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Solution of a class of nonlinear matrix equations
Accepted Manuscript Solution of a class of nonlinear matrix equations
Snehasish Bose, Sk Monowar Hossein, Kallol Paul
PII: DOI: Reference:
S0024-3795(17)30294-X http://dx.doi.org/10.1016/j.laa.2017.05.006 LAA 14154
To appear in:
Linear Algebra and its Applications
Received date: Accepted date:
26 July 2016 2 May 2017
Please cite this article in press as: S. Bose et al., Solution of a class of nonlinear matrix equations, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.006
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Solution of a class of nonlinear matrix equations Snehasish Bosea , Sk Monowar Hosseinb , Kallol Paula,∗ a Department
b Department
of Mathematics, Jadavpur University, Jadavpur-32, West Bengal, India of Mathematics, Aliah University, Sector-V, Kolkata-91, West Bengal, India
Abstract In this paper we solve nonlinear matrix equations of the form Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri
i=1
and Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri +
i=1
q
(Bj ∗ Gj (X)Bj )qj ,
j=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1], Q ∈ P(n), the collection of all n × n Hermitian positive definite matrices and Ai , Bj ’s are n × n matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). Examples are given to illustrate that the equations can not be solved by previously known theorems. Keywords: Matrix Equation, Fixed point, Thompson metric. 2010 MSC: 15A24, 47H10, 47H09
1. Introduction and Preliminaries Let H(n) be the set of all n × n Hermitian matrices and P(n) be the set of all n × n Hermitian positive definite matrices. We consider nonlinear matrix equations of the form Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri
(1.1)
i=1
and Xδ = Q +
p i=1
(Ai ∗ Fi (X)Ai )ri +
q
(Bj ∗ Gj (X)Bj )qj ,
(1.2)
j=1
∗ Corresponding
author Email addresses: [email protected] (Snehasish Bose), sami [email protected] (Sk Monowar Hossein), [email protected] (Kallol Paul)
Preprint submitted to Elsevier
May 4, 2017
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1], Q ∈ P(n) and Ai , Bj ’s are n × n matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). Particular instances of such nonlinear matrix equations occur in many problems in control theory, dynamical programming, ladder network, stochastic filtering and statistics. Over the years many authors have studied these type of matrix equations to compute their positive definite solution using different iterative method which allowed them to identify the solution easily. Engwerda [10] and Engwarda et al. [11] first discussed the solution of the type of nonlinear matrix equations X +AT X −1 A = I and X +A∗ X −1 A = Q respectively. Ferrante and Levy [12] showed that the equation X − A∗ X −1 A = Q (δ = 1, p = r1 = 1 and F1 (X) = X −1 in (1.1)) admits a largest solution which corresponds to the unique positive definite solution and a smallest solution which corresponds to the unique negative definite solution if and only if √ A is nonsingular. Ivanov et al. [16] solved nonlinear matrix equation X − A∗ X −1 A = I 1 (δ = 1, p = r1 = 1, Q = I and F1 (X) = X − 2 in (1.1)) using iteration method. They obtained a sufficient condition for existence of it’s positive definite solution. 1 Later El-Syad and Ramadan [8] investigated the solution of X −A∗ X − 2m A = I, while Meini [20] proposed an efficient method for computing the extreme solution of X − A∗ X −1 A = Q (here δ = 1, p = r1 = 1 and F1 (X) = X −1 ). El-Sayed and Ran [9] first used monotonicity methods, and discussed a wider class of nonlinear matrix equations of the form X + A∗ F (X)A = Q. In [19] Liu and Gao, studied the equation X s − AT X −t A = I. Later Hasanov [14] and Zhou et al. [25] generalized their method to solve X − A∗ X −q A = Q and X s − A∗ F (X)A = Q accordingly. mIn [7] and [6] Duan and Liao used different iterative method to solve X − i=1 Ai ∗ X δi Ai = Q, 0 < |δi | < 1 (δ = 1 and m Fi (X) = X δi in (1.1)) and X − i=1 Ai ∗ X r Ai = Q, r ∈ [−1, 0) ∪ (0, 1) (here r Fi (X) = X ). Fixed point theory plays an important role in solving nonlinear matrix equation. Ran and Reurings [22] using Ky Fan norm in H(n) and analogous result of Banach fixed point theorem in partially ordered metric space m solved X = Q+ i=1 Ai ∗ F (X)Ai (δ = 1 and Fi (X) = F (X) in (1.1)). Reurings [23] also solved the matrix equation X = I + A∗ f (X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). In [18] m Lim showed that equation X − i=1 Mi X δi Mi ∗ = Q, 0 < |δi | < 1 has a unique positive definite solution using Thompson metric and Banach fixed point method while Huang et al. [15] and Duan et al. [7] used Hilbert projective metric and fixed point theorems for monotone and mixed monotone operators respectively. Later Berzig and Samet [3] used similar method of Lim to solve a system of nonlinear matrix equations. A mapping F : X × X → X has a mixed monotone property [1] if F (x, y) is monotone non-decreasing in x and is monotone non-increasing in y, i.e, for any x, y ∈ X, x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y1 ) ≥ F (x, y2 ).
2
A pair (x, y) ∈ X×X is called a coupled fixed point [1] of a mapping F : X×X → X if F (x, y) = x and F (y, x) = y. Coupled fixed point theory have been studied by many mathematicians like Guo and Lakshmikantham [13], Bhaskar and Lakshmikantham [4], Berinde [1], Bota et al. [5] and many more. In [4] Bhaskar and Lakshmikantham presented a coupled fixed point theorem using the mixed monotone property in partial ordered metric spaces and applied it to solve periodic boundary value problem. Later in 2011, Berinde [1] generalized the result of Bhaskar-Lakshmikantham. He proved the following result. Theorem 1.1. [1] Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mixed monotone mapping for which there exists a constant k ∈ [0, 1) such that for each u ≥ x, v ≤ y, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k[d(x, u) + d(y, v)].
(1.3)
If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ) and y0 ≥ F (y0 , x0 ),
(1.4)
x0 ≥ F (x0 , y0 ) and y0 ≤ F (y0 , x0 ),
(1.5)
or then there exist x ¯, y¯ ∈ X such that x ¯ = F (¯ x, y¯) and y¯ = F (¯ y, x ¯). Using the technique of coupled fixed point [2] solved a nonlinm theory, Berzig m ear matrix equation of the form X = Q + i=1 Ai ∗ XAi − i=1 Bi ∗ XBi , where Q is a positive definite matrix and Ai , Bi are n × n matrices. In this paper we consider matrix equations (1.1) and (1.2). In section 2 we derive an improved sufficient condition for existence of a unique Hermitian positive definite solution of equation (1.1). An example is given to show that this result does not follow from Lim [18] and Berzig and Samet [3]. In section 3 we establish a generalized coupled fixed point theorem for mixed monotone operator and use it to obtain a sufficient condition for a unique Hermitian positive definite solution of (1.2). We illustrate the scenario with examples. Throughout this paper we write A ≥ B (or A > B) for A, B ∈ H(n) if A − B ∈ K(n) (or A − B ∈ P(n)), where K(n) is set of all n × n Hermitian positive semidefinite matrices. In particular A ≥ 0 (or A > 0) implies A ∈ K(n) (or A ∈ P(n)). We use d( , ) as Thompson metric [18] on P(n) which is defined by d(A, B) = log max α, β , (1.6) where α = inf δ : A ≤ δB and β = inf μ : B ≤ μA . 1 1 For A, B ∈ P(n), α = inf δ : A ≤ δB = λ+ (B − 2 AB − 2 ), maximum eigen 1 1 1 1 value of B − 2 AB − 2 and β = inf μ : B ≤ μA = λ+ (A− 2 BA− 2 ), maximum − 12 − 12 eigenvalue of A BA . Thompson metric has great importance in characterization of P(n). We have listed some its important properties (see [18], [24]) which we will use in this paper. 3
• d(X, Y ) = d(X −1 , Y −1 ) = d(M XM ∗ , M Y M ∗ ), for any X, Y ∈ P(n) and non-singular matrix M . • d(X r , Y r ) ≤ |r|d(X, Y ), for any X, Y ∈ P(n) and r ∈ [−1, 1]. • For all A, B, C, D ∈ P(n), we have d(A + B, C + D) ≤ max d(A, C), d(B, D) . In particular d(A + B, A + C) ≤ d(B, C). • P(n) is complete with respect to Thompson metric d( , ). Also if (Xn ) is a Cauchy sequence in P(n) w.r.t. Thompson metric d( , ) then it is a Cauchy sequence w.r.t. the norm ||.|| too, for which K(n), ||.|| is complete, and it converges to same point in both spaces. Therefore if a non-decreasing (non-increasing) sequence (Xn ) converges to X ∈ P(n) w.r.t. Thompson metric d( , ) then Xn ≤ X (X ≤ Xn ) for all n ∈ N.
