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axioms Article On Some Coupled Fixed Points of Generalized T-Contraction Mappings in A b (s)-Metric Space and Its Application 1,2, 3, 4 Reny George *, Zoran D. Mitrovic ´ * and Stojan Radenovic ´ Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia Department of Mathematics and Computer Science, St. Thomas College, Bhilai, Chhattisgarh 491022, India Faculty of Electrical Emgineering, University of Banja Luka, Patre 5, 78000 Banja Luka, Bosnia and Herzegovina Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Beograd, Serbia; radens@beotel.net * Correspondence: r.kunnelchacko@psau.edu.sa (R.G.); zoran.mitrovic@etf.unibl.org (Z.D.M.) Received: 16 October 2020; Accepted: 7 November 2020; Published: 9 November 2020 Abstract: Common coupled ﬁxed point theorems for generalized T-contractions are proved for a pair of mappings S : X X ! X and g : X ! X in a b (s)-metric space, which generalize, extend, and improve some recent results on coupled ﬁxed points. As an application, we prove an existence and uniqueness theorem for the solution of a system of nonlinear integral equations under some weaker conditions and given a convergence criteria for the unique solution, which has been properly veriﬁed by using suitable example. Keywords: common coupled fixed point; b (s)-metric space; T-contraction; weakly compatible mapping 1. Introduction In the last three decades, the deﬁnition of a metric space has been altered by many authors to give new and generalized forms of a metric space. In 1989, Bakhtin [1] introduced one such generalization in the form of a b-metric space and in the year 2000 Branciari [2] gave another generalization in the form a rectangular metric space and generalized metric space. Thereafter, using the above two concepts, many generalizations of a metric space appeared in the form of rectangular b-metric space [3], hexagonal b-metric space [4], pentagonal b-metric space [5], etc. The latest such generalization was given by Mitrovic ´ and Radenovic ´ [6] in which the authors deﬁned a b (s)-metric space which is a generalization of all the concepts told above. Some recent ﬁxed point theorems in such generalized metric spaces can be found in [6–9]. In [10–12], one can ﬁnd some interesting coupled ﬁxed point theorems and their applications proved in some generalized forms of a metric space. In the present note, we have given coupled ﬁxed point results for a pair of generalized T-contraction mappings in a b (s)-metric space. Our results are new and it extends, generalize, and improve some of the coupled ﬁxed point theorems recently dealt with in [10–12]. In recent years, ﬁxed point theory has been successfully applied in establishing the existence of solution of nonlinear integral equations (see [11–15] ). We have applied one of our results to prove the existence and convergence of a unique solution of a system of nonlinear integral equations using some weaker conditions as compared to those existing in literature. 2. Preliminaries Deﬁnition 1. [6] Let X be a nonempty set. Assume that, for all x, y,2 X and distinct u , , u 2 Xfx, yg, 1 v d : X X ! R satisﬁes : Axioms 2020, 9, 129; doi:10.3390/axioms9040129 www.mdpi.com/journal/axioms Axioms 2020, 9, 129 2 of 13 1. d (x, y) 0 and d (x, y) = 0 if and only if x = y, v v 2. d (x, y) = d (y, x), v v 3. d (x, y) s[d (x, u ) + d (u , u ) + + d (u , u ) + d (u , y)], for some s 1. v v 1 v 1 2 v v1 v v v Then, (X, d ) is a b (s)-metric space. v v Deﬁnition 2. [6] In the b (s)-metric space (X, d ), the sequence < u > v v n (a) converges to u 2 X if d (u , u) ! 