Existence of Best Proximity Point in O-CompleteMetric Spaces

Citation: Poonguzali, G.; Pragadeeswarar, V.; De la Sen, M. Existence of Best Proximity Point in O-Complete Metric Spaces. Mathematics 2023,11, 3453. https://doi.org/10.3390/ math11163453 Academic Editors: Mircea Balaj, Vasile Berinde and Massimiliano Giuli Received: 31 May 2023 Revised: 2 August 2023 Accepted: 7 August 2023 Published: 9 August 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Existence of Best Proximity Point in O-CompleteMetric Spaces G. Poonguzali 1, V. Pragadeeswarar 1,* and Manuel De la Sen 2,* 1Department of Mathematics, Amrita School of Physical Sciences, Amrita Vishwa Vidyapeetham, Coimbatore 641112, India; [email protected] 2 Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa, Bizkaia, Spain *Correspondence: [email protected] (V.P.); [email protected] (M.D.l.S.) Abstract: In this work, we prove the existence of the best proximity point results for ⊥ -contraction (orthogonal-contraction) mappings on an O -complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide suitable examples to demonstrate the validity of our results. Keywords: best proximity point; O -complete metric space; O -closed set; P-property; weakly proximally ⊥-preserving; ⊥-continuous MSC: 37C25 1. Introduction and Preliminaries Over the past 100 years, fixed point theory has been an active area of research, due to its significance in applications. Simultaneously, in the theory of functional analysis, the idea of proximity pairs for two sets was briefly discussed. Many researchers contributed their vision on when and where we can have the best proximity points for sets. Another group of researchers who were active on fixed point results wanted to analyze the case when we do not have an exact solution to the equation of the form T(x) = x . Researchers such as Ky Fan, Segal, Singh, and Prolla [ 1 – 3 ] have provided a wealth of valuable results in best approximation theory. These findings shed light on situations where fixed points are absent, and under certain smooth conditions, we can obtain approximate solutions to the equations. Notably, Ky Fan [ 1 ] proved the existence of the best approximation for a continuous function on a compact convex subset of a normed space. In a subsequent study in 1989, Segal et al. [2] proved the existence of the best approximation for an approximately compact subset of a normed space. Furthermore, Prolla et al. [ 3 ] extended this concept to multifunctions. Around the end of the 1990s and the start of 2000, a group of researchers used the idea of the best proximity point for mappings, which unifies the fixed point and best approximation results [ 4 – 6 ]. Later, many generalizations were made by many researchers; refer to [7–11]. On the other hand, the Banach contraction principle is a significant mathematical discovery in fixed point theory. It has been expanded and applied to various types of metric spaces, such as semi-metrics, quasi-metrics, pseudo-metrics, fuzzy metric spaces, and partial metric spaces, among others (see [9,10,12–18]). In that line, in 2017, Gordji et al. [19,20] introduced a new type of metric space called an orthogonal metric space and proved the fixed point results. They also demonstrated the application of these results in establishing the existence and uniqueness of solutions for first-order ordinary differential equations, where the