2. Solution of X δ = Q +
p
i=1 (Ai
∗
Fi (X)Ai )ri
In this section we consider the equation Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri ,
(2.1)
i=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri ∈ [−1, 1], Q is a positive definite matrix, Ai ’s are non-singular matrices and Fi ’s are monotone mappings from P(n) into P(n). We state the following two theorems which we will use to establish the existence and uniqueness of solution of (2.1). Theorem 2.1. Let (X, , d) be an ordered complete metric space and let f : X → X be a non-decreasing map such that x0 f (x0 ) for some x0 ∈ X. Suppose there exists a pair of control function ψ and φ such that for every two comparable elements x, y ∈ X ψ(d(f (x), f (y))) ≤ ψ(M (x, y, f )) − φ(M (x, y, f )),
(2.2)
where 1 M (x, y, f ) = max d(x, y), d(x, f (x)), d(y, f (y)), (d(x, f (y)) + d(y, f (x))) . 2 (2.3) Then in each of the following cases f has a fixed point: (i) f is continuous, or (ii) if for any non-decreasing sequence (xn ) converges to x ∈ X, then xn x for all n. If further for any x, y ∈ f (X) there exists a z ∈ X such that both x and y are comparable with z, and (f n (z)) is a convergent sequence in (X, d) or z is comparable with f (z) then the fixed point of f is unique. 4
The existence of fixed point of the above theorem due to Radenovi´c and Kadelburg [21]. Theorem 2.2. Let (X, , d) be an ordered complete metric space and let f : X → X be a continuous non-increasing map such that x0 f (x0 ) for some x0 ∈ X. Suppose there exists a constant k ∈ [0, 1) such that for every two comparable elements x, y ∈ X d(f (x), f (y)) ≤ kM (x, y, f ),
(2.4)
where 1 M (x, y, f ) = max d(x, y), d(x, f (x)), d(y, f (y)), (d(x, f (y)) + d(y, f (x))) . 2 Then f has a fixed point. If further for any x, y ∈ f (X) there exists a z ∈ X such that both x and y are comparable with z, and (f n (z)) is a convergent sequence in (X, d) or z is comparable with f (z) then the fixed point of f is unique. p 1 Denoting the map X → (Q + i=1 (Ai ∗ Fi (X)Ai )ri ) δ by f , we prove the following theorems. Theorem 2.3. Let Fi : P(n) → P(n) be an order-preserving mapping for each i = 1, 2, .., p and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri ≤ 1 and for all X, Y ∈ P(n) with X ≤ Y we have |ri |d(Fi (X), Fi (Y )) ≤ k.|δ|M (X, Y, f ), i = 1, 2, .., p,
(2.5)
where k ∈ [0, 1), then in each of the following cases the equation (2.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p, δ.ri ≥ 0. (ii) For each i = 1, 2, .., p, δ.ri ≤ 0 and Fi is continuous. Proof. Let δ.ri ≥ 0 for each i = 1, 2, .., p. Consider X, Y ∈ P(n) such that X ≤ Y . Then f (X) ≤ f (Y ), as each Fi is order-preserving. Also p p 1 1 (Ai ∗ Fi (X)Ai )ri ) δ , (Q + (Ai ∗ Fi (Y )Ai )ri ) δ d f (X), f (Y ) = d (Q + i=1
i=1
p p 1 d Q+ (Ai ∗ Fi (X)Ai )ri , Q + (Ai ∗ Fi (Y )Ai )ri ≤ |δ| i=1 i=1
p p 1 ∗ d (Ai Fi (X)Ai )ri , (Ai ∗ Fi (Y )Ai )ri |δ| i=1 i=1
1 ∗ max d (Ai Fi (X)Ai )ri , (Ai ∗ Fi (Y )Ai )ri ≤ |δ| i
1 max |ri |d Ai ∗ Fi (X)Ai , Ai ∗ Fi (Y )Ai ≤ |δ| i
1 max |ri |d Fi (X), Fi (Y ) = |δ| i ≤ kM (X, Y, f ).
≤
5
Hence using Theorem 2.1 we conclude that
the equation (2.1) has a solution in P(n). Now since f maps P(n) into X ∈ P(n)|X ≥ Q and Q ≤ f (Q), therefore using contraction in (2.5) we conclude that (f n (Q)) is convergent in (P(n), d). Thus the solution of (2.1) is unique. Next let δ.ri ≤ 0 for each i = 1, 2, .., p. Then for all X, Y ∈ P(n) with X ≤ Y we get f (X) ≥ f (Y ). Also continuity of Fi ’s implies continuity of f . Therefore using Theorem 2.2 we conclude the result. Similarly if each Fi is order-reversing then f is order-reversing when δ.ri ≥ 0 or order-preserving when δ.ri ≤ 0. Also continuity of Fi ’s implies continuity of f . Therefore we have the following result. Theorem 2.4. Let Fi : P(n) → P(n) be an order-reversing mapping for each i = 1, 2, .., p and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri ≤ 1 and for all X, Y ∈ P(n) with X ≤ Y we have |ri |d(Fi (X), Fi (Y )) ≤ k.|δ|M (X, Y, f ), i = 1, 2, .., p, where k ∈ [0, 1), then in each of the following cases the equation (2.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p, δ.ri ≥ 0 and Fi is continuous. (ii) For each i = 1, 2, .., p, δ.ri ≤ 0. Now we give an example to show our result is more general. Example 1. Let g : (0, ∞) → (0, ∞) be defined by ⎧ 1 if 0 < t ≤ 1 ⎨ t2 + 2 g(t) = if 1 < t ≤ 2 t2 + 2 ⎩ 1 16 if t > 2. 3t + 3
(2.6)
Then g is continuous in (0, ∞). Now for each X ∈ P(3) there exists a unitary matrix PX and a diagonal matrix DX such that PX ∗ XPX = DX . If λ1 , λ⎛ 2 , λ3 are eigenvalues ⎞ of X with λ1 ≤ λ2 ≤ λ3 then we can take 0 λ1 0 DX = ⎝ 0 λ2 0 ⎠. 0 0 λ3 Define F : P(3) → P(3) by F (X) = F (PX DX PX ∗ ) = PX G(DX )PX ∗ , (2.7) ⎛ ⎞ g(λ1 ) 0 0 0 ⎠. g(λ2 ) where G(DX ) = ⎝ 0 0 0 g(λ3 ) Let SX be another unitary matrix such that PX DX PX ∗ = SX DX SX ∗ . Then 6
PX G(DX )PX ∗ = SX G(DX )SX ∗ also. Therefore F (X) is well defined. Now consider a nonlinear matrix equation ⎛
0.1 where Q = ⎝ 0.01 0.001 ¯ ∈ P(3) Suppose X
0.01 0.1 0.01 is a
X = Q + A∗ F (X)A, (2.8) ⎞ 0.001 0.01 ⎠ and A = I3 . 0.1 ¯ = QX, ¯ as XF ¯ (X) ¯ = solution of (2.8). Then XQ
Figure 1: Convergence history of (2.8)
¯ X. ¯ Therefore there is a basis consisting of vectors which are simultaneously F (X) ¯ Hence there exists a unitary matrix TQ such eigenvectors for both Q and X. ¯ Q = DX¯ . that TQ ∗ QTQ = DQ and TQ ∗ XT
Let X = P(3, PQ , Q) = X ∈ P(3) | PQ ∗ XPQ is diagonal , where PQ ∗ QPQ = DQ . Then X ⊆ P(3) is complete with respect to Thompson metric d( , ). Since for any X ∈ X, F (X) ∈ X thus for any X ∈ X, f (X) = Q + F (X) is also in X. Now for any X ≤ Y in X, F (X) ≤ F (Y ) and d(F (X), F (Y )) ≤ 0.96M (X, Y, f ).
(2.9)
Therefore using Theorem 2.3 we conclude that the equation (2.8) has a unique ¯ = lim f n (Q) ¯ = lim f n (Q) in P(3, PQ , Q). Since the solution X solution X n→∞
n→∞
of (2.8) is the same for any PQ , therefore the solution of the equation (2.8) is 7
unique in whole P(3). By taking X0 = Q and Xk+1 = f (Xk ), at 30th step we find the unique solution ⎛ ⎞ 8.1499999999999 0.0149999999999 0.0015000000000 ¯ = X30 = ⎝ 0.0149999999999 8.1499999999999 0.0149999999999 ⎠ of the X 0.0015000000000 0.0149999999999 8.1499999999999 equation (2.8). ¯ f (X)) ¯ = 2.087219286295273 × 10−14 . The residual error is R30 = d(X, The convergence history is given by Figure 1, where Curve 1 corresponds to d(Xk , f (Xk )). Note that if X = 2I3 and Y = I3 then d(X, Y ) = d(F (X), F (Y )) = 2 kd(X, Y ) for any k ∈ [0, 1). Therefore Lim [18] or Berzig and Samet’s [3] process is not applicable here. 3. Solution of X δ = Q +
p
i=1 (Ai
∗
Fi (X)Ai )ri +
q
j=1 (Bj
∗
Gj (X)Bj )qj
In this section we consider the equation Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri +
i=1
q
(Bj ∗ Gi (X)Bj )qj ,
(3.1)
j=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1] and Q ∈ P(n), Ai , Bj ’s are n × n non-singular matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). To solve this equation we prove a generalized coupled fixed point theorem for mixed monotone function in a metric space (X, d). Theorem 3.1. Let (X, ≤, d) be a complete partially ordered metric space. Let F : X2 → X be a continuous mixed monotone mapping for which there exists a constant k ∈ [0, 1) such that for each x ≤ u, y ≥ v, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k.M [x, y, u, v, F ],
(3.2)
where
M [x, y, u, v, F ] = max d(x, u) + d(y, v), d(x, F (x, y)) + d(y, F (y, x)), 1 (3.3) {d(x, F (u, v))+ 2 d(u, F (x, y))} + {d(y, F (v, u)) + d(v, F (y, x))} .
