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Applications in Integral Equations through Common Results in C*-Algebra-Valued Sb-Metric Spaces

Applications in Integral Equations through Common Results in C*-Algebra-Valued Sb-Metric Spaces axioms Article Applications in Integral Equations through Common Results in C*-Algebra-Valued S -Metric Spaces 1 1, 2 S. S. Razavi , H. P. Masiha * and Manuel De La Sen Faculty of Mathematics, K. N. Toosi University of Technology, Tehran 16315-1618, Iran; srazavi@mail.kntu.ac.ir Institute of Research and Development of Processes of the Basque Country, 48940 Leioa, Spain; manuel.delasen@ehu.eus * Correspondence: masiha@kntu.ac.ir Abstract: We study some common results in C*-algebra-valued S -metric spaces. We also present an interesting application of an existing and unique result for one type of integral equation. Keywords: integral equation; C* algebra; S -metric space; common fixed point; compatible; weakly compatible MSC: 34A12; 47H10; 54H25 1. Introduction A metric space is suitable for those interested in analysis, mathematical physics, or applied sciences. Thus, various extensions of metric spaces have been studied, and several results related to the existence of fixed points were obtained (see [1–3]). In 2014, Ma et al. introduced C*-algebra-valued metric spaces [4], and in 2015, they introduced the concept of C*-algebra-valued b-metric spaces and studied some results in Citation: Razavi, S.S.; Masiha, H.P.; this space [5]. In addition, Razavi and Masiha investigated some common principles in De La Sen, M. Applications in C*-algebra-valued b-metric spaces [6]. Integral Equations through Common Recently, Sedghi et al. defined the concept of an S-metric space [7]. Additionally, Ege Results in C*-Algebra-Valued and Alaca introduced the concept of C*-algebra-valued S-metric spaces [8]. S -Metric Spaces. Axioms 2023, 12, Inspired by the work of Souayah and Mlaiki in [9], we introduced the C*-algebra- 413. https://doi.org/10.3390/ valued S -metric space in [10]. In this paper, we study some common fixed-point principles axioms12050413 in this space. We also investigate the existence and uniqueness of the result for one type of Academic Editors: Behzad integral equation. Djafari-Rouhani and Hsien-Chung Wu 2. Preliminaries Received: 16 February 2023 This section provides a short introduction to some realities about the theory of Revised: 2 April 2023 C* algebras [11]. First, suppose that A is a unital C* algebra with the unit 1 . Set Accepted: 20 April 2023 A = ft 2 A : t = t g. The element t 2 A is said to be positive, and we write Published: 24 April 2023 t  0 if and only if t = t and s(t)  [0, ¥), in which 0 in A is the zero element and the A A spectrum of t is s(t). OnA , we can find a natural partial ordering given by u  v if and only if v u  0 . h A We denote with A and A the sets of ft 2 A : t  0 g and ft 2 A : tk = kt , 8k 2 Ag, Copyright: © 2023 by the authors. respectively. Licensee MDPI, Basel, Switzerland. In 2015, Ma et al. [5] introduced the notion of C*-algebra-valued b-metric spaces This article is an open access article as follows: distributed under the terms and conditions of the Creative Commons Definition 1. Let X be a nonempty set and A be a C* algebra. Suppose that k 2 A such that Attribution (CC BY) license (https:// jjkjj  1. A function d : X X ! A is called a C*-algebra-valued b metric on X if for all creativecommons.org/licenses/by/ b u, v, t 2 A, the following apply: 4.0/). Axioms 2023, 12, 413. https://doi.org/10.3390/axioms12050413 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 413 2 of 12 (1) d (u, v)  0 for every u and v in X , and d (u, v) = 0 if and only if u = v; b b (2) d (u, v) = d (v, u); b b (3) d (u, v)  k[d (u, t) + d (t, v)]. b b b Therefore, (X ,A, d ) is a C*-algebra-valued b-metric space (in short, a C*-AV-BM space) with a coefficient k. In 2015, Kalaivani et al. [12] presented the notion of a C*-algebra-valued S-metric space: Definition 2. Assume that X is a nonempty set and A is a C* algebra. A function s : X X X ! A is called a C*-algebra-valued S metric on X if for all u, v, t, a 2 X , the following apply: (1) s(u, v, t)  0 ; (2) s(u, v, t) = 0 if and only if u = v = t; (3) s(u, v, t)  s(u, u, a) + s(v, v, a) + s(t, t, a). Then, (X ,A, s) is a C*-algebra-valued S-metric space (in short, a C*-AV-SM space). In fact, in 2016, Souayah et al. [9] presented the notion of an S -metric space: Definition 3. Assume that X is a nonempty set and s  1 is a given number. A function g : X X X ! [0, ¥) is an S metric on X if for every u, v, t, a 2 X , the following apply: b b (1) g (u, v, t) = 0 if and only if u = v = t; (2) g (u, v, t)  s[g (u, u, a) + g (v, v, a) + g (t, t, a)]. b b b b Then, (X , g ) is called an S -metric space (in short, an S M space) with a coefficient s. b b b Definition 4. An S -metric g is called symmetric if b b g (u, u, v) = g (v, v, u), 8u, v 2 X . b b Razavi and Masiha [10] introduced the notion of a C*-algebra-valued S -metric space as follows: Definition 5. Assume that X is a nonempty set and k 2 A such that jjkjj  1. A function s : X X X ! A is called a C*-algebra-valued S metric on X if for every u, v, t, a 2 X , the b b following apply: (1) s (u, v, t)  0 ; b A (2) s (u, v, t) = 0 if and only if u = v = t; (3) s (u, v, t)  k[s (u, u, a) + s (v, v, a) + s (t, t, a)]. b b b b Then, (X ,A, s ) is called a C*-algebra-valued S -metric space (in short, a C*-AV-S M space) b b b with a coefficient k. Definition 6. A C*-AV-S M s is symmetric if b b s (u, u, v) = s (v, v, u), 8u, v 2 X . b b Under the above definitions, we give an example in a C*-AV-S M space: Example 1. Let X = R and A = M (R) be all 2 2 matrices with the usual operations of addition, scalar multiplication, and matrix multiplication. It is clear that jj Ajj = ( ja j ) å i j i,j=1 Axioms 2023, 12, 413 3 of 12 defines a norm on A, where A = (a ) 2 A.  : A ! A defines an involution on A and where i j A = A. Then, A is a C algebra. For A = (a ) and B = (b ) in A, a partial order on A can be i j i j given as follows: A  B , (a b )  0 8i, j = 1, 2 i j i j Let (X , d) be a b-metric space where, jjkjj  1 and s : X X X ! M (R), fulfilling b 2 d(u, v) + d(v, t) + d(u, t) 0 s (u, v, t) = 0 d(u, v) + d(v, t) + d(u, t) Then, this is a C*-AV-S M space. Now, we check condition (3) of Definition 5: d(u, v) + d(v, t) + d(u, t) 0 s (u, v, t) = 0 d(u, v) + d(v, t) + d(u, t) 2d(u, a) 0 2d(v, a) 0 2d(t, a) 0 k + k + k 0 2d(u, a) 0 2d(v, a) 0 2d(t, a) d(u, a) 0 d(v, a) 0 d(t, a) 0 = k[2 + 2 + 2 ] 0 d(u, a) 0 d(v, a) 0 d(t, a) = k[s (u, u, a) + s (v, v, a) + s (t, t, a)] b b b Thus, for all u, v, t, a 2 X , (X ,A, s ) is a C*-AV-S M space. b b 3. Definitions and Basic Properties We define some concepts in a C*-AV-S M space and present some lemmas which will be needed in the follow-up: Definition 7. Let (X ,A, s ) be a C*-AV-S M space and fu g be a sequence in X : b b n (1) If jjs (u , u , u)jj ! 0, where n ! ¥, then fu g converges to u, and we present it with n n n lim u = u. n!¥ n (2) If for all p 2 N, jjs (u , u , u )jj ! 0, where n ! ¥, then fu g is a Cauchy sequence n+ p n+ p n n in X . (3) If every Cauchy sequence is convergent inX , then (X ,A, s ) is a complete C*-AV-S M space. b b Definition 8. Suppose that (X ,A, s ) and (X ,A , s ) are C*-AV-S M spaces, and let 1 1 b b b f : (X ,A, s ) ! (X ,A , s ) be a function. Then, f is continuous at a point u 2 X if, for every b 1 1 b sequence, fu g in X , s (u , u , u) ! 0 , (n ! ¥) implies s ( f (u ), f (u ), f (u)) ! 0 , n n n n n b A b A where n ! ¥. A function f is continuous at X if and only if it is continuous at all u 2 X . The next lemmas will be used tacitly in the follow-up: Lemma 1 ([13]). Suppose that A is a unital C* algebra with a unit 1 : 1) If fu g  A and lim u = 0 , then for any u 2 A, lim u u u = 0 . n n!¥ n n!¥ n A A n=1 0 0 0 2) If u, v 2 A and t 2 A , then u  v yields tu  tv, in which A = A \A . h + + + 3) If u 2 A with jjujj < , then 1 u is invertible, and jju(1 u) jj < 1. + A A 4) If u, v 2 A such that uv = vu, then uv  0 . Lemma 2. Let (X ,A, s ) be a symmetric C*-AV-S M space and fu g be a sequence in X . If b b n fu g converges to u and v, then u = v. n Axioms 2023, 12, 413 4 of 12 Proof. Let lim u = u and lim u = v. Under condition (3) of Definitions 5 and 6, n!¥ n n!¥ n we have s (u, u, v)  k[s (u, u, u ) + s (u, u, u ), s (v, v, u )] n n n b b b b = k[s (u , u , u) + s (u , u , u) + s (u , u , v)] n n n n n n b b b = 2ks (u , u , u) + ks (u , u , v) b n n b n n ! 0 , (n ! ¥). as jjs (u, u, v)jj = 0 if and only if u = v. Due to the following definition, we extend the concept of compatible mappings of Jungck [14] to C*-algebra-valued metric spaces: Definition 9. Let (X ,A, s ) be a C*-AV-S M space. A pairfy, jg is called compatible if and only b b if s (yju , yju , jyu ) ! 0 whenever fu g is a sequence in X such that lim yu = n n n n n!¥ n b A lim ju = u for some u 2 X . n!¥ n Definition 10. A point u 2 X is a coincidence point of y and j if and only if yu = ju. Herein, t = yu = ju is a point of coincidence of y and j. If y and j commute at all of their coincidence points, then they are weakly compatible, but the converse is not true. If mappings T and S are compatible, then they are weakly compatible in metric spaces. Provided that the converse is not true [15], the same holds for the C*-algebra-valued S -metric spaces: Theorem 1. If mappings y and j on the C*-AV-S M space (X ,A, s ) are compatible, then they b b are weakly compatible. Proof. Let yu = ju for some u 2 X . It suffices to present that yju = jyu. By setting u  u for all n 2 N, then lim yu = lim ju . Since y and j are compatible, we n n!¥ n n!¥ n achieve lim s (yju , yju , jyu ) ! 0 as n ! ¥; that is, jjs (yju , yju , jyu )jj n!¥ b n n n A b n n n ! 0, where n ! ¥. Hence, s (yju , yju , jyu ) = 0 , which means yju = jyu. n n n b A The subsequent lemma can be seen in [15]: Lemma 3 ([15]). Let y and j be weakly compatible mappings of a setX . If y and j have a unique point of coincidence, then it is the unique common fixed point (FP) of y and j. 4. Main Results Here, we present an extension of the common principles for the mappings which applies to variant contractive conditions in complete symmetric C*-valued S -metric spaces: Theorem 2. Suppose that (X ,A, s ) is a a complete symmetric C*-AV-S M space and b b y, j : X ! X satisfies s (yu, yu, jv)  a s (u, u, v)a, (1) b b for all u, v 2 X , where a 2 A in which jjajj < 1. Hence, y and j have a unique common FP in X . Axioms 2023, 12, 413 5 of 12 Proof. Suppose that u 2 X andfu g is a sequence inX such that u = yu , u = 0 n 2n+1 2n 2n+2 ju . From Equation (1), we have 2n+1 s (u , u , u ) = s (ju , ju , yu ) b 2n+2 2n+2 2n+1 b 2n+1 2n+1 2n a s (u , u , u )a 2n+1 2n+1 2n 2 2 (a ) s (u , u , u )(a) 2n 2n 2n1 2n+1 2n+1 (a ) s (u , u , u )(a) , 1 1 0 By remembering the property where if t, k 2 A , then t  k yields u tu  u ku, we see the following for each n 2 N: 2n 2n s (u , u , u )  (a ) s (u , u , u )(a) . b 2n+1 2n+1 2n b 1 1 0 Similarly, we have n n s (u , u , u )  (a ) s (u , u , u )(a) . b n+1 n+1 n b 1 1 0 Let s (u , u , u ) = B for some B 2 A . For any p 2 N, we achieve b 1 1 0 0 0 + s (u , u , u )  b[s (u , u , u ) + s (u , u , u ) b n+ p n+ p n b n+ p n+ p n+ p1 b n+ p n+ p n+ p1 + s (u , u , u )] n n b n+ p1 = 2bs (u , u , u ) + bs (u , u , u ) n+ p n+ p n n b n+ p1 b n+ p1 = 2bs (u , u , u ) + bs (u , u , u ) b n+ p n+ p n+ p1 b n+ p1 n+ p1 n 2bs (u , u , u ) b n+ p n+ p n+ p1 + 2b s (u , u , u ) b n+ p1 n+ p1 n+ p2 + b s (u , u , u ) n+ p2 n+ p2 n 2bs (u , u , u ) b n+ p n+ p n+ p1 + 2b s (u , u , u ) b n+ p1 n+ p1 n+ p2 + 2b s (u , u , u ) n+ p2 n+ p2 n+ p3 + + 2b s (u , u , u ) b n+1 n+1 n n+ p1 n+ p1 2b(a ) s (u , u , u )(a) b 1 1 0 2  n+ p2 n+ p2 + 2b (a ) s (u , u , u )(a) b 1 1 0 3  n+ p3 n+ p3 + 2b (a ) s (u , u , u )(a) b 1 1 0 p  n n + + 2b (a ) s (u , u , u )(a) b 1 1 0 p1 k  n+ pk n+ pk 2 b (a ) s (u , u , u )(a) å b 1 1 0 k=1 p1 k  n+ pk n+ pk = 2 b (a ) B (a) å 0 k=1 p1 1 1 k k n+ pk n+ pk 2 2 2 2 = 2 ((a ) b B )(B b a ) 0 0 k=1 Axioms 2023, 12, 413 6 of 12 p1 1 1 k k 2 n+ pk  2 n+ pk 2 2 2 (B b a ) (B b a ) 0 0 k=1 p1 n+ pk 2 2 jjB b a jj 1 å A k=1 p1 2 2(n+ pk) k 2jjB jj jjajj jjbjj 1 å A k=1 p 2(n+1) jjbjj jjajj 2jjB jj 1 0 A jjbjjjjajj ! 