Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 DOI 10.1186/s13663-015-0343-0 RESEARCH Open Access Some generalized fixed point results in a b-metric space and application to matrix equations Hemant Kumar Pathak1, Reny George2,3*, Hossam A Nabwey2,4, Mahdy S El-Paoumy2,5 and Kaivatath Puthalath Reshma6,7 * Correspondence: renygeorge02@yahoo.com 2 Department of Mathematics, College of Science, Salman bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia 3 Department of Mathematics and Computer Science, St. Thomas College, Bhilai, Chhattisgarh, India Full list of author information is available at the end of the article Abstract We have proved a generalized Presic-Hardy-Rogers contraction principle and Ciric-Presic type contraction principle for two mappings in a b-metric space. As an application, we derive some convergence results for a class of nonlinear matrix equations. Numerical experiments are also presented to illustrate the convergence algorithms. MSC: coincident point; common fixed point; b-metric space; matrix equation Keywords: 47H10 1 Introduction There appears in literature several generalizations of the famous Banach contraction principle. One such generalization was given by Presic, as follows. Theorem. Let (X, d) be a metric space, k be a positive integer, T : X k X be a mapping satisfying the following condition: d T(x, x,..., xk ), T(x, x,..., xk+ ) q d(x, x ) + q d(x, x ) + + qk d(xk, xk+ ), (. ) where x, x,..., xk+ are arbitrary elements in X and q, q,..., qk are nonnegative constants such that q +q + +qk <. Then there exists some x X such that x = T(x, x,..., x). Moreover, if x, x,..., xk are arbitrary points in X and for n N, xn+k = T(xn, xn+,..., xn+k ), then the sequence xn is convergent and lim xn = T(lim xn, lim xn,..., lim xn ). Note that for k = the above theorem reduces to the well-known Banach contraction principle. Ciric and Presic generalizing the above theorem proved the following. Theorem. Let (X, d) be a metric space, k be a positive integer, T : X k X be a mapping satisfying the following condition: d T(x, x,..., xk ), T(x, x,..., xk+ ) λ max d(x, x ), d(x, x ),..., d(xk, xk+ ), (. ) 2015 Pathak et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 2 of 17 where x 1, x 2,...,x k+1 are arbitrary elements in X and λ (0, 1). Then there exists some x Xsuchthatx= T(x, x,...,x). Moreover, if x 1, x 2,...,x k are arbitrary points in X and for n N, x n+k = T(x n, x n+1,...,x n+k 1 ), then the sequence x n is convergent and lim x n = T(lim x n, lim x n,...,lim x n ). If in addition T satisfies D(T(u, u,...,u), T(v, v,...,v)) < d(u, v) for all u, v X, then x is the unique point satisfying x = T(x, x,...,x). In 4, 5 Pacurar gave a classic generalization of the above results. Later the above results were further extended and generalized by many authors (see 6 14). Generalizing the concept of metric space, Bakhtin 15 introduced the concept of b-metric space which is not necessarily Hausdorff and proved the Banach contraction principle in the setting of a b-metric space. Since