Approximating Solutions of Matrix Equations via Fixed Point Techniques

: The principal goal of this work is to investigate new sufﬁcient conditions for the existence and convergence of positive deﬁnite solutions to certain classes of matrix equations. Under speciﬁc assumptions, the basic tool in our study is a monotone mapping, which admits a unique ﬁxed point in the setting of a partially ordered Banach space. To estimate solutions to these matrix equations, we use the Krasnosel’ski˘ı iterative technique. We also discuss some useful examples to illustrate our results.


Introduction
Matrix equations are often used in the study of ladder networks, control theory, stochastic filtering, dynamic programming, statistics, and other fields, according to Anderson [1]. Consider the linear matrix equation below [2].
where A 1 , . . . , A m are arbitrary matrices of order n × n, for each i, A * i is adjoint of A i and Q is a positive definite matrix of order n × n. Next, consider the following nonlinear matrix equation: where F is continuous mapping in the set of all positive definite matrices to itself, under certain assumptions on F (order-preserving or order reversing). Ran and Reurings [2] obtained positive definite solutions of matrix Equations (1) and (2) using the aid of the Banach contraction principle in partially ordered sets. Nieto and Rodríguez-López [3] also used partially ordered spaces and fixed point theorems to find solutions of some differential equations [4]. The advantage of this strategy is that the mapping requirements only need to be satisfied for comparable elements, and the relevance of this viewpoint is to govern the essence of the solutions, whether they are negative or positive, which leads to a variety of interesting applications. For more details on the applications of fixed point theory in partially ordered spaces, one may refer to [5][6][7][8] and references therein.
Berinde [9], on the other hand, recently developed a new form of contraction mappings called as (b, θ)-enriched contraction mappings, which generalize contraction and nonexpansive mappings.
The purpose of this work is to investigate the existence and convergence of solutions of matrix equations. To accomplish this, we use the idea of monotone enriched contraction mapping in partially ordered Banach spaces. More specifically, we extend the concept of (b, θ)-enriched contraction mapping in the setting of partially ordered Banach spaces and establish some existence and convergence results. Thereafter, we use these findings to solve the matrix Equations (1) and (2). To approximate the solutions of these matrix equations, we use the Krasnosel'skiȋ iterative technique. Some useful examples discussed herein illustrate our results.

Preliminaries
Let B be a Banach space and is a partial order on B. We say that ϑ, ν ∈ B are comparable whenever ϑ ν or ν ϑ. Let partial order be compatible with the linear structure of B, that is, for every ϑ, ν, ζ ∈ B and λ ≥ 0, we have A sequence {ϑ n } is monotone increasing if ϑ n ϑ n+1 for all n ∈ N. We shall utilize the observation considered in [5] (Lemma 3.1). Assume that {ϑ n } is a monotone sequence that has a cluster point, that is, there is a subsequence {ϑ n j } that converges to g. Since the order intervals are closed, it follows g ∈ [ϑ n , →) for each n, that is, g is an upper bound for {ϑ n }. If g 1 is another upper bound for {ϑ n }, then ϑ n ∈ (←, g 1 ] for each n, and hence g g 1 . It implies that {ϑ n } converges to g = sup{ϑ n }. If {ϑ n } is a monotone increasing (resp. monotone decreasing) sequence that converges to p, then ϑ n p (resp. p ϑ n ).
where F(ξ) is the set of all fixed points of ξ.
It is well-known that a nonexpansive mapping with a fixed point is quasi-nonexpansive. However, the converse need not be true.

Main Results
Berinde [9] recently introduced a new type of contraction mapping, which is described below:

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The class of nonexpansive mappings and the class of (b, θ)-enriched contraction mappings are independent in nature.
Since ϑ 1 S(ϑ 1 ), by the monotonicity of S, and successively, we can write ϑ n ϑ n+1 = S(ϑ n ) for all n ∈ N. Take ϑ = ϑ n and ν = ϑ n−1 in (9), we have Successively from (11), we can obtain for all n ≥ 2 and m ∈ N. It implies that {ϑ n } is a Cauchy sequence and must converge to a point in Banach space B. Take First, we assume that ξ is continuous, then from (10) Therefore, ξ(u) = u, and ξ has a fixed point in B. If we assume that (ii) is true, then it can be seen that ϑ n u for all n ∈ N.
Take ϑ = ϑ n and ν = u in (9), we get Thus S(u) = u, from Lemma 1, F(S) = F(ξ), u is a fixed point of ξ. Case 2. If b = 0, then ξ is a monotone contraction mapping and following the similar steps for ξ in place of S, we can complete the proof.
In the next theorem, we prove the uniqueness of the fixed point and the global convergence of the Krasnosel'skiȋ iterative method. Then ξ has a unique fixed point. Moreover, if (X1) is true then the sequence {ϑ n } defined by converges to a point in F(ξ) for any initial guess ϑ 1 ∈ B.
Proof. Let ϑ ∈ B be another fixed point of ξ. First, we suppose that hypothesis (X1) is true. We follow the same technique as in [3]. Let for given ϑ 1 ∈ B (given point as in Theorem 1). We consider the following two cases: If ϑ is not comparable to ν, from (X1) there exists either a lower or an upper bound of ϑ and ν, that is, there is a z ∈ B such that z is comparable to ϑ and ν. Since S is a monotone, S n (z) is comparable to ϑ = S n (ϑ) and ν = S n (ν) for all n ∈ N ∪ {0}. Now Therefore, lim n→∞ S n (p) = ν. Again p is not comparable to ν, then from (X1), there exists z ∈ B such that z is comparable to p and ν. Since S is a monotone, S n (z) is comparable to S n (p) and ν = S n (ν) for all n ∈ N ∪ {0}. Thus Hence lim n→∞ S n (p) = ν. If F(ξ) is totally ordered, then following the same technique as in Case 1, we can complete the proof.