0 (n ! ¥), in which 1 is the unit element in A. As fu g is a Cauchy sequence in X , and X is complete, there exists u 2 X such n=1 that lim u = u. n!¥ n By using condition (3) of Definitions 5 and 6 as well as Equation (1), we have s (u, u, ju)  b[s (u, u, u ) + s (u, u, u ) + s (u , u , ju)] 2n+1 2n+1 2n+1 2n+1 b b b b = 2bs (u, u, u ) + bs (u , u , ju) b 2n+1 b 2n+1 2n+1 = 2bs (u , u , u) + bs (yu , yu , ju) 2n+1 2n+1 2n 2n b b 2bs (u , u , u) + ba s (u , u , u)a b 2n+1 2n+1 b n n ! 0 (n ! ¥). Hence, ju = u. Again, we note that 0  s (yu, yu, u) = s (yu, yu, ju)  a s (u, u, u)a = 0 , A b b b A In other words, s (yu, yu, u) = 0 , and hence yu = u. b A For the uniqueness of the common FP in X , let there be another point v 2 X such that yv = jv = v. From Equation (1), we achieve 0  s (u, u, v) = s (yu, yu, yv)  a s (u, u, v)a A b b b which, together with jjajj < 1, yields that 0  jjs (u, u, v)jj  jja s (u, u, v)ajj b b jja jjjjs (u, u, v)jjjjajj jjajj jjs (u, u, v)jj jjs (u, u, v)jj Thus, jjs (u, u, v)jj = 0 and s (u, u, v) = 0 , which gives u = v. Hence, y and j have b b a unique common FP in X . With the proof of Theorem 2, the relevant results are as follows: Corollary 1. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space, and suppose b b that y, j : X ! X represent two mappings such that jjs (yu, yu, jv)  jjajjjjs (u, u, v)jj, b b for all u, v 2 X , where a 2 A and jjajj < 1. Then, y and j have a unique common FP in X . Axioms 2023, 12, 413 7 of 12 Corollary 2. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and the mapping b b y : X ! X satisfies m m n s (y u, y u, y v)  a s (u, u, v)a, b b for all u, v 2 X , in which a 2 A and jjajj < 1, and m and n are fixed positive integers. Thus, y has a unique FP in X . m n Proof. Set y = y and j = y in Equation (1). The result is obtained using Theorem 2. Remark 1. By substituting y = j into Equation (1), we have s (yu, yu, yv)  a s (u, u, v)a, b b for all u, v 2 X , where a 2 A and jjajj < 1. Thus, we conclude the next corollary. Corollary 3. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and the mapping b b y : X ! X satisfies s (yu, yu, yv)  a s (u, u, v)a, b b for all u, v 2 X , where a 2 A and jjajj < 1. Then, y has a unique FP in X . Theorem 3. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  a s (u, u, v)a, (2) b b for all u, v 2 X , where a 2 A and jjajj < 1. If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . Additionally, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. Suppose that u 2 X is arbitrary. Choose u 2 X such that ju = yu . This is 0 1 1 0 correct because R(y)  R(j). Let u 2 X such that ju = yu . In the same way, we obtain 2 2 1 a sequence fu g in X satisfying ju = yu . Therefore, with Equation (2), we have n n n1 n=1 s (ju , ju , ju ) = s (yu , yu , yu ) n+1 n+1 n n n n1 b b a s (ju , ju , ju )a b n n n1 n n (a ) s (ju , ju , ju )(a) , 1 1 0 which shows that fju g is a Cauchy sequence in R(j). Since R(j) is complete in X , n=1 there exists q 2 X such that lim ju = jq, and thus n!¥ n s (ju , ju , yq) = s (yu , yu , yq) b n n b n1 n1 a s (ju , ju , jq)a, b n1 n1 From lim ju = jq and Lemma 1, we obtain a s (ju , ju , jq)a ! 0 as n!¥ n b n1 n1 A n ! ¥, and then lim ju = yq. Lemma 2 yields that jq = yq. If there is an element w n!¥ n in X such that yw = jw, then Equation (2) yields s (jq, jq, jw) = s (yq, yq, yw)  a s (jq, jq, jw)a, b b b In the same way as in Theorem 2, we obtain jq = jw because 0  jjs (jq, jq, jw)jj  jjajj jjs (jq, jq, jw)jj b b ) jjs (jq, jq, jw)jj = 0 ) s (jq, jq, jw) = 0 ) jq = jw. b b A Axioms 2023, 12, 413 8 of 12 Hence, y and j have a unique point of coincidence in X . Through Lemma 3, we conclude that y and j have a unique common FP in X . Theorem 4. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  as (yu, yu, ju) + as (yv, yv, jv), (3) b b b 0 1 for all u, v 2 X , where a 2 A and jjajj < . If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . In addition, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. As in Theorem 3, we select fu g in X and set ju = yu . Therefore, through n n n1 n=1 Equation (3), we have s (ju , ju , ju ) = s (yu , yu , yu ) b n+1 n+1 n b n n n1 as (yu , yu , ju ) + as (yu , yu , ju ) n n n b b n1 n1 n1 = as (ju , ju , ju ) + as (ju , ju , ju ) b n+1 n+1 n b n n n1 Thus, we obtain (1 a)s (ju , ju , ju )  as (ju , ju , ju ) b n+1 n+1 n b n n n1 1 ¥ 1 n Since jjajj < , then 1 a is invertible, and (1 a) = a which, together with n=0 0 1 0 a 2 A , yields (1 a) a 2 A . Lemma 1’s condition (2) leads to + + s (ju , ju , ju )  ts (ju , ju , ju ), (4) n n n b n+1 n+1 b n1 1 0 where t = (1 a) a 2 A and jjtjj < 1. Now, by induction and the use of Lemma 1’s condition (2), we obtain s (ju , ju , ju )  t s (ju , ju , ju ). n+1 n+1 n 1 1 0 b b For each m  1, p  1, and b 2 A where jjbjj > 1, we have s (ju , ju , ju )  b[s (ju , ju , ju ) m+ p m+ p m m+ p m+ p m+ p1 b b + s (ju , ju , ju ) b m+ p m+ p m+ p1 + s (ju , ju , ju )] b m+ p1 m+ p1 = 2bs (ju , ju , ju ) m+ p m+ p b m+ p1 + s (ju , ju , ju ) m+ p1 m+ p1 m 2bs (ju , ju , ju ) b m+ p m+ p m+ p1 + 2b s (ju , ju , ju ) b m+ p1 m+ p1 m+ p2 + b s (ju , ju , ju ) m+ p2 m+ p2 m 2bs (ju , ju , ju ) b m+ p m+ p m+ p1 + 2b s (ju , ju , ju ) b m+ p1 m+ p1 m+ p2 + 2b s (ju , ju , ju ) m+ p2 m+ p2 m+ p3 + + 2b s (ju , ju , ju ) b m+1 m+1 m Axioms 2023, 12, 413 9 of 12 m+ p1 2bt s (ju , ju , ju ) 1 1 0 2 m+ p2 + 2b t s (ju , ju , ju ) 1 1 0 3 m+ p3 + 2b t s (ju , ju , ju ) 1 1 0 p m + + 2b t s (ju , ju , ju ) b 1 1 0 m+ p1 2 m+ p2 = 2bt B + 2b t B 0 0 3 m+ p3 p m + 2b t B + + 2b t B 0 0 k m+ pk = 2 b t B k=1 m+ pk 2 2 = 2 jB t b j k=1 k m+ pk 2jjB jj jjbjj jjtjj 1 0 å A k=1 p m+1 jjbjj jjtjj 2jjB jj 1 jjtjjjjbjj ! 0, (m ! ¥), where B = s (ju , ju , ju ). Hence, fju g is a Cauchy sequence in R(j). Since 0 0 n b 1 1 n=0 R(j) is complete, there exists q 2 X such that lim ju = jq. Again, according to n!¥ n Equation (4), we have s (ju , ju , yq) = s (yu , yu , yq)  ts (ju , ju , jq) n n b b n1 n1 b n1 n1 This implies that lim ju = yq. Under Lemma 2, yq = jq. Therefore, y and j n!