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Article

Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

1
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
2
Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
3
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4
Department of Mathematics and Computer Science, St. Thomas College, Bhilai 490001, Chhattisgarh, India
*
Authors to whom correspondence should be addressed.
Mathematics 2021, 9(18), 2199; https://doi.org/10.3390/math9182199
Submission received: 31 July 2021 / Revised: 3 September 2021 / Accepted: 6 September 2021 / Published: 8 September 2021

Abstract

:
We discuss a pair of nonlinear matrix equations (NMEs) of the form X = R 1 + i = 1 k A i * F ( X ) A i , X = R 2 + i = 1 k B i * G ( X ) B i , where R 1 , R 2 P ( n ) , A i , B i M ( n ) , i = 1 , , k , and the operators F , G : P ( n ) P ( n ) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

1. Introduction and Preliminaries

Nonlinear matrix equations have long been a popular topic in nonlinear analysis. Control theory, dynamical programming, stochastic filtering, queuing theory, statistics, and a variety of other mathematical and physical fields use these equations regularly. Using partially ordering and the traditional Banach contraction theorem, Ran and Reurings [1,2] investigated the existence of solutions to matrix problems (in short, BCT). Turinici in [3] is a good place to start for early results in this regard. Indeed, Ran and Reurings [1] and Nieto and Rodriguez-Lopez [4,5] reinvestigated and examined this field (see also [6]). Abbas et al. [7] examined the occurrence of fixed points of T-Ćirić-type mappings in partially ordered spaces. For nonlinear contractions under symmetric closure of an arbitrary relation, Samet and Turinici [8] achieved some findings. Ahmadullah et al. [9,10,11,12] and Alam and Imdad [13] have recently utilized a binary relation to show a relation-theoretic counterpart of the BCT that unifies certain well-known relevant order-theoretic fixed point findings. Suzuki [14], on the other hand, discovered a surprising generalization of the BCT that describes metric completeness, i.e., a metric space is complete if and only if every Suzuki-type mapping has a fixed point in it.
Motivated by the preceding work, we present the concept of joint Suzuki-implicit type pair of mappings on arbitrary binary relation and establish certain existence findings under weaker assumptions. We provide some instance cases to show the veracity of our findings. We then apply these findings to nonlinear matrix equations. We also explore the convergence behavior of various initializations with graphical representations using MATLAB.
For the purpose of thoroughness, we will now review some notation, definitions, and outcomes. Throughout this article, the notations Z , N , R , R + have their usual meanings, and N * = N { 0 } .
We call ( Ξ , R ) a relational set if (i) Ξ is a set and (ii) R is a binary relation on Ξ .
In addition, if ( Ξ , d ) is a metric space, we call ( Ξ , d , R ) a relational metric space (RMS, for short).
The following are some standard terms used in the theory of relational sets (see, e.g., [8,13,15,16,17]).
Let ( Ξ , R ) be a relational set, ( Ξ , d , R ) be an RMS, and let T , S be self-mappings on Ξ . Then:
  • ν Ξ is R -related to ϑ Ξ if and only if ( ν , ϑ ) R ;
  • The set ( Ξ , R ) is said to be complete if for all ν , ϑ Ξ , [ ν , ϑ ] R , where [ ν , ϑ ] R means that either ( ν , ϑ ) R or ( ϑ , ν ) R ;
  • A sequence ( ν n ) in Ξ is said to be R -preserving if ( ν n , ν n + 1 ) R , n N * ;
  • ( Ξ , d , R ) is said to be R -complete if every R -preserving Cauchy sequence converges in X;
  • R is said to be T -closed if ( ν , ϑ ) R ( T ν , T ϑ ) R . It is said to be weakly T -closed if ( ν , ϑ ) R [ T ν , T ϑ ] R ;
  • R is said to be d-self-closed if for every R -preserving sequence with ν n ν , there is a subsequence ( ν n k ) of ( ν n ) , such that [ ν n k , ν ] R , for all k N * ;
  • A subset Z of Ξ is called R -directed if for each ν , ϑ Z , there exists μ Ξ such that ( ν , μ ) R and ( ϑ , μ ) R . It is called ( T , R ) -directed if for each ν , ϑ Z , there exists μ Ξ such that ( ν , T μ ) R and ( ϑ , T μ ) R ;
  • T is said to be R -continuous at ν if for every R -preserving sequence ( ν n ) converging to ν , we get T ( ν n ) T ( ν ) as n . Moreover, T is said to be R -continuous if it is R -continuous at every point of Ξ ;
  • For ν , ϑ Ξ , a path of length k (where k is a natural number) in R from ν to ϑ is a finite sequence { μ 0 , μ 1 , μ 2 , , μ k } Ξ satisfying the following conditions:
    (i)
    z 0 = ν and μ k = ϑ ,
    (ii)
    ( μ i , μ i + 1 ) R for each i ( 0 i k 1 ) ,
    then this finite sequence is called a path of length k joining ν to ϑ in R ;
  • If for a pair of ν , ϑ Ξ , there is a finite sequence { μ 0 , μ 1 , μ 2 , , μ k } Ξ satisfying the following conditions:
    (i)
    T μ 0 = ν and T μ k = ϑ ,
    (ii)
    ( T μ i , T μ i + 1 ) R for each i  ( 0 i k 1 ) ,
    then this finite sequence is called a T -path of length k joining ν to ϑ in R .
    Notice that a path of length k involves k + 1 elements of Ξ , although they are not necessarily distinct;
  • R on Ξ is called ( T , S ) -closed if ( T ν , T ϑ ) R whenever ( S ν , S ϑ ) R , for all ν , ϑ X .
  • A subset Z of Ξ is called ( S , R ) -directed if for each ν , ϑ Z , there exists μ Ξ such that ( ν , S μ ) R and ( ϑ , S μ ) R ;
  • T is said to be ( S , R ) -continuous at ν if for every R -preserving sequence ( S ν n ) converging to S ν , we get T ( ν n ) T ( ν ) as n . Moreover, T is said to be ( S , R ) -continuous if it is ( S , R ) -continuous at every point of Ξ ;
  • T and S are said to be R -compatible if lim n d ( S ( T ν n ) , T ( S ν n ) ) = 0 , whenever lim n S ( ν n ) = lim n T ( ν n ) , for any sequence { ν n } Ξ such that the sequences { T ν n } and { S ν n } are R -preserving;
  • For a pair of ν , ϑ Ξ , there is a finite sequence { ω 0 , ω 1 , ω 2 , , ω k } Ξ satisfying the following conditions:
    (i)
    S ω 0 = ν and S ω k = ϑ ,
    (ii)
    ( S ω i , S ω i + 1 ) R for each i ( 0 i k 1 ) ,
    then this finite sequence is called a S -path of length k joining ν to ϑ in R .
Proposition 1
([13]). Let R be a binary relation defined on a non-empty set Ξ. Then, ( ν , ϑ ) R s [ ν , ϑ ] R .
We fix following notations for ( Ξ , R ) a relational space, self-mappings T , S on Ξ and a R -directed subset D of Ξ ,
(i)
F i x ( T ) : = the set of all fixed points of T ;
(ii)
X ( T , R ) : = { ν Ξ : ( ν , T ν ) R } ;
(iii)
P ( ν , ϑ , R ) : = the class of all paths in R from ν to ϑ in R , where ν , ϑ Ξ ;
(iv)
P S ( ν , ϑ , R ) : = the class of all S -paths in R from ν to ϑ in R , where ν , ϑ Ξ ;
(v)
P S ( ν , ϑ , T , R ) : = the class of all S -paths { ω 0 , ω 1 , ω 2 , , ω } joining ν to ϑ in R such that [ S ω i , T ω i ] R for each i { 1 , 2 , 3 , , 1 } ;
(vi)
C F i x ( T , S ) : = the set of all common fixed points of T and S ;
(vii)
C O P ( T , S ) : = the set of all coincidence points of T and S ;
(viii)
X ( T , S , R ) : = { ν Ξ : ( S ν , T ν ) R } ;
(ix)
Δ ( D , S , R ) : = ν , ϑ D { μ Ξ : ( ν , S μ ) R and ( ϑ , S μ ) R } .
To complete the main result, we need:
Lemma 1
([18]). Let g be a self-mapping defined on a set Ξ . Then, there exists a subset Z Ξ with g ( Z ) = g ( Ξ ) and g : Z Ξ is one-one.