0 as n ! ¥; v n (b) is a Cauchy sequence if d (u , u ) ! 0 as n, m ! +¥. v n m Clearly, b (1)-metric space is the usual metric space, whereas b (s), b (1), b (s), and b (1)-metric 1 1 2 2 spaces are, respectively, the b-metric space ([1]), rectangular metric space ([2]), rectangular b-metric space ([3]), and v-generalized metric space ([2]). Lemma 1. [6] If (X, d ) is a b (s)-metric space, then (X, d ) is a b (s )-metric space. v v v 2v Deﬁnition 3. An element (u, v) 2 X X is called a coupled coincidence point of S : X X ! X and g : X ! X if g(u) = S(u, v) and g(v) = S(v, u). In this case, we also say that (g(u), g(v)) is the point of coupled coincidence of S and g. If u = g(u) = S(u, v) and v = g(v) = S(v, u), then we say that (u, v) is a common coupled ﬁxed point of S and g. We will denote by COCPfS, gg and CCOFPfS, gg respectively the set of all coupled coincidence points and the set of all common coupled ﬁxed points of S and g. Deﬁnition 4. S : X X ! X and g : X ! X are said to be weakly compatible if and only if S(g(u), g(v)) = g(S(u, v)) for all (u, v) 2 COCPfS, gg. 3. Main Results We will start this section by proving the following lemma which is an extension of Lemma 1.12 of [6] to two sequences: Lemma 2. Let (X, d ) be a b (s)-metric space and let < u > and < v > be two sequences in X such that v v n n u 6= u , v 6= v (n 0). Suppose that l 2 [0, 1) and c , c are real nonnegative numbers such that n n+1 n n+1 1 2 m n K lK + c l + c l , for all m, n 2 N, (1) m,n m1,n1 1 2 where K = maxfd (u , u ), d (v , v )g or K = d (u , u ) + d (v , v ). Then, < u > and m,n v m n v m n m,n v m n v m n n < v > are Cauchy sequences. Proof. From (1), we have n n+1 K lK + c l + c l n,n+1 n1,n 1 2 (2) n n n+1 l K + c nl + c nl 0,1 1 2 n n l K + C nl . 0,1 0 Axioms 2020, 9, 129 3 of 13 For m, n, k 2 N, by (1), we have m+k1 n+k1 K l maxfK , c l + c l )g m+k,n+k m+k1,n+k1 1 2 m+k n+k lK + c l + c l ) m+k1,n+k1 1 2 (3) k k m n l K + kC l (l + l ). m,n Since 0 < l < 1, we can ﬁnd a positive integer q such that 0 < l < . Now, suppose v 2. Then, by using condition 3. of a b (s)-metric and inequalities (2) and (3), we have K s[K + K + + K + K + K + K ] m,n m+v3,m+v2 m+v2,m+q m+q ,n+q n+q ,n m,m+1 m+1,m+2 k k k k m m+1 m+v3 m m+1 m+v2 s[l + l + + l ]K + sC [ml + (m + 1)l + + (m + v 3)l ] 0 0 m m v2 q +s[l K + ml (l + l )K ] v2,q 0 q q m n n n q k k k +s[l K + q l (l + l )K ] + s[l K + nl (l + 1)K ]. m,n k 0 q ,0 0 Then, m m sl s(m + v 3)l K K + m,n 0,1 q q k k (1 sl )(1 l) (1 l)(1 sl ) m m v2 q + [l K + ml (l + l )K ] v2,q 0,1 q k 1 sl s s q m n n n q k k + [q l (l + l )K ] + [l K + nl (l + 1)K ]. 0,1 q ,0 0,1 q q k k k 1 sl 1 sl Thus, from the deﬁnition of K , we see that, as m, n ! +¥, d (u , u ) ! 0 and d (v , v ) ! 0 m,n v m n v m n and thus < u > and < v > are Cauchy sequences. n n 3.1. Coupled Fixed Point Theorems We now present our main theorems as follows: Theorem 1. Let (X, d ) be a b (s)-metric space , T : X ! X be a one to one mapping, S : X X ! X and v v g : X ! X be mappings such that S(X X) g(X), T g(X) is complete. If there exist real numbers l, m, n with 0 l < 1, 0 m, n 1, minflm, lng < such that, for all u, v, w, z 2 X d (TS(u, v), TS(w, z)) l maxfd (T gu, T gw), d (T gv, T gz), md (T gu, TS(u, v)), md (T gv, TS(v, u), v v v v v (4) nd (T gw, TS(w, z)), nd (T gz, TS(z, w))g v v then the following holds : 1. There exist w , w in X, such that sequences < T gu > and < T gv > converge to T gw and T gw x y n n x y 0 0 0 0 respectively, where the iterative sequences < gu > and < gv > are deﬁned by gu = S(u , v ) n n n n1 n1 and gv = S(v , u ) for some arbitrary (u , v ) 2 X X. n1 n1 0 0 2. (w , w ) 2 COCPfS, gg . x y 0 0 3. If S and g are weakly compatible, then S and g have a unique common coupled ﬁxed point. Proof. 