d(u, F (u, v)) + d(v, F (v, u)),
If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ), y0 ≥ F (y0 , x0 ),
(3.4)
x0 ≥ F (x0 , y0 ), y0 ≤ F (y0 , x0 ),
(3.5)
or
8
then F has a coupled fixed point. If further for any two points x = (x1 , x2 ), y = (y1 , y2 ) ∈ (F (X2 ))2 there exists z = (z1 , z2 ) ∈ X2 such that z1 comparable with both x1 and y1 , and z2 is also comparable with x2 and y2 with reverse order of x1 , y1 and z1 , and (F n (z1 , z2 )), (F n (z2 , z1 )) are convergent in (X, d). Then the coupled fixed point of F is unique. Proof. Define a partial order on X2 as (x, y) (u, v) ⇔ x ≤ u, y ≥ v, for all (x, y), (u, v) ∈ X2 , ¯ : X2 → [0, ∞) as and a functional d ¯ d((x, y), (u, v)) = d(x, u) + d(y, v), ¯ is a complete partially ordered metric for all (x, y), (u, v) ∈ X2 . Then (X2 , , d) space. Now consider a function f : X2 → X2 defined by f (x, y) = (F (x, y), F (y, x)), for all (x, y) ∈ X2 . Since F has mixed monotone property, for all (x, y) (u, v) i.e, x ≤ u, y ≥ v we have (F (x, y), F (y, x)) (F (u, v), F (v, u)). Therefore f is ¯ non-decreasing. Also continuity of F in (X, d) implies continuity of f in (X2 , d). Assume that (3.4) condition holds. Therefore there exists z0 = (x0 , y0 ) ∈ X2 such that (x0 , y0 ) f (x0 , y0 ). Now for all (x, y) (u, v) we have, ¯ f (x, y), f (u, v) = d ¯ F (x, y), F (y, x) , F (u, v), F (v, u) d = d F (x, y), F (u, v) + d F (y, x), F (v, u) ≤ kM [x, y, u, v, F ], where
M [x, y, u, v, F ] = max d(x, u) + d(y, v), d(x, F (x, y)) + d(y, F (y, x)), 1 {d(x, F (u, v))+ 2 d(u, F (x, y))} + {d(y, F (v, u)) + d(v, F (y, x))}
¯ (x, y), (u, v) , d ¯ (x, y), (F (x, y), F (y, x)) , = max d d(u, F (u, v)) + d(v, F (v, u)),
¯ (x, y), (F (u, v), F (v, u)) ¯ (u, v), (F (u, v), F (v, u)) , 1 d d 2 ¯ + d (u, v), (F (x, y), F (y, x))
¯ (x, y), (u, v) , d ¯ (x, y), f (x, y)) , d ¯ (u, v), f (u, v) , = max d 1 ¯ ¯ (u, v), f (x, y)) d (x, y), f (u, v) + d . 2 9
Let x = (x1 , x2 ), y = (y1 , y2 ) ∈ f (X2 ). Then by our assumption there exists z = (z1 , z2 ) ∈ X2 such that x1 ≤ z1 , y1 ≤ z1 and z2 ≤ x2 , z2 ≤ y2 (in other case the proof is similar). Therefore x z and y z. Also since (F n (z1 , z2 )) and ¯ Therefore (F n (z2 , z1 )) are convergent in (X, d ), (f n (z)) is convergent in (X2 , d). using Theorem 2.1 we conclude that f has a unique fixed point (¯ x , y ¯ ) ∈ X2 . Now (¯ x, y¯) = F (¯ x, y¯), F (¯ y, x ¯) implies that x ¯ = F (¯ x, y¯) and y¯ = F (¯ y, x ¯). Thus, F has a unique coupled fixed point. Remark 1. It is easy to see that Theorem (3.1) generalizes the Theorem (1.1). Now we present a condition for which the components of the coupled fixed point will be equal. Theorem 3.2. In addition to the hypothesis of Theorem 3.1, suppose that if there exists a z in X such that both x0 , y0 are comparable with z and (F n (x0 , z)), (F n (z, x0 )) are convergent in (X, d) then F has a unique coupled fixed point (¯ x, y¯) with x ¯ = y¯. Proof. If x0 , y0 are comparable, then we have either x0 ≤ y0 or y0 ≤ x0 . So let x0 ≤ y0 (in the other case proof is similar). Then x1 = F (x0 , y0 ) ≤ F (y0 , x0 ) = y1 , as F has mixed monotone property. Continuing this process we get xn ≤ yn ⇒ (xn , yn ) (yn , xn ) for all n ∈ N. Therefore using (3.2) we have
d(xn , yn ) ≤ k. max d(xn−1 , yn−1 ),d(xn−1 , xn ), d(yn−1 , yn ), (3.6) 1 [d(xn , yn−1 ) + d(xn−1 , yn )] 2 ¯ and limn→∞ yn = y¯, letting n → ∞ in (3.6) we have Now, since limn→∞ xn = x ¯ x, y¯) ≤ k.d(¯ ¯ x, y¯), d(¯ ¯ x, y¯) = 0, that is, x where k ∈ [0, 1), implies d(¯ ¯ = y¯. Now let x0 , y0 are not comparable then by our assumption there exists z ∈ X comparable to both x0 and y0 . Suppose x0 ≤ z and y0 ≤ z (in other case proof is similar). Then (x0 , y0 ) (x0 , z), (x0 , z) (z, x0 ) and (z, x0 ) (y0 , x0 ). Therefore f (x0 , y0 ) f (x0 , z), f (x0 , z) f (z, x0 ) and f (z, x0 ) f (y0 , x0 ), as f is non-decreasing in (X2 , ). Also limn→∞ f n (x0 , y0 ) = (¯ x, y¯), y, x ¯) and limn→∞ F n (x0 , z) = z¯1 , limn→∞ F n (z, x0 ) = z¯2 limn→∞ f n (y0 , x0 ) = (¯ as (F n (z, x0 )) and (F n (x0 , z)) are convergent. Therefore using (3.2) we have ¯ n (x0 , y0 ), f n (x0 , z)) ≤ k.M (f n−1 (x0 , y0 ), f n−1 (x0 , z), f ), d(f ¯ n (x0 , z), f n (z, x0 )) ≤ k.M (f n−1 (x0 , z), f n−1 (z, x0 ), f ), d(f n
n
¯ (z, x0 ), f (y0 , x0 )) ≤ k.M (f d(f
n−1
(z, x0 ), f
n−1
(y0 , x0 ), f ),
where ¯ x, y¯), (z¯1 , z¯2 )), lim M (f n−1 (x0 , y0 ), f n−1 (x0 , z), f ) = d((¯
n→∞
¯ z¯1 , z¯2 ), (z¯2 , z¯1 )), lim M (f n−1 (x0 , z), f n−1 (z, x0 ), f ) = d((
n→∞
¯ z¯2 , z¯1 ), (¯ lim M (f n−1 (z, x0 ), f n−1 (y0 , x0 ), f ) = d(( y, x ¯)).
n→∞
10
(3.7)
Therefore taking limit into both side of (3.7) we get ¯ z¯1 , z¯2 ), (z¯2 , z¯1 )) = d(( ¯ z¯2 , z¯1 ), (¯ ¯ x, y¯), (z¯1 , z¯2 )) = d(( y, x ¯)) = 0 d((¯ Hence x ¯ = y¯. If F : X2 → X is a non-decreasing map (x ≤ u, y ≤ v ⇒ F (x, y) ≤ F (u, v)) or a non-increasing map (x ≤ u, y ≤ v ⇒ F (u, v) ≤ F (x, y)), similar results can be obtained using the method of Theorem 3.1 and Theorem 3.2. Theorem 3.3. Let (X, ≤, d) be a complete partially ordered metric space. Let F : X2 → X be a non-decreasing, or non-increasing mapping for which there exists a constant k ∈ [0, 1) such that for each x ≤ u, y ≤ v, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k.M [x, y, u, v, F ], where M [x, y, u, v, F ] is defined in the equation (3.3). If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ), y0 ≤ F (y0 , x0 ), or x0 ≥ F (x0 , y0 ), y0 ≥ F (y0 , x0 ), then in each of the following cases F has a coupled fixed point. (i) F is continuous. (ii) If for any non-decreasing sequence (xn ) converges to x ∈ X, then xn ≤ x for all n and if for any non-increasing sequence (yn ) converges to y ∈ X, then y ≤ yn for all n. (only in case of F is non-decreasing) If further for any two points x = (x1 , x2 ), y = (y1 , y2 ) ∈ (F (X2 ))2 there exists z = (z1 , z2 ) ∈ X2 such that z1 comparable with both x1 and y1 and z2 is also comparable with x2 and y2 with same order of x1 , y1 and z1 and (F n (z1 , z2 )), (F n (z2 , z1 )) are convergent in (X, d). Then the coupled fixed point of F is unique. Theorem 3.4. In addition to the hypothesis of Theorem 3.3, suppose that if there exists a z in X such that both x0 , y0 are comparable with z and (F n (z, z)) is convergent in (X, d) or z is comparable with F (z, z), then F has a unique coupled fixed point (¯ x, y¯) with x ¯ = y¯. q p 1 Denoting the map (X, Y ) → (Q+ i=1 (Ai ∗ Fi (X)Ai )ri + j=1 (Bj ∗ Gj (Y )Bj )qj ) δ by f , we prove the following theorems. Theorem 3.5. Let Fi : P(n) → P(n) be a continuous order-preserving mapping for each i = 1, 2, .., p while Gj : P(n) → P(n) be a continuous order-reversing mapping for each j = 1, 2, .., q and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X, Y, U, V ∈ P(n) with X ≤ U and Y ≥ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
(3.8) + max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a positive definite solution. 11
(i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≥ 0, and there exists 1 1 an X0 ∈ P(n) such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≤ 0, and there exists 1 1 a Y0 ∈ P(n) such that Y0 ≥ Q δ and Y0 ≥ f (Q δ , Y0 ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (Q δ , αm .In ) for all m ∈ N. Proof. Let δ.ri ≥ 0, δ.qj ≥ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Consider X, Y, U, V ∈ P(n) such that X ≤ U and Y ≥ V . Then f (X, Y ) ≤ f (U, V ) and f (Y, X) ≥ f (V, U ) i.e, f has mixed monotone property. Also d f (X, Y ), f (U, V ) + d f (Y, X), f (V, U ) p q 1 = d (Q + (Ai ∗ Fi (X)Ai )ri + (Bj ∗ Gj (Y )Bj )qj ) δ , i=1
