An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

We propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form U =Q + ∑k i=1Ai G(U )Ai , whereQ is an n× n Hermitian positive definite matrix,A1,A2, . . . ,Am are n× n matrices and G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.


Introductory notes
Several mathematicians have established fixed point findings for contraction type mappings in metric spaces with partial order in recent years. Turinici established some early results in this approach in [21,22]; it should be noted that their beginning points were amorphous contributions in the area due to Matkowski [10,11]. Ran and Reurings [16], as well as Nieto and Ródríguez-López [13,14], have looked at these kinds of findings. Turinici's findings were expanded upon and refined in articles [13,14]. Fixed point theorems for nonlinear contraction under symmetric closure of an arbitrary relation were recently developed by Samet and Turinici [18]. Alam and Imdad used an amorphous relation to show a relation-theoretic counterpart of the Banach contraction principle, which combines a number of well-known relevant order-theoretic fixed point theorems. According to Mizoguchi and Takahashi [12], Kada et al. [4] developed the idea of ω-distance on a metric space in 1996 and proved a modified Caristi fixed point theorem, Ekeland's -variational principle and the nonconvex minimisation theorem. Rouzkard et al. [17] used the ω-distance function to find a generalised (ψ, φ)-weakly contractive condition for orbital continuous maps. Lakzian et al. [6,7] established various fixed point theorems for (α, ψ)-contractive mappings and Kannan contraction in the presence of a ω-distance. Senapati and Dey [19] recently published a modified version of the ω-distance function and demonstrated that Alam and Imdad [1]'s conclusion is false. For more details about the ω-distance functional, see [3].

Motivation
One of the most visually attractive applications of contraction mapping is found in nonlinear matrix equations. The question now is whether the previously described contraction can be improved and generalised. In relational metric spaces with w-distance functional, we investigate a novel class of implicit relations and implicit type contractions. We establish fixed point findings by justifying the applicability and usefulness of the provided results. Contraction and rational type contraction mappings are included in the proposed implicit contraction. We use these findings to examine the necessary conditions for the existence of a unique positive definite solution to the nonlinear matrix equation (NME) of the form where Q is an n × n Hermitian positive definite matrix, A 1 , A 2 , . . . , A m are n × n matrices and G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis, and visualisation of solutions in a surface plot.

Relational metric spaces
Throughout this article, the notations Z, N, R, R + have their usual meanings, and N * = N ∪ {0}.
We call (W, R) a relational set if (i) W = ∅ is a set and (ii) R is a binary relation on W.
In addition, if (W, d) is a metric space, we call (W, d, R) a relational metric space (RMS, for short).
Let (W, R) be a relational set, (W, d, R) be an RMS, and let K be a self-mapping on W. Then: 6 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 ∪ {0}. 7 A subset Z of W is called R-directed if for each ν, ϑ ∈ Z there exists μ ∈ W such that (ν, μ) ∈ R and (ϑ, μ) ∈ R. It is called (K, R)-directed if for each ν, ϑ ∈ Z there exists μ ∈ W such that (ν, Kμ) ∈ R and (ϑ, Kμ) ∈ R.
8 K is said to be R-continuous at ν if for every R-preserving sequence (ν n ) converging to ν we get K(ν n ) → K(ν) as n → ∞. Moreover, K is said to be R-continuous if it is R-continuous at every point of W.
9 For ν, ϑ ∈ W, 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 } ⊂ W 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 ∈ W, there is a finite sequence {μ 0 , μ 1 , μ 2 , . . . , μ k } ⊂ W satisfying the following conditions: then this finite sequence is called a K-path of length k joining ν to ϑ in R. Notice that a path of length k involves k + 1 elements of W, although they are not necessarily distinct. We fix the following notation for a relational space (W, R), a self-mapping K on W and an R-directed subset D of W: (i) Fix(K) := the set of all fixed points of K,

Relational metric spaces with w-distance
The corresponding definitions and lemmas, in the setting of metric spaces endowed with an arbitrary binary relation R, are as follows.

New class of implicit relations
In this section we introduce a modified version of implicit relations and examples discussed in [2,15].