1. We shall start the proof by showing that the sequences < T gu > and < T gv > are Cauchy n n sequences, where < gu > and < gv > are as mentioned in the hypothesis. n n Axioms 2020, 9, 129 4 of 13 By (4), we have d (T gu , T gu ) = d (TS(u , v ), TS(u , v )) v n n+1 v n1 n1 n n l maxfd (T gu , T gu ), d (T gv , T gv ), md (T gu , TS(u , v )), v n1 n v n1 n v n1 n1 n1 md (T gv , TS(v , u )), nd (T gu , TS(u , v )), nd (T gv , TS(v , u ))g (5) v n1 n1 n1 v n n n v n n n l maxfd (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu ), v n1 n v n1 n v n1 n d (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv )g. v n1 n v n n+1 v n n+1 Similarly, we get d (T gv , T gv ) l maxfd (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv ), v n n+1 v n1 n v n1 n v n1 n d (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu )g. (6) v n1 n v n n+1 v n n+1 Let K = maxfd (T gu , T gu ), d (T gv , T gv )g. By (5) and (6), we get n v n v n n+1 n+1 K l maxfd (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu )g. (7) n v n1 n v n1 n v n n+1 v n n+1 If maxfd (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu )g v n v n v n v n n1 n1 n+1 n+1 = d (T gv , T gv ) or d (T gu , T gu ), v n n+1 v n n+1 then (7) will yield a contradiction. Thus, we have maxfd (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu )g v n1 n v n1 n v n n+1 v n n+1 = maxfd (T gv , T gv ), d (T gu , T gu )g, v n v n n1 n1 and then (7) gives 2 n K l maxfd (T gv , T gv ), d (T gu , T gu )g = lK l K l K . (8) n v n1 n v n1 n n1 n2 0 For any m, n 2 N, we have d (T gu , T gu ) = d (TS(u , v ), TS(u , v ) v m n v m1 m1 n1 n1 l maxfd (T gu , T gu ), d (T gv , T gv ), v m1 n1 v m1 n1 md (T gu , TS(u , v )), md (T gv , TS(v , u )), v v m1 m1 m1 m1 m1 m1 nd (T gu , TS(u , v )), nd (T gv , TS(v , u ))g v n1 n1 n1 v n1 n1 n1 l maxfd (T gu , T gu ), d (T gv , T gv ), d (T gu , T gu ), v v v m m1 n1 m1 n1 m1 d (T gv , T gv ), d (T gu , T gu ), d (T gv , T gv )g. v m1 m v n1 n v n1 n Then, by using (8), we get d (T gu , T gu ) l maxfd (T gu , T gu ), d (T gv , T gv )g v m n v v m1 n1 m1 n1 m n +(l + l )K g. (9) 0 Axioms 2020, 9, 129 5 of 13 Similarly, we have d (T gv , T gv ) l maxfd (T gu , T gu ), d (T gv , T gv )g v m n v m1 n1 v m1 n1 m n +(l + l )K g. (10) Let K = maxfd (T gu , T gu ), d (T gv , T gv )g. By (9) and (10), we get m,n v m n v m n m n K lK + (l + l )K . m,n m1,n1 0 Thus, we see that inequality (1) is satisﬁed with c = c = K . Hence, by Lemma 2, < T gu > 1 2 0 n and < T gv > are Cauchy sequences. For v = 1, the same follows from Lemma 1. Since (T g(X), d) is complete, we can ﬁnd w , w 2 X such that x y 0 0 lim T gu = T gw and lim T gv = T gw . n x n y 0 0 n!¥ n!¥ 2. Now, d (TS(w , w ), T gw ) s[d (TS(w , w ), TS(u , v ) + d (TS(u , v ), TS(u , v )) v x y x v x y n n v n n n+1 n+1 0 0 0 0 0 + + d (TS(u , v ), TS(u , v ) + d (TS(u , v ), T gw ) v n+v2 n+v2 n+v1 n+v1 v n+v1 n+v1 x s[lmaxfd (T gw , T gu ), d (T gw , T gv ), md (T gw , TS(w , w )), v x n v y n v x x y 0 0 0 0 0 md (T gw , TS(w , w ), nd (T gu , TS(u , v )), nd (T gv , TS(v , u ))g v y y x v n n n v n n n 0 0 0 (11) +d (T gu , T gu ) + + d (T gu , T gu ) + d (T gu , T gw ) v n+1 n+2 v n+v1 n+v v n+v x s[lmaxfd (T gw , T gu ), d (T gw , T gv ), md (T gw , TS(w , w )), v x n v y n v x x y 0 0 0 0 0 md (T gw , TS(w , w ), nd (T gu , T gu ), nd (T gv , T gv )g v y y x v n n+1 v n n+1 0 0 0 +d (T gu , T gu ) + + d (T gu , T gu + d (T gu , T gw ). v n+1 n+2 v n+v1 n+v v n+v x Note that, since < T gu > and < T gv > are Cauchy sequences, by deﬁnition, n n d (T gu , T gu ) ! 