Banach contraction mapping principle is not applicable. Motivated by the aforementioned results [ 19 , 20 ], in this paper, we extend the results from the fixed point to the best proximity point for non-self-mappings in the context of an orthogonal set. Using these existence results, we prove a common best proximity point Mathematics 2023,11, 3453. https://doi.org/10.3390/math11163453 https://www.mdpi.com/journal/mathematics Mathematics 2023,11, 3453 2 of 9 result. Finally, we provide suitable examples to demonstrate the validity of our results, which cannot be achieved through other best proximity point techniques. Furthermore, in the literature of fixed point theory, we have enormous results on the complete metric space and partially ordered metric space, but not many on the orthogonal metric space. In [ 21 ], the existence of the best proximity points was provided for a map that is a continuous and proximal contraction, or it has to be a contraction map on an approximately compact set. In this paper, we provide the existence of the best proximity point for a weaker condition called ⊥- continuity on an O-closed set. Research on the concept of an orthogonal space is worth analyzing as it represents a more general space that cannot be compared with a partially ordered space. The upcoming examples will explain the necessity of having an Orthogonal space. Example 1 ([ 20 ]) . Consider M=R2 .Define ⊥ as u⊥v if <u , v>= 0on M .Then, (M , ⊥) is an O-set, since u= ( 0, 0 )⊥v ,for all v∈M .However, (M , ⊥) is not a partial order set. Choose u= (1, 0),v= (0, 1),r= (−1, 0);it is clear that u ⊥v,v⊥r, but u 6⊥ r. Example 2. Consider (M=R , ≤) . Then, M is a partially ordered set. but not an O -set with the ≤relation, because we cannot find any u ∈M such that u ≤p or p ≤u for all p ∈R. Throughout this paper, the following notions are used: Let Aand Bbe any two nonempty subsets of a metric space X. d(A,B):=inf{d(a,b):a∈Aand b∈B}, A0={a∈A:d(a,b) = d(A,B)for some b∈B}, B0={b∈B:d(a,b) = d(A,B)for some a∈A}. Definition 1. Let A and B be any two nonempty subsets of a metric space X . Then, a point p∈A is called a best proximity point of a mapping T :A→B, if the following holds true: d(p,Tp) = d(A,B). Definition 2 ([ 20 ]) . Let M6=∅ , and let ⊥⊆ M×M be any binary relation. We call (M , ⊥) an O-set (orthogonal set) if ⊥satisfies the following condition: ∃u0∈M:(∀v,v⊥u0)or (∀v,u0⊥v). We usually use (M , ⊥) to represent an O -set. Furthermore, note that this orthogonal relation is not a transitive relation. Example 3 ([ 20 ]) . Take M= [ 0, ∞) , and if uv ∈ {u , v} ,then u⊥v .It is clear to see that if u0=0or u0=1,(M,⊥)is an orthogonal set. Definition 3 ([ 20 ]) . Consider any O -set (M , ⊥) .Let (un) be any sequence, then we say that (un) is an O-sequence if un⊥un+1or un+1⊥unfor all n ∈N. Example 4. Let M=R , and define u⊥v by uv ≤u or v .Take un= 1 /n ,then un is an O-sequence, since ∀n,un⊥un+1. Definition 4 ([ 20 ]) . Let (X , ⊥) be any O -set. Let A be any subset of X .Then, A is orthogonal closed set (O-closed set) if, when any O-sequence xn→x, then x ∈A. Example 5. Let X= [ 0, ∞) . Choose the usual order on X ,then (X , ≤) is an O -set. Consider A= [0, 1], then A is an orthogonal closed set. Mathematics 2023,11, 3453 3 of 9 Every closed set is an orthogonal closed set, but an orthogonal closed set need not be a closed set. Example 6. Let X = [0, 1]and p ∈(0, 1), and