obtain some new best proximity point results in partially order metric spaces, as an application of our results.
Definition 4.
A mapping F: A B is said to be proximally order-preserving if and only if it satisfies the condition
x 1 x 2 , d ( u 1 , F x 1 ) = d ( A , B ) , d ( u 2 , F x 2 ) = d ( A , B ) . u 1 u 2
for all x 1 , x 2 , u 1 , u 2 A .
Definition 5.
Let ( X , ) be a partially ordered set. A sequence { x n } X is said to be nondecreasing with respect to ⪯ if x n x n + 1 for all n.
Theorem 5.
Let A and B be two closed subsets of a complete partially ordered metric space ( X , d , ) with A 0 ϕ and let F: A B be a given non-self mapping such that
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 d ( A , B ) , d ( x , F y ) + d ( y , F x ) 2 d ( A , B ) }
for all x , y A with x y , ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(ii) 
F is continuous;
(iii) 
there exists x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) satisfies x 0 x 1 .
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Define α : A × A [ 0 , + ) by
α ( x , y ) = 1 , i f x y , 0 , o t h e r w i s e .
Now, we prove that F is a α-proximal admissible mapping. For this, assume
α ( x , y ) 1 , d ( u , T x ) = d ( A , B ) , d ( v , T y ) = d ( A , B ) ,
so
x y , d ( u , T x ) = d ( A , B ) , d ( v , T y ) = d ( A , B ) .
Now, since F is proximally order-preserving, u v . Thus, α ( u , v ) 1 . Furthermore, by assumption that the comparable elements x 0 and x 1 in A 0 with d ( x 1 , T x 0 ) = d ( A , B ) satisfies α ( x 0 , x 1 ) 1 . Finally, for all comparable x , y A , we have α ( x , y ) 1 and hence by (24), we have
α ( x , y ) Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k .
That is, F is C ´ iri c ´ type α - ψ - Θ -contraction. Hence, all the conditions of Theorem 2 are satisfied. Thus, F has a best proximity point. □
H : If { x n } is a non-decreasing sequence in A such that x n u A as n , then there exists a subsequence { x n ( p ) } of { x n } such that x n ( p ) u .
Theorem 6.
Let A and B be two closed subsets of a partially ordered complete metric space ( X , d , ) with A 0 ϕ and let F: A B be a non self mapping such that
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 d ( A , B ) , d ( x , F y ) + d ( y , F x ) 2 d ( A , B ) }
for all comparable x , y A , where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(ii) 
there exist x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) satisfied x 0 x 1 ;
(iii) 
condition H holds.
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Following the definition of α : A × A [ 0 , ) as in the proof of Theorem 5, one can easily observe that F is an α-proximal admissible mapping and C ´ iri c ´ type α - ψ - Θ contraction. Suppose that α ( x n , x n + 1 ) 1 for all n N such that x n x as n , then x n x n + 1 for all n N . Hence, by property H , we have a subsequence { x n ( p ) } of x n such that x n ( p ) x for all n N and so α ( x n ( p ) , x ) 1 for all n N . Thus, all the conditions of Theorem 3 are satisfied and F has a best proximity point: □
U : For all x , y B P P ( F ) , x y .
Theorem 7.
Adding condition U to the hypotheses if Theorem 5 (resp., Theorem 6), one obtains a unique u in A such that d ( u , F u ) = d ( A , B ) .
Proof. 
Define α : A × A [ 0 , + ) as in Theorem 5, we observe that F is an α-proximal admissible mapping and C ´ iri c ´ type α - ψ - Θ contraction. For uniqueness, suppose that u and v are two best proximity points of F with u v , that is, d ( u , F u ) = d ( A , B ) = d ( v , F v ) . Then, by U , u v , which implies by the definition of α that α ( u , v ) 1 . Thus, by Theorem 4, we have the uniqueness of the best proximity point. □
If we take M ( x , y ) = d ( x , y ) in Theorem 5, then we have following corollary:
Corollary 6.
Let A and B be two closed subsets of a partially ordered complete metric space ( X , d , ) with A 0 ϕ and let F: A B be a given non-self mapping such that
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( d ( x , y ) ) ) ] k
for all comparable x , y A , where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F ( A 0 ) B 0 and the pair ( A , B ) satisfies the weak P-property;
(ii) 
F is continuous;
(iii) 
there exists x 0 , x 1 A 0 with d ( x 1 , F x 0 ) = d ( A , B ) satisfies x 0 x 1 .
Then, there exists u A such that d ( u , F u ) = d ( A , B ) .

