A relational-theoretic approach to get solution of nonlinear matrix equations

In this study, we consider a nonlinear matrix equation of the form X = Q + (cid:2) mi =1 A ∗ i G ( X ) A i , where Q is a Hermitian positive deﬁnite matrix, A ∗ i stands for the conjugate transpose of an n × n matrix A i , and G is an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive deﬁnite matrices such that G ( O ) = O . We discuss suﬃcient conditions that ensure the existence of a unique positive deﬁnite solution of the given matrix equation. For this, we derive some ﬁxed point results for Suzuki-FG contractive mappings on metric spaces (not necessarily complete) endowed with arbitrary binary relation (not necessarily a partial order). We provide adequate examples to validate the ﬁxed-point results and the importance of related work, and the convergence analysis of nonlinear matrix equations through an illustration with graphical representations.


Introduction
The study of nonlinear matrix equations (NME) appeared first in the literature concerned with an algebraic Riccati equation. These equations occur in a large number of problems in control theory, dynamical programming, ladder network, stochastic filtering, queuing theory, statistics, and many other applicable areas.
Let H(n) (resp. K(n), P(n)) denote the set of all n × n Hermitian (resp. positive semidefinite, positive definite) matrices over C and M(n) the set of all n × n matrices over C. In [1], Ran and Reurings discussed the existence of solutions of the following equation: in K(n), where B ∈ M(n), Q is positive definite, and F is a mapping from K(n) into M(n).
Note that X is a solution of (1) if and only if it is a fixed point of the mapping G(X ) = Q -B * F(X )B. In [2], authors used the notion of partial ordering and established a modification of the Banach contraction principle, which they applied for solving a class of NMEs of the form X = Q + m i=1 B * i F(X )B i using the Ky Fan norm in M(n).

Theorem 1.1 ([2]) Let F : H(n) → H(n)
be an order-preserving, continuous mapping which maps P(n) into itself and Q ∈ P(n). If B i , B * i ∈ P(n) and m i=1 B i B * i < M · I n for some M > 0 (I n -the unit matrix in M(n)) and if | tr(F(Y) -F(X ))| ≤ 1 M | tr(Y -X )| for all X , Y ∈ H(n) with X ≤ Y, then the equation X = Q + m i=1 B * i F(X )B i has a unique positive definite solution (PDS).
In recent years, a number of mathematicians have obtained fixed point results for contraction type mappings in metric spaces equipped with partial order. Some early results in this direction were established by Turinici in [3,4]; one may note that their starting points were "amorphous" contributions in the area due to Matkowski [5,6]. These types of results have been reinvestigated by Ran and Reurings [1] and also by Nieto and Ródríguez-López [7,8]. Samet and Turinici [9] established fixed point theorem for nonlinear contraction under symmetric closure of an arbitrary relation. Ahmadullah et al. [10][11][12] and Alam and Imdad [13] employed an amorphous relation to prove a relation-theoretic analogue of the Banach contraction principle which in turn unifies a lot of well-known relevant order-theoretic fixed point theorems. Recently, Hasanuzzaman and Imdad [14] used the concept of simulation function and proved the relation theoretic metrical fixed point results for Suzuki type Z R -contraction and discussed application in solving nonlinear matrix equations.
Motivated by the above reference work, we introduce the notion of Suzuki-FG contractive mapping on metric spaces endowed with an arbitrary binary relation (not necessarily partial order), and then we prove existence and uniqueness fixed point results under weaker conditions. We justify our work by some illustrative examples and demonstrate the genuineness of Suzuki-FG contraction over Suzuki contraction, generalized Suzuki contraction, and implicit type contraction mapping. Further, we apply this result to NMEs and discuss its convergence behavior with respect to three different initial values with graphical representations and solutions by the surface plot. The experiment was run on a macOS Mojave version 10.14.6 CPU @1.6 GHz intel core i5 8GB with MATLAB R2020b as the programming language (Online).