define x⊥y⇐⇒ (x≤y≤p x=0otherwise. Here, choose A= [ 0, q) with q∈(p , 1 ) .Then, A is an O -closed set. Furthermore, it is not a closed set. Definition 5. Let (A , B) be a pair of nonempty subsets of a metric space (X , d) . The pair (A , B) satisfies the P-property if, whenever a1,a2∈A and b1,b2∈B with, d(a1,b1) = d(A,B) d(a2,b2) = d(A,B))=⇒d(a1,a2) = d(b1,b2). Definition 6 ([ 20 ]) . Let (X , ⊥ , d) be an orthogonal metric space ( (X , ⊥) is an O-set, and (X , d) is a metric space). Then, T:X→X is said to be orthogonally continuous (or ⊥ -continuous) in a∈X if, for each O-sequence {an}n∈N in X with an→a , we have T(an)→T(a) . Furthermore, T is said to be ⊥-continuous on X if T is ⊥-continuous in each a ∈X. Every continuous mapping is ⊥-continuous, but the converse is not true. Definition 7 ([ 20 ]) . Let (X , ⊥ , d) be an orthogonal metric space and 0 <k< 1. A mapping T:X→X is called an orthogonal-contraction (briefly, ⊥ -contraction) with Lipschitz constant k if, for all x,y∈X with x ⊥y, d(Tx,Ty)≤kd(x,y). Every contraction is a ⊥-contraction, but the converse is not true. 2. Main Results Now, we will prove the lemma that will be used to establish the existence of the best proximity point results. Lemma 1. Let A be an orthogonal closed subset of an O -complete metric space X ,then A is an O-complete metric space. Proof. Let (xn) be any O -Cauchy sequence in A . Then, (xn)⊆X . Since X is an O -complete metric space, there exists x∈X such that xn→x . Furthermore, (xn) is an O -sequence, which converges to x∈X. Hence, x∈A. Definition 8. Let A and B be any two nonempty subsets of a metric space (X , d) .A map T:A→ B is said to be proximally ⊥-preserving if d(a1,Tb1) = d(A,B) d(a2,Tb2) = d(A,B))=⇒a1⊥a2if b1⊥b2, for all a1,a2,b1,b2∈A. Theorem 1. Let A and B be two nonempty O -closed subsets of an O -complete metric space (X , ⊥ , d) such that A06=∅ .If (A , B) has the P-property and also T:A→B satisfies the following: 1. T is ⊥-continuous and a ⊥-contraction mapping; Mathematics 2023,11, 3453 4 of 9 2. T(A0)⊆B0; 3. T is proximally ⊥-preserving; 4. A0is an O-set. Then, d(u,Tu) = d(A,B),for some u ∈A. Proof. Since A0 is an O -set, there exists p∈A0 such that u⊥p , or p⊥u for all u∈A0 . Without loss of generality, assume that u⊥p . From Condition 2, we have Tp ∈B0 , and hence, there exists u1∈A0 such that d(u1 , Tp) = d(A , B) . Furthermore, note that Tu1∈B0 , and hence, d(u2 , Tu1) = d(A , B) . By the proximally ⊥ -preserving property of T , we obtain u1⊥u2 . Applying a similar argument, we construct an O -sequence u1⊥u2⊥u3⊥ ··· ⊥ ur⊥ ··· with d(ur+1 , Tur) = d(A , B) for all r∈N . Using the P-property of (A,B), we have d(ur,ur+1) = d(Tur−1,Tur). Consider, d(ur,ur+1) = d(Tur−1,Tur) ≤kd(ur−1,ur) . . . ≤krd(u0,u1). (1) Since k<1, limr→∞kr=0. Hence, limr→∞d(ur,ur+1) = 0. If r,s∈Nand s<r, then d(us,ur)≤d(us,us+1) + d(us+1,us+2)···+d(ur−1,ur) ≤ksd(u0,u1) + ks+1d(u0,u1) + ···+kr−1d(u0,u1) (by (1)) ≤ks[1+k+···+kr−s−1]d(u0,u1) ≤kn 1−kd(u0,u1). As s , r→∞ , d(us , ur)→ 0, which means that (ur) is an O -Cauchy sequence. Here, A is an O -closed subset of an O -complete metric space. By Lemma 1, A is an O -complete metric space (X , ⊥ , d) . Therefore, there exists u∗∈A such that limr→∞ur=u∗ . Since T is ⊥ -continuous, limr→∞Tur−1=Tu∗ , which implies d(ur , Tur)→d(u∗ , Tu∗) as r→∞ . Hence, d(u∗,Tu∗) = d(A,B). Theorem 2. Let (X , ⊥ , d) be any O -complete metric space. Let