2. Joint Suzuki-Implicit Type Results on Relational Metric Spaces

Let Φ be the class of functions φ : R + R + satisfying the following conditions:
(i)
φ is increasing and φ ( 0 ) = 0 ;
(ii)
n = 1 φ n ( ζ ) < , for ζ > 0 ; where φ n is n th -iterate.
It should be noted that φ ( ζ ) < ζ and the family Φ .
Example 1.
Consider ( Ξ , d ) with usual metric, where Ξ = [ 0 , 1 ] . Define the mapping φ ( ζ ) = 2 λ ζ 9 , where 0 < λ < 1 . Then, we have φ n ( ζ ) 2 n λ n ζ 9 n . Therefore, n = 1 φ n ( ζ ) = n = 1 2 n λ n ζ 9 n < and hence Φ .
Let G : = { G : R + 6 R is lower semi - continuous ( l . s . c . ) real valued function } satisfy the following conditions:
( G 1 )
G ( ζ , ξ , ξ , ζ , μ , 0 ) 0 for all ζ , ξ , μ 0 , implies that there exists φ Φ such that ζ φ ( ξ ) ;
( G 2 )
G ( ζ , 0 , 0 , ζ , ζ , 0 ) > 0 , for all ζ > 0 .
Let G : = { G G } and following additional conditions hold:
( G 3 )
G ( ζ , ζ , 0 , 0 , ζ , ζ ) > 0 , for all ζ > 0 .
Obviously, G is more general than G .
The following examples are inspired from [19,20].
Example 2.
Let G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 a max { r 2 , r 3 , r 4 } + b ( r 5 + r 6 ) , 0 a < 1 and b > 0 .
( G 1 )
For ζ , ξ , μ 0 , we have
G ( ζ , ξ , ξ , ζ , μ , 0 ) = ζ a max { ξ , ξ , ζ } + b μ 0 .
If ξ ζ , then ( 1 a ) ζ < 0 , a contradiction as a [ 0 , 1 ) . Therefore, ζ φ ( ξ ) , where φ ( ξ ) = h ξ ( 0 < h < 1 ).
( G 2 )
G ( ζ , 0 , 0 , ζ , ζ , 0 ) = ( 1 a + b ) ζ > 0 for all ζ > 0 .
Example 3.
Let G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 2 a r 2 2 b r 3 2 + r 4 2 r 5 + r 6 + 1 , 0 < a , b < 1 and a + 2 b < 1 .
( G 1 )
For ζ , ξ , μ 0 , we have
G ( ζ , ξ , ξ , ζ , μ , 0 ) = ζ 2 a ξ 2 b ζ 2 + ξ 2 1 + μ 0 ,
which implies that ζ 2 a ξ 2 + b ( ζ 2 + ξ 2 ) . Then, ζ 2 a + b 1 b ξ 2 . Hence, ζ φ ( ξ ) , where φ ( ξ ) = h ξ , h = a + b 1 b < 1 .
( G 2 )
G ( ζ , 0 , 0 , ζ , ζ , 0 ) = ( 1 b ) ζ 2 + ζ 3 1 + ζ > 0 .
Example 4.
Let G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 2 a r 2 2 b r 3 2 + r 4 2 r 5 2 + r 6 2 + 1 , 0 < a , b < 1 and a + 2 b < 1 ,similar to Example 3.
Example 5.
Let G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 a r 2 b r 3 c r 4 r 5 r 5 + r 6 + 1 , 0 < a , b , c < 1 and a + b + c < 1 .
( G 1 )
For ζ , ξ , μ 0 , we have
G ( ζ , ξ , ξ , ζ , μ , 0 ) = ζ a ξ b ξ c ζ μ 1 + μ 0
which implies that ζ ( a + b ) ξ c ζ , that is, ζ φ ( ξ ) , where φ ( ξ ) = h ξ , h = a + b 1 c < 1 .
( G 2 )
G ( ζ , 0 , 0 , ζ , ζ , 0 ) = ζ + ( 1 c ) ζ 2 1 + ζ > 0 for all ζ > 0 .
Example 6.
Let G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 a r 2 b r 3 r 6 r 5 + r 6 + 1 c r 4 , 0 < a , b , c < 1 and a + b + c < 1 .
( G 1 )
For ζ , ξ , μ 0 , we have
G ( ζ , ξ , ξ , ζ , μ , 0 ) = ζ a ξ c ζ 0
which implies that ζ a ξ c ζ , that is, ζ φ ( ξ ) , where φ ( ξ ) = h ξ , h = a 1 c < 1 .
( G 2 )
G ( ζ , 0 , 0 , ζ , ζ , 0 ) = ( 1 c ) ζ > 0 for all ζ > 0 .
Definition 1.
Let ( Ξ , d , R ) be an RMS and T , S : Ξ Ξ be given mappings. A pair of mappings ( T , S ) is said to be a joint Suzuki-implicit type mapping if there exists G G such that for ( S ν , S ϑ ) R
1 2 d ( S ν , T ν ) d ( S ν , S ϑ ) G ( d ( T ν , T ϑ ) , d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ν , T ϑ ) , d ( S ϑ , T ν ) ) 0 .
We denote by Υ the collection of all joint Suzuki-implicit type mappings on ( Ξ , d , R ) .
Theorem 1.
Let ( Ξ , d , R ) be an RMS and Y an R -complete subspace of Ξ. Let T , S : Ξ Ξ . Suppose that the following conditions hold:
( C 1 )
X ( T , S , R ) ;
( C 2 )
T ( Ξ ) Y S ( Ξ ) ;
( C 3 )
R is ( T , S ) -closed;
( C 4 )
( T , S ) Υ ;
( C 5 )
(a)
Y S ( Ξ ) ,
(b)
T is ( S , R ) -continuous or T and S are continuous or R | Y is d-self-closed.