R ω -implicit contractive condition under w-distance
We define an R ω -implicit contractive mapping in the metric space under w-distance using the above introduced implicit relation. Definition 4.1 Let (W, d, R) be a relational metric space with w-distance w and K : W → W be a given mapping. We say that K is an R ω -implicit contractive mapping if there exists a function H ∈ such that for all (ϑ, ν) ∈ R. Now, we are equipped to state and prove our first main result as follows.

Theorem 4.2 Let (W, d, R)
be an RMS with w-distance ω and K : W → W. Suppose that the following conditions hold: Proof Let θ 0 ∈ X(K, R) be a point as given in (A 1 ). If K n θ 0 = K n+1 θ 0 for some n ∈ N ∪ {0}, then there is nothing to prove. Construct a sequence {θ n } of Picard iterates θ n = K n (θ 0 ) for all n ∈ N * .
Using (A 1 )-(A 2 ), we have that (Kθ 0 , K 2 θ 0 ) ∈ R. Continuing this process inductively, we obtain Using (1) for ϑ = θ n-1 , ν = θ n , Denoting n = ω(K n θ 0 , K n+1 θ 0 ) for all n ∈ N * and applying (H 1 ) in the fifth variables, we have that is, the sequence { n } is a nonincreasing sequence of real numbers. Therefore there exists ζ such that Applying the limit in (4), by the continuity of H, we get a contradiction, and therefore ζ = 0. Thus we conclude that lim n→∞ n = lim n→∞ ω(K n θ 0 , K n+1 θ 0 ) = 0. Similarly, from (i) (Kθ 0 , θ 0 ) ∈ R, using condition (ii), we get (θ n+1 , θ n ) ∈ R for all n ∈ N * . Using this conclusion and above arguments, it can be shown that Next, we show that {K n θ 0 } is a Cauchy sequence in W. For this we show that On the contrary, suppose that condition (6) does not hold. Then we can find δ > 0 and By (3), there exists k 0 ∈ N such that n k > k 0 implies that In view of the two last inequalities, we observe that m k = n k+1 . We may assume that m k is the minimal index such that (7) holds, so that Now, making use of (7), we get Thus, Using the triangle inequality, we have Taking the limit on both sides and making use of (3), (5) and (8), we obtain Again, using the triangle inequality, we have Taking the limit on both sides and making use of (3), (5) and (8), we obtain Combining (9) and (10), we have Now, since R is transitive and since {θ n } is an R-preserving sequence, we must have (K n k θ 0 , K m k θ 0 ) ∈ R for all r ∈ N. Therefore, on applying condition (1), we get Now applying (H 1 ) in the fifth and sixth variables, we have Applying the limit and using the continuity of H, we get H δ, δ, 0, 0, δ, δ) ≤ 0, a contradiction to (H 3 ). Hence, {K n θ 0 } must be a Cauchy sequence in W.
From R-completeness of W, there exists a point ϑ * ∈ W such thatlim n→∞ K n θ 0 = ϑ * . We shall show that ϑ * is a fixed point of K.
Next, we have the following result.

Theorem 4.3
The conclusion of Theorem 4.2 remains true if condition (A 5 ) is replaced with the following one: Proof Following the proof of Theorem 4.2, we observe that the sequence {K n θ 0 } is a Cauchy sequence, and so there exists a point ϑ * in W such that lim n→∞ K n θ 0 = ϑ * . Since lim m,n→∞ ω(K n θ 0 , K m θ 0 ) = 0 for each > 0, there exists N ∈ N such that n > N implies ω(K N θ 0 , K n θ 0 ) < . Since lim n→∞ K n θ 0 = ϑ * and ω(ϑ, ·) is lower semi-continuous, Assume that Kϑ * = ϑ * . Then, by hypothesis (v'), we have which contradicts our assumption. Therefore, Kϑ * = ϑ * . The last conclusion is derived as in the proof of Theorem 4.2.
In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 4.2 and 4.3.
It follows that the sequence {w(K n z, ν)} is nonincreasing. As earlier, we have lim n→∞ w K n z, ν = 0.
Also, since (z, ϑ) ∈ R, proceeding as earlier, we can prove that lim n→∞ w K n z, ϑ = 0, and by using Lemma 2.2 we infer that ν = ϑ; i.e. the fixed point of K is unique. • Assumption (II). For any two fixed points ϑ, ν of K, there must be an element z ∈ K(W) such that (z, ϑ) ∈ R and (z, ν) ∈ R.