¥ n have a point of coincidence in X . Here, we prove the uniqueness of points of coincidence. For this, let there be p 2 X such that y p = j p. By applying Equation (3), we have s (j p, j p, jq) = s (y p, y p, yq)  as (y p, y p, j p) + as (yq, yq, jq), b b b b This implies thatjjs (j p, j p, jq)jj = 0, and thus j p = jq. Therefore, under Lemma 3, y and j have a unique common FP in X . Theorem 5. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  as (yu, yu, jv) + as (ju, ju, yv), (5) b b b 0 1 for every u, v 2 X , in which a 2 A and jjabjj < . If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . Additionally, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. As in Theorem 3, we select fu g in X and set ju = yu . Therefore, under n n n1 n=1 Equation (5), we have s (ju , ju , ju ) = s (yu , yu , yu ) n n n b n+1 n+1 b n1 as (yu , yu , ju ) + as (ju , ju , yu ) b n n n1 b n n n1 = as (ju , ju , ju ) + as (ju , ju , ju ) n n n b n+1 n+1 n1 b ab[2s (ju , ju , ju ) + s (ju , ju , ju )] n+1 n+1 n n n n1 b b = 2abs (ju , ju , ju ) + abs (ju , ju , ju ) n n n b n+1 n+1 b n1 Axioms 2023, 12, 413 10 of 12 Thus, we obtain (1 2ab)s (ju , ju , ju )  abs (ju , ju , ju ), n+1 n+1 n n n n1 b b Therefore, we have s (ju , ju , ju )  (1 2ab) abs (ju , ju , ju ), n+1 n+1 n n n n1 b b and consequently s (ju , ju , ju )  ts (ju , ju , ju ), n+1 n+1 n n n n1 b b 1 0 where t = (1 2ab) ab 2 A and jjtjj < 1. Similar to the process in Theorem 4, we find that y and j have a point of coincidence yq in X . Here, we prove the uniqueness of the points of coincidence. For this, let there be p 2 X such that y p = j p. By applying Equation (5), we obtain s (j p, j p, jq) = s (y p, y p, yq) b b as (y p, y p, jq) + as (j p, j p, yq) b b = as (j p, j p, jq) + as (j p, j p, jq), b b In other words, we have s (j p, j p, jq)  (1 a) as (j p, j p, jq). b b Since jj(1 a) ajj < 1, this implies that jjs (j p, j p, jq)jj = 0, and thus j p = jq. Therefore, Lemma 3 implies that y and j have a unique common FP in X . If we choose j = id in Theorem 5, then we obtain R(j) = X , and y is weakly compatible with j. We also have the following result: Corollary 4. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and b b y : X ! X satisfies s (yu, yu, yv)  as (yu, yu, v) + as (yv, yv, u), b b b 0 1 for all u, v 2 X , where a 2 A and jjabjj < . Hence, y has a unique FP in X . + 3 5. Application in Integral Equations Let us use the following equations: ll x(m) = (T (m, n, x(n))dn + J(m), m 2 E Z (6) x(m) = (T (m, n, x(n))dn + J(m), m 2 E in which E is a Lebesgue measurable set where m(E) < ¥. In fact, we suppose that X = L (E) presents the class of essentially bounded measur- able functions on E , where E is a Lebesgue measurable set such that m(E) < ¥. One may consider the functions T , T , a, b to fulfill the following assumptions: 1 2 (i) T , T : E E  R ! R are integrable. In addition, an integrable function a is from 0 ¥ E E to R , and J 2 L (E). (ii) There exists ` 2 (0, 1) such that jT (m, n, x) T (m, n, y)j  `ja(m, n)jjx yj, 1 2 for m, n 2 E and x, y 2 R. Axioms 2023, 12, 413 11 of 12 (iii) sup ja(m, n)jdn  1. m2E E Theorem 6. Let assumptions (i–iii) hold. Hence, the integral in Equation (6) has a unique common solution in L (E). ¥ 2 Proof. Suppose that X = L (E) and B(L (E)) is a set of bounded linear operators on a 2 2 Hilbert space L (E). We equipX with the S metric s : X X X ! B(L (E)), which is b b ascertained by s (a, b, g) = M p , b (jagj+jbgj) where M p is the multiplication operator on L (E) ascertained by (jagj+jbgj) M (a) = h.a ; a 2 L (E). Therefore, (X , B(L (E)), s ) is a complete C*-AV-S M space. We can describe the b b self-mappings Y, F : X ! X as follows: Yx(m) = T (m, n, x(n))dn + J(m), Fx(m) = T (m, n, x(n))dn + J(m), for each m 2 E . Therefore, we have s (Yx, Yx, Fy) = M p . (jYxFyj+jYxFyj) We can obtain jjs (Yx, Yx, Fy)jj = sup h M p h, hi (jYxFyj+jYxFyj) jjhjj=1 = sup h M p h, hi (2jYxFyj) jjhjj=1 = sup h2 M p h, hi jYxFyj jjhjj=1 p p = sup (2 jYx Fyj )h(t)h(t)dt jjhjj=1 Z Z p p 2 2 sup [ jT (m, n, x(n)) T (m, n, y(n))j] jh(t)j dt 1 2 E E jjhjj=1 Z Z p p 2 2 sup [ `ja(m, n)(x(n) y(n))jdn] jh(t)j dt E E jjhjj=1 Z Z p p p 2 2 ` sup [ ja(m, n)jdn] jh(t)j dt.jjx yjj E E jjhjj=1 Z Z 2 p ` sup ja(m, n)jdn. sup jh(t)j dt2 jjx yjj E E m2E jjhjj=1 2 `jjx yjj = `jj2(x y)jj = `jj M pjj (jxyj+jxyj) = jjajjjjs (x, x, y)jj By setting a = `1 , then a 2 B(L (E)) and jjajj = ` < 1. Therefore, Corollary 1 B(L (E)) implies the result. Axioms 2023, 12, 413 12 of 12 Author Contributions: S.S.R.: Investigation, writing draft and final articles; H.P.M.: Investigation, writing draft; M.D.L.S.: checking draft and final version, funding, Project direction. All authors have read and agreed to the published version of the manuscript. Funding: Basque Government, Grant IT1155-22. Conflicts of Interest: The authors declare no conflict of interest. References 1. Fagin, R.; Stockmeyer, L. Relaxing the triangle inequality in pattern matching. Int. J. Comput. Vis. 1998, 30, 219–231. [CrossRef] 2. Jankovic, ´ S.; Golubovic, ´ Z.; Radenovic, ´ S. Compatible and weakly compatible mappings in cone metric spaces. Math. Comput. Model. 2010, 52, 1728–1738. [CrossRef] 3. Kirk, W.; Shahzad, N. Fixed Point Theory in Distance Spaces; Springer: Cham, Switzerland, 2014. 4. Ma, Z.H.; Jiang, L.N.; Sun, H.K. C*-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2014, 2014, 206. [CrossRef] 5. Ma, Z.H.; Jiang, L.N. C*-algebra-valued b-metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2015, 2015, 222. [CrossRef] 6. Razavi, S.S.; Masiha, H.P. Common fixed point theorems in C*-algebra valued b-metric spaces with applications to integral equations. Fixed Point Theory 2019, 20, 649–662. [CrossRef] 7. Sedghi, S.; Shobe, N.; Aliouche, A. A generalization of fixed point theorem in S-metric spaces. Mat. Vesnik. 2012, 64, 258–266. 8. Ege, M.E.; Alaca, C. C -algebra-valued S-metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2018, 66, 165–177. 9. Souayah, N.; Mlaiki, N. A fixed point theorem in S -metric space. J. Math. Comput. Sci. 2016, 16, 131–139. [CrossRef] 10. Razavi, S.S.; Masiha, H.P. C*-Algebra-Valued S -Metric Spaces and Applications to Integral Equations. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran, 2023, in progress. 11. Murphy, G.J. C*-Algebras and Operator Theory; Academic Press: London, UK, 1990. 12. Kalaivani, C.; Kalpana, G. Fixed point theorems in C*-algebra-valued S-metric spaces with some applications. UPB Sci. Bull. Ser. A 2018, 80, 93–102. 13. Lakshmikantham, V.; Ciric, ´ L.B. Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70, 4341–4349. [CrossRef] 14. Jungck, G. Compatible mappings and common fixed points. Int. J. Math. Sci. 1986, 9, 771–779. [CrossRef] 15. Abbas, M.; Jungck, G. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 2008, 341, 416–420. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Axioms Multidisciplinary Digital Publishing Institute