A and B be two nonempty subsets of X.Let T :A→B satisfy the following conditions: 1. T is ⊥-continuous and a ⊥-contraction; 2. T(A0)⊆B0and (A,B)satisfy the P-property; 3. T is proximally ⊥-preserving; 4. There exists u0,u1∈A0such that d(u1,Tu0) = d(A,B)and u0⊥u1. Then, there exists an element u ∈A such that d(u,Tu) = d(A,B). Proof. By the hypothesis, there exists u0and u1in A0such that d(u1,Tu0) = d(A,B)and u0⊥u1. Since u1∈A0 , this implies Tu1∈B0 , and hence, there exists u2∈A0 such that d(u2 , Tu1) = d(A , B) , by the proximally ⊥ -preserving condition of T , we obtain u1⊥u2 . Proceeding like this, we obtain u1⊥u2⊥ ··· ⊥ ur⊥ur+1⊥ ··· . Then, (ur) is an O -sequence with d(ur+1,Tur) = d(A,B)for all r∈N. Since (A,B)has the P-property, we have d(ur,ur+1) = d(Tur−1,Tur)≤kd(ur−1,ur)≤krd(u0,u1). Since k<1, kr→0, limr→∞d(ur,ur+1) = 0. Mathematics 2023,11, 3453 5 of 9 Claim: (ur)is an O-Cauchy sequence. If s,r∈Nand r<s, then d(ur,us)≤[d(ur,ur+1) + ···+d(us−1,us)] ≤krd(u0,u1) + ···+ks−1d(u0,u1) ≤kr 1−kd(u0,u1). Therefore, d(ur , ur)→ 0 as s , r→∞ . Therefore, (ur) is an O -Cauchy sequence. Hence, limr→∞ur=u∗ . Since T is ⊥ -continuous, limr→∞Tur−1=Tu∗ , which implies d(ur , Tur)→ d(u∗,Tu∗). Therefore, u∗is a best proximity point. Example 7. Consider X:=R2 with ⊥ defined as u⊥v if <u , v>= 0. Now, define T: {0}×R→ {1}×Rby T(0, x) = ((1, x/2):x∈Q∩R (1, 0):x∈QC∩R. Here, observe that T is ⊥ -continuous and a ⊥ -contraction. It is easy to observe that A0=A and B0=B ; therefore, T(A0)⊆B0 .Furthermore, (A , B) has the P-property. It is evident that the above map T satisfies all the conditions of Theorem 2.Clearly, ( 0, 0 ) is the best proximity point for T. Theorem 3. Let (X , ⊥ , d) be an O -complete metric space. Let A and B be two nonempty O - closed subsets of X such that A06=∅ .Furthermore, assume that (A , B) has the P-property. Let T:A→B satisfy the following conditions: 1. T is a ⊥-contraction mapping and proximally ⊥-preserving; 2. T(A0)⊆B0; 3. If (ur)is any O-sequence with ur→u,then ur⊥u for all r ∈N; 4. A0is an O-set. Then, there exists u ∈A such that d(u,Tu) = d(A,B). Proof. By using the same technique as in Theorem 2, we can construct an O -Cauchy sequence (ur) with d(ur+1 , Tur) = d(A , B) , and there exists u∈A , such that ur→u . Thus, for any e/ 2 > 0, there exists N1∈N such that d(ur , u)≤e/ 2, for all r≥N1 . Similarly, for any e/ 2 k> 0, there exists N2∈N such that d(us , u)≤e/ 2 k , where k is the contraction constant of Tand for all s≥N2. Choosing, N=max{N1,N2}, we obtain d(u,Tu)≤d(u,uN) + d(uN,TuN) + d(TuN,Tu) ≤e/2 +d(A,B) + kd(uN,u) (Since uN⊥u&Tis ⊥ − contraction) ≤e/2 +d(A,B) + e/2 ≤d(A,B) + e. Since,eis arbitrary, we can conclude that d(u,Tu) = d(A,B). Let us denote the new notion called weakly proximally ⊥- preserving as follows. Definition 9. Two maps T,S:A→B are said to be weakly proximally ⊥- preserving if: 1. For all a∈A ,there exist v1 , v2∈A with d(v1 , Ta) = d(A , B) , d(v2 , Sv1) = d(A , B) and v1⊥v2. 