or, alternatively,
( C 5 ' )
(c)
T and S are R -compatible,
(d)
T is R -continuous and either S is R -continuous or ( Ξ , d , R ) is ( S , R ) -regular;
(e)
S is continuous and either T is R -continuous or R is ( S , d ) -self-closed.
Then, C O P ( T , S ) .
Proof. 
By ( C 2 ) , we can define a mapping g on Ξ satisfying S g ν = T ν for all ν Ξ . By ( C 1 ) , ( S ν , S g ν ) R , therefore, from ( C 4 ) , 1 2 d ( S ν , T ν ) = 1 2 d ( S ν , S g ν ) d ( S ν , S g ν ) holds, which implies that
G ( d ( T ν , T g ν ) , d ( S ν , S g ν ) , d ( S ν , T ν ) , d ( S g ν , T g ν ) , d ( S ν , T g ν ) , d ( S g ν , T ν ) ) 0 G ( d ( S g ν , S g g ν ) , d ( S ν , S g ν ) , d ( S ν , S g ν ) , d ( S g ν , S g g ν ) , d ( S ν , S g g ν ) , d ( S g ν , S g ν ) ) 0 G ( d ( S g ν , S g g ν ) , d ( S ν , S g ν ) , d ( S ν , S g ν ) , d ( S g ν , S g g ν ) , d ( S ν , S g g ν ) , 0 ) ) 0 .
It follows from ( G 1 ) that φ Φ , such that
d ( S g ν , S g g ν ) φ ( d ( S ν , S g ν ) )
holds for all ν Ξ . Let ϱ Ξ . Put ϱ 0 = ϱ and ϱ n = g n ϱ for all n N . Then, ϱ n + 1 = g ϱ n and ( C 1 ) implies that
S ϱ n + 1 = T ϱ n for all n N .
Using ( C 3 ), we have that ( T ϱ 0 , T 2 ϱ 0 ) R . Continuing this process inductively, we obtain
( T n ϱ 0 , T n + 1 ϱ 0 ) R
for any n N * . In addition,
( S ϱ n , S ϱ n + 1 ) R
for any n N * so that { S ν n } is an R -preserving sequence. By (2), we have
d ( S ϱ n , S ϱ n + 1 ) = d ( S g ϱ n 1 , S g g ϱ n 1 ) φ ( d ( S ϱ n 1 , S g ϱ n 1 ) ) = φ ( d ( S ϱ n 1 , S ϱ n ) ) φ 2 ( d ( S ϱ n 2 , S ϱ n 1 ) ) = φ n ( d ( S ϱ 0 , S ϱ 1 ) ) .
Using Equation (5) and the triangular inequality, for all n , m N * with m > n ,
d ( S ϱ n , S ϱ m ) d ( S ϱ n , S ϱ n + 1 ) + d ( S ϱ n + 1 , S ϱ n + 2 ) + + d ( S ϱ m 1 , S ϱ m ) k = n m φ k ( d ( S ϱ 0 , S ϱ 1 ) k n φ k ( d ( S ϱ 0 , S ϱ 1 ) 0 as n .
Therefore, { S ϱ n } is an R -preserving Cauchy sequence in Ξ . As { S ϱ n } S ( Ξ ) and { S ϱ n } Y S ( Ξ ) (due to C 5 ( a ) ), therefore { S ϱ n } is an R -preserving Cauchy sequence in Y . Owing to the R -completeness of Ξ , the existence of ω Ξ with lim n S ϱ n = ω is ensured. Assume ( C 5 ) ( a ) . As Y S ( Ξ ) , there exists some ρ Ξ such that
lim n S ϱ n = ρ = S ω .
Assume ( C 5 ) ( b ) . First, we assume that T is ( S , R ) -continuous. By using Equations (4) and (6), we obtain
lim n S ϱ n + 1 = lim n T ϱ n = T ω .
By the uniqueness of the limit, we have T ω = S ω , so that ω C O P ( S , T ) .
Assume ( C 5 ) ( b ) . Suppose that T and S are continuous. From Lemma 1, there exists a subset Z Ξ such that S ( Z ) = S ( Ξ ) and S : Z Ξ is one-one. Now, define h : S ( Z ) S ( Ξ ) by
h ( S η ) = T η , for all S η S ( Z ) where η Z .
Since S is one-one and T ( Ξ ) S ( Z ) , h is well defined. As T and S are continuous, so is h. On using the fact S ( Z ) = S ( Ξ ) and the conditions ( C 2 ) and ( C 5 ) ( a ) , we have T ( Ξ ) S ( Z ) Y and Y S ( Ξ ) which ensures the availability of a sequence { ν n } Z satisfying Equation (3). Take ϱ Z . Using Equations (6) and (7) and the continuity of h, we obtain
T ϱ = h ( S ϱ ) = h ( lim n S ν n ) = lim n h ( S ν n ) = lim n T ν n = S ϱ ,
so that ϱ C O P ( S , T ) .
Next, assume that R | Y is d-self-closed. Since { S ϱ n } is R -preserving in Y and S ϱ n S ϱ , there is a subsequence { S ϱ n k } of { S ϱ n } with [ S ϱ n k , S ϱ ] R | Y R , for all k N * . Notice that, for all k N * , [ S ϱ n k , S ϱ ] R implies that either ( S ϱ n k , S ϱ ) R or ( S ϱ , S ϱ n k ) R .
Now, we assert that for all k N *
1 2 d ( S ϱ n k 1 , S ϱ n k ) d ( S ϱ n k 1 , S ϱ ) or 1 2 d ( S ϱ n k , S ϱ n k + 1 ) d ( S ϱ n k , S ϱ ) .