then several papers have dealt with fixed point theory or the variational principle for single-valued and multi-valued operators in b-metric spaces (see 16 23 and the references therein). In this paper we have proved common fixed point theorems for the generalized Presic-Hardy-Rogers contraction and Ciric-Presic contraction for two mappings in a b-metric space. Our results extend and generalize many wellknown results. As an application, we have derived some convergence results for a class of nonlinear matrix equations. Numerical experiments are also presented to illustrate the convergence algorithms. 2 Preliminaries Definition 2.1 15 LetX be a nonempty set and d : X X 0, )satisfy: (bm1) d(x, y)=0if and only if x = y for all x, y X; (bm2) d(x, y)=d(y, x) for all x, y X; (bm3) there exists a real number s 1 such that d(x, y) sd(x, z)+d(z, y) for all x, y, z X. Then d is called a b-metric on X and (X, d) is called a b-metric space (in short bms) with coefficient s. Convergence, Cauchy sequence and completeness in b-metric space are defined as follows. Definition 2.2 15 Let(X, d) beab-metric space, {x n } be a sequence in X and x X. Then: (a) The sequence {x n } is said to be convergent in (X, d),anditconvergestox if for every ε >0there exists n 0 N such that d(x n, x)<ε for all n > n 0, and this fact is represented by lim n x n = x or x n x as n. (b) The sequence {x n } is said to be Cauchy sequence in (X, d) if for every ε >0there exists n 0 N such that d(x n, x n+p )<ε for all n > n 0, p >0or, equivalently, if lim n d(x n, x n+p )=0for all p >0. (c) (X, d) is said to be a complete b-metric space if every Cauchy sequence in X converges to some x X. Definition 2.3 9 Let(X, d) beametricspace,k be a positive integer, T : X k X and f : X X be mappings. (a) An element x X is said to be a coincidence point of f and T if and only if f (x)=t(x, x,...,x).ifx = f (x)=t(x, x,...,x),thenwesaythatx is a common fixed point of f and T.Ifw = f (x)=t(x, x,...,x),thenw is called a point of coincidence of f and T.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 3 of 17 (b) Mappings f and T are said to be commuting if and only if f (T(x, x,...,x)) = T(fx, fx,...,fx) for all x X. (c) Mappings f and T are said to be weakly compatible if and only if they commute at their coincidence points. Remark 2.4 For k = 1 the above definitions reduce to the usual definition of commuting and weakly compatible mappings in a metric space. The set of coincidence points of f and T is denoted by C(f, T). Lemma 2.5 24 Let X be a nonempty set, k be a positive integer and f : X k X, g : X X be two weakly compatible mappings. If f and g have a unique point of coincidence y = f (x, x,..., x) =g(x), then y is the unique common fixed point of f and g. Khan et al. 8definedthesetfunctionθ :0, ) 4 0, ) as follows: 1. θ is continuous, 2. for all t 1, t 2, t 3, t 4 0, ), θ(t 1, t 2, t 3, t 4 )=0 t 1 t 2 t 3 t 4 =0. 