(Q +
j=1 p
(Ai ∗ Fi (U )Ai )ri +
i=1
d (Q +
p i=1
(Q +
q
1
(Bj ∗ Gj (X)Bj )qj ) δ ,
j=1 q
(Ai ∗ Fi (V )Ai )ri +
i=1
≤
1 (Bj ∗ Gj (V )Bj )qj ) δ +
j=1
(Ai ∗ Fi (Y )Ai )ri +
p
q
1
(Bj ∗ Gj (U )Bj )qj ) δ
j=1
p q
1 max d( (Ai ∗ Fi (X)Ai )ri , (Ai ∗ Fi (U )Ai )ri ), |δ| i=1 i=1
d(
p
q (Bj Gj (Y )Bj ) , (Bj ∗ Gj (V )Bj )qj ) ∗
qj
j=1
j=1
p q
1 max d( (Ai ∗ Fi (Y )Ai )ri , (Ai ∗ Fi (V )Ai )ri ), + |δ| i=1 i=1
d(
p
(Bj ∗ Gj (X)Bj )qj ,
j=1
q
(Bj ∗ Gj (U )Bj )qj )
j=1
1 max max |ri |d(Fi (X), Fi (U )) , max |qj |d(Gj (Y ), Gj (V )) ≤ i j |δ|
1 + max max |ri |d(Fi (Y ), Fi (V )) , max |qj |d(Gj (X), Gj (U )) i j |δ| ≤ kM [X, Y, U, V, f ]. 1
Now for any X, Y ∈ P(n), Q δ ≤ f (X, Y ). So if there exists some X0 ∈ P(n) 1 1 such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). Also continuity of Fi ’s and Gi ’s 12
implies continuity of f . Therefore using Theorem 3.1 and Theorem 3.2 we get ¯ = limn→∞ f n (X0 , Q δ1 ) of equation (3.1) in P(n). Now for each a solution X X, Y ∈ f (P(n)2 ), there exists some scalar α such that X, Y ≤ α.In . Therefore by our assumption if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N, then for any (X1 , X2 ) and 1 1 2 (Y1 , Y2 ) in f (P(n)2 ) , (X1 , X2 ), (Y1 , Y2 ) (αm .In , Q δ ) and (αm .In , Q δ ) 1 1 f (αm .In , Q δ ), f (Q δ , αm .In ) for some m ∈ N. Therefore, by using Theorem 3.1 and Theorem 3.2 we conclude that the equation (3.1) has a unique positive definite solution. Next let δ.ri ≤ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Then for all X, Y, U, V ∈ P(n) with X ≤ U and Y ≥ V implies f (X, Y ) ≥ f (U, V ) and f (Y, X) ≤ f (V, U ). Therefore using above method, in this case we also get the same result. Similarly if δ.ri ≥ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q, then using Theorem 3.3 and Theorem 3.4 we have the following result. Theorem 3.6. Let Fi : P(n) → P(n) be an order-preserving mapping for each i = 1, 2, .., p while Gj : P(n) → P(n) be an order-reversing mapping for each j = 1, 2, .., q and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X ≤ U and Y ≤ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
(3.9) + max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≤ 0. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0 and Fi , Gj ’s are continuous. Proof. Let δ.ri ≥ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Consider X, Y, U, V ∈ P(n) such that X ≤ U and Y ≤ V . Then f (X, Y ) ≤ f (U, V ) and f (Y, X) ≤ f (V, U ) i.e, f is non-decreasing. Therefore following the same calculation of Theorem 3.5 we get d f (X, Y ), f (U, V ) + d f (Y, X), f (V, U ) ≤ k.M [X, Y, U, V, f ]. 1
1
Also since Q δ ≤ f (X, Y ) for any X, Y ∈ P(n) therefore by taking x0 = y0 = Q δ and using Theorem 3.3 and Theorem 3.4 we get our desired result. Similarly if for each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0 and Fi , Gj ’s are continuous, then f is non-increasing and continuous. So in this case also using Theorem 3.3 and Theorem 3.4 we get our desired result. 13
Now if Fi , Gj ’s are both order-preserving mappings for each i = 1, 2, .., p and j = 1, .., q, then we have the following result. Theorem 3.7. Let both Fi : P(n) → P(n) and Gj : P(n) → P(n) be continuous order-preserving mappings for each i = 1, 2, .., p and j = 1, 2, .., q also let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X ≤ U and Y ≥ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
+ max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} (3.10) i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a positive definite solution. (i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≤ 0, and there exists 1 1 an X0 ∈ P(n) such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0, and there exists 1 1 a Y0 ∈ P(n) such that Y0 ≥ Q δ and Y0 ≥ f (Q δ , Y0 ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (Q δ , αm .In ) for all m ∈ N. One can also obtain similar results if both Fi ’s and Gj ’s are order-reversing mappings. We now give an example. Example 2. Consider the nonlinear matrix equation 1
1
1
X 2 = Q + (A∗ XA) 2 + B ∗ X 2 B + (C ∗ XC)− 3 , ⎛ ⎞ ⎛ ⎞ 1 0.1 0.01 1 0.5 0.5 1 0.001 ⎠ , A = ⎝ 0.5 1 0.5 ⎠, where Q = ⎝ 0.1 0.01 0.001 1 0.5 0.5 1 ⎛
1 0.2 B = ⎝ 0.2 0 0.2 0.2
(3.11)
⎞ ⎛ ⎞ 0.2 1.4142 −0.4714 −0.5773 0.2 ⎠ , C = ⎝ −0.4714 1.4142 −0.4714 ⎠. −1 −0.5773 −0.4714 1.7321
Then for all X ≤ U and Y ≥ V in P(3) we have
1 1 1 1 max max{ d(X, U ), d(X 2 , U 2 )}, | − |d(Y, V )} 2 3
1 1 1 1 2 + max max{ d(Y, V ), d(Y , V 2 )}, | − |d(X, U )} 2 3 5 < 2. M [X, Y, U, V, f ], 6 14
(3.12)
Figure 2: Convergence history of (3.11)
where f : P(3) × P(3) → P(3) is defined by 1 1 1 1 f (X, Y ) = Q + (A∗ XA) 2 + B ∗ X 2 B + (C ∗ Y C)− 3 2 .
(3.13)
Also for any α ≥ 14 in R we have 1
α.I3 ≥ f (α.I3 , Q 2 ).
(3.14)
Therefore using Theorem 3.7 we conclude that equation (3.11) has a unique positive definite solution. 1 By taking X0 = 14.I3 , Y0 = Q 2 and Xk+1 = f (Xk , Yk ), Yk+1 = f (Yk , Xk ), at 20th step we find the unique solution ⎛ ⎞ 2.1872533415096 0.3828578135719 0.2340628340690 ¯ = X20 = Y20 = ⎝ 0.3828578135719 1.7839885977502 0.1783011164762 ⎠ X 0.2340628340690 0.1783011164762 2.1528423137242 of the equation (3.11). The residual error is R20 = 7.016609515630743 × 10−14 . The convergence history is given by Figure 2, where Curve 1 corresponds to d(Xk , f (Xk , Yk )) and Curve 2 corresponds to d(Yk , f (Yk , Xk )).
15
Acknowledgement First author would like to thank CSIR, Govt. of India for providing financial support in the form of research fellowship. References [1] V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), no. 18, 7347–7355. [2] M. Berzig, Solving a class of matrix equations via the BhaskarLakshmikantham coupled fixed point theorem, Appl. Math. lett. 25 (2012), no. 11, 1638–1643. [3] M. Berzig and B. Samet, Solving systems of nonlinear matrix equations involving Lipshitzian mappings, Fixed Point Theory Appl. 2011 (2011), 89. [4] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), no. 7, 1379–1393. [5] M. Bota, A. Petrusel, G. Petrusel, and B. Samet, Coupled fixed point theorems for single-valued operators in b-metric spaces, Fixed Point Theory Appl. 2015 (2015), no. 1, 231. [6] X. Duan and A. On Hermitian positive definite solution of the matrix Liao, m equation X − i=1 Ai ∗ X r Ai = Q, J. Comput. Appl. Math. 229 (2009), 27–36. [7] X. Liao and B. Tang, On the nonlinear matrix equation X − mDuan,∗ A. δi i=1 Ai X Ai = Q, Linear Algebra Appl. 429 (2008) 110–121. [8] S.M. El-Sayed, M.A. Ramadan, On the√existence of a positive definite so2m X −1 A = I, Int. J. Comp. Math. lution of the matrix equation X − A∗ 76 (2001), 331–338. [9] S.M. El-Sayed and A.C.M. Ran, On an iteration method for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001), 632– 645. [10] J.C. Engwerda, On the existence of a positive definite solution of the matrix equation X + AT X −1 A = I, Linear Algebra Appl. 194 (1993), 91–108. [11] J.C. Engwerda, A.C.M. Ran, and A.L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X + A∗ X −1 A = Q, Linear Algebra Appl. 186 (1993), 255–275.