On the contrary, let there exist N such that
1 2 d ( S ϱ n 1 , S ϱ n ) > d ( S ϱ n 1 , S ϱ ) and 1 2 d ( S ϱ n , S ϱ n + 1 ) > d ( S ϱ n , S ϱ ) .
Now, from Equations (5) and (9), we have
d ( S ϱ n 1 , S ϱ n ) d ( S ϱ n 1 , S ϱ ) + d ( S ϱ n , S ϱ ) < 1 2 [ d ( S ϱ n 1 , S ϱ n ) + d ( S ϱ n , S ϱ n + 1 ) ] < 1 2 [ d ( S ϱ n 1 , S ϱ n ) + d ( S ϱ n 1 , S ϱ n ) ] = d ( S ϱ n 1 , S ϱ n ) ,
a contradiction and, therefore, Equation (8) remains true. This implies that either
1 2 d ( S ϱ 2 n 1 , S ϱ 2 n ) d ( S ϱ 2 n 1 , S ϱ ) or 1 2 d ( S ϱ 2 n , S ϱ 2 n + 1 ) d ( S ϱ 2 n , S ϱ )
holds for n N . Thus, there exists a subsequence { n k } of { n } such that 1 2 d ( S ϱ n k , S ϱ n k + 1 ) d ( S ϱ n k , S ϱ ) for k N . Then, 1 2 d ( S ϱ n k , T ϱ n k ) d ( S ϱ n k , S ϱ ) for k N .
Applying the condition ( C 4 ) to ( S ϱ n k , S ϱ ) R , for all k N * , 1 2 d ( S ϱ n k , T ϱ n k ) d ( S ϱ n k , S ϱ ) for n r and, therefore, for n N with n r
G ( d ( T ϱ n k , T ϱ ) , d ( S ϱ n k , S ϱ ) , d ( S ϱ n k , T ϱ n k ) , d ( S ϱ , T ϱ ) , d ( S ϱ n k , T ϱ ) , d ( S ϱ , T ϱ n k ) ) 0
or
G ( d ( S ϱ n k + 1 , T ϱ ) , d ( S ϱ n k , S ϱ ) , d ( S ϱ n k , S ϱ n k + 1 ) , d ( S ϱ , T ϱ ) , d ( S ϱ n k , T ϱ ) , d ( S ϱ , S ϱ n k + 1 ) ) 0 .
Taking lim inf k and using S ϱ n k d S ϱ , the lower semi-continuity of G and continuity of d, we obtain
G ( d ( S ϱ , T ϱ ) , 0 , 0 , d ( S ϱ , T ϱ ) , d ( S ϱ , T ϱ ) , 0 ) 0 .
It follows from ( G 2 ) that d ( S ϱ , T ϱ ) = 0 , that is, ϱ C O P ( T , S ) .
Alternatively, we suppose that ( C 5 ) holds. Firstly, assume that ( C 2 ) holds.
As { S ϱ n } T ( Ξ ) Y (in view (3.1)), we notice that S ϱ n is an R -preserving Cauchy sequence in Y . Since Y is R -complete, there exists ϱ Y such that
lim n S ϱ n = ϱ and lim n T ϱ n = ϱ .
As { S ϱ n } and { T ϱ n } are R -preserving sequences (due to Equations (3) and (4)), utilizing the condition ( C 3 ) and Equation (10), we obtain
lim n d ( TS ϱ n , ST ϱ n ) = 0 .
Using Equations (4) and (10), and due to the R -continuity of T and S , we have
lim n S ( T ϱ n ) = S ( lim n T ϱ n ) = S ϱ
and
lim n T ( S ϱ n ) = T ( lim n S ϱ n ) = T ϱ .
Using Equations (12) and (13) and the continuity of d, we have T ϱ = S ϱ . Hence, ϱ C O P ( T , S ) . □
Theorem 2.
Let all the conditions of Theorem 1 hold and
( C 6 )
P S ( ν , ϑ , T , R | S ( Ξ ) s ) ,
then, T and S have a unique point of coincidence if G G . In addition, if
( C 7 )
( T , S ) is weakly compatible,
then C F i x ( T , S ) is unique.
Proof. 
Following Theorem 1, C O P ( T , S ) . Consider two arbitrary elements ϱ , σ C ( T , S ) , so that
T ϱ = S ϱ = x ¯ and T σ = S σ = y ¯
Claim 1. x ¯ = y ¯ .
In view of the hypothesis ( C 6 ) , there exists an S -path (say, { ω 0 , ω 1 , ω 2 , , ω } ) of length in R | S ( Ξ ) s from T ϱ to T σ , with
S ω 0 = T ϱ , S ω l = T σ , [ S ω j , S ω i + 1 ] R | S ( Ξ ) R , for each i { 0 , 1 , 2 , , 1 }
and
[ S ω j , T ω j ] R | S ( Ξ ) R , for each i { 1 , 2 , , l 1 } .
Define two constant sequences
ω n 0 = ϱ and ω n = σ .
Then, using Equation (14), for all n N *
T ω n 0 = T ϱ = x ¯ , and T ω n = T σ = y ¯ .
Setting,
ω 0 i = ω i for each i { 0 , 1 , 2 , , } ,
we construct the joint sequence { ω n i } , i.e., T ω n i = S ω n + 1 i corresponding to each ω j . Since [ S ω 0 i , S ω 1 i ] R (in view of Equations (15) and (16)), then, using Equation (5) and ( T , S ) -closedness of R , we obtain
lim n d ( S ω n i , S ω n + 1 i ) = 0 , for each i { 1 , 2 , , 1 } .
Using [ S ω 0 i , S ω 0 i + 1 ] R (in view of Equations (15) and (5)) and the ( T , S ) -closedness of R , we obtain
[ T ω n i , T ω n i + 1 ] R , for each i { 0 , 1 , 2 , , l 1 } and for all n N * ,
or,