Preliminaries
Throughout this article, the notations Z, N, R, R + have their usual meanings, and N * = N ∪ {0}.
We call (E, R) a relational set if (i) E = ∅ is a set and (ii) R is a binary relation on E.
In addition, if (E, d) is a metric space, we call (E, d, R) a relational metric space (RMS, for short).
Let (E, R) be a relational set, (E, d, R) be an RMS, and let be a self-mapping on E. Then: . is said to be R-continuous at ν if for every R-preserving sequence (ν n ) converging to ν, we get (ν n ) → (ν) as n → ∞. Moreover, is said to be R-continuous if it is R-continuous at every point of E. 9. For ν, ϑ ∈ E, 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 } ⊂ E satisfying the following conditions: then this finite sequence is called a path of length k joining ν to ϑ in R. 10. If, for a pair of ν, ϑ ∈ E, there is a finite sequence {μ 0 , μ 1 , μ 2 , . . . , μ k } ⊂ E satisfying the following conditions: then this finite sequence is called a -path of length k joining ν to ϑ in R. Notice that a path of length k involves k + 1 elements of E although they are not necessarily distinct. We fix the following notation for a relational metric space (E, d, R), a self-mapping on E, and an R-directed subset D of E: 3 Results on Suzuki-FG contractive mappings Definition 3.1 ([18]) The collection of all functions F : R + → R satisfying: (F 1 ) F is continuous and strictly increasing; (F 2 ) for each {ξ n } ⊆ R + , lim n→∞ ξ n = 0 iff lim n→∞ F(ξ n ) = -∞, will be denoted by F.
The collection of all pairs of mappings (G, β), where G : R + → R, β : R + → [0, 1), satisfying: Definition 3.2 Let (E, d, R) be an RMS and P : E → E be a given mapping. A mapping P is said to be a Suzuki-FG contractive mapping if there exist F ∈ F and (G, β) ∈ G β such that, for (ν, ϑ) ∈ E with (ν, ϑ) ∈ R * , where We denote by (SFG) R the collection of all Suzuki-FG contractive mappings on (E, d, R).
Now, we are equipped to state and prove our first main result as follows.

Theorem 3.3
Let (E, d, R) be an RMS and P : E → E. Suppose that the following conditions hold: Proof Starting with ν 0 ∈ E given by (C 1 ), we construct a sequence {ν n } of Picard iterates ν n+1 = P n (ν 0 ) for all n ∈ N * .
The R-completeness of E implies that there exists ν * ∈ E such that lim n→∞ ν n = ν * . Now, first by (C5), we have and hence ν * is a fixed point of P.

Theorem 3.4
In addition to the assumptions of Theorem 3.3, let P(ν, ϑ; R| P(E) ) = ∅ for all ν, ϑ ∈ P(E). Then P has a unique fixed point.