0, d (T gv , T gv ) ! 0 as n ! ¥. Thus, from (11), as n ! ¥, we get v n v n n+1 n+1 d (TS(w , w ), T gw ) sl maxfmd (T gw , TS(w , w )), md (T gw , TS(w , w ))g. v x y x v x x y v y y x 0 0 0 0 0 0 0 0 0 Similarly, we get d (TS(w , w ), T gw ) sl maxfmd (T gw , TS(w , w )), md (T gw , TS(w , w )g. v y x y v x x y v y y x 0 0 0 0 0 0 0 0 0 Thus, we have maxfd (TS(w , w ), T gw ), d (TS(w , w ), T gw )g v x y x v y x y 0 0 0 0 0 0 slm maxfd (T gw , TS(w , w )), d (T gw , TS(w , w )g. (12) v x x y v y y x 0 0 0 0 0 0 Proceeding along the same lines as above, we also have maxfd (T gw , TS(w , w )), d (T gw , TS(w , w ))g v x x y v y y x 0 0 0 0 0 0 sln maxfd (T gw , TS(w , w )), d (T gw , TS(w , w )g. (13) v x x y v y y x 0 0 0 0 0 0 Using (12) and (13) along with the condition minflm, lng < , we get TS(w , w ) = T gw x y x s 0 0 0 and TS(w , w ) = T gw . As T is one to one, we have S(w , w ) = gw and S(w , w ) = gw . y x y x y x y x y 0 0 0 0 0 0 0 0 0 Therefore, (w , w ) 2 COCPfS, gg . x y 0 0 Axioms 2020, 9, 129 6 of 13 3. Suppose S and g are weakly compatible. First, we will show that, if (w , w ) 2 COCPfS, gg, x y 0 0 then gw = gw and gw = gw , or in other words the point of coupled coincidence of S and g is x y x y 0 0 0 0 unique. By (5), we have d (T gw , T gw ) = d (TS(w , w ), TS(w , w )) v x v x y x 0 x y 0 0 0 0 0 lmaxfd (T gw , T gw ), d (T gw , T gw ), md (T gw , TS(w , w )), v x v y v x y x x y 0 0 0 0 0 0 0 md (T gw , TS(w , w ), nd (T gw , TS(w , w )), nd (T gw , TS(w , w ))g v v x x y v y y x y y x 0 0 0 0 0 0 0 0 0 lmaxfd (T gw , T gw ), d (T gw , T gw )g. v x v y x y 0 0 0 0 Similarly, we have d (T gw , T gw ) lmaxfd (T gw , T gw ), d (T gw , T gw )g. v y v x v y y 0 x 0 y 0 0 0 0 Thus, from the above two inequalities, we get maxfd (T gw , T gw ), d (T gw , T gw ) lmaxfd (T gw , T gw ), d (T gw , T gw )g v x v y v x v y x y x y 0 0 0 0 0 0 0 0 which implies that T gw = T gw and T gw = T gw . Since T is one to one, we get gw = gw x y x x y x 0 0 0 0 0 0 and gw = gw , which is the point of coupled coincidence of S and g is unique. Since S and g are y 0 weakly compatible and, since (w , w ) 2 COCPfS, gg, we have x y 0 0 ggw = gS(w , w ) = S(gw , gw ) x x y x y 0 0 0 0 0 and ggw = gS(w , w ) = S(gw , gw ) y y x y x 0 0 0 0 0 which shows that (gw , gw ) 2 COCPfS, gg. By the uniqueness of the point of coupled coincidence, x y 0 0 we get ggw = gw and ggw = gw and thus (gw , gw ) 2 CCOFPfS, gg. Uniqueness of the x x y y x y 0 0 0 0 0 0 coupled ﬁxed point follows easily from (4). Our next result is a generalized version of Theorem 2.1 of Gu [10]. Theorem 2. Let (X, d ), T, S and g be as in Theorem 1 and suppose there exist b , b , b in the interval [0,1), 1 2 3 such that b + b + b < 1, minimumfb , b g < and for all u, v, w, z 2 X 1 2 3 2 3 d (TS(u, v), TS(w, z) + d (TS(v, u), TS(z, w) b (d (T gu, T gw) + d (T gv, T gz)) + v v 1 v v b (d (T gu, TS(u, v)) + d (T gv, TS(v, u)) + b (d (T gw, TS(w, z)) + d (T gz, TS(z, w))). (14) 2 v v 3 v v Then, conclusions 1, 2, and 3 of Theorem 1 are true. 0 0 Proof. Let K = d (T gu , T gu ) + d (T gv , T gv ) and K = d (T gu , T gu ) + v n v n v m n n n+1 n+1 m,n d (T gv , T gv ). From condition (14), we obtain v m n d (T gu , T gu ) + d (T gv , T gv ) = d (TS(u , v ), TS(u , v )) + v n n+1 v n n+1 v n1 n1 n n d (TS(v , u ), TS(v , u )) v n n n1 n1 b [d (T gu , T gu ) + d (T gv , T gv )] + b [d (T gu , TS(u , v )) 1 v n1 n v n1 n 2 v n1 n1 n1 +d (T gv , TS(v , u ))] + b [d (T gu , TS(u , v )) + d (T gv , TS(v , u ))] v n1 n1 n1 3 v n n n v n n n (b + b )[d (T gu , T gu ) + d (T gv , T gv )] 1 2 v n1 n v n1 n +b [d (T gu , T gu ) + d (T gv , T gv )]. 