Applications in Integral Equations through Common Results in C*-Algebra-Valued Sb-Metric Spaces

Razavi, S. S.; Masiha, H. P.; De La Sen, Manuel
Axioms , Volume 12 (5) – Apr 24, 2023
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axioms Article Applications in Integral Equations through Common Results in C*-Algebra-Valued S -Metric Spaces 1 1, 2 S. S. Razavi , H. P. Masiha * and Manuel De La Sen Faculty of Mathematics, K. N. Toosi University of Technology, Tehran 16315-1618, Iran; srazavi@mail.kntu.ac.ir Institute of Research and Development of Processes of the Basque Country, 48940 Leioa, Spain; manuel.delasen@ehu.eus * Correspondence: masiha@kntu.ac.ir Abstract: We study some common results in C*-algebra-valued S -metric spaces. We also present an interesting application of an existing and unique result for one type of integral equation. Keywords: integral equation; C* algebra; S -metric space; common fixed point; compatible; weakly compatible MSC: 34A12; 47H10; 54H25 1. Introduction A metric space is suitable for those interested in analysis, mathematical physics, or applied sciences. Thus, various extensions of metric spaces have been studied, and several results related to the existence of fixed points were obtained (see [1–3]). In 2014, Ma et al. introduced C*-algebra-valued metric spaces [4], and in 2015, they introduced the concept of C*-algebra-valued b-metric spaces and studied some results in Citation: Razavi, S.S.; Masiha, H.P.; this space [5]. In addition, Razavi and Masiha investigated some common principles in De La Sen, M. Applications in C*-algebra-valued b-metric spaces [6]. Integral Equations through Common Recently, Sedghi et al. defined the concept of an S-metric space [7]. Additionally, Ege Results in C*-Algebra-Valued and Alaca introduced the concept of C*-algebra-valued S-metric spaces [8]. S -Metric Spaces. Axioms 2023, 12, Inspired by the work of Souayah and Mlaiki in [9], we introduced the C*-algebra- 413. https://doi.org/10.3390/ valued S -metric space in [10]. In this paper, we study some common fixed-point principles axioms12050413 in this space. We also investigate the existence and uniqueness of the result for one type of Academic Editors: Behzad integral equation. Djafari-Rouhani and Hsien-Chung Wu 2. Preliminaries Received: 16 February 2023 This section provides a short introduction to some realities about the theory of Revised: 2 April 2023 C* algebras [11]. First, suppose that A is a unital C* algebra with the unit 1 . Set Accepted: 20 April 2023 A = ft 2 A : t = t g. The element t 2 A is said to be positive, and we write Published: 24 April 2023 t  0 if and only if t = t and s(t)  [0, ¥), in which 0 in A is the zero element and the A A spectrum of t is s(t). OnA , we can find a natural partial ordering given by u  v if and only if v u  0 . h A We denote with A and A the sets of ft 2 A : t  0 g and ft 2 A : tk = kt , 8k 2 Ag, Copyright: © 2023 by the authors. respectively. Licensee MDPI, Basel, Switzerland. In 2015, Ma et al. [5] introduced the notion of C*-algebra-valued b-metric spaces This article is an open access article as follows: distributed under the terms and conditions of the Creative Commons Definition 1. Let X be a nonempty set and A be a C* algebra. Suppose that k 2 A such that Attribution (CC BY) license (https:// jjkjj  1. A function d : X X ! A is called a C*-algebra-valued b metric on X if for all creativecommons.org/licenses/by/ b u, v, t 2 A, the following apply: 4.0/). Axioms 2023, 12, 413. https://doi.org/10.3390/axioms12050413 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 413 2 of 12 (1) d (u, v)  0 for every u and v in X , and d (u, v) = 0 if and only if u = v; b b (2) d (u, v) = d (v, u); b b (3) d (u, v)  k[d (u, t) + d (t, v)]. b b b Therefore, (X ,A, d ) is a C*-algebra-valued b-metric space (in short, a C*-AV-BM space) with a coefficient k. In 2015, Kalaivani et al. [12] presented the notion of a C*-algebra-valued S-metric space: Definition 2. Assume that X is a nonempty set and A is a C* algebra. A function s : X X X ! A is called a C*-algebra-valued S metric on X if for all u, v, t, a 2 X , the following apply: (1) s(u, v, t)  0 ; (2) s(u, v, t) = 0 if and only if u = v = t; (3) s(u, v, t)  s(u, u, a) + s(v, v, a) + s(t, t, a). Then, (X ,A, s) is a C*-algebra-valued S-metric space (in short, a C*-AV-SM space). In fact, in 2016, Souayah et al. [9] presented the notion of an S -metric space: Definition 3. Assume that X is a nonempty set and s  1 is a given number. A function g : X X X ! [0, ¥) is an S metric on X if for every u, v, t, a 2 X , the following apply: b b (1) g (u, v, t) = 0 if and only if u = v = t; (2) g (u, v, t)  s[g (u, u, a) + g (v, v, a) + g (t, t, a)]. b b b b Then, (X , g ) is called an S -metric space (in short, an S M space) with a coefficient s. b b b Definition 4. An S -metric g is called symmetric if b b g (u, u, v) = g (v, v, u), 8u, v 2 X . b b Razavi and Masiha [10] introduced the notion of a C*-algebra-valued S -metric space as follows: Definition 5. Assume that X is a nonempty set and k 2 A such that jjkjj  1. A function s : X X X ! A is called a C*-algebra-valued S metric on X if for every u, v, t, a 2 X , the b b following apply: (1) s (u, v, t)  0 ; b A (2) s (u, v, t) = 0 if and only if u = v = t; (3) s (u, v, t)  k[s (u, u, a) + s (v, v, a) + s (t, t, a)]. b b b b Then, (X ,A, s ) is called a C*-algebra-valued S -metric space (in short, a C*-AV-S M space) b b b with a coefficient k. Definition 6. A C*-AV-S M s is symmetric if b b s (u, u, v) = s (v, v, u), 8u, v 2 X . b b Under the above definitions, we give an example in a C*-AV-S M space: Example 1. Let X = R and A = M (R) be all 2 2 matrices with the usual operations of addition, scalar multiplication, and matrix multiplication. It is clear that jj Ajj = ( ja j ) å i j i,j=1 Axioms 2023, 12, 413 3 of 12 defines a norm on A, where A = (a ) 2 A.  : A ! A defines an involution on A and where i j A = A. Then, A is a C algebra. For A = (a ) and B = (b ) in A, a partial order on A can be i j i j given as follows: A  B , (a b )  0 8i, j = 1, 2 i j i j Let (X , d) be a b-metric space where, jjkjj  1 and s : X X X ! M (R), fulfilling b 2 d(u, v) + d(v, t) + d(u, t) 0 s (u, v, t) = 0 d(u, v) + d(v, t) + d(u, t) Then, this is a C*-AV-S M space. Now, we check condition (3) of Definition 5: d(u, v) + d(v, t) + d(u, t) 0 s (u, v, t) = 0 d(u, v) + d(v, t) + d(u, t) 2d(u, a) 0 2d(v, a) 0 2d(t, a) 0 k + k + k 0 2d(u, a) 0 2d(v, a) 0 2d(t, a) d(u, a) 0 d(v, a) 0 d(t, a) 0 = k[2 + 2 + 2 ] 0 d(u, a) 0 d(v, a) 0 d(t, a) = k[s (u, u, a) + s (v, v, a) + s (t, t, a)] b b b Thus, for all u, v, t, a 2 X , (X ,A, s ) is a C*-AV-S M space. b b 3. Definitions and Basic Properties We define some concepts in a C*-AV-S M space and present some lemmas which will be needed in the follow-up: Definition 7. Let (X ,A, s ) be a C*-AV-S M space and fu g be a sequence in X : b b n (1) If jjs (u , u , u)jj ! 0, where n ! ¥, then fu g converges to u, and we present it with n n n lim u = u. n!¥ n (2) If for all p 2 N, jjs (u , u , u )jj ! 0, where n ! ¥, then fu g is a Cauchy sequence n+ p n+ p n n in X . (3) If every Cauchy sequence is convergent inX , then (X ,A, s ) is a complete C*-AV-S M space. b b Definition 8. Suppose that (X ,A, s ) and (X ,A , s ) are C*-AV-S M spaces, and let 1 1 b b b f : (X ,A, s ) ! (X ,A , s ) be a function. Then, f is continuous at a point u 2 X if, for every b 1 1 b sequence, fu g in X , s (u , u , u) ! 0 , (n ! ¥) implies s ( f (u ), f (u ), f (u)) ! 0 , n n n n n b A b A where n ! ¥. A function f is continuous at X if and only if it is continuous at all u 2 X . The next lemmas will be used tacitly in the follow-up: Lemma 1 ([13]). Suppose that A is a unital C* algebra with a unit 1 : 1) If fu g  A and lim u = 0 , then for any u 2 A, lim u u u = 0 . n n!¥ n n!¥ n A A n=1 0 0 0 2) If u, v 2 A and t 2 A , then u  v yields tu  tv, in which A = A \A . h + + + 3) If u 2 A with jjujj < , then 1 u is invertible, and jju(1 u) jj < 1. + A A 4) If u, v 2 A such that uv = vu, then uv  0 . Lemma 2. Let (X ,A, s ) be a symmetric C*-AV-S M space and fu g be a sequence in X . If b b n fu g converges to u and v, then u = v. n Axioms 2023, 12, 413 4 of 12 Proof. Let lim u = u and lim u = v. Under condition (3) of Definitions 5 and 6, n!¥ n n!¥ n we have s (u, u, v)  k[s (u, u, u ) + s (u, u, u ), s (v, v, u )] n n n b b b b = k[s (u , u , u) + s (u , u , u) + s (u , u , v)] n n n n n n b b b = 2ks (u , u , u) + ks (u , u , v) b n n b n n ! 0 , (n ! ¥). as jjs (u, u, v)jj = 0 if and only if u = v. Due to the following definition, we extend the concept of compatible mappings of Jungck [14] to C*-algebra-valued metric spaces: Definition 9. Let (X ,A, s ) be a C*-AV-S M space. A pairfy, jg is called compatible if and only b b if s (yju , yju , jyu ) ! 0 whenever fu g is a sequence in X such that lim yu = n n n n n!¥ n b A lim ju = u for some u 2 X . n!¥ n Definition 10. A point u 2 X is a coincidence point of y and j if and only if yu = ju. Herein, t = yu = ju is a point of coincidence of y and j. If y and j commute at all of their coincidence points, then they are weakly compatible, but the converse is not true. If mappings T and S are compatible, then they are weakly compatible in metric spaces. Provided that the converse is not true [15], the same holds for the C*-algebra-valued S -metric spaces: Theorem 1. If mappings y and j on the C*-AV-S M space (X ,A, s ) are compatible, then they b b are weakly compatible. Proof. Let yu = ju for some u 2 X . It suffices to present that yju = jyu. By setting u  u for all n 2 N, then lim yu = lim ju . Since y and j are compatible, we n n!¥ n n!¥ n achieve lim s (yju , yju , jyu ) ! 0 as n ! ¥; that is, jjs (yju , yju , jyu )jj n!¥ b n n n A b n n n ! 0, where n ! ¥. Hence, s (yju , yju , jyu ) = 0 , which means yju = jyu. n n n b A The subsequent lemma can be seen in [15]: Lemma 3 ([15]). Let y and j be weakly compatible mappings of a setX . If y and j have a unique point of coincidence, then it is the unique common fixed point (FP) of y and j. 4. Main Results Here, we present an extension of the common principles for the mappings which applies to variant contractive conditions in complete symmetric C*-valued S -metric spaces: Theorem 2. Suppose that (X ,A, s ) is a a complete symmetric C*-AV-S M space and b b y, j : X ! X satisfies s (yu, yu, jv)  a s (u, u, v)a, (1) b b for all u, v 2 X , where a 2 A in which jjajj < 1. Hence, y and j have a unique common FP in X . Axioms 2023, 12, 413 5 of 12 Proof. Suppose that u 2 X andfu g is a sequence inX such that u = yu , u = 0 n 2n+1 2n 2n+2 ju . From Equation (1), we have 2n+1 s (u , u , u ) = s (ju , ju , yu ) b 2n+2 2n+2 2n+1 b 2n+1 2n+1 2n a s (u , u , u )a 2n+1 2n+1 2n 2 2 (a ) s (u , u , u )(a) 2n 2n 2n1 2n+1 2n+1 (a ) s (u , u , u )(a) , 1 1 0 By remembering the property where if t, k 2 A , then t  k yields u tu  u ku, we see the following for each n 2 N: 2n 2n s (u , u , u )  (a ) s (u , u , u )(a) . b 2n+1 2n+1 2n b 1 1 0 Similarly, we have n n s (u , u , u )  (a ) s (u , u , u )(a) . b n+1 n+1 n b 1 1 0 Let s (u , u , u ) = B for some B 2 A . For any p 2 N, we achieve b 1 1 0 0 0 + s (u , u , u )  b[s (u , u , u ) + s (u , u , u ) b n+ p n+ p n b n+ p n+ p n+ p1 b n+ p n+ p n+ p1 + s (u , u , u )] n n b n+ p1 = 2bs (u , u , u ) + bs (u , u , u ) n+ p n+ p n n b n+ p1 b n+ p1 = 2bs (u , u , u ) + bs (u , u , u ) b n+ p n+ p n+ p1 b n+ p1 n+ p1 n 2bs (u , u , u ) b n+ p n+ p n+ p1 + 2b s (u , u , u ) b n+ p1 n+ p1 n+ p2 + b s (u , u , u ) n+ p2 n+ p2 n 2bs (u , u , u ) b n+ p n+ p n+ p1 + 2b s (u , u , u ) b n+ p1 n+ p1 n+ p2 + 2b s (u , u , u ) n+ p2 n+ p2 n+ p3 + + 2b s (u , u , u ) b n+1 n+1 n n+ p1 n+ p1 2b(a ) s (u , u , u )(a) b 1 1 0 2  n+ p2 n+ p2 + 2b (a ) s (u , u , u )(a) b 1 1 0 3  n+ p3 n+ p3 + 2b (a ) s (u , u , u )(a) b 1 1 0 p  n n + + 2b (a ) s (u , u , u )(a) b 1 1 0 p1 k  n+ pk n+ pk 2 b (a ) s (u , u , u )(a) å b 1 1 0 k=1 p1 k  n+ pk n+ pk = 2 b (a ) B (a) å 0 k=1 p1 1 1 k k n+ pk n+ pk 2 2 2 2 = 2 ((a ) b B )(B b a ) 0 0 k=1 Axioms 2023, 12, 413 6 of 12 p1 1 1 k k 2 n+ pk  2 n+ pk 2 2 2 (B b a ) (B b a ) 0 0 k=1 p1 n+ pk 2 2 jjB b a jj 1 å A k=1 p1 2 2(n+ pk) k 2jjB jj jjajj jjbjj 1 å A k=1 p 2(n+1) jjbjj jjajj 2jjB jj 1 0 A jjbjjjjajj ! 