Case (A):
We have (u, v) ∈ R, then P n u = u and P n v = v such that (P n u, P n v) ∈ R * for n = 0, 1, . . . . Now, we assert that 1 2 d P n u, P n+1 u < d P n u, P n v or 1 2 d P n+1 u, P n+2 u < d P n+1 u, P n v . (20) Let, to the contrary, there exist ς ∈ N such that and These imply that and so Now, from (5) and using (21)-(23), we have a contradiction, and therefore (20) remains true. Therefore, using condition (2), where N P n u, P n v ) = max d P n u, P n v , d P n u, P n+1 u , d P n v, P n+1 v , d(P n u,P n+1 v)+d(P n v,P n+1 u) 2 .
Since u and v are fixed points of P, we have and so we get which gives G(β (d(u, v))) ≥ 0, and so β (d(u, v)) ≥ 1, a contradiction. Therefore the fixed point is unique. Case (B): By assumption (I), there exists z ∈ E satisfying condition (19). Due to the P-closedness of R, we get P n-1 z, u ∈ R, P n-1 z, v ∈ R. Now, we assert that 1 2 d P n-1 z, P n z < d P n-1 z, u or 1 2 d P n z, P n+1 z < d P n z, u .
Let, to the contrary, there exist ς ∈ N such that and These imply that which implies that (using (26)) Now, from (5) and using (25)-(27), we have a contradiction, and therefore (24) remains true. Therefore, using condition (2), where Using (z, Pz) ∈ R, similarly as in the proof of Theorem 3.3, it can be shown that d(P n-1 z, P n z) → 0 as n → ∞. Therefore, for n sufficiently large, max d P n-1 z, u , d P n-1 z, P n z , d(u, Pu) = d P n-1 z, u and from (28) we have As in the proof of Theorem 3.3, it can be shown that d(P n z, u) ≤ d (P n-1 z, u). It follows that the sequence {d(P n z, u)} is nonincreasing. As earlier, we have lim n→∞ d P n z, u = 0.
Also, since (z, v) ∈ R, proceeding as earlier, we can prove that lim n→∞ d P n z, v = 0, and by using limit uniqueness, we infer that u = v; i.e., the fixed point of P is unique. • Assume (II). For any two fixed points u, v of P, there must be an element z ∈ P(E) such that As R is P-closed, so for all n ∈ N ∪ {0}, P n z, u ∈ R and P n z, v ∈ R.
In the line of proof of Case(B) (I), we obtain u = v, i.e., P has a unique fixed point. • Assume (III). Suppose that u, v are two fixed points of P. Then we must have (u, v) ∈ R, and since u = Pv, we have (v, u) ∈ R * . Also we can get 1 2 d(u, Pu) ≤ d(u, v) following the lines of the proof of Case A (I). Therefore, using condition (2), which gives G(β (d(u, v)) ≥ 0, and so β(d(u, v) ≥ 1, a contradiction. Therefore the fixed point is unique. In a similar way, if (v, u) ∈ R, we have u = v. • Assume (IV). Suppose that u, v are two fixed points of P. Let {z 0 , z 1 , . . . , z k } be an R s -path in Fix(P) connecting u and v. As in Case(A) (I), it must be z i-1 = z i for each i = 1, 2, . . . , k, and it follows that u = v. If we take R = {(ν, ν) ∈ E × E | ν ν}, then we have more new results as consequences of Theorem 3.3. (E, d, ) be an ordered complete metric space. Let P : E → E be increasing and (SFG) R on E . Suppose that there exists ν 0 ∈ E such that ν 0 Pν 0 . If P is E -continuous or E is d-self-closed, then ν * ∈ Fix(P). Moreover, for each ν 0 ∈ E with ν 0 Pν 0 , the Picard sequence P n (ν 0 ) for all n ∈ N converges to a ν * ∈ Fix(P). (E, d, R) be an RMS and P : E → E. Suppose that the following conditions hold:

Application to nonlinear matrix equations
For a matrix B ∈ H(n), we will 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 tr = s + (B). For C, D ∈ H(n), C D (resp. C D) will mean that the matrix C -D is positive semi-definite (resp. positive definite).
The following lemmas are needed in the subsequent discussion. We establish the existence and uniqueness of the solution of the nonlinear matrix equation (NME) where Q is a Hermitian positive definite matrix, A * i stands for the conjugate transpose of an n × n matrix A i , and G is an order-preserving continuous mapping from the set of all Hermitian matrices to the set of all positive definite matrices such that G(O) = O.

Theorem 5.3
Consider NME (31). Assume that there exists a positive real number η such that holds, then for τ > 0 we have Then NME (31) has a unique solution. Moreover, the iteration where X 0 ∈ P(n) satisfies converges in the sense of trace norm · tr to the solution of matrix equation (31).
Proof Define a mapping T : P(n) → P(n) by and a binary relation Then a fixed point of the mapping T is a solution of matrix equation (31). Notice that T is well defined, R-continuous, and R is T -closed. Since for some Q ∈ P(n), we have (Q, T (Q)) ∈ R, and hence P(n)(T ; R) = ∅.
Then To see the convergence of the sequence {X n } defined in (32), we start with three different initial values