3 v n n+1 v n n+1 Axioms 2020, 9, 129 7 of 13 Therefore, d (T gu , T gu ) + d (T gv , T gv ) l [d (T gu , T gu ) + d (T gv , T gv )], v n n+1 v n n+1 v n1 n v n1 n b + b 1 2 where l = < 1. Thus, we get 1 b 0 0 0 0 n 0 K l K l K . (15) n1 0 For any m, n 2 N, we have d (T gu , T gu ) + d (T gv , T gv ) = d (TS(u , v ), TS(u , v ) + v m n v m n v m1 m1 n1 n1 d (TS(v , u ), TS(v , u ) m1 m1 n1 n1 b [d (T gu , T gu ) + d (T gv , T gv )] 1 v m1 n1 v m1 n1 +b [d (T gu , TS(u , v )) + d (T gv , TS(v , u ))] v v 2 m1 m1 m1 m1 m1 m1 +b [d (T gu , TS(u , v )) + d (T gv , TS(v , u ))] 3 v n1 n1 n1 v n1 n1 n1 b d (T gu , T gu ) + d (T gv , T gv )] + b [d (T gu , T gu ) v v v m [ m1 n1 m1 n1 2 m1 +d (T gv , T gv )] + b [d (T gu , T gu ) + d (T gv , T gv )]. v m1 m 3 v n1 n v n1 n Then, by using (15), we get d (T gu , T gu ) + d (T gv , T gv ) b [d (T gu , T gu ) + d (T gv , T gv )] v m n v m n 1 v m1 n1 v m1 n1 m n 0 0 0 +(b l + b l )K g. 2 3 0 That is, 0 0 0 m n K lK + (l + l )K m,n m1,n1 0 where l = b + b + b < 1. Now for m, n, r 2 N. Thus, we see that inequality (1) is satisﬁed with 1 2 3 c = c = K . Hence, by Lemma 2, < T gu > and < T gv > are Cauchy sequences. For v = 1, 1 2 0 n n the same follows from Lemma 1. Since (T g(X), d) is complete, we can ﬁnd w , w 2 X such that x y 0 0 lim T gu = T gw and lim T gv = T gw . n x n y 0 0 n!¥ n!¥ Again, from condition 3 in Deﬁnition 1, we have d (TS(w , w ), T gw )) s[d (TS(w , w ), TS(u , v )) + d (TS(u , v ), TS(u , v )) + + v x y x v x y n n v n n 0 0 0 0 0 n+1 n+1 +d (TS(u , v ), TS(u , v ))+ v n+v2 n+v2 n+v1 n+v1 d (TS(u , v ), T gw ))] v n+v1 n+v1 x and d (TS(w , w ), T gw )) s[d (TS(w , w ), TS(v , u )) + d (TS(v , u ), TS(v , u )) + + v y x y v y x n n v n n n+1 n+1 0 0 0 0 0 d (TS(v , u ), TS(v , u ))+ n+v2 n+v2 n+v1 n+v1 d (TS(v , u ), T gw ))]. v x n+v1 n+v1 0 Axioms 2020, 9, 129 8 of 13 Therefore, d (TS(w , w ), T gw ) + d (TS(w , w ), T gw ) s[d (TS(w , w ), TS(u , v ) v x y x v y x y v x y n n 0 0 0 0 0 0 0 0 +d (TS(w , w ), TS(v , u ) v y x n n 0 0 +d (TS(u , v ), TS(u , v )) + + d (TS(u , v ), TS(u , v )) v n n n+1 n+1 v n+v2 n+v2 n+v1 n+v1 +d (TS(v , u ), TS(v , u )) + + d (TS(v , u ), TS(v , u )) v n n v n+v2 n+v2 n+1 n+1 n+v1 n+v1 +d (TS(u , v ), T gw ) + d (TS(v , u ), T gw )] v n+v1 n+v1 x v n+v1 n+v1 y 0 0 s[b (d (T gw , T gu ) + d (T gw , T gv )) + b (d (T gw , TS(w , w )) + v x n v y n 2 v x x y 0 0 0 0 0 d (T gw , TS(w , w )) + b (d (T gu , TS(u , v )) + d (T gv , TS(v , u )))g v y y x 3 v n n n v n n n 0 0 0 +d (T gu , T gu ) + + d (T gu , T gu ) + +d (T gv , T gv ) + + d (T gv , T gv ) v n v n v n v n n+1 n1 n+1 n1 +d (T gu , T gw ) + d (T gv , T gw )]. v n+v1 x v n+v1 y 0 0 As n ! ¥, we get d (TS(w , w ), T gw ) + d (TS(w , w ), T gw ) v x y x v y x y 0 0 0 0 0 0 sb [d (T gw , TS(w , w )) + d (T gw , TS(w , w ))]. (16) 2 v x x y v y y x 0 0 0 0 0 0 Similarly, we can show that d (T gw , TS(w , w )) + d (T gw , TS(w , w )) v x x y v y y x 0 0 0 0 0 0 sb [d (T gw , TS(w , w )) + d (T gw , TS(w , w )] (17) 3 v x x y v y y x 0 0 0 0 0 0 Using (16) and (17) along with the condition minfb , b g < , we get d (T gw , TS(w , w )) + 2 3 v x x y s 0 0 0 d (T gw , TS(w , w )) = 0, i.e., TS(w , w ) = T gw and TS(w , w ) = T gw . As T is one to v y y x x y x y x y 0 0 0 0 0 0 0 0 0 one, we have S(w , w ) = gw and S(w , w ) = gw . Therefore, (w , w ) 2 COCPfS, gg . x y x y x y x y 0 0 0 0 0 0 0 0 If (w , w ) 2 COCPfS, gg, then, by (14), we have x y 0 0 d (T gw , T gw ) + d (T gw , T gw ) = d (TS(w , w ), TS(w , w )) + d (TS(w , w ), TS(w , w )) v x x v y y v x y x y v y x y x 0 0 0 0 0 0 0 0 0 0 0 0 b [d (T gw , T gw ) + d (T gw , T gw )] + b [d (T gw , TS(w , w )) 1 v x v y 2 v x 0 y 0 x x y 0 0 0 0 0 +d (T gw , TS(w , w )] + b [d (T gw , TS(w , w )) + d (T gw , TS(w , w ))] v 3 v x x y v y y x y y x 0 0 0 0 0 0 0 0 0 b [d (T gw , T gw ) + d (T gw , T gw )]. 