0 (n ! ¥), in which 1 is the unit element in A. As fu g is a Cauchy sequence in X , and X is complete, there exists u 2 X such n=1 that lim u = u. n!¥ n By using condition (3) of Definitions 5 and 6 as well as Equation (1), we have s (u, u, ju)  b[s (u, u, u ) + s (u, u, u ) + s (u , u , ju)] 2n+1 2n+1 2n+1 2n+1 b b b b = 2bs (u, u, u ) + bs (u , u , ju) b 2n+1 b 2n+1 2n+1 = 2bs (u , u , u) + bs (yu , yu , ju) 2n+1 2n+1 2n 2n b b 2bs (u , u , u) + ba s (u , u , u)a b 2n+1 2n+1 b n n ! 0 (n ! ¥). Hence, ju = u. Again, we note that 0  s (yu, yu, u) = s (yu, yu, ju)  a s (u, u, u)a = 0 , A b b b A In other words, s (yu, yu, u) = 0 , and hence yu = u. b A For the uniqueness of the common FP in X , let there be another point v 2 X such that yv = jv = v. From Equation (1), we achieve 0  s (u, u, v) = s (yu, yu, yv)  a s (u, u, v)a A b b b which, together with jjajj < 1, yields that 0  jjs (u, u, v)jj  jja s (u, u, v)ajj b b jja jjjjs (u, u, v)jjjjajj jjajj jjs (u, u, v)jj jjs (u, u, v)jj Thus, jjs (u, u, v)jj = 0 and s (u, u, v) = 0 , which gives u = v. Hence, y and j have b b a unique common FP in X . With the proof of Theorem 2, the relevant results are as follows: Corollary 1. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space, and suppose b b that y, j : X ! X represent two mappings such that jjs (yu, yu, jv)  jjajjjjs (u, u, v)jj, b b for all u, v 2 X , where a 2 A and jjajj < 1. Then, y and j have a unique common FP in X . Axioms 2023, 12, 413 7 of 12 Corollary 2. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and the mapping b b y : X ! X satisfies m m n s (y u, y u, y v)  a s (u, u, v)a, b b for all u, v 2 X , in which a 2 A and jjajj < 1, and m and n are fixed positive integers. Thus, y has a unique FP in X . m n Proof. Set y = y and j = y in Equation (1). The result is obtained using Theorem 2. Remark 1. By substituting y = j into Equation (1), we have s (yu, yu, yv)  a s (u, u, v)a, b b for all u, v 2 X , where a 2 A and jjajj < 1. Thus, we conclude the next corollary. Corollary 3. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and the mapping b b y : X ! X satisfies s (yu, yu, yv)  a s (u, u, v)a, b b for all u, v 2 X , where a 2 A and jjajj < 1. Then, y has a unique FP in X . Theorem 3. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  a s (u, u, v)a, (2) b b for all u, v 2 X , where a 2 A and jjajj < 1. If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . Additionally, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. Suppose that u 2 X is arbitrary. Choose u 2 X such that ju = yu . This is 0 1 1 0 correct because R(y)  R(j). Let u 2 X such that ju = yu . In the same way, we obtain 2 2 1 a sequence fu g in X satisfying ju = yu . Therefore, with Equation (2), we have n n n1 n=1 s (ju , ju , ju ) = s (yu , yu , yu ) n+1 n+1 n n n n1 b b a s (ju , ju , ju )a b n n n1 n n (a ) s (ju , ju , ju )(a) , 1 1 0 which shows that fju g is a Cauchy sequence in R(j). Since R(j) is complete in X , n=1 there exists q 2 X such that lim ju = jq, and thus n!¥ n s (ju , ju , yq) = s (yu , yu , yq) b n n b n1 n1 a s (ju , ju , jq)a, b n1 n1 From lim ju = jq and Lemma 1, we obtain a s (ju , ju , jq)a ! 0 as n!¥ n b n1 n1 A n ! ¥, and then lim ju = yq. Lemma 2 yields that jq = yq. If there is an element w n!¥ n in X such that yw = jw, then Equation (2) yields s (jq, jq, jw) = s (yq, yq, yw)  a s (jq, jq, jw)a, b b b In the same way as in Theorem 2, we obtain jq = jw because 0  jjs (jq, jq, jw)jj  jjajj jjs (jq, jq, jw)jj b b ) jjs (jq, jq, jw)jj = 0 ) s (jq, jq, jw) = 0 ) jq = jw. b b A Axioms 2023, 12, 413 8 of 12 Hence, y and j have a unique point of coincidence in X . Through Lemma 3, we conclude that y and j have a unique common FP in X . Theorem 4. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  as (yu, yu, ju) + as (yv, yv, jv), (3) b b b 0 1 for all u, v 2 X , where a 2 A and jjajj < . If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . In addition, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. As in Theorem 3, we select fu g in X and set ju = yu . Therefore, through n n n1 n=1 Equation (3), we have s (ju , ju , ju ) = s (yu , yu , yu ) b n+1 n+1 n b n n n1 as (yu , yu , ju ) + as (yu , yu , ju ) n n n b b n1 n1 n1 = as (ju , ju , ju ) + as (ju , ju , ju ) b n+1 n+1 n b n n n1 Thus, we obtain (1 a)s (ju , ju , ju )  as (ju , ju , ju ) b n+1 n+1 n b n n n1 1 ¥ 1 n Since jjajj < , then 1 a is invertible, and (1 a) = a which, together with n=0 0 1 0 a 2 A , yields (1 a) a 2 A . Lemma 1’s condition (2) leads to + + s (ju , ju , ju )  ts (ju , ju , ju ), (4) n n n b n+1 n+1 b n1 1 0 where t = (1 a) a 2 A and jjtjj < 1. Now, by induction and the use of Lemma 1’s condition (2), we obtain s (ju , ju , ju )  t s (ju , ju , ju ). n+1 n+1 n 1 1 0 b b For each m  1, p  1, and b 2 A where jjbjj > 1, we have s (ju , ju , ju )  b[s (ju , ju , ju ) m+ p m+ p m m+ p m+ p m+ p1 b b + s (ju , ju , ju ) b m+ p m+ p m+ p1 + s (ju , ju , ju )] b m+ p1 m+ p1 = 2bs (ju , ju , ju ) m+ p m+ p b m+ p1 + s (ju , ju , ju ) m+ p1 m+ p1 m 2bs (ju , ju , ju ) b m+ p m+ p m+ p1 + 2b s (ju , ju , ju ) b m+ p1 m+ p1 m+ p2 + b s (ju , ju , ju ) m+ p2 m+ p2 m 2bs (ju , ju , ju ) b m+ p m+ p m+ p1 + 2b s (ju , ju , ju ) b m+ p1 m+ p1 m+ p2 + 2b s (ju , ju , ju ) m+ p2 m+ p2 m+ p3 + + 2b s (ju , ju , ju ) b m+1 m+1 m Axioms 2023, 12, 413 9 of 12 m+ p1 2bt s (ju , ju , ju ) 1 1 0 2 m+ p2 + 2b t s (ju , ju , ju ) 1 1 0 3 m+ p3 + 2b t s (ju , ju , ju ) 1 1 0 p m + + 2b t s (ju , ju , ju ) b 1 1 0 m+ p1 2 m+ p2 = 2bt B + 2b t B 0 0 3 m+ p3 p m + 2b t B + + 2b t B 0 0 k m+ pk = 2 b t B k=1 m+ pk 2 2 = 2 jB t b j k=1 k m+ pk 2jjB jj jjbjj jjtjj 1 0 å A k=1 p m+1 jjbjj jjtjj 2jjB jj 1 jjtjjjjbjj ! 