2. For all a∈A ,there exist w1 , w2∈A with d(w1 , Sa) = d(A , B) , d(w2 , Tw1) = d(A , B) and w1⊥w2. Theorem 4. Let A and B be two nonempty O -closed subsets of an O -complete metric space (X , ⊥ , d) with A06=∅ ,and also, assume that (A , B) has the P-property. Let T , S:A→B be two non-self-mappings satisfying the following conditions: Mathematics 2023,11, 3453 6 of 9 1. (T,S)is weakly proximally ⊥-preserving; 2. T or S is ⊥-continuous; 3. For all u,v with u ⊥v,d(Tu,Sv)≤kd(u,v)for some k ∈[0, 1), 4. If any O-sequence (un)converges, then un⊥u for all n,where u =limn→∞un. Then, there exists u ∈A such that d(u,Tu) = d(u,Su) = d(A,B). Proof. Since A06=∅ , choose any u0∈A0 . Applying T on u0 , then Tu0∈B0 . As (T , S) is weakly proximally ⊥ -preserving, we have d(u1 , Tu0) = d(A , B) , d(Tu2 , Su1) = d(A , B) , and u1⊥u2 . Continuing the same way using the weakly proximally ⊥ -preserving condition of (T , S) , we can construct an O -sequence (ur) with d(u2r+1 , Tu2r) = d(A , B) , d(u2r+2 , Su2r+1)= d(A , B) and ur+1⊥u2r+2 . Now, it is time for our usual technique of proving this (ur)to be a Cauchy sequence. For that, observe d(u2r+1,u2r+2) = d(Tu2r,Su2r+1)(By P-Property) ≤kd(u2r,u2r+1) =kd(Tu2r−1,Su2r) ≤k2d(u2r−1,u2r) . . . ≤k2r+1d(u0,u1). Since k< 1, k2r+1→ 0, this implies limr→∞d(u2r+1 , u2r+2) = 0. Now, for r , s∈N with s>r, we have d(ur,us)≤d(ur,ur+1) + d(ur+1,ur+2) + ···+d(us−1,us) ≤krd(u0,u1) + kr+1d(u0,u1) + ···ks−1d(u0,u1) ≤kr[1+k+k2+···+ks−r−1]d(u0,u1). By the above inequality, it is evident that (ur) is an O -Cauchy sequence. Since our space is O -complete, (ur) converges, say u , which implies ur⊥u for all r∈N . Without loss of generality, assume that T is ⊥ -continuous, then it is easy to conclude that d(u2r+1 , Tu2r)→ d(u , Tu) . Furthermore, note that d(u , Tu) = d(A , B) . Thus, u is the best proximity point for T. Next, our claim is to show that u is the best proximity point for S . By the convergence of (ur) , for e/ 2 > 0, there exists N1∈N , such that d(ur , u)≤e/ 2 for all r≥N1 ; furthermore, for e/ 2 k> 0, there exists N2∈N , such that d(ur , u)≤e/ 2 for all r≥N2 . By choosing N=max{N1,N2}, consider d(u,Su)≤d(u,u2N+1) + d(u2N+1,Tu2N) + d(Tu2N,Su) ≤e/2 +d(u2N+1,Tu2N) + kd(u2N,u) ≤e/2 +d(u2N+1,Tu2N) + e/2 ≤e+d(u2N+1,Tu2N). We obtain d(u , Su)≤d(A , B) + e . It is easy to conclude that d(u , Su) = d(A , B) , since e is arbitrary. Hence, d(u,Tu) = d(u,Su) = d(A,B). Till now, in the literature on thew best proximity point, the existence of a common best proximity point in metric spaces or partially ordered metric spaces requires a stronger condition called the continuity of a map or the approximate compactness of a set. In the following example, one can easily observe that T is not a continuous map. Nevertheless, a common best proximity point exists. Mathematics 2023,11, 3453 7 of 9 Example 8. Consider X=R2 with ⊥ defined as (u1 , u2)⊥(v1 , v2) , if u1≤v1 and u2≤v2 . Furthermore, choose d(u , v) = |u1−v1|+|u2−v2| .Then, (X , ⊥ , d) is an O -complete metric space. Let us consider A:={( 0, a):a∈R} and B:={( 1, b):b∈R} .Then, d(A , B) = 1. Now, define T:A→B by T( 0, a) = ((1, −a/2):a∈Q∩R (1, −a/4):a∈QC∩R and S:A→B as S( 0, b) = (1, −b/4).We are now ready to verify the conditions of Theorem 4. Condition 1. (T,S)is weakly proximally ⊥-preserving: Let u ∈A,then u = (0, u1), where u1∈R. Case (i): If u1∈Q∩R ,then Tu = ( 1, −u1/ 2 ) .It is easy to see that, if we take v= (0, −u1/2)and w = (0, −u1/8),then d(u,Tv) = d(A,B) = d(v,Sw)and also v ⊥w. Case (ii): If u1∈QC∩R ,then Tu = ( 1, −u1/ 4 ) .It is easy to see that, if we take v= ( 0, −u1/ 4 ) and w= ( 0, −u1/ 16 ) .Then, d(u , Tv) = d(A , B) = d(v , Sw) and also v⊥w . Similarly, for all u∈A ,we can find w , w0∈A with d(w , Su) = d(A , B) , d(w0 , Tw) = d(A , B) , which also implies w ⊥w0. Condition 2. T or S is ⊥-continuous: Here, S