3 Main results Throughout this paper we assume that the b-metric d : X X 0, ) is continuous on X 2. Theorem 3.1 Let (X, d) be a b-metric space with coefficient s 1. For any positive integer k, let f : X k Xandg: X X be mappings satisfying the following conditions: f ( X k) g(x), (3.1) d ( f (x 1, x 2,...,x k ), f (x 2, x 3,...,x k+1 ) ) k α i d(gx i, gx i+1 )+ d ( gx i, f (x j, x j,...,x j ) ) + L θ ( d ( gx 1, f (x k+1, x k+1, x k+1,...,x k+1 ) ), d ( gx k+1, f (x 1, x 1, x 1,...,x 1 ) ), d ( gx 1, f (x 1, x 1,...,x 1 ) ), d ( gx k+1, f (x k+1, x k+1,...,x k+1 ) )), (3.2) where x 1, x 2,...,x k+1 are arbitrary elements in X and α i, β ij, L are nonnegative constants such that k n=1 sk+3 n α n + k+1 k+1 <1and g(x) is complete. (3.3) Then f and g have a unique coincidence point, i.e., C(f, g). In addition, if f and g are weakly compatible, then f and g have a unique common fixed point. Moreover, for any x 1 X, the sequence {y n } defined by y n = g(x n )=f (x n 1, x n 1,...,x n 1 )=Fx n 1 converges to the common fixed point of f and g. Proof Let x 1 X, thenf (x 1, x 1,...,x 1 ) f (X k ) g(x). So there exists x 2 X such that f (x 1, x 1,...,x 1 )=g(x 2 ). Now f (x 2, x 2,...,x 2 ) f (X k ) g(x)andsothereexistsx 3 X such that f (x 2, x 2,...,x 2 )=g(x 3 ). Continuing this process we define the sequence y n in g(x)as
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 4 of 17 y n = g(x n )=f (x n 1, x n 1,...,x n 1 )=Fx n 1, n =1,2,...,k +1,whereF is the associate operator for f.letd n = d(y n, y n+1 )=d(gx n, gx n+1 )andd ij = d(gx i, f (x j, x j,...,x j )). Then we have d n+1 = d ( g(x n+1 ), g(x n+2 ) ) = d(fx n, Fx n+1 ) = d ( f (x n, x n,...,x n ), f (x n+1, x n+1,...,x n+1 ) ) sd ( f (x n, x n,...,x n ), f (x n, x n,...,x n+1 ) ) + s 2 d ( f (x n, x n,...,x n+1 ), f (x n, x n,...,x n+1, x n+1 ) ) + s 3 d ( f (x n, x n,...,x n+1, x n+1 ), f (x n,...,x n+1, x n+1, x n+1 ) ) + + s k d ( f (x n, x n+1,...,x n+1, x n+1 ), f (x n+1,...,x n+1, x n+1, x n+1 ) ). Using (3.2)weget { k k k d n+1 s α k d n + β 1,j + β 2,j + + β kj D n,n k k } + β i,k+1 D n,n+1 + β k+1,j D n+1,n + β k+1,k+1 D n+1,n+1 + s 2 {α k 1 d n + + + k 1 k 1 k 1 k 1 β 1,j + β 2,j + + β k 1,j D n,n k 1 β i,k + β i,k+1 D n,n+1 + k+1 k+1 } β k,j + β k+1,j D n+1,n+1 j=k j=k + + s k {α 1 d n + β 1,1 D n,n + + k+1 k+1 k 1 k 1 β k,j + β k+1,j D n+1,n β 1,j D n,n+1 + } β 2,j + β 3,j + + β k+1,j D n+1,n+1 k+1 β i,1 D n+1,n + L θ ( d ( gx n,(fx n+1, x n+1, x n+1,...,x n+1 ) ), d ( gx n+1, f (x n, x n, x n,...,x n ) ), d ( gx n, f (x n, x n,...,x n ) ), d ( gx n+1, f (x n+1, x n+1,...,x n+1 ) )), i=2 i.e., d n+1 { sα k + s 2 α k 1 + s 3 α k 2 + + s k k k α 1 dn + s D n,n k k } + β i,k+1 D n,n+1 + β k+1,j D n+1,n + β k+1,k+1 D n+1,n+1
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 5 of 17 { k 1 k 1 + s 2 D n,n + + + i.e., k+1 k 1 D n,n+1 + j=k } D n+1,n+1 + + s {β k 1,1 D n,n + i=k j=k k+1 β i,1 D n+1,n + k+1 i=2 i=2 k+1 k+1 k 1 D n+1,n i=k β 1,j D n,n+1 } D n+1,n+1 + L 0, d n+1 sα k + s 2 α k 1 + s 3 α k 2 + + s k α 1 dn k k k 1 k 1 + s + s 2 + + s k 1 + s + s + k k 1 β i,k+1 + s 2 + + s k 1 i=k j=k 2 2 + s k β 1,1 D n,n 2 + s k β 1,j D n,n+1 k k 1 β k+1,j + s 2 + + s k 1 s k