[ S ω n i , S ω n i + 1 ] R , for each i { 0 , 1 , 2 , , 1 } and for all n N * .
Define ϖ n i : = d ( S ω n i , S ω n i + 1 ) , for all n N * and for each i { 0 , 1 , 2 , , 1 } . We assert that lim n ϖ n i = 0 . Suppose on the contrary that lim n ϖ n i = ϖ > 0 . Since [ S ω n i , S ω n i + 1 ] R , either ( S ω n i , S ω n i + 1 ) R or, ( S ω n i + 1 , S ω n i ) R . For ( S ω n i , S ω n i + 1 ) R , 1 2 d ( S ω n i , S ω n i + 1 ) = 1 2 d ( S ω n i , T ω n i ) d ( S ω n i , S ω n i + 1 ) , then applying condition (1), we have
G ( d ( T ω n i , T ω n i + 1 ) , d ( S ω n i , S ω n i + 1 ) , d ( S ω n i , T ω n i ) , d ( S ω n i + 1 , T ω n i + 1 ) , d ( S ω n i , T ω n i + 1 ) , d ( S ω n i + 1 , T ω n i ) ) 0 ,
or,
G ( d ( S ω n + 1 i , S ω n + 1 i + 1 ) , d ( S ω n i , S ω n i + 1 ) , d ( S ω n i , S ω n + 1 i ) , d ( S ω n i + 1 , S ω n + 1 i + 1 ) , d ( S ω n i , S ω n + 1 i + 1 ) , d ( S ω n i + 1 , SS ω n + 1 i ) ) 0 .
Taking lim inf as n and using lim n ϖ n i = ϖ along with the lower semi-continuity of G and Equation (18), we obtain
G ( ϖ , ϖ , 0 , 0 , ϖ , ϖ ) 0 ,
a contradiction (in view of ( G 3 ) ) and, hence, (for each i { 0 , 1 , 2 , , 1 } )
lim n ϖ n i = ϖ = 0 .
Similarly, if ( S ω n i + 1 , S ω n i ) R , then as above, we obtain (for each i { 0 , 1 , 2 , , 1 } )
lim n ϖ n i = ϖ = 0 .
Thus,
lim n ϖ n i : = lim n d ( S ω n i , S ω n i + 1 ) = 0 , for each i { 0 , 1 , 2 , , 1 } .
Using Equation (14), lim n ϖ n i = 0 and the triangular inequality, we have
d ( x ¯ , y ¯ ) = d ( S ω n 0 , S ω n l ) i = 0 1 d ( S ω n i , S ω n i + 1 ) = i = 0 1 ϖ n i 0 as n ,
so that d ( x ¯ , y ¯ ) = 0 implying thereby x ¯ = y ¯ . Therefore, S x = S y .
Claim 2. To proof C F i x ( T , S ) . Let ϱ C O P ( T , S ) , i.e., T ϱ = S ϱ . Since T and S commute at their coincidence points, we have
T ( S ϱ ) = S ( T ϱ ) = S ( S ϱ ) .
Set S ϱ = μ . Then, from Equation (19), T μ = S μ . Hence, μ C O P ( T , S ) . From Claim I, we have
μ = S ϱ = S μ = T μ ,
so that μ C F i x ( T , S ) .
Claim 3. To proof C F i x ( T , S ) is a singleton set.   □
On the contrary, assume ω is another element in C F i x ( T , S ) . Then, ω C O P ( T , S ) , and ω = S ω = S μ = μ . Thus, C F i x ( T , S ) is a singleton set. □
Theorem 3.
If one replaces ( C 6 ) of Theorem 2 by one of the following conditions
( C 6 ’)
R | S ( Ξ ) is complete;
( C 6 ”)
T ( Ξ ) is ( S , R | S ( Ξ ) s ) -directed and Δ ( T ( Ξ ) , S , R s ) X ( T , S , R s ) ,
then Theorem 2 is concluded.
Proof. 
Let us assume that the condition ( C 6 ) holds. Take an arbitrary pair of points θ 1 , θ 2 in T ( Ξ ) . Owing to the hypothesis T ( Ξ ) S ( Ξ ) , there exist ν , ϑ Ξ such that θ 1 = S ν , θ 2 = S ϑ . As R | S ( Ξ ) is complete, [ S ν , S ϑ ] R | S ( Ξ ) which shows that { ν , ϑ } is an S -path of length 1 from θ 1 to θ 2 in R | S ( Ξ ) s , so that P S ( θ 1 , θ 2 , T , R | S ( Ξ ) s ) . The rest of the proof follows from Theorem 2.
Now, if the condition ( C 6 ) holds, then for any θ 1 , θ 2 in T ( Ξ ) , there is μ in Ξ such that [ θ 1 , S μ ] R and [ θ 2 , S μ ] R . As T ( Ξ ) S ( Ξ ) , ν , ϑ Ξ so that θ 1 = S ν , θ 2 = S ϑ and, hence, { ν , μ , ϑ } is an S -path of length 2 joining θ 1 to θ 2 in R | S ( Ξ ) s . As μ Δ ( T ( Ξ ) , S , R | S ( Ξ ) s ) X ( T , S , R | S ( Ξ ) s ) , therefore [ S μ , T μ ] R | S ( Ξ ) . Hence, for each θ 1 , θ 2 in T ( Ξ ) , P S ( θ 1 , θ 2 , T , R | S ( Ξ ) s ) . The rest of the proof follows from Theorem 2. □
Example 7.
Let Ξ = [ 0 , ) be endowed with the usual metric d . Define a binary relation R on Ξ by ( ν , ϑ ) R if and only if ν ϑ . Then, ( Ξ , d , R ) is a relational metric space. Next, define mappings T , S : Ξ Ξ by