1 v x x v y y 0 0 0 0 Thus, d (T gw , T gw ) + d (T gw , T gw ) = 0, which implies that T gw = T gw and v x v y x x 0 y 0 x 0 0 0 0 T gw = T gw . Since T is one to one, we get gw = gw and gw = gw , which is the point of y x y y x y 0 0 0 0 0 0 coupled coincidence of S, and g is unique. The remaining part of the proof is the same as in the proof of Theorem 1. The next results can be proved as in Theorems 1 and 2 and so we will not give the proof. Theorem 3. Theorem 1 holds if we replace condition (4) with the following condition: 6 1 There exist b 2 [0, 1), i 2 f1, . . . , 6g such that b < 1, minfb + b , b + b g < and for all i i 3 4 5 6 i=1 u, v, w, z 2 X, d (TS(u, v), TS(w, z)) b d (T gu, T gw) + b d (T gv, T gz) + b d (T gu, TS(u, v)) v 1 v 2 v 3 v +b d (T gv, TS(v, u) + b d (T gw, TS(w, z)) + b d (T gz, TS(z, w)). (18) 4 v 5 v 6 v Taking T to be the identity mapping in Theorems 1–3, we have the following: Axioms 2020, 9, 129 9 of 13 Corollary 1. Let (X, d ), S, g, l, m and n be as in Theorem 1 such that, for all u, v, w, z 2 X, the following holds : d (S(u, v), S(w, z) lmaxfd (gu, gw), d (gv, gz), md (gu, S(u, v)), md (gv, S(v, u), v v v v v nd (gw, S(w, z)), nd (gz, S(z, w))g. (19) v v Then, COCPfS, gg 6= f. Furthermore, if S and g are weakly compatible, then S and g has a unique common coupled ﬁxed point. Moreover, for some arbitrary (u , v ) 2 X X, the iterative sequences (< gu > 0 0 , < gv >) deﬁned by gu = S(u , v ) and gv = S(v , u ) converge to the unique common n n n1 n1 n n1 n1 coupled ﬁxed point of S and g. Corollary 2. Corollary 1 holds if the condition (19) is replaced with the following condition: There exist b , b , b in the interval [0,1), such that b + b + b < 1, minfb , b g < and for all 1 2 3 1 2 3 2 3 u, v, w, z 2 X d (S(u, v), S(w, z) + d (S(v, u), S(z, w) b (d (gu, gw) + d (gv, gz)) + v v 1 v v b (d (gu, S(u, v)) + d (gv, S(v, u)) + b (d (gw, S(w, z)) + d (gz, S(z, w))). (20) v v v v 2 3 Corollary 3. Corollary 1 holds if the condition (19) is replaced with the following condition: 6 1 There exist b 2 [0, 1), i 2 f1, . . . 6g such that b < 1, minfb + b , b + b g < and, for all å 3 5 6 i i 4 i=1 s u, v, w, z 2 X, d (S(u, v), S(w, z)) b d (gu, gw) + b d (gv, gz) + v 1 v 2 v b d (gu, S(u, v)) + b d (gv, S(v, u) + b d (gw, S(w, z)) + b d (gz, S(z, w)). (21) 3 v 4 v 5 v 6 v Remark 1. Since every b-metric space is a b (s) metric space, we note that Theorem 1 is a substantial generalization of Theorem 2.2 of Ramesh and Pitchamani [11]. In fact, we do not require continuity and sub sequential convergence of the function T. Remark 2. Note that condition (2.1) of Gu [10] implies (20) and hence Corollary 2 gives an improved version of Theorem 2.1 of Gu [10]. Remark 3. Condition (3.1) of Hussain et al. [12] implies (18) and hence Theorem 3 is an extended and generalized version of Theorem 3.1 of [12]. 3.2. Application to a System of Integral Equations In this section, we give an application of Theorem 1 to study the existence and uniqueness of solution of a system of nonlinear integral equations. Let X = C[0, A] be the space of all continuous real valued functions deﬁned on [0, A], A > 0. Our problem is to ﬁnd (u(t), v(t)) 2 X X, t 2 [0, A] such that, for f : [0, A] R R ! R and G : [0, A] [0, A] ! R and K 2 C([0, A], the following holds: u(t) = G(t, r) f (t, u(r), v(r))dr + K(t) v(t) = G(t, r) f (t, v(r), u(r))dr + K(t). (22) Now, suppose F : X X ! X is given by F(u(t), v(t)) = G(t, r) f (t, u(r), v(r))dr + K(t). 