4. Fixed Point Results for C ´ iri c ´ Type α - ψ - Θ -Contraction

As an application of results proven in above sections, we deduce new fixed point results for C ´ iri c ´ type α - ψ - Θ -contraction in the frame work of metric and partially ordered metric spaces.
If we take A = B = X in Theorems 2 and 3, we obtain the following fixed point results:
Theorem 8.
Let ( X , d ) be a complete metric space and let F: X X be a self mapping satisfying
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 , d ( x , F y ) + d ( y , F x ) 2 }
for all x , y X , where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F is α-admissible;
(ii) 
F is continuous;
(iii) 
there exists x 0 X such that α ( x 0 , F x 0 ) 1 .
Then, F has a fixed point.
Theorem 9.
Let ( X , d ) be a complete metric space and let F: X X be a self mapping satisfying
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 , d ( x , F y ) + d ( y , F x ) 2 }
for all x , y X , where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F is α-admissible;
(ii) 
there exists x 0 X such that α ( x 0 , F x 0 ) 1 .
(iii) 
condition H is satisfied.
Then, T has a fixed point.
U : For all x , y F i x ( F ) , α ( x , y ) 1 .
Theorem 10.
Adding condition U to the hypotheses of Theorem 8 (res., Theorem 9), we obtain a unique x in X such that F x = x .
By taking α ( x , y ) = 1 and using ψ ( t ) < t , for t > 0 , in Theorem 8, we obtain the following result presented in [4]:
Corollary 7
([4], Corollary 2.1). Let ( X , d ) be a complete metric space and F: X X be a given map. Suppose that there exist Θ Ω and k ( 0 , 1 ) such that
d ( F x , F y ) 0 Θ ( d ( F x , F y ) ) [ Θ ( d ( x , y ) ) ] k
for all x , y X . Then, F has a unique fixed point.
If we take A = B = X in Theorems 5 and 6, we obtain the following fixed point results for complete partially ordered metric spaces:
Theorem 11.
Let ( X , d , ) be a partially ordered complete metric space and let F: X X be a non decreasing self mapping satisfying
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 , d ( x , F y ) + d ( y , F x ) 2 }
for all comparable x , y X where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
F is continuous,
(ii) 
there exists x 0 X such that x 0 F x 0 .
Then, F has a fixed point.
Theorem 12.
Let ( X , d , ) be a partially ordered complete metric space and let F: X X be a non decreasing self mapping satisfying
Θ [ d ( F x , F y ) ] [ ψ ( Θ ( M ( x , y ) ) ) ] k ,
where
M ( x , y ) = m a x { d ( x , y ) , d ( x , F x ) + d ( y , F y ) 2 , d ( x , F y ) + d ( y , F x ) 2 }
for all comparable x , y X , where ψ Ψ , Θ Ω and k ( 0 , 1 ) . Suppose that
(i) 
there exists x 0 X such that x 0 F x 0 .
(ii) 
condition H is satisfied.
Then, F has a fixed point.
U : For all x , y F i x ( F ) , x y .
Theorem 13.
Adding condition U to the hypotheses of Theorem 11 (res., Theorem 12), we obtain a unique x in X such that F x = x .
If we take ψ ( t ) = k t for k ( 0 , 1 ) , Θ ( t ) = e t and M ( x , y ) = d ( x , y ) in Theorem 11, we obtain the following main results of Nieto et al. [39]:
Corollary 8
([39], Theorem 2.1). Let ( X , d , ) be a partially ordered complete metric space and let F: X X be a non decreasing self mapping satisfying
d ( F x , F y ) k d ( x , y )
for all comparable x , y X and k ( 0 , 1 ) . Suppose that
(i) 
F is continuous;
(ii) 
there exists x 0 X such that x 0 F x 0 .
Then, F has a fixed point.
Removing the condition of continuity of the mapping F in Corollary 8 and using an extra condition on X, we have the following corollary:
Corollary 9
([39], Theorem 2.2). Let ( X , d , ) be a partially ordered complete metric space and let F: X X be a non decreasing self mapping satisfying
d ( F x , F y ) k d ( x , y )
for all comparable x , y X and k ( 0 , 1 ) . Suppose that
(i) 
if a nondcreasing sequence x n x in X, then x n x , for all n;
(ii) 
there exists x 0 X such that x 0 F x 0 .
Then, F has a fixed point.