T ν = ν 2 , i f ν < 1 ; 1 2 , i f ν 1 , and S ν = 2 ν , f o r a l l ν Ξ .
Then, it is easy to verify that X ( T , S , R ) , since 0 X ( T , S , R ) ; T ( Ξ ) Y S ( Ξ ) for Y = [ 0 , 1 ] ; R is ( T , S ) -closed. Let φ be as in Example 2 and define G as follows:
G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 a max { r 2 , r 3 , r 4 , r 6 } + b r 5 ,
where a < 1 and b > 0 . It is easy to verify that G G . Taking a = 9 10 and b = 1 10 . Then, Equation (1) reduces to
1 2 d ( S ν , T ν ) d ( S ν , S ϑ )
and implies
d ( T ν , T ϑ ) + 1 10 d ( S ν , T ϑ ) 9 10 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } .
We show that ( T , S ) satisfy Equation (20). We divide the proof in three parts.
Let ν , ϑ Ξ be such that ( S ν , S ϑ ) R and either 1 2 d ( S ν , T ν ) d ( S ϑ , T ϑ ) or 1 2 d ( S ϑ , T ϑ ) d ( S ϑ , T ϑ ) .
 Case 1. 
ν , ϑ < 1 and ν ϑ . Then,
d ( T ν , T ϑ ) + 1 10 d ( S ν , T ϑ ) = 1 2 | ν ϑ | + 1 10 2 ν ϑ 2 9 10 max 2 | ν ϑ | , 2 ν ν 2 , 2 ϑ ϑ 2 , 2 ϑ ν 2 = 9 10 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } .
 Case 2. 
ν , ϑ 1 . Then,
d ( T ν , T ϑ ) + 1 10 d ( S ν , T ϑ ) = 0 + 1 10 2 ν 1 2 9 10 max | 2 ν 2 ϑ | , 2 ν 1 2 , 2 ϑ 1 2 , 2 ϑ 1 2 = 9 10 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } .
 Case 3. 
ν < 1 and ϑ 1 . Then,
d ( T ν , T ϑ ) + 1 10 d ( S ν , T ϑ ) = 1 2 | ν 1 | + 2 ν 1 2 9 10 max | 2 ν 2 ϑ | , 2 ν ν 2 , 2 ϑ 1 2 , 2 ϑ ν 2 = 9 10 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } .
Therefore, in all the cases, we have
d ( T ν , T ϑ ) 9 10 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } + 1 10 d ( S ν , T ϑ ) 0 ,
which implies that
G ( d ( T ν , T ϑ ) , d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ν , T ϑ ) , d ( S ϑ , T ν ) ) 0 .
Thus, ( T , S ) Υ . In addition, it is easy to check that T and S are R -compatible; T is R -continuous and S is continuous. Therefore, by Theorem 1, it follows that C O P ( T , S ) . Indeed, we see that 0 C O P ( T , S ) . In addition, the pair ( T , S ) commutes at 0 C O P ( T , S ) . Clearly, P S ( ν , ϑ , T , R | S ( Ξ ) s ) , for each ν , ϑ T ( Ξ ) . Clearly 0 C F i x ( T , S ) is unique.
Example 8 (Example 2 [14])] Let Ξ = { 1 , 0 , 1 , 2 } be endowed with the usual metric d. Then, ( Ξ , d ) is a complete metric space. Define T : Ξ Ξ a mapping by
T ν = 0 if ν 2 , 1 if ν = 2 .
Now, since θ ( r ) d ( 1 , T 1 ) 1 = d ( 1 , 2 ) and d ( T 1 , T 2 ) = 1 > r d ( 1 , 2 ) for every r [ 0 , 1 ) , the mapping T does not satisfy Theorem 2 [14].
Now, we define a binary relation R on Ξ by ( ν , ϑ ) R if and only if ν ϑ . Then, ( Ξ , d , R ) is a complete relational metric space. Let Y = Ξ , and G and φ be as in Example 7. Define S : Ξ Ξ be S ν = ν for all ν Ξ . Then, it is easy to verify that X ( T , S , R ) , since 0 X ( T , S , R ) ; T ( Ξ ) Y S ( Ξ ) and R is ( T , S ) -closed.
For all ν , ϑ Ξ , a 2 3 and b 1 2 , we have
1 2 d ( S ν , T ν ) 3 and
d ( T ν , T ϑ ) + 1 2 d ( S ν , T ϑ ) 2 2 3 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } .
Therefore,
d ( T ν , T ϑ ) 2 3 max { d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ϑ , T ν ) } + 1 2 d ( S ν , T ϑ ) 0 G ( d ( T ν , T ϑ ) , d ( S ν , S ϑ ) , d ( S ν , T ν ) , d ( S ϑ , T ϑ ) , d ( S ν , T ϑ ) , d ( S ϑ , T ν ) ) 0 .
Thus, ( T , S ) Υ . In addition, it is easy to check that T and S are R -compatible; T is R -continuous and S is continuous. Therefore, by Theorem 1, it follows that C O P ( T , S ) . Indeed, we see that 0 C O P ( T , S ) . In addition, the pair ( T , S ) commutes at 0 C O P ( T , S ) . Clearly, P S ( ν , ϑ , T , R | S ( Ξ ) s ) , for each ν , ϑ T ( Ξ ) . Clearly, 0 C F i x ( T , S ) is unique.