is a continuous function, and hence, S is ⊥ -continuous. Furthermore, observe that T is not ⊥ -continuous, since O -sequence xn= ( 0, − 1 −√2/n) converges to x= ( 0, − 1 ) .However, T(xn) = 1, −(−1−√2/n) 4!converges to (1, 1/4), which is not equal to Tx = (1, 1/2). Condition 3. If u⊥v ,then d(Tu , Sv)≤kd(u , v) for some k∈[ 0, 1 ) . Let u= ( 0, u1) , v= (0, v1)∈A. Case (i): If u1∈Q,then d(Tu,Sv) = d((1, −u1/2),(1, −v1/4)) =|−u1/2 +v1/4| ≤ |−u1/2 +v1/2|(Since u1≤v1) ≤1 2d(u,v). Case (ii): If u1∈QC,then d(Tu,Sv) = d((1, −u1/4),(1, −v1/4)) =|−u1/4 +v1/4| ≤1 4d(u,v) ≤1 2d(u,v). By choosing k =1/2, it is evident that, for all x ⊥y,d(Tu,Sv)≤d(u,v). Condition 4. If (xn)is an O-sequence with xn→x,then xn⊥x for all n: Since (xn) is an O -sequence, we have xn= ( 0, an)≤xn+1= ( 0, an+1) , which implies an≤an+1 .Hence, (xn) is a monotonically increasing sequence, which converges to the supremum, say x := (0, a).It is clear that xn⊥x for all n ∈N.Furthermore, it is easy to observe that (A,B) has the P-property. Here, u∗= (0, 0)satisfies d(u∗,Tu∗) = d(u∗,Su∗) = d(A,B). Theorem 5. Let A and B be two nonempty closed subsets of an O -complete metric space (X , ⊥ , d) with A06=∅ ,and also, assume that (A , B) has the P-property. Let T , S:A→B be two non-self-mappings satisfying the following conditions: 1. (T,S)is weakly proximally ⊥-preserving; 2. T or S is ⊥-continuous; 3. For all u,v with u ⊥v,d(Tu,Sv)≤kd(u,v)for some k ∈[0, 1); Mathematics 2023,11, 3453 8 of 9 4. If u is a best proximity point of either T or S, then u ⊥u. Then, there exists u ∈A such that d(u,Tu) = d(u,Su) = d(A,B). Proof. Following the same technique that we used in Theorem 4, we can easily construct the O -Cauchy sequence (un) such that d(u2n+1 , Tu2n) = d(A , B) , and d(u2n+1 , Su2n+2) = d(A , B) . As usual, O -completeness provides the convergence of (un) , that is there exists u∈A such that un→u . Without loss of generality, assume that S is ⊥ -continuous, then it is easy to conclude that d(u2n+1 , Su2n+2)→d(u , Su) . Furthermore, note that d(u , Su) = d(A,B). Hence, uis the best proximity point for S; thus u⊥u. Consider d(u,Tu)≤d(u,Su) + d(Su,Tu) ≤d(u,Su) + kd(u,u) ≤d(u,Su). Similarly, consider d(u,Su)≤d(u,Tu) + d(Tu,Su) ≤d(u,Tu) + kd(u,u) ≤d(u,Tu). Hence, d(u,Tu) = d(u,Su), which means that d(u,Tu) = d(u,Su) = d(A,B). 3. Conclusions The fixed point and best proximity point results ensure the existence of solutions to many problems in non-linear analysis. In our paper, we have given the existence of the best proximity point and common best proximity point in a more general metric space called the O -metric space, which fails to satisfy the transitivity condition. Furthermore, we provided an example where our map fails to be continuous and fails to be a contraction; still, we can find the best proximity point and common best proximity points. Author Contributions: Conceptualization, G.P. and V.P.; methodology, G.P., V.P. and M.D.l.S.; validation, G.P. and V.P.; writing—original draft preparation, G.P., V.P. and M.D.l.S.; writing—review and editing, G.P., V.P. and M.D.l.S.; funding acquisition, M.D.l.S. All authors have read and agreed to the published version of the manuscript. Funding: This work has been partially funded by the Basque Government through Grant IT1207-19 and Grant IT1155-22. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. 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