5. Applications to Nonlinear Matrix Equations

In this section, an illustration of Theorem 13 to guarantee the existence of a positive definite solution of nonlinear matrix equations is given. We shall use the following notations: Let M ( n ) be the set of all n × n complex matrices, H ( n ) M ( n ) be the class of all n × n Hermitian matrices, P ( n ) H ( n ) be the set of all n × n Hermitian positive definite matrices, H + ( n ) H ( n ) be the set of all n × n positive semidefinite matrices. Instead of X P ( n ) , we will write X 0 . Furthermore, X 0 means X H + ( n ) . In addition, we will use X Y ( X Y ) instead of X Y 0 ( Y X 0 ) . Furthermore, for every X , Y H ( n ) , there is a greatest lower bound and a least upper bound. The symbol | | . | | denotes the spectral norm of the matrix A, that is, | | A | | = λ + ( A A ) such that λ + ( A A ) is the largest eigenvalue of A A , where A is the conjugate transpose of A. We denote by | | . | | τ the Ky Fan norm defined by | | A | | τ = i = 1 n s i ( A ) = t r ( ( A A ) 1 2 ) , where s i ( A ) , i = 1 , n , are the singular values of A M ( n ) and t r ( A ) for (Hermitian) nonnegative matrices. For a given Q P ( n ) , we denote the modified norm | | . | | τ , Q by | | A | | τ , Q = | | Q 1 2 A Q 1 2 | | τ . The set H ( n ) equipped with the metric induced by | | . | | 1 , Q is a complete metric space for any positive definite matrix Q. Moreover, H ( n ) is a partially ordered set with partial order ⪯ where X Y Y X .
In this section, denote d ( X , Y ) = | | Y X | | τ , Q = t r ( Q 1 2 ( Y X ) Q 1 2 ) . We consider the following class of nonlinear matrix equation:
X = Q ± i = 1 m A i γ ( X ) A i ,
where Q P ( n ) , A i , i = 1 , 2 , m , are arbitrary n × n matrices and a continuous mapping γ : H ( n ) H ( n ) which maps P ( n ) ) into P ( n ) . Assume that γ is an order-preserving ( γ is order preserving if A , B H ( n ) with A B implies that γ ( A ) γ ( B ) ) mapping.
Lemma 1
([40]). Let A 0 and B 0 be n × n matrices. Then, 0 t r ( A B ) | | A | | . t r ( B ) .
Now, we prove the following result:
Theorem 14.
Let F : H ( n ) H ( n ) be an order-preserving continuous mapping which maps P ( n ) into P ( n ) and and Q P ( n ) . Assume that
(a) 
0 i = 1 m A i γ ( Q ) A i Q ;
(b) 
for all X Y and M > 1
d ( γ ( X ) , γ ( Y ) ) d ( F ( X ) , F ( Y ) ) Θ ( t r ( M ( X , Y ) ) ) M 1 2 Θ ( t r ( F ( X ) F ( Y ) ) ) ( Θ ( t r ( M ( X , Y ) ) ) ) 1 2 ,
where
M ( X , Y ) = m a x { d ( X , Y ) , d ( X , F ( X ) ) + d ( Y , F ( Y ) ) 2 , d ( X , F ( Y ) ) + d ( Y , F ( X ) ) 2 }
holds. Then, (26) has a positive definite solution X ^ in P ( n ) .
Proof. 
Define F : H ( n ) H ( n ) by
F ( X ) = Q + i = 1 m A i γ ( X ) A i
and ψ ( t ) = t M , M > 1 . Then, a fixed point of F is a solution of (26). Let X , Y H ( n ) with X Y , then F ( X ) F ( Y ) . Thus, for d ( X , Y ) > 0 , we have
d ( F ( X ) , F ( Y ) ) = | | F ( Y ) F ( X ) | | τ , Q , = t r ( Q 1 2 ( F ( Y ) F ( X ) ) Q 1 2 ) = t r ( i = 1 m Q 1 2 A i ( γ ( Y ) γ ( X ) ) A i Q 1 2 ) = i = 1 m t r ( Q 1 2 A i ( γ ( Y ) γ ( X ) ) Q 1 2 ) = i = 1 m t r ( A i Q A i ( γ ( Y ) γ ( X ) ) ) = i = 1 m t r ( A i Q A i Q 1 2 Q 1 2 ( γ ( Y ) γ ( X ) ) Q 1 2 Q 1 2 ) = i = 1 m t r ( Q 1 2 A i Q A i Q 1 2 Q 1 2 ( γ ( Y ) γ ( X ) ) Q 1 2 ) = t r ( ( i = 1 m Q 1 2 A i Q A i Q 1 2 ) ( Q 1 2 ( γ ( Y ) γ ( X ) ) Q 1 2 ) ) | | i = 1 m Q 1 2 A i Q A i Q 1 2 | | . | | γ ( Y ) γ ( X ) | | τ , Q .
The inequality follows from Lemma 1. From condition (a) and (b), we have that
d ( F ( X ) , F ( Y ) ) d ( F ( X ) , F ( Y ) Θ ( t r ( M ( X , Y ) ) ) ) M 1 2 Θ ( t r ( F ( X ) F ( Y ) ) ) ( Θ ( t r ( M ( X , Y ) ) ) ) 1 2
and Q F ( Q ) . This implies
Θ ( t r ( F ( X ) F ( Y ) ) ) 1 M 1 2 ( Θ ( t r ( M ( X , Y ) ) ) ) 1 2 = 1 M Θ ( t r ( M ( X , Y ) ) ) 1 2 = ( ψ ( Θ ( t r ( M ( X , Y ) ) ) ) ) 1 2 .
Thus, using Theorem 13, we conclude that F has a unique fixed point and hence the matrix Equation (26) has a unique solution X ^ in P ( n ) . □
Example 3.
Consider the matrix equation
X = Q + A 1 X A 1 + A 2 X A 2 ,
where Q , A 1 and A 2 are given by
Q = 9 3 1 1 3 9 3 1 1 3 9 3 1 1 3 9 , A 1 = 0.0325 0.057 0.057 0.0325 0.057 0.0325 0.0325 0.057 0.057 0.0325 0.0325 0.057 0.0325 0.057 0.0325 0.057 ,
A 2 = 0.68 0.0871 0.68 0.0871 0.0871 0.68 0.0871 0.68 0.68 0.0871 0.68 0.0871 0.0871 0.68 0.0871 0.68 .
Define Θ ( t ) = t + 1 and F ( X ) = X 8 . Then, conditions (a) and (b) of Theorem 14 are satisfied for M = 2 . By using the iterative sequence,
X n + 1 = Q + i = 1 2 A i X n A i
with
X 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .
After 18 iterations, we get the unique solution
X ^ = 16.774 3.03286 1.87143 1.00872 3.03286 16.774 3.02615 1.87143 1.87143 3.02615 16.774 3.03286 1.00872 1.87143 3.03286 16.774
of the matrix Equation (29). The residual error is R 18 = | | X ^ i = 1 2 A i X ^ A i | | = 3.50316 × 10 5 and the convergence history is given in the Figure 1:

6. Conclusions

This paper is concerned with the existence and uniqueness of the best proximity point results for C ´ iri c ´ type contractive conditions via auxiliary functions ψ Ψ and Θ Ω in the framework of complete metric spaces and complete partially ordered metric spaces. In addition, as a consequence, some fixed point results as a special case of our best proximity point results of the relevant contractive conditions in such spaces are studied. To illustrate the existence results, some examples are constructed. Finally, as an application of our fixed point result for partially ordered metric space, the existence of positive definite solution for nonlinear matrix equation is investigated and a numerical example is presented. Our results generalized the results of Jleli et al. [4,32], Suzuki [33] and Nieto et al. [39].

Author Contributions

These authors contributed equally to this work.

Funding

This paper was funded by the scientific research foundation of Education Bureau of Hebei Province (Grant No. QN2016191) and Doctoral Fund of Hebei University of Architecture, China (Grant No. B201801).

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments. Second and third author would like to thanks UOS for the project No. UOS/ORIC/2016/54.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Convergence history for (29).
Figure 1. Convergence history for (29).
Symmetry 11 00093 g001

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Ma, Z.; Hussain, A.; Adeel, M.; Hussain, N.; Savas, E. Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations. Symmetry 2019, 11, 93. https://doi.org/10.3390/sym11010093

AMA Style

Ma Z, Hussain A, Adeel M, Hussain N, Savas E. Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations. Symmetry. 2019; 11(1):93. https://doi.org/10.3390/sym11010093

Chicago/Turabian Style

Ma, Zhenhua, Azhar Hussain, Muhammad Adeel, Nawab Hussain, and Ekrem Savas. 2019. "Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations" Symmetry 11, no. 1: 93. https://doi.org/10.3390/sym11010093

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