0 Axioms 2020, 9, 129 10 of 13 F(v(t), u(t)) = G(t, r) f (t, v(r), u(r))dr + K(t). Then, (22) is equivalent to the coupled fixed point problem F(u(t), v(t)) = u(t), F(v(t), u(t)) = v(t). Theorem 4. The system of Equation (22) has a unique solution provided the following holds: (i) G : [0, A] [0, A] ! R and f : [0, A] R R ! R are continuous functions. (ii) K 2 C([0, A]. (iii) For all x, y, u, v 2 X and t 2 [0, A], we can ﬁnd a function g : X ! X and real numbers p 1, l, m, n with 0 l < 1, 0 m, n 1, minimum flm, lng < satisfying s1 p p p p (iii a) :j f (t, u(r), v(r))) f (t, x(r), y(r))) j l maxfj g(u(r)) g(x(r)) j ,j g(v(r)) g(y(r)) j , p p m j g(u(r)) F(u(r), v(r)) j , m j g(v(r)) F(v(r), u(r)) j , p p n j g(x(r)) F(x(r), y(r)) j , n j g(y(r)) F(y(r), x(r)) j g. (iii-b) F(g(u(t)), g(v(t))) = g(F(u(t), v(t))) . p 1 (iv) su p j G(t, r) j dr . t2[0, A] p1 Moreover, for some arbitrary u (t), v (t) in X, the sequence (< gu (t) >, < gv (t) >) deﬁned by 0 0 n n gu (t) = G(t, r) f (t, u (r), v (r))dr + K(t) n n1 n1 gv (t) = G(t, r) f (t, v (r), u (r))dr + K(t) (23) n1 n1 converges to the unique solution. Proof. Deﬁne d : X X ! R such that for all u, v 2 X, d (u, v) = su p j u(t) v(t) j . (24) t2[0,A] s1 Clearly, d is a b ((v + 1) )-metric space. v v For some r 2 [0, A], we have j F(u(t), v(t)) F(x(t), y(t)) j R R A A = j G(t, r) f (t, u(r), v(r))dr + g(t) G(t, r) f (t, x(r), y(r))dr + g(t) j 0 0 p p j G(t, r) j j f (t, u(r), v(r)) f (t, x(r), y(r)) j dr p p p p ( j G(t, r) j dr)l [maxfj g(u(r)) g(x(r)) j ,j g(v(r)) g(y(r)) j , p p m j g(u(r)) F(u(r), v(r)) j , m j g(v(r)) F(v(r), u(r)) j , p p n j g(x(r)) F(x(r), y(r)) j , n j g(y(r)) F(y(r), x(r)) j g. p p ( j G(t, r) j dr)l [maxfd (g(u), g(x)), d (g(v), g(y)), md (g(u), F(u, v)), md (g(v), F(v, u)), v v v v nd (g(x), F(x, y)), nd (g(y), F(y, x))g. v v Thus, using condition (iv), we have d (F(u, v), F(x, y)) = su p j F(u(t), v(t)) F(x(t), y(t)) j t2[0, A] l[maxfd (g(u), g(x)), d (g(v), g(y)), md (g(u), F(u, v)), md (g(v), F(v, u)), v v v v nd (g(x), F(x, y)), nd (g(y), F(y, x))g. v v Thus, all the conditions of Corollary 1 are satisﬁed and so F has a unique coupled ﬁxed point 0 0 (u , v ) 2 C([0, A] C([0, A], which is the unique solution of (22) and the sequence (< gu (t) >, < gv (t) >) deﬁned by (23) converges to the unique solution of (22). n Axioms 2020, 9, 129 11 of 13 Example 1. Let X = C[0, 1] be the space of all continuous real valued functions deﬁned on [0, 1] and deﬁne d : X X ! R such that, for all u, v 2 X, d (u, v) = su p j u(t) v(t) j . (25) t2[0,1] Clearly, d is a b (3)-metric. Now, consider the functions f : [0, 1] R R ! R given by 3 2 45(t+r) 2 9 8 f (t, u, v) = t + u + v, G : [0, 1] [0, 1] ! R given by G(t, r) = , K 2 C([0, 1] given 20 20 10 by K(t) = t. Then, Equation (22) becomes 45(t + r) 9 8 u(t) = t + (t + u(r) + v(r))dr 10 20 20 45(t + r) 9 8 v(t) = t + (t + v(r) + u(r))dr. (26) 10 20 20 Then, 9 8 2 2 j f (t, u, v) f (t, x, y) j = j (u x) + (v y) j 20 20 9 8 j Maxf (u x), (v y)g j 10 10 2 2 Maxfj u x j ,j v y) j g. In addition, Z Z 1 1 2 2 su p j G(t, r) j dr = (t + r) dr = 1.05. t2[0,1] 0 0 We see that all the conditions of Theorem 4 are satisﬁed, with l = , m = 0, n = 0, p = 2 and g = I (Identity mapping). Hence, Theorem 4 ensures a unique solution of (26). Now, for u (t) = 1 and v (t) = 0, we construct 0 0 the sequence (< u (t) >, < v (t) >g given by n n 45(t + r) 9 8 u (t) = t + (t + u (r) + v (r))dr n1 n1 10 20 20 45(t + r) 9 8 v (t) = t + (t + v (r) + u (r))dr. (27) n1 n1 