k+1 i=3 j=3 2 + s k β i,1 D n+1,n + s k 1 + + s 2 + sβ k+1,k+1 D n+1,n+1 i=2 i=3 j=3 = Ad n + BD n,n + CD n,n+1 + ED n+1,n + FD n+1,n+1, where A, B, C, E and F are the coefficients of d n, D n,n, D n,n+1, D n+1,n and D n+1,n+1 respectively in the above inequality. By the definition, D n,n = d(gx n, f (x n, x n,...,x n )) = d(gx n, gx n+1 )= d n, D n,n+1 = d(gx n, f (x n+1, x n+1,...,x n+1 )) = d(gx n, gx n+2 ), D n+1,n = d(gx n+1, f (x n, x n,...,x n )) = d(gx n+1, gx n+1 )=0,D n+1,n+1 = d(gx n+1, f (x n+1, x n+1,...,x n+1 )) = d(gx n+1, gx n+2 )=d n+1 ;therefore, i=k j=k i=2 d n+1 Ad n + Bd n + Cd(gx n, gx n+2 )+Fd n+1 Ad n + Bd n + Csd(gx n, gx n+1 )+Csd(gx n+1, gx n+2 )+Fd n+1 =(A + B + Cs)d n +(Cs + F)d n+1, i.e., (1 Cs F)d n+1 (A + B + Cs)d n. Again, interchanging the role of x n and x n+1 and repeating the above process, we obtain (1 Es B)d n+1 (A + F + Es)d n. It follows that ( 2 (C + E)s F B ) dn+1 ( 2A + B + F + s(c + E) ) d n d n+1 d n+1 λd n, where λ = 2A+B+F+s(C+E) 2 B F (C+E)s.Thuswehave 2A + B + F + s(c + E) 2 B F (C + E)s d n d n+1 λ n+1 d 0 for all n 0. (3.4)
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 6 of 17 We will show that λ <1andsλ <1.Wehave A + B + F + s(c + E) sa + B + C + E + F = s sα k + s 2 α k 1 + s 3 α k 2 + + s k α 1 k k k 1 k 1 + s s + s 2 + + s k 1 + s s + s s k k 1 β i,k+1 + s 2 + + s k 1 i=k j=k 2 2 + s k β 1,1 2 + s k β 1,j k k 1 β k+1,j + s 2 + + s k 1 i=3 j=3 2 + s k β i,1 + s sβ k+1,k+1 + s 2 + + s k 1 + s k i=k j=k = s 2 α k + s 3 α k 1 + s 4 α k 2 + + s k+1 α 1 + s 2 + s 3 + s 4 + + s k+1 = s 2 α k + s 3 α k 1 + s 4 α k 2 + + s k+1 α 1 + s 2 + s 3 + s 4 + + s k+1 k+1 s 3 α k + s 4 α k 1 + s 5 α k 2 + + s k+2 α 1 = + s 3 + s 4 + s 5 + + s k+2 k s α k+3 n n + <1, n=1 k+2 and so λ <1.WealsohavesA + sb + sf + s(c + E)=s(A + B + F + C + E) < 1 (proved above) and sa + B + F + s 2 (C + E) s 2 A + B + C + E + F = s 2 sα k + s 2 α k 1 + s 3 α k 2 + + s k α 1 + s 2 s + s 2 s k i=3 k k 1 k 1 + s 2 + + s k 1 k k 1 β i,k+1 + s 2 + + s k 1 j=k j=3 2 i=2 i=2 2 + s k β 1,1 2 + s k β 1,j j=3
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 7 of 17 + s 2 s k k 1 β k+1,j + s 2 + + s k 1 i=k i=3 2 + s k β i,1 + s 2 sβ k+1,k+1 + s 2 + + s k 1 + s k i=k j=k = s 3 α k + s 4 α k 1 + s 5 α k 2 + + s k+2 α 1 + s 3 + s 4 + s 5 + + s k+2 = s 3 α k + s 4 α k 1 + s 5 α k 2 + + s k+2 α 1 = + s 3 + s 4 + s 5 + + s k+2 k s α k+3 n n + <1, n=1 and so sλ <1. Thus, for all n, p N, k+1 i=3 j=3 i=2 i=2 d(gx n, gx n+p ) sd(gx n, gx n+1 )+s 2 d(gx n+1, gx n+2 )+ + s p 1 d(gx n+(p 1), gx n+p ) = sd n + s 2 d n+1 + + s p 1 d n+(p 1) sλ n d 0 + s 2 λ n+1 d 0 + + s p 1 λ n+(p 1) d 0 sλn 1 sλ d 0 0 asn. Thus {gx n } is a Cauchy sequence. By completeness of g(x), there exists u X such that lim gx n = u and there exists p X such that g(p)=u. (3.5) n We shall show that u is the fixed point of f and g. Using a similar process as the one used in the calculation of d n+1,weobtain d ( g(p), f (p,...,p) ) s d ( ) ( g(p), y n+1 + d yn+1, f (p, p,...,p) ) s d ( ) g(p), y n+1 + d(fxn, Fp) s d ( ) g(p), y n+1 + Ad(gxn, gp)+bd ( gx n, f (x n, x n,...,x n ) ) + Cd ( gx n, f (p, p,...,p) ) + Ed ( gp, f (x n, x n,...,x n ) ) + Fd ( gp, f (p, p,...,p) ). It follows from (3.5)that d ( g(p), f (p,...,p) ) s(c + F)d ( gp, f (p, p,...,p) ). (3.6)