Illustrations
In this part, we show how important our results are when compared to metric distances.
Consider the self-mapping K on W given by Considering Example 3.2, we can define H ∈ as where 1/2 < a < 1, 0 < b < 1/2. We show that K satisfies (18). We divide the proof into three parts for ϑ ≤ 1. Similarly, we can show for ν ≤ 1.
• Let ϑ ≤ 1 and ν > 1. Then It can be easily checked that the above cases hold true for a = 9/10, b = 1/10. Thus K is an R ω -implicit contractive mapping. Note that K is not R-continuous at ϑ = 1 and ϑ = 2 3 . Finally, we will show that condition (A 5 ') of Theorem 4.3 holds true. If ν > 0, we have ν = Kν so that Thus, all the conditions of Theorem 4.3 are satisfied and Fix(K) = {0}.
Remark 5.6 It is observed that for 0 = K0 in the metric space That is, condition (A 5 ) of Theorem 4.3 is also not satisfied in the metric space.

Application to nonlinear matrix equations
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) be the set of all n × n matrices over C. Denote by s(U ) any singular value of a matrix U , and its trace norm by s + (U ) = U . For C, D ∈ H(n), C D (resp. C D) will mean that the matrix C -D is positive semi-definite (resp. positive definite).

Theorem 6.1 Consider the equation
where Q ∈ P(n), A i ∈ M(n), i = 1, . . . , k, and the operator G : P(n) → P(n) is continuous in the trace norm. Let, for some M, N 1 ∈ R and for any U ∈ P(n) with U ≤ M, s(G(U )) ≤ N 1 hold for all singular values of G(U). Assume that: (III) There exist 1/2 < a < 1, 0 < b < 1/2 such that, for any s(Q), Then NME (23) has a unique solution U ∈ P(n) with U ≤ M. Further, the solution can be obtained as the limit of the iterative sequence {U n }, where for j ≥ 0, and U 0 is an arbitrary element of P(n) satisfying U 0 ≤ M.
Proof Denote by := {U ∈ P(n) : U ≤ M} a closed subset of P(n). According to (II), any solution of (23) in has to be positive definite. We have, for any U ∈ , Since all singular values of U satisfy s(G(U )) ≤ N 1 , it follows that G(U) ≤ N 1 n. Thus, (27) implies Define now an operator K : → by Also, define a binary relation It is clear that finding positive definite solution(s) of equation (23) is equivalent to finding fixed point(s) of K. Notice that K is well defined, R-continuous and R is K-closed. Since for some Q ∈ , we have (Q, K(Q)) ∈ R or (K(Q), Q) ∈ R, and hence (K; R) = ∅. Now, for (U , V) ∈ R, we have Thus, for any U, V ∈ with U V, we have For some fixed U, V ∈ with (U , V) ∈ R, from (24) and (25), we have The above relation holds for every singular value of Q, so adding up, we obtain Therefore, from (28) we get Let ω : × → R + be defined by Then ( , · , ω) is a complete relational metric space with the above ω-distance. It follows from (29) that [ω(U , K(U))] 2 + [ω(V, K(V))] 2 1 + ω(U , K(V)) + ω(K(U ), V) .
Taking H ∈ as Example 3.1, (30) can be rewritten as H ω(K(U ), K(V)), ω(U , V), ω(U , K(U)), ω(V, K(V)), ω(U , K(V)), ω(K(U ), V) ≤ 0. Now, all the hypotheses of Theorem 4.2 are satisfied, and therefore there existsX ∈ P(n) such that K(X ) =X . Hence, matrix equation (23) has a solution in P(n). Furthermore, due to the existence of the least upper bound and the greatest lower bound for each pair U, V ∈ K(P(n)), we have Y(U, V; R| K(P(n)) ) = ∅ for all U, V ∈ K(P(n)). Hence, using Theorem 4.4, K has a unique fixed point, and hence we conclude that matrix equation (23) has a unique solution in P(n).