10 20 20 3 2 Using MATLAB, we see that above sequence converges to f0.6708t + 0.3354t + 2.2339t + 3 2 0.7677, 0.6708t + 0.3354t + 2.2339t + 0.7677g, and this is the unique solution of the system of nonlinear integral Equation (26). The convergence table is given in Table 1 below. Axioms 2020, 9, 129 12 of 13 Table 1. Convergence of sequences < u (t) > and < v (t) >. n n u (t) = v (t) = n n p p n R R 1 45(t+r) 1 45(t+r) 2 9 8 2 9 8 t + (t + u (r) + v (r))dr t + (t + v (r) + u (r))dr n1 n1 n1 n1 0 10 20 20 0 10 20 20 2 2 u (t) = t + .0167(2t + 1)(20t + 9)) v (t) = t + .0671(2t + 1)(5t + 2)) 1 1 3 2 3 2 2 u (t) = 0.6708t + 0.3354t + 1.3t + 0.5007 v (t) = 0.6708t + 0.3354t + 1.29t + 0.5115 2 2 3 2 3 2 3 u (t) = 0.6708t + 0.3354t + 1.8210t + 0.5174 v (t) = 0.6708t + 0.3354t + 1.8208t + 0.5171 3 3 3 2 3 2 4 u (t) = 0.6708t + 0.3354t + 1.9734t + 0.6179 v (t) = 0.6708t + 0.3354t + 1.9734t + 0.6178 4 4 3 2 3 2 5 u (t) = 0.6708t + 0.3354t + 2.0743t + 0.6755 v (t) = 0.6708t + 0.3354t + 2.0743t + 0.6755 5 5 3 2 3 2 6 u (t) = 0.6708t + 0.3354t + 2.1359t + 0.7111 v (t) = 0.6708t + 0.3354t + 2.1359t + 0.7111 6 6 3 2 3 2 7 u (t) = 0.6708t + 0.3354t + 2.1737t + 0.73298 v (t) = 0.6708t + 0.3354t + 2.1737t + 0.73298 7 7 3 2 3 2 u (t) = 0.6708t + 0.3354t + 2.19699t + 0.7464 v (t) = 0.6708t + 0.3354t + 2.19699t + 0.7464 8 8 3 2 3 2 9 u (t) = 0.6708t + 0.3354t + 2.2113t + 0.7547 v (t) = 0.6708t + 0.3354t + 2.2113t + 0.7547 9 9 3 2 3 2 10 u (t) = 0.6708t + 0.3354t + 2.2200t + 0.7597 v (t) = 0.6708t + 0.3354t + 2.2200t + 0.7597 10 10 3 2 3 2 u (t) = 0.6708t + 0.3354t + 2.2254t + 0.7628 v (t) = 0.6708t + 0.3354t + 2.2254t + 0.7628 11 11 3 2 3 2 12 u (t) = 0.6708t + 0.3354t + 2.2287t + 0.7647 v (t) = 0.6708t + 0.3354t + 2.2287t + 0.7647 12 12 3 2 3 2 13 u (t) = 0.6708t + 0.3354t + 2.2308t + 0.7658 v (t) = 0.6708t + 0.3354t + 2.2308t + 0.7658 13 13 3 2 3 2 u (t) = 0.6708t + 0.3354t + 2.23199t + 0.7666 v (t) = 0.6708t + 0.3354t + 2.23199t + 0.7666 14 14 3 2 3 2 15 u (t) = 0.6708t + 0.3354t + 2.2328t + 0.7671 v (t) = 0.6708t + 0.3354t + 2.2328t + 0.7671 15 15 3 2 3 2 16 u (t) = 0.6708t + 0.3354t + 2.2333t + 0.7674 v (t) = 0.6708t + 0.3354t + 2.2333t + 0.7674 16 16 3 2 3 2 17 u (t) = 0.6708t + 0.3354t + 2.2336t + 0.7675 v (t) = 0.6708t + 0.3354t + 2.2336t + 0.7675 17 17 3 2 3 2 18 u (t) = 0.6708t + 0.3354t + 2.2338t + 0.7676 v (t) = 0.6708t + 0.3354t + 2.2338t + 0.7676 18 18 3 2 3 2 19 u (t) = 0.6708t + 0.3354t + 2.2339t + 0.7677 v (t) = 0.6708t + 0.3354t + 2.2339t + 0.7677 19 19 3 2 3 2 20 u (t) = 0.6708t + 0.3354t + 2.2339t + 0.7677 v (t) = 0.6708t + 0.3354t + 2.2339t + 0.7677 20 20 Remark 4. Condition (iv) of Theorem 4 above is weaker than the corresponding conditions used in similar theorems of [11,13,14]. R R 1 1 2 45 2 Remark 5. In example 1 above, we see that su p j G(t, r) j dr = (t + r) dr = 1.05 > 1 and t2[0,1] 0 0 100 thus condition (v) of Theorem 3.1 of [11], condition (30) of Theorem 3.1 of [13] and condition (iii) of Theorem 3.1 of [14] are not satisﬁed. Author Contributions: Investigation, R.G., Z.D.M., and S.R.; Methodology, R.G.; Software, Z.D.M.; Supervision, R.G., Z.D.M., and S.R. All authors have read and agreed to the published version of the manuscript. Acknowledgments: 1. The authors are thankful to the Deanship of Scientiﬁc Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research. 2. The authors are thankful to the learned reviewers for their valuable comments which helped in improving this paper. Funding: This research received no external funding. 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Published: Nov 9, 2020
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