16
[12] A. Ferrante and B.C. Levy, Hermitian solutions of the equation X = Q + N X −1 N ∗ , Linear Algebra Appl. 247 (1996), 359–373. [13] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), no. 5, 623–632. [14] V.I. Hasanov, Positive definite solutions of the matrix equations X ± A∗ X −q A = Q, Linear Algebra Appl. 404 (2005), 166–182. [15] M. Huang, C. Huang, T. Tsai, Applications of Hilbert’s projective metric to a class of positive nonlinear operators, Linear Algebra Appl. 413 (2006), 202–211. [16] I.G. Ivanov, B.V. √ Minchev, V.I. Hasanov, Positive definite solutions of the equation X − A∗ X −1 A = I, Application of Mathematics in Engineering’24, Proceedings of the XXIV Summer School Sozopol’98, Heron Press, Sofia, 1999, pp. 113–116. [17] A. Liao, G. Yao and X. Duan, Thompson metric method for solving a class of nonlinear matrix equation, Appl Math Comput. 216 (2010), 1831–1836. m [18] Y. Lim, Solving the nonlinear matrix equation X − i=1 Mi X δi Mi ∗ = Q via a contraction principle, Linear Algebra Appl. 430 (2009), 1380–1383. [19] X-G. Liu, H. Gao, On the positive definite solutions of the equation X s ± AT X −t A = I, Linear Algebra Appl. 368 (2003), 83–97. [20] B. Meini, Efficient computation of the extreme solutions of X + A∗ X −1 A = Q and X − A∗ X −1 A = Q, Math. Comput. 71 (2002), 1189–1204. [21] S. Radenovi´c and Z. Kadelburg, Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl. 60 (2010), no. 6, 1776–1783. [22] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Soc. 132 (2004), 1435-1443. [23] M.C.B. Reurings, Contractive maps on normed linear spaces and their applications to nonlinear matrix equations, Linear Algebra Appl. 418 (2006), 292–311. [24] A.C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), no. 3, 438–443. [25] D. Zhou, G. Chen, G. Wu and X. Zhang, On the nonlinear matrix equation X s + A∗ F (X)A = Q with S ≥ 1, J. Comput. Math. 2 (2013), 209–220.
17
Snehasish Bose, Sk Monowar Hossein, Kallol Paul
PII: DOI: Reference:
S0024-3795(17)30294-X http://dx.doi.org/10.1016/j.laa.2017.05.006 LAA 14154
To appear in:
Linear Algebra and its Applications
Received date: Accepted date:
26 July 2016 2 May 2017
Please cite this article in press as: S. Bose et al., Solution of a class of nonlinear matrix equations, Linear Algebra Appl. (2017), http://dx.doi.org/10.1016/j.laa.2017.05.006
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Solution of a class of nonlinear matrix equations Snehasish Bosea , Sk Monowar Hosseinb , Kallol Paula,∗ a Department
b Department
of Mathematics, Jadavpur University, Jadavpur-32, West Bengal, India of Mathematics, Aliah University, Sector-V, Kolkata-91, West Bengal, India
Abstract In this paper we solve nonlinear matrix equations of the form Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri
i=1
and Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri +
i=1
q
(Bj ∗ Gj (X)Bj )qj ,
j=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1], Q ∈ P(n), the collection of all n × n Hermitian positive definite matrices and Ai , Bj ’s are n × n matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). Examples are given to illustrate that the equations can not be solved by previously known theorems. Keywords: Matrix Equation, Fixed point, Thompson metric. 2010 MSC: 15A24, 47H10, 47H09
1. Introduction and Preliminaries Let H(n) be the set of all n × n Hermitian matrices and P(n) be the set of all n × n Hermitian positive definite matrices. We consider nonlinear matrix equations of the form Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri
(1.1)
i=1
and Xδ = Q +
p i=1
(Ai ∗ Fi (X)Ai )ri +
q
(Bj ∗ Gj (X)Bj )qj ,
(1.2)
j=1
∗ Corresponding
author Email addresses: [email protected] (Snehasish Bose), sami [email protected] (Sk Monowar Hossein), [email protected] (Kallol Paul)
Preprint submitted to Elsevier
May 4, 2017
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1], Q ∈ P(n) and Ai , Bj ’s are n × n matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). Particular instances of such nonlinear matrix equations occur in many problems in control theory, dynamical programming, ladder network, stochastic filtering and statistics. Over the years many authors have studied these type of matrix equations to compute their positive definite solution using different iterative method which allowed them to identify the solution easily. Engwerda [10] and Engwarda et al. [11] first discussed the solution of the type of nonlinear matrix equations X +AT X −1 A = I and X +A∗ X −1 A = Q respectively. Ferrante and Levy [12] showed that the equation X − A∗ X −1 A = Q (δ = 1, p = r1 = 1 and F1 (X) = X −1 in (1.1)) admits a largest solution which corresponds to the unique positive definite solution and a smallest solution which corresponds to the unique negative definite solution if and only if √ A is nonsingular. Ivanov et al. [16] solved nonlinear matrix equation X − A∗ X −1 A = I 1 (δ = 1, p = r1 = 1, Q = I and F1 (X) = X − 2 in (1.1)) using iteration method. They obtained a sufficient condition for existence of it’s positive definite solution. 1 Later El-Syad and Ramadan [8] investigated the solution of X −A∗ X − 2m A = I, while Meini [20] proposed an efficient method for computing the extreme solution of X − A∗ X −1 A = Q (here δ = 1, p = r1 = 1 and F1 (X) = X −1 ). El-Sayed and Ran [9] first used monotonicity methods, and discussed a wider class of nonlinear matrix equations of the form X + A∗ F (X)A = Q. In [19] Liu and Gao, studied the equation X s − AT X −t A = I. Later Hasanov [14] and Zhou et al. [25] generalized their method to solve X − A∗ X −q A = Q and X s − A∗ F (X)A = Q accordingly. mIn [7] and [6] Duan and Liao used different iterative method to solve X − i=1 Ai ∗ X δi Ai = Q, 0 < |δi | < 1 (δ = 1 and m Fi (X) = X δi in (1.1)) and X − i=1 Ai ∗ X r Ai = Q, r ∈ [−1, 0) ∪ (0, 1) (here r Fi (X) = X ). Fixed point theory plays an important role in solving nonlinear matrix equation. Ran and Reurings [22] using Ky Fan norm in H(n) and analogous result of Banach fixed point theorem in partially ordered metric space m solved X = Q+ i=1 Ai ∗ F (X)Ai (δ = 1 and Fi (X) = F (X) in (1.1)). Reurings [23] also solved the matrix equation X = I + A∗ f (X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). In [18] m Lim showed that equation X − i=1 Mi X δi Mi ∗ = Q, 0 < |δi | < 1 has a unique positive definite solution using Thompson metric and Banach fixed point method while Huang et al. [15] and Duan et al. [7] used Hilbert projective metric and fixed point theorems for monotone and mixed monotone operators respectively. Later Berzig and Samet [3] used similar method of Lim to solve a system of nonlinear matrix equations. A mapping F : X × X → X has a mixed monotone property [1] if F (x, y) is monotone non-decreasing in x and is monotone non-increasing in y, i.e, for any x, y ∈ X, x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y) ≤ F (x2 , y) and y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y1 ) ≥ F (x, y2 ).
2
A pair (x, y) ∈ X×X is called a coupled fixed point [1] of a mapping F : X×X → X if F (x, y) = x and F (y, x) = y. Coupled fixed point theory have been studied by many mathematicians like Guo and Lakshmikantham [13], Bhaskar and Lakshmikantham [4], Berinde [1], Bota et al. [5] and many more. In [4] Bhaskar and Lakshmikantham presented a coupled fixed point theorem using the mixed monotone property in partial ordered metric spaces and applied it to solve periodic boundary value problem. Later in 2011, Berinde [1] generalized the result of Bhaskar-Lakshmikantham. He proved the following result. Theorem 1.1. [1] Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mixed monotone mapping for which there exists a constant k ∈ [0, 1) such that for each u ≥ x, v ≤ y, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k[d(x, u) + d(y, v)].
(1.3)
If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ) and y0 ≥ F (y0 , x0 ),
(1.4)
x0 ≥ F (x0 , y0 ) and y0 ≤ F (y0 , x0 ),
(1.5)
or then there exist x ¯, y¯ ∈ X such that x ¯ = F (¯ x, y¯) and y¯ = F (¯ y, x ¯). Using the technique of coupled fixed point [2] solved a nonlinm theory, Berzig m ear matrix equation of the form X = Q + i=1 Ai ∗ XAi − i=1 Bi ∗ XBi , where Q is a positive definite matrix and Ai , Bi are n × n matrices. In this paper we consider matrix equations (1.1) and (1.2). In section 2 we derive an improved sufficient condition for existence of a unique Hermitian positive definite solution of equation (1.1). An example is given to show that this result does not follow from Lim [18] and Berzig and Samet [3]. In section 3 we establish a generalized coupled fixed point theorem for mixed monotone operator and use it to obtain a sufficient condition for a unique Hermitian positive definite solution of (1.2). We illustrate the scenario with examples. Throughout this paper we write A ≥ B (or A > B) for A, B ∈ H(n) if A − B ∈ K(n) (or A − B ∈ P(n)), where K(n) is set of all n × n Hermitian positive semidefinite matrices. In particular A ≥ 0 (or A > 0) implies A ∈ K(n) (or A ∈ P(n)). We use d( , ) as Thompson metric [18] on P(n) which is defined by d(A, B) = log max α, β , (1.6) where α = inf δ : A ≤ δB and β = inf μ : B ≤ μA . 1 1 For A, B ∈ P(n), α = inf δ : A ≤ δB = λ+ (B − 2 AB − 2 ), maximum eigen 1 1 1 1 value of B − 2 AB − 2 and β = inf μ : B ≤ μA = λ+ (A− 2 BA− 2 ), maximum − 12 − 12 eigenvalue of A BA . Thompson metric has great importance in characterization of P(n). We have listed some its important properties (see [18], [24]) which we will use in this paper. 3
• d(X, Y ) = d(X −1 , Y −1 ) = d(M XM ∗ , M Y M ∗ ), for any X, Y ∈ P(n) and non-singular matrix M . • d(X r , Y r ) ≤ |r|d(X, Y ), for any X, Y ∈ P(n) and r ∈ [−1, 1]. • For all A, B, C, D ∈ P(n), we have d(A + B, C + D) ≤ max d(A, C), d(B, D) . In particular d(A + B, A + C) ≤ d(B, C). • P(n) is complete with respect to Thompson metric d( , ). Also if (Xn ) is a Cauchy sequence in P(n) w.r.t. Thompson metric d( , ) then it is a Cauchy sequence w.r.t. the norm ||.|| too, for which K(n), ||.|| is complete, and it converges to same point in both spaces. Therefore if a non-decreasing (non-increasing) sequence (Xn ) converges to X ∈ P(n) w.r.t. Thompson metric d( , ) then Xn ≤ X (X ≤ Xn ) for all n ∈ N.