0, (m ! ¥), where B = s (ju , ju , ju ). Hence, fju g is a Cauchy sequence in R(j). Since 0 0 n b 1 1 n=0 R(j) is complete, there exists q 2 X such that lim ju = jq. Again, according to n!¥ n Equation (4), we have s (ju , ju , yq) = s (yu , yu , yq)  ts (ju , ju , jq) n n b b n1 n1 b n1 n1 This implies that lim ju = yq. Under Lemma 2, yq = jq. Therefore, y and j n!¥ n have a point of coincidence in X . Here, we prove the uniqueness of points of coincidence. For this, let there be p 2 X such that y p = j p. By applying Equation (3), we have s (j p, j p, jq) = s (y p, y p, yq)  as (y p, y p, j p) + as (yq, yq, jq), b b b b This implies thatjjs (j p, j p, jq)jj = 0, and thus j p = jq. Therefore, under Lemma 3, y and j have a unique common FP in X . Theorem 5. Assume that (X ,A, s ) is a complete symmetric C*-AV-S M space and y, b b j : X ! X satisfies s (yu, yu, yv)  as (yu, yu, jv) + as (ju, ju, yv), (5) b b b 0 1 for every u, v 2 X , in which a 2 A and jjabjj < . If R(y), contained in R(j) and R(j), is complete in X , then y and j have a unique point of coincidence in X . Additionally, if y and j are weakly compatible, then y and j have a unique common FP in X . Proof. As in Theorem 3, we select fu g in X and set ju = yu . Therefore, under n n n1 n=1 Equation (5), we have s (ju , ju , ju ) = s (yu , yu , yu ) n n n b n+1 n+1 b n1 as (yu , yu , ju ) + as (ju , ju , yu ) b n n n1 b n n n1 = as (ju , ju , ju ) + as (ju , ju , ju ) n n n b n+1 n+1 n1 b ab[2s (ju , ju , ju ) + s (ju , ju , ju )] n+1 n+1 n n n n1 b b = 2abs (ju , ju , ju ) + abs (ju , ju , ju ) n n n b n+1 n+1 b n1 Axioms 2023, 12, 413 10 of 12 Thus, we obtain (1 2ab)s (ju , ju , ju )  abs (ju , ju , ju ), n+1 n+1 n n n n1 b b Therefore, we have s (ju , ju , ju )  (1 2ab) abs (ju , ju , ju ), n+1 n+1 n n n n1 b b and consequently s (ju , ju , ju )  ts (ju , ju , ju ), n+1 n+1 n n n n1 b b 1 0 where t = (1 2ab) ab 2 A and jjtjj < 1. Similar to the process in Theorem 4, we find that y and j have a point of coincidence yq in X . Here, we prove the uniqueness of the points of coincidence. For this, let there be p 2 X such that y p = j p. By applying Equation (5), we obtain s (j p, j p, jq) = s (y p, y p, yq) b b as (y p, y p, jq) + as (j p, j p, yq) b b = as (j p, j p, jq) + as (j p, j p, jq), b b In other words, we have s (j p, j p, jq)  (1 a) as (j p, j p, jq). b b Since jj(1 a) ajj < 1, this implies that jjs (j p, j p, jq)jj = 0, and thus j p = jq. Therefore, Lemma 3 implies that y and j have a unique common FP in X . If we choose j = id in Theorem 5, then we obtain R(j) = X , and y is weakly compatible with j. We also have the following result: Corollary 4. Suppose that (X ,A, s ) is a complete symmetric C*-AV-S M space and b b y : X ! X satisfies s (yu, yu, yv)  as (yu, yu, v) + as (yv, yv, u), b b b 0 1 for all u, v 2 X , where a 2 A and jjabjj < . Hence, y has a unique FP in X . + 3 5. Application in Integral Equations Let us use the following equations: ll x(m) = (T (m, n, x(n))dn + J(m), m 2 E Z (6) x(m) = (T (m, n, x(n))dn + J(m), m 2 E in which E is a Lebesgue measurable set where m(E) < ¥. In fact, we suppose that X = L (E) presents the class of essentially bounded measur- able functions on E , where E is a Lebesgue measurable set such that m(E) < ¥. One may consider the functions T , T , a, b to fulfill the following assumptions: 1 2 (i) T , T : E E  R ! R are integrable. In addition, an integrable function a is from 0 ¥ E E to R , and J 2 L (E). (ii) There exists ` 2 (0, 1) such that jT (m, n, x) T (m, n, y)j  `ja(m, n)jjx yj, 1 2 for m, n 2 E and x, y 2 R. Axioms 2023, 12, 413 11 of 12 (iii) sup ja(m, n)jdn  1. m2E E Theorem 6. Let assumptions (i–iii) hold. Hence, the integral in Equation (6) has a unique common solution in L (E). ¥ 2 Proof. Suppose that X = L (E) and B(L (E)) is a set of bounded linear operators on a 2 2 Hilbert space L (E). We equipX with the S metric s : X X X ! B(L (E)), which is b b ascertained by s (a, b, g) = M p , b (jagj+jbgj) where M p is the multiplication operator on L (E) ascertained by (jagj+jbgj) M (a) = h.a ; a 2 L (E). Therefore, (X , B(L (E)), s ) is a complete C*-AV-S M space. We can describe the b b self-mappings Y, F : X ! X as follows: Yx(m) = T (m, n, x(n))dn + J(m), Fx(m) = T (m, n, x(n))dn + J(m), for each m 2 E . Therefore, we have s (Yx, Yx, Fy) = M p . (jYxFyj+jYxFyj) We can obtain jjs (Yx, Yx, Fy)jj = sup h M p h, hi (jYxFyj+jYxFyj) jjhjj=1 = sup h M p h, hi (2jYxFyj) jjhjj=1 = sup h2 M p h, hi jYxFyj jjhjj=1 p p = sup (2 jYx Fyj )h(t)h(t)dt jjhjj=1 Z Z p p 2 2 sup [ jT (m, n, x(n)) T (m, n, y(n))j] jh(t)j dt 1 2 E E jjhjj=1 Z Z p p 2 2 sup [ `ja(m, n)(x(n) y(n))jdn] jh(t)j dt E E jjhjj=1 Z Z p p p 2 2 ` sup [ ja(m, n)jdn] jh(t)j dt.jjx yjj E E jjhjj=1 Z Z 2 p ` sup ja(m, n)jdn. sup jh(t)j dt2 jjx yjj E E m2E jjhjj=1 2 `jjx yjj = `jj2(x y)jj = `jj M pjj (jxyj+jxyj) = jjajjjjs (x, x, y)jj By setting a = `1 , then a 2 B(L (E)) and jjajj = ` < 1. Therefore, Corollary 1 B(L (E)) implies the result. Axioms 2023, 12, 413 12 of 12 Author Contributions: S.S.R.: Investigation, writing draft and final articles; H.P.M.: Investigation, writing draft; M.D.L.S.: checking draft and final version, funding, Project direction. All authors have read and agreed to the published version of the manuscript. Funding: Basque Government, Grant IT1155-22. 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Journal

Axioms – Multidisciplinary Digital Publishing Institute

Published: Apr 24, 2023

Keywords: integral equation; C* algebra; Sb-metric space; common fixed point; compatible; weakly compatible

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