Solutions to Linear Matrix Equation
In this section, we discuss the solution of the matrix Equation (1). We define a mapping G on H(n) (the set of all Hermitian matrices of order n × n) as follows: where A 1 , . . . , A m , A * i (for each i) and Q are the same as in (1). It can be seen that solutions of (1) are the fixed points of G. Let A ∈ M(n) (set of all matrices of order n × n), then , where s j (A) are the singular values of A for j = 1, 2, . . . , n. For given Q + ∈ P (n) (the set of all positive definite matrices of order n × n), the following norm can be defined: It is can be seen that H(n) equipped with the norm . 1,Q + is a partially ordered Banach space for any Q + ∈ P (n). We write We denote by I an identity matrix of order n × n, and . , the spectral norm, that is, Theorem 3. Let Q ∈ P (n) and for some Q + ∈ P (n), we have where b ∈ [0, ∞) and θ ∈ [0, b + 1). Then (1) Mapping G admits a unique fixed point in H(n).
(2) For given U 0 ∈ H(n), the sequence {U k } defined by converges to the unique solution of (1), which is in P (n).
Proof. It can be seen that for all U, V ∈ H(n), there exist a lower bound or an upper bound.
Thus from Lemma 2, we have Therefore, from Theorems 1 and 2, mapping G has a unique fixed point and the sequence {U k } converges to the solution of (1). It is evident that G maps P (n) into the set {U ∈ H(n)|Q ≤ U}; therefore, the solution lies in this set and is positive definite. (13) for m = 3, n = 4, i.e., It can also be verified that the elements of each sequence are order-preserving. The convergence behavior is shown in Figure 1.

Solutions to Nonlinear Matrix Equations
In this section, we consider the following nonlinear matrix equations: where F : P (n) → P (n) is a continuous mapping. For more details of these class of equations, see [14]. In view of different conditions on mapping F, we consider the following cases: Case 1. If F is order-preserving and considering the following equation: We can define The mapping G is well-defined on P (n) and order-preserving. For all U ∈ P (n), Q ≤ G(U). In particular, Q ≤ G(Q). Since G is order-preserving Thus, {G j (Q)} is an increasing sequence.

Proposition 1.
Suppose that there exists an U 0 such that G(U 0 ) ≤ U 0 . Then G maps the set {U : Q ≤ U ≤ U 0 } into itself. The sequence {G j (Q)} converges to a point U − which is the smallest solution of (17). Further, the sequence {G j (U 0 )} is a decreasing sequence, which is the largest solution in the set [Q, U 0 ].
Thus, {G j (Q)} is an increasing sequence and bounded above by G p (U 0 ) for any p ∈ N. Further, the sequence {G j (U 0 )} is bounded below the decreasing sequence. Let Suppose U is any solution of (17), then Hence U ≤ U + .
The following theorem ensures the uniqueness of the solution of (17).

Theorem 4.
Assume that for all U, V ∈ H(n) with U ≤ V, we have where θ ∈ [0, b + 1). Then (17) has a unique solution which is positive definite. Moreover, for converges (in sense of norm . 1 ) to the solution of (17).
From the assumptions in the theorem, all the hypotheses of Theorem 2 are fulfilled and we obtain the desired result. Example 6. Consider the nonlinear matrix Equation (17) for m = 3, n = 3, F(U) = U 1/3 , i.e.,

Case 2. Consider the following equation
We can define Assume that F is order-reversing in (20), then G is order-preserving. Assume that there exists U 0 ≤ Q such that U 0 ≤ G(U 0 ). Then One can easily see that [U 0 , Q] is mapped into itself.

Proposition 2.
Suppose that there exists a U 0 such that U 0 ≤ G(U 0 ). Then G maps the set {U : U 0 ≤ U ≤ Q} into itself. The sequence {G j (Q)} converges to a point U + which is the largest solution of (20). Further, the sequence {G j (U 0 )} is an increasing sequence and converges to a point U − , which is the smallest solution in the set [U 0 , Q].
Theorem 5. Let Q ∈ P (n) and suppose that there exists U 0 ≤ Q such that U 0 ≤ G(U 0 ). Further, assume that for U 0 ≤ U ≤ V ≤ Q, we have . Then (20) has a unique solution, which is positive definite. Moreover, given converges (in the sense of norm . 1 ) to the solution of (20).

Conclusions
In this paper, we studied new existence and convergence conditions for solutions of linear and nonlinear matrix equations.