2. Solution of X δ = Q +
p
i=1 (Ai
∗
Fi (X)Ai )ri
In this section we consider the equation Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri ,
(2.1)
i=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri ∈ [−1, 1], Q is a positive definite matrix, Ai ’s are non-singular matrices and Fi ’s are monotone mappings from P(n) into P(n). We state the following two theorems which we will use to establish the existence and uniqueness of solution of (2.1). Theorem 2.1. Let (X, , d) be an ordered complete metric space and let f : X → X be a non-decreasing map such that x0 f (x0 ) for some x0 ∈ X. Suppose there exists a pair of control function ψ and φ such that for every two comparable elements x, y ∈ X ψ(d(f (x), f (y))) ≤ ψ(M (x, y, f )) − φ(M (x, y, f )),
(2.2)
where 1 M (x, y, f ) = max d(x, y), d(x, f (x)), d(y, f (y)), (d(x, f (y)) + d(y, f (x))) . 2 (2.3) Then in each of the following cases f has a fixed point: (i) f is continuous, or (ii) if for any non-decreasing sequence (xn ) converges to x ∈ X, then xn x for all n. If further for any x, y ∈ f (X) there exists a z ∈ X such that both x and y are comparable with z, and (f n (z)) is a convergent sequence in (X, d) or z is comparable with f (z) then the fixed point of f is unique. 4
The existence of fixed point of the above theorem due to Radenovi´c and Kadelburg [21]. Theorem 2.2. Let (X, , d) be an ordered complete metric space and let f : X → X be a continuous non-increasing map such that x0 f (x0 ) for some x0 ∈ X. Suppose there exists a constant k ∈ [0, 1) such that for every two comparable elements x, y ∈ X d(f (x), f (y)) ≤ kM (x, y, f ),
(2.4)
where 1 M (x, y, f ) = max d(x, y), d(x, f (x)), d(y, f (y)), (d(x, f (y)) + d(y, f (x))) . 2 Then f has a fixed point. If further for any x, y ∈ f (X) there exists a z ∈ X such that both x and y are comparable with z, and (f n (z)) is a convergent sequence in (X, d) or z is comparable with f (z) then the fixed point of f is unique. p 1 Denoting the map X → (Q + i=1 (Ai ∗ Fi (X)Ai )ri ) δ by f , we prove the following theorems. Theorem 2.3. Let Fi : P(n) → P(n) be an order-preserving mapping for each i = 1, 2, .., p and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri ≤ 1 and for all X, Y ∈ P(n) with X ≤ Y we have |ri |d(Fi (X), Fi (Y )) ≤ k.|δ|M (X, Y, f ), i = 1, 2, .., p,
(2.5)
where k ∈ [0, 1), then in each of the following cases the equation (2.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p, δ.ri ≥ 0. (ii) For each i = 1, 2, .., p, δ.ri ≤ 0 and Fi is continuous. Proof. Let δ.ri ≥ 0 for each i = 1, 2, .., p. Consider X, Y ∈ P(n) such that X ≤ Y . Then f (X) ≤ f (Y ), as each Fi is order-preserving. Also p p 1 1 (Ai ∗ Fi (X)Ai )ri ) δ , (Q + (Ai ∗ Fi (Y )Ai )ri ) δ d f (X), f (Y ) = d (Q + i=1
i=1
p p 1 d Q+ (Ai ∗ Fi (X)Ai )ri , Q + (Ai ∗ Fi (Y )Ai )ri ≤ |δ| i=1 i=1
p p 1 ∗ d (Ai Fi (X)Ai )ri , (Ai ∗ Fi (Y )Ai )ri |δ| i=1 i=1
1 ∗ max d (Ai Fi (X)Ai )ri , (Ai ∗ Fi (Y )Ai )ri ≤ |δ| i
1 max |ri |d Ai ∗ Fi (X)Ai , Ai ∗ Fi (Y )Ai ≤ |δ| i
1 max |ri |d Fi (X), Fi (Y ) = |δ| i ≤ kM (X, Y, f ).
≤
5
Hence using Theorem 2.1 we conclude that
the equation (2.1) has a solution in P(n). Now since f maps P(n) into X ∈ P(n)|X ≥ Q and Q ≤ f (Q), therefore using contraction in (2.5) we conclude that (f n (Q)) is convergent in (P(n), d). Thus the solution of (2.1) is unique. Next let δ.ri ≤ 0 for each i = 1, 2, .., p. Then for all X, Y ∈ P(n) with X ≤ Y we get f (X) ≥ f (Y ). Also continuity of Fi ’s implies continuity of f . Therefore using Theorem 2.2 we conclude the result. Similarly if each Fi is order-reversing then f is order-reversing when δ.ri ≥ 0 or order-preserving when δ.ri ≤ 0. Also continuity of Fi ’s implies continuity of f . Therefore we have the following result. Theorem 2.4. Let Fi : P(n) → P(n) be an order-reversing mapping for each i = 1, 2, .., p and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri ≤ 1 and for all X, Y ∈ P(n) with X ≤ Y we have |ri |d(Fi (X), Fi (Y )) ≤ k.|δ|M (X, Y, f ), i = 1, 2, .., p, where k ∈ [0, 1), then in each of the following cases the equation (2.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p, δ.ri ≥ 0 and Fi is continuous. (ii) For each i = 1, 2, .., p, δ.ri ≤ 0. Now we give an example to show our result is more general. Example 1. Let g : (0, ∞) → (0, ∞) be defined by ⎧ 1 if 0 < t ≤ 1 ⎨ t2 + 2 g(t) = if 1 < t ≤ 2 t2 + 2 ⎩ 1 16 if t > 2. 3t + 3
(2.6)
Then g is continuous in (0, ∞). Now for each X ∈ P(3) there exists a unitary matrix PX and a diagonal matrix DX such that PX ∗ XPX = DX . If λ1 , λ⎛ 2 , λ3 are eigenvalues ⎞ of X with λ1 ≤ λ2 ≤ λ3 then we can take 0 λ1 0 DX = ⎝ 0 λ2 0 ⎠. 0 0 λ3 Define F : P(3) → P(3) by F (X) = F (PX DX PX ∗ ) = PX G(DX )PX ∗ , (2.7) ⎛ ⎞ g(λ1 ) 0 0 0 ⎠. g(λ2 ) where G(DX ) = ⎝ 0 0 0 g(λ3 ) Let SX be another unitary matrix such that PX DX PX ∗ = SX DX SX ∗ . Then 6
PX G(DX )PX ∗ = SX G(DX )SX ∗ also. Therefore F (X) is well defined. Now consider a nonlinear matrix equation ⎛
0.1 where Q = ⎝ 0.01 0.001 ¯ ∈ P(3) Suppose X
0.01 0.1 0.01 is a
X = Q + A∗ F (X)A, (2.8) ⎞ 0.001 0.01 ⎠ and A = I3 . 0.1 ¯ = QX, ¯ as XF ¯ (X) ¯ = solution of (2.8). Then XQ
Figure 1: Convergence history of (2.8)
¯ X. ¯ Therefore there is a basis consisting of vectors which are simultaneously F (X) ¯ Hence there exists a unitary matrix TQ such eigenvectors for both Q and X. ¯ Q = DX¯ . that TQ ∗ QTQ = DQ and TQ ∗ XT
Let X = P(3, PQ , Q) = X ∈ P(3) | PQ ∗ XPQ is diagonal , where PQ ∗ QPQ = DQ . Then X ⊆ P(3) is complete with respect to Thompson metric d( , ). Since for any X ∈ X, F (X) ∈ X thus for any X ∈ X, f (X) = Q + F (X) is also in X. Now for any X ≤ Y in X, F (X) ≤ F (Y ) and d(F (X), F (Y )) ≤ 0.96M (X, Y, f ).
(2.9)
Therefore using Theorem 2.3 we conclude that the equation (2.8) has a unique ¯ = lim f n (Q) ¯ = lim f n (Q) in P(3, PQ , Q). Since the solution X solution X n→∞
n→∞
of (2.8) is the same for any PQ , therefore the solution of the equation (2.8) is 7
unique in whole P(3). By taking X0 = Q and Xk+1 = f (Xk ), at 30th step we find the unique solution ⎛ ⎞ 8.1499999999999 0.0149999999999 0.0015000000000 ¯ = X30 = ⎝ 0.0149999999999 8.1499999999999 0.0149999999999 ⎠ of the X 0.0015000000000 0.0149999999999 8.1499999999999 equation (2.8). ¯ f (X)) ¯ = 2.087219286295273 × 10−14 . The residual error is R30 = d(X, The convergence history is given by Figure 1, where Curve 1 corresponds to d(Xk , f (Xk )). Note that if X = 2I3 and Y = I3 then d(X, Y ) = d(F (X), F (Y )) = 2 kd(X, Y ) for any k ∈ [0, 1). Therefore Lim [18] or Berzig and Samet’s [3] process is not applicable here. 3. Solution of X δ = Q +
p
i=1 (Ai
∗
Fi (X)Ai )ri +
q
j=1 (Bj
∗
Gj (X)Bj )qj
In this section we consider the equation Xδ = Q +
p
(Ai ∗ Fi (X)Ai )ri +
i=1
q
(Bj ∗ Gi (X)Bj )qj ,
(3.1)
j=1
where δ ∈ (−∞, −1] ∪ [1, ∞), ri , qj ∈ [−1, 1] and Q ∈ P(n), Ai , Bj ’s are n × n non-singular matrices, also Fi , Gj ’s are monotone mappings from P(n) into P(n). To solve this equation we prove a generalized coupled fixed point theorem for mixed monotone function in a metric space (X, d). Theorem 3.1. Let (X, ≤, d) be a complete partially ordered metric space. Let F : X2 → X be a continuous mixed monotone mapping for which there exists a constant k ∈ [0, 1) such that for each x ≤ u, y ≥ v, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k.M [x, y, u, v, F ],
(3.2)
where
M [x, y, u, v, F ] = max d(x, u) + d(y, v), d(x, F (x, y)) + d(y, F (y, x)), 1 (3.3) {d(x, F (u, v))+ 2 d(u, F (x, y))} + {d(y, F (v, u)) + d(v, F (y, x))} .