3. Application to Nonlinear Matrix Equations

Let H ( n ) stand for the set of all n × n Hermitian matrices over C , K ( n ) H ( n ) stand for the set of all n × n positive semi-definite matrices, P ( n ) K ( n ) stand for the set of n × n positive definite matrices, M ( n ) stand for the set of all n × n matrices over C .
For a matrix B H ( n ) , we denote by s ( B ) any of its singular values and by s + ( B ) the sum of all of its singular values, that is, the trace norm B = s + ( B ) . For C , D H ( n ) , C D (resp. C D ) means that the matrix C D is positive semi-definite (resp. positive definite).
The following lemmas are needed in the subsequent discussion.
 Lemma 2 ([1]) 
If A O and B O are n × n matrices, then
0 t r ( A B ) A t r ( B ) .
 Lemma 3 ([1]) 
If A H ( n ) such that A I n , then A < 1 .
Consider the NME
X = R 1 + i = 1 k A i * F ( X ) A i , X = R 2 + i = 1 k B i * G ( X ) B i ,
where R 1 , R 2 P ( n ) , A i , B i M ( n ) , i = 1 , , k , and the operators F , G : P ( n ) P ( n ) are continuous in the trace norm.
 Theorem 4. 
Consider the problem described by Equation (21). Assume that:
(H1)
There exist R 1 , R 2 P ( n ) , such that i = 1 m A i * F ( R 1 ) A i 0 , i = 1 m B i * G ( R 2 ) B i 0 ;
(H2)
i = 1 m A i A i * η I n , i = 1 m B i B i * η I n ;
(H3)
There exists X 0 P ( n ) such that
R 1 + i = 1 m A i * F ( X 0 ) A i R 2 + i = 1 m B i * G ( X 0 ) B i ;
(H4)
If { W n } P ( n ) is a sequence such that W n W with W n W n + 1 for all n N * , then there exists a subsequence { W n k } of { W n } with W n k W (or W n k W ), for all k N * ;
(H5)
For every X , Y P ( n ) such that X Y with
i = 1 m B i * G ( X ) B i i = 1 m B i * G ( Y ) B i implies i = 1 m A i * F ( X ) A i i = 1 m A i * F ( Y ) A i ;
(H6)
For every X , Y P ( n ) such that X Y with i = 1 m A i * F ( X ) A i i = 1 m A i * F ( Y ) A i , if
t r R 2 + i = 1 m B i * G ( X ) B i R 1 i = 1 m B A i * F ( X ) A i < 2 t r i = 1 m B i * G ( X ) B i i = 1 m B i * G ( Y ) B i ,
holds, then, for a [ 0 , 1 ) , b > 0
t r ( F ( X ) F ( Y ) ) a η max t r i = 1 m B i * G ( X ) B i i = 1 m B i * G ( Y ) B i , t r R 2 + i = 1 m B i * G ( X ) B i R 1 i = 1 m A i * F ( X ) A i , t r R 2 + i = 1 m B i * G ( Y ) B i R 1 i = 1 m A i * F ( Y ) A i b η t r R 2 + i = 1 m B i * G ( X ) B i R 1 i = 1 m A i * F ( Y ) A i + t r R 2 + i = 1 m B i * G ( Y ) B i R 1 i = 1 m A i * F ( X ) A i .
Then, the matrix equation (21) has a unique solution.
 Proof. 
Let us consider the set C = { X P ( n ) : X M } , which is a closed subset of P ( n ) .
Define now the operators S , T : C C by
T ( X ) = R 1 + i = 1 m A i * F ( X ) A i , S ( X ) = R 2 + i = 1 m B i * G ( X ) B i ,
for X C . It is clear that finding positive definite solution(s) of the system (21) is equivalent to finding common fixed point(s) of S and T .
Define a binary relation
R = { ( X , Y ) P ( n ) × P ( n ) : X Y } .
Notice that T and S are well defined. From assumption ( H 3 ) , X ( T , S , R ) , and from ( H 5 ) , R is ( T , S ) -closed. Following assumption ( H 4 ) , R is . t r -self-closed.
Now, for ( SX , SY ) R , from assumption ( H 6 ) , we have
1 2 SX T ( X ) t r < SX SY t r .
Then,
T ( X ) T ( Y ) t r = t r ( T ( X ) T ( Y ) ) = t r ( i = 1 m A i * ( F ( X ) F ( Y ) ) A i ) = i = 1 m t r ( A i * ( F ( X ) F ( Y ) ) A i ) = i = 1 m t r ( A i A i * ( F ( X ) F ( Y ) ) ) = t r ( ( i = 1 m A i A i * ) ( F ( X ) F ( Y ) ) ) i = 1 m A i A i * × ( F ( X ) F ( Y ) ) t r i = 1 m A i A i * η × a max SX SY t r , SX T X t r , SY T Y t r b SX T Y t r + SY T X t r a max SX SY t r , SX T X t r , SY T Y t r b SX T Y t r + SY T X t r .
Consider G G given by G ( r 1 , r 2 , r 3 , r 4 , r 5 , r 6 ) = r 1 a max { r 2 , r 3 , r 4 } + b [ r 5 + r 6 ] where a [ 0 , 1 ) , b > 0 . Thus, all the hypotheses of Theorem 1 are satisfied and, therefore, there exists X ^ P ( n ) such that T ( X ^ ) = S ( X ^ ) , and hence, the matrix equation (21) has a solution in P ( n ) . □
Example 9.Consider the following nonlinear equations:
T ( X ) = R 1 + A 1 * × F ( X ) × A 1 + A 2 * × F ( X ) × A 2 S ( X ) = R 2 + B 1 * × G ( X ) × B 1 + B 2 * × G ( X ) × B 2
Consider matrices A 1 , A 2 , B 1 , B 2 , R 1 , R 2 as
A 1 = r a n d ( r ) + d i a g ( 1 + r a n d ( r , 1 ) ) , B 1 = 5 × r a n d ( r ) + 5 × d i a g ( 1 + r a n d ( r , 1 ) ) A 2 = r a n d ( r ) + d i a g ( 1 + r a n d ( r , 1 ) ) , B 2 = 5 × r a n d ( r ) + 5 × d i a g ( 1 + r a n d ( r , 1 ) ) X = A 1 × A 1 * + d i a g ( 1 + r a n d ( r , 1 ) ) , R 1 = R 2 = A 2 × A 2 * + d i a g ( 1 + r a n d ( r , 1 ) )
A 1 = 2.3969 0.0842 0.0117 0.6311 0.8319 1.7174 0.5399 0.8593 0.1343 0.3242 1.6108 0.9742 0.0605 0.3017 0.1465 1.9015 , A 2 = 1.5621 0.6201 0.5170 0.6948 0.4539 2.1496 0.5567 0.4265 0.4279 0.7202 1.5429 0.8363 0.9661 0.3469 0.5621 2.5069
B 1 = 9.6692 0.3232 3.0674 0.9539 2.4590 10.2450 4.0932 1.2929 0.3552 4.1331 13.5283 4.4893 4.4387 1.9727 4.6556 10.6263 , B 2 = 12.0166 3.9212 2.9545 4.1709 2.1514 11.0289 2.2969 0.0782 3.4688 0.5467 6.3417 4.3186 4.7261 1.9497 1.1434 8.2484
R 1 = R 2 = 4.6724 2.6260 2.4937 3.7566 2.6260 7.2266 2.9578 2.5662 2.4937 2.9578 4.8896 3.6269 3.7566 2.5662 3.6269 9.1713 .
The initial matrices are
U 0 = 7.9831 2.6873 0.9830 1.3722 2.6873 6.2889 2.3754 2.2816 0.9830 2.3754 5.1872 2.1945 1.3722 2.2816 2.1945 5.5958 , V 0 = 10 4 × 1.7183 1.5016 0.9708 1.0817 1.5016 1.5514 1.0849 1.1748 0.9708 1.0849 0.8228 0.8681 1.0817 1.1748 0.8681 0.9634 ,
W 0 = 765.8160 588.3909 351.6007 400.9216 588.3909 645.2194 440.0624 469.7429 351.6007 440.0624 370.5008 364.7142 400.9216 469.7429 364.7142 427.7357 .
To test our algorithm, we take r = 4 , η = 1.1356 e + 03 , a = 0.9 , b = 0.05 , tolerance: tol=1e-14 and F ( X ) = X 2 , G ( X ) = X 5 . The numerical results are given in Table 1.
After 8 successive iterations, we obtain the following coincidence point
X ^ = 0.1078 0.0641 0.0509 0.1010 0.0641 0.0412 0.0298 0.0570 0.0509 0.0298 0.0244 0.0484 0.1010 0.0570 0.0484 0.0980 ,
S ( X ^ ) = T ( X ^ ) = 4.6724 2.6260 2.4937 3.7566 2.6260 7.2266 2.9578 2.5662 2.4937 2.9578 4.8896 3.6269 3.7566 2.5662 3.6269 9.1713 . The graphical view of convergence and surface graph of X ^ are shown in Figure 1 and Figure 2, respectively:

Author Contributions

Writing—review and editing, H.K.N., R.P. and R.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by the Foundation for Science and Technology Development of Ton Duc Thang University (FOSTECT), website: http://fostect.tdtu.edu.vn, under Grant FOSTECT.2019.14.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

This research is supported by the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia. We would like to express our gratitude to the editor for his generous assistance. We are also grateful to the educated referee for providing us with helpful comments that enabled us to make some improvements to the text.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1.  Convergence behavior. 
Figure 1.  Convergence behavior. 
Mathematics 09 02199 g001
Figure 2.  Surface plot. 
Figure 2.  Surface plot. 
Mathematics 09 02199 g002
Table 1. Three analyses based on initialization.
Table 1. Three analyses based on initialization.
Initial. Matj F ( X ) G ( X ) Iter No.CPUError
U 0 U 2 U 5 80.0120640
V 0 V 2 V 5 80.0113780
W 0 W 2 W 5 80.0105200
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Nashine, H.K.; Pant, R.; George, R. Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results. Mathematics 2021, 9, 2199. https://doi.org/10.3390/math9182199

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Nashine HK, Pant R, George R. Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results. Mathematics. 2021; 9(18):2199. https://doi.org/10.3390/math9182199

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Nashine, Hemant Kumar, Rajendra Pant, and Reny George. 2021. "Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results" Mathematics 9, no. 18: 2199. https://doi.org/10.3390/math9182199

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Nashine, H. K., Pant, R., & George, R. (2021). Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results. Mathematics, 9(18), 2199. https://doi.org/10.3390/math9182199

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