d(u, F (u, v)) + d(v, F (v, u)),
If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ), y0 ≥ F (y0 , x0 ),
(3.4)
x0 ≥ F (x0 , y0 ), y0 ≤ F (y0 , x0 ),
(3.5)
or
8
then F has a coupled fixed point. If further for any two points x = (x1 , x2 ), y = (y1 , y2 ) ∈ (F (X2 ))2 there exists z = (z1 , z2 ) ∈ X2 such that z1 comparable with both x1 and y1 , and z2 is also comparable with x2 and y2 with reverse order of x1 , y1 and z1 , and (F n (z1 , z2 )), (F n (z2 , z1 )) are convergent in (X, d). Then the coupled fixed point of F is unique. Proof. Define a partial order on X2 as (x, y) (u, v) ⇔ x ≤ u, y ≥ v, for all (x, y), (u, v) ∈ X2 , ¯ : X2 → [0, ∞) as and a functional d ¯ d((x, y), (u, v)) = d(x, u) + d(y, v), ¯ is a complete partially ordered metric for all (x, y), (u, v) ∈ X2 . Then (X2 , , d) space. Now consider a function f : X2 → X2 defined by f (x, y) = (F (x, y), F (y, x)), for all (x, y) ∈ X2 . Since F has mixed monotone property, for all (x, y) (u, v) i.e, x ≤ u, y ≥ v we have (F (x, y), F (y, x)) (F (u, v), F (v, u)). Therefore f is ¯ non-decreasing. Also continuity of F in (X, d) implies continuity of f in (X2 , d). Assume that (3.4) condition holds. Therefore there exists z0 = (x0 , y0 ) ∈ X2 such that (x0 , y0 ) f (x0 , y0 ). Now for all (x, y) (u, v) we have, ¯ f (x, y), f (u, v) = d ¯ F (x, y), F (y, x) , F (u, v), F (v, u) d = d F (x, y), F (u, v) + d F (y, x), F (v, u) ≤ kM [x, y, u, v, F ], where
M [x, y, u, v, F ] = max d(x, u) + d(y, v), d(x, F (x, y)) + d(y, F (y, x)), 1 {d(x, F (u, v))+ 2 d(u, F (x, y))} + {d(y, F (v, u)) + d(v, F (y, x))}
¯ (x, y), (u, v) , d ¯ (x, y), (F (x, y), F (y, x)) , = max d d(u, F (u, v)) + d(v, F (v, u)),
¯ (x, y), (F (u, v), F (v, u)) ¯ (u, v), (F (u, v), F (v, u)) , 1 d d 2 ¯ + d (u, v), (F (x, y), F (y, x))
¯ (x, y), (u, v) , d ¯ (x, y), f (x, y)) , d ¯ (u, v), f (u, v) , = max d 1 ¯ ¯ (u, v), f (x, y)) d (x, y), f (u, v) + d . 2 9
Let x = (x1 , x2 ), y = (y1 , y2 ) ∈ f (X2 ). Then by our assumption there exists z = (z1 , z2 ) ∈ X2 such that x1 ≤ z1 , y1 ≤ z1 and z2 ≤ x2 , z2 ≤ y2 (in other case the proof is similar). Therefore x z and y z. Also since (F n (z1 , z2 )) and ¯ Therefore (F n (z2 , z1 )) are convergent in (X, d ), (f n (z)) is convergent in (X2 , d). using Theorem 2.1 we conclude that f has a unique fixed point (¯ x , y ¯ ) ∈ X2 . Now (¯ x, y¯) = F (¯ x, y¯), F (¯ y, x ¯) implies that x ¯ = F (¯ x, y¯) and y¯ = F (¯ y, x ¯). Thus, F has a unique coupled fixed point. Remark 1. It is easy to see that Theorem (3.1) generalizes the Theorem (1.1). Now we present a condition for which the components of the coupled fixed point will be equal. Theorem 3.2. In addition to the hypothesis of Theorem 3.1, suppose that if there exists a z in X such that both x0 , y0 are comparable with z and (F n (x0 , z)), (F n (z, x0 )) are convergent in (X, d) then F has a unique coupled fixed point (¯ x, y¯) with x ¯ = y¯. Proof. If x0 , y0 are comparable, then we have either x0 ≤ y0 or y0 ≤ x0 . So let x0 ≤ y0 (in the other case proof is similar). Then x1 = F (x0 , y0 ) ≤ F (y0 , x0 ) = y1 , as F has mixed monotone property. Continuing this process we get xn ≤ yn ⇒ (xn , yn ) (yn , xn ) for all n ∈ N. Therefore using (3.2) we have
d(xn , yn ) ≤ k. max d(xn−1 , yn−1 ),d(xn−1 , xn ), d(yn−1 , yn ), (3.6) 1 [d(xn , yn−1 ) + d(xn−1 , yn )] 2 ¯ and limn→∞ yn = y¯, letting n → ∞ in (3.6) we have Now, since limn→∞ xn = x ¯ x, y¯) ≤ k.d(¯ ¯ x, y¯), d(¯ ¯ x, y¯) = 0, that is, x where k ∈ [0, 1), implies d(¯ ¯ = y¯. Now let x0 , y0 are not comparable then by our assumption there exists z ∈ X comparable to both x0 and y0 . Suppose x0 ≤ z and y0 ≤ z (in other case proof is similar). Then (x0 , y0 ) (x0 , z), (x0 , z) (z, x0 ) and (z, x0 ) (y0 , x0 ). Therefore f (x0 , y0 ) f (x0 , z), f (x0 , z) f (z, x0 ) and f (z, x0 ) f (y0 , x0 ), as f is non-decreasing in (X2 , ). Also limn→∞ f n (x0 , y0 ) = (¯ x, y¯), y, x ¯) and limn→∞ F n (x0 , z) = z¯1 , limn→∞ F n (z, x0 ) = z¯2 limn→∞ f n (y0 , x0 ) = (¯ as (F n (z, x0 )) and (F n (x0 , z)) are convergent. Therefore using (3.2) we have ¯ n (x0 , y0 ), f n (x0 , z)) ≤ k.M (f n−1 (x0 , y0 ), f n−1 (x0 , z), f ), d(f ¯ n (x0 , z), f n (z, x0 )) ≤ k.M (f n−1 (x0 , z), f n−1 (z, x0 ), f ), d(f n
n
¯ (z, x0 ), f (y0 , x0 )) ≤ k.M (f d(f
n−1
(z, x0 ), f
n−1
(y0 , x0 ), f ),
where ¯ x, y¯), (z¯1 , z¯2 )), lim M (f n−1 (x0 , y0 ), f n−1 (x0 , z), f ) = d((¯
n→∞
¯ z¯1 , z¯2 ), (z¯2 , z¯1 )), lim M (f n−1 (x0 , z), f n−1 (z, x0 ), f ) = d((
n→∞
¯ z¯2 , z¯1 ), (¯ lim M (f n−1 (z, x0 ), f n−1 (y0 , x0 ), f ) = d(( y, x ¯)).
n→∞
10
(3.7)
Therefore taking limit into both side of (3.7) we get ¯ z¯1 , z¯2 ), (z¯2 , z¯1 )) = d(( ¯ z¯2 , z¯1 ), (¯ ¯ x, y¯), (z¯1 , z¯2 )) = d(( y, x ¯)) = 0 d((¯ Hence x ¯ = y¯. If F : X2 → X is a non-decreasing map (x ≤ u, y ≤ v ⇒ F (x, y) ≤ F (u, v)) or a non-increasing map (x ≤ u, y ≤ v ⇒ F (u, v) ≤ F (x, y)), similar results can be obtained using the method of Theorem 3.1 and Theorem 3.2. Theorem 3.3. Let (X, ≤, d) be a complete partially ordered metric space. Let F : X2 → X be a non-decreasing, or non-increasing mapping for which there exists a constant k ∈ [0, 1) such that for each x ≤ u, y ≤ v, d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ k.M [x, y, u, v, F ], where M [x, y, u, v, F ] is defined in the equation (3.3). If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ), y0 ≤ F (y0 , x0 ), or x0 ≥ F (x0 , y0 ), y0 ≥ F (y0 , x0 ), then in each of the following cases F has a coupled fixed point. (i) F is continuous. (ii) If for any non-decreasing sequence (xn ) converges to x ∈ X, then xn ≤ x for all n and if for any non-increasing sequence (yn ) converges to y ∈ X, then y ≤ yn for all n. (only in case of F is non-decreasing) If further for any two points x = (x1 , x2 ), y = (y1 , y2 ) ∈ (F (X2 ))2 there exists z = (z1 , z2 ) ∈ X2 such that z1 comparable with both x1 and y1 and z2 is also comparable with x2 and y2 with same order of x1 , y1 and z1 and (F n (z1 , z2 )), (F n (z2 , z1 )) are convergent in (X, d). Then the coupled fixed point of F is unique. Theorem 3.4. In addition to the hypothesis of Theorem 3.3, suppose that if there exists a z in X such that both x0 , y0 are comparable with z and (F n (z, z)) is convergent in (X, d) or z is comparable with F (z, z), then F has a unique coupled fixed point (¯ x, y¯) with x ¯ = y¯. q p 1 Denoting the map (X, Y ) → (Q+ i=1 (Ai ∗ Fi (X)Ai )ri + j=1 (Bj ∗ Gj (Y )Bj )qj ) δ by f , we prove the following theorems. Theorem 3.5. Let Fi : P(n) → P(n) be a continuous order-preserving mapping for each i = 1, 2, .., p while Gj : P(n) → P(n) be a continuous order-reversing mapping for each j = 1, 2, .., q and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X, Y, U, V ∈ P(n) with X ≤ U and Y ≥ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
(3.8) + max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a positive definite solution. 11
(i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≥ 0, and there exists 1 1 an X0 ∈ P(n) such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≤ 0, and there exists 1 1 a Y0 ∈ P(n) such that Y0 ≥ Q δ and Y0 ≥ f (Q δ , Y0 ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (Q δ , αm .In ) for all m ∈ N. Proof. Let δ.ri ≥ 0, δ.qj ≥ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Consider X, Y, U, V ∈ P(n) such that X ≤ U and Y ≥ V . Then f (X, Y ) ≤ f (U, V ) and f (Y, X) ≥ f (V, U ) i.e, f has mixed monotone property. Also d f (X, Y ), f (U, V ) + d f (Y, X), f (V, U ) p q 1 = d (Q + (Ai ∗ Fi (X)Ai )ri + (Bj ∗ Gj (Y )Bj )qj ) δ , i=1
(Q +
j=1 p
(Ai ∗ Fi (U )Ai )ri +
i=1
d (Q +
p i=1
(Q +
q
1
(Bj ∗ Gj (X)Bj )qj ) δ ,
j=1 q
(Ai ∗ Fi (V )Ai )ri +
i=1
≤
1 (Bj ∗ Gj (V )Bj )qj ) δ +
j=1
(Ai ∗ Fi (Y )Ai )ri +
p
q
1
(Bj ∗ Gj (U )Bj )qj ) δ
j=1
p q
1 max d( (Ai ∗ Fi (X)Ai )ri , (Ai ∗ Fi (U )Ai )ri ), |δ| i=1 i=1
d(
p
q (Bj Gj (Y )Bj ) , (Bj ∗ Gj (V )Bj )qj ) ∗
qj
j=1
j=1
p q
1 max d( (Ai ∗ Fi (Y )Ai )ri , (Ai ∗ Fi (V )Ai )ri ), + |δ| i=1 i=1
d(
p
(Bj ∗ Gj (X)Bj )qj ,
j=1
q
(Bj ∗ Gj (U )Bj )qj )
j=1
1 max max |ri |d(Fi (X), Fi (U )) , max |qj |d(Gj (Y ), Gj (V )) ≤ i j |δ|
1 + max max |ri |d(Fi (Y ), Fi (V )) , max |qj |d(Gj (X), Gj (U )) i j |δ| ≤ kM [X, Y, U, V, f ]. 1
Now for any X, Y ∈ P(n), Q δ ≤ f (X, Y ). So if there exists some X0 ∈ P(n) 1 1 such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). Also continuity of Fi ’s and Gi ’s 12
implies continuity of f . Therefore using Theorem 3.1 and Theorem 3.2 we get ¯ = limn→∞ f n (X0 , Q δ1 ) of equation (3.1) in P(n). Now for each a solution X X, Y ∈ f (P(n)2 ), there exists some scalar α such that X, Y ≤ α.In . Therefore by our assumption if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N, then for any (X1 , X2 ) and 1 1 2 (Y1 , Y2 ) in f (P(n)2 ) , (X1 , X2 ), (Y1 , Y2 ) (αm .In , Q δ ) and (αm .In , Q δ ) 1 1 f (αm .In , Q δ ), f (Q δ , αm .In ) for some m ∈ N. Therefore, by using Theorem 3.1 and Theorem 3.2 we conclude that the equation (3.1) has a unique positive definite solution. Next let δ.ri ≤ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Then for all X, Y, U, V ∈ P(n) with X ≤ U and Y ≥ V implies f (X, Y ) ≥ f (U, V ) and f (Y, X) ≤ f (V, U ). Therefore using above method, in this case we also get the same result. Similarly if δ.ri ≥ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q, then using Theorem 3.3 and Theorem 3.4 we have the following result. Theorem 3.6. Let Fi : P(n) → P(n) be an order-preserving mapping for each i = 1, 2, .., p while Gj : P(n) → P(n) be an order-reversing mapping for each j = 1, 2, .., q and let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X ≤ U and Y ≤ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
(3.9) + max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a unique positive definite solution. (i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≤ 0. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0 and Fi , Gj ’s are continuous. Proof. Let δ.ri ≥ 0 and δ.qj ≤ 0 for each i = 1, 2, .., p and j = 1, 2, .., q. Consider X, Y, U, V ∈ P(n) such that X ≤ U and Y ≤ V . Then f (X, Y ) ≤ f (U, V ) and f (Y, X) ≤ f (V, U ) i.e, f is non-decreasing. Therefore following the same calculation of Theorem 3.5 we get d f (X, Y ), f (U, V ) + d f (Y, X), f (V, U ) ≤ k.M [X, Y, U, V, f ]. 1
1
Also since Q δ ≤ f (X, Y ) for any X, Y ∈ P(n) therefore by taking x0 = y0 = Q δ and using Theorem 3.3 and Theorem 3.4 we get our desired result. Similarly if for each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0 and Fi , Gj ’s are continuous, then f is non-increasing and continuous. So in this case also using Theorem 3.3 and Theorem 3.4 we get our desired result. 13
Now if Fi , Gj ’s are both order-preserving mappings for each i = 1, 2, .., p and j = 1, .., q, then we have the following result. Theorem 3.7. Let both Fi : P(n) → P(n) and Gj : P(n) → P(n) be continuous order-preserving mappings for each i = 1, 2, .., p and j = 1, 2, .., q also let Q ∈ P(n). If δ ∈ (−∞, −1] ∪ [1, ∞), −1 ≤ ri , qj ≤ 1 and for all X ≤ U and Y ≥ V we have
max max{|ri |d(Fi (X), Fi (U ))}, max{|qj |d(Gj (Y ), Gj (V ))} i j
+ max max{|ri |d(Fi (Y ), Fi (V ))}, max{|qj |d(Gj (X), Gj (U ))} (3.10) i
j
≤ k|δ|M [X, Y, U, V, f ], where k ∈ [0, 1), then in each of the following cases the equation (3.1) has a positive definite solution. (i) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≥ 0, δ.qj ≤ 0, and there exists 1 1 an X0 ∈ P(n) such that X0 ≥ Q δ and X0 ≥ f (X0 , Q δ ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (αm .In , Q δ ) for all m ∈ N. (ii) For each i = 1, 2, .., p and j = 1, 2, .., q, δ.ri ≤ 0, δ.qj ≥ 0, and there exists 1 1 a Y0 ∈ P(n) such that Y0 ≥ Q δ and Y0 ≥ f (Q δ , Y0 ). In this case the solution is unique, if there exists an increasing unbounded sequence (αm ) 1 such that αm .In ≥ f (Q δ , αm .In ) for all m ∈ N. One can also obtain similar results if both Fi ’s and Gj ’s are order-reversing mappings. We now give an example. Example 2. Consider the nonlinear matrix equation 1
1
1
X 2 = Q + (A∗ XA) 2 + B ∗ X 2 B + (C ∗ XC)− 3 , ⎛ ⎞ ⎛ ⎞ 1 0.1 0.01 1 0.5 0.5 1 0.001 ⎠ , A = ⎝ 0.5 1 0.5 ⎠, where Q = ⎝ 0.1 0.01 0.001 1 0.5 0.5 1 ⎛
1 0.2 B = ⎝ 0.2 0 0.2 0.2
(3.11)
⎞ ⎛ ⎞ 0.2 1.4142 −0.4714 −0.5773 0.2 ⎠ , C = ⎝ −0.4714 1.4142 −0.4714 ⎠. −1 −0.5773 −0.4714 1.7321
Then for all X ≤ U and Y ≥ V in P(3) we have
1 1 1 1 max max{ d(X, U ), d(X 2 , U 2 )}, | − |d(Y, V )} 2 3
1 1 1 1 2 + max max{ d(Y, V ), d(Y , V 2 )}, | − |d(X, U )} 2 3 5 < 2. M [X, Y, U, V, f ], 6 14
(3.12)
Figure 2: Convergence history of (3.11)
where f : P(3) × P(3) → P(3) is defined by 1 1 1 1 f (X, Y ) = Q + (A∗ XA) 2 + B ∗ X 2 B + (C ∗ Y C)− 3 2 .
(3.13)
Also for any α ≥ 14 in R we have 1
α.I3 ≥ f (α.I3 , Q 2 ).
(3.14)
Therefore using Theorem 3.7 we conclude that equation (3.11) has a unique positive definite solution. 1 By taking X0 = 14.I3 , Y0 = Q 2 and Xk+1 = f (Xk , Yk ), Yk+1 = f (Yk , Xk ), at 20th step we find the unique solution ⎛ ⎞ 2.1872533415096 0.3828578135719 0.2340628340690 ¯ = X20 = Y20 = ⎝ 0.3828578135719 1.7839885977502 0.1783011164762 ⎠ X 0.2340628340690 0.1783011164762 2.1528423137242 of the equation (3.11). The residual error is R20 = 7.016609515630743 × 10−14 . The convergence history is given by Figure 2, where Curve 1 corresponds to d(Xk , f (Xk , Yk )) and Curve 2 corresponds to d(Yk , f (Yk , Xk )).
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