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 8 of 17 As s(c + F) < 1, weobtainf(p) = g(p) = f (p, p,...,p) = u Thus, u is a point of coincidence of f and g. Ifu is another point of coincidence of f and g, then there exists p X such that F(p )=g(p )=f (p, p,...,p )=u. Then we have d ( u, u ) = d ( Fp, Fp ) Ad ( gp, gp ) + Bd ( gp, f (p, p,...,p) ) + Cd ( gp, f ( p, p,...,p )) + Ed ( gp, f (p, p,...,p) ) + Fd ( gp, f ( p, p,...,p )) = Ad ( u, u ) + Bd(u, u)+cd ( u, u ) + Ed ( u, u ) + Fd ( u, u ) =(A + C + E + F)d ( u, u ). As A + C + E + F < 1, we obtain from the above inequality that d(u, u )=0,thatis,u = u. Thus the point of coincidence u is unique. Further, if f and g are weakly compatible, then by Lemma 2.5, u is the unique common fixed point of f and g. Remark 3.2 Taking s =1,g = I and θ(t 1, t 2, t 3, t 4 )=0inTheorem3.1, wegettheorem4 of Shukla et al. 13. Remark 3.3 For s =1,g = I, i = j, β ij = δ k+1, i, L = 1, we obtain Theorem 2.1 of Khan et al. 8. Remark 3.4 For s =1,g = I, β ij =0, i, j {1,2,...,k +1} and θ(t 1, t 2, t 3, t 4 )=min{(t 1, t 2, t 3, t 4 )}, we obtain the result of Pacurar 5. Remark 3.5 For s =1,g = I, α i =0,i = j, β ij = a, L = 0, we obtain the result of Pacurar 4. Remark 3.6 For s =1,g = I, β ij =0, i, j {1,2,...,k +1}, L = 0, we obtain the result of Presic 2. Next we prove a generalized Ciric-Presic type fixed point theorem in a b-metric space. Consider a function φ : R k R such that 1. φ is an increasing function, i.e., x 1 < y 1, x 2 < y 2,...,x k < y k implies φ(x 1, x 2,...,x k )<φ(y 1, y 2,...,y k ); 2. φ(t, t, t,...) t for all t X; 3. φ is continuous in all variables. Theorem 3.7 Let (X, d) be a b-metric space with s 1. For any positive integer k, let f : X k Xandg: X X be mappings satisfying the following conditions: f ( X k) g(x), (3.7) d ( f (x 1, x 2,...,x k ), f (x 2, x 3,...,x k+1 ) ) λφ ( d(gx 1, gx 2 ), d(gx 2, gx 3 ), d(gx 3, gx 4 ),...,d(gx k, gx k+1 ) ), (3.8)
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 9 of 17 where x 1, x 2,...,x k+1 are arbitrary elements in X, λ (0, 1 s k ), g(x) is complete (3.9) and d ( f (u, u,...,u), f (v, v,...,v) ) < d(gu, gv) (3.10) for all u, v X. Then f and g have a coincidence point, i.e., C(f, g). In addition, if f and g are weakly compatible, then f and g have a unique common fixed point. Moreover, for any x 1 X, the sequence {y n } defined by y n = g(x n )=f (x n, x n+1,...,x n+k 1 ) converges to the common fixed point of f and g. Proof For arbitrary x 1, x 2,...,x k in X,let ( d(gx1, gx 2 ) R = max, d(gx 2, gx 3 ),..., d(gx ) k, f (x 1, x 2,...,x k )), (3.11) θ θ 2 θ k where θ = λ 1 k.by(3.7)wedefinethesequence yn in g(x)asy n = gx n for n =1,2,...,k and y n+k = g(x n+k )=f (x n, x n+1,...,x n+k 1 ), n =1,2,... Let α n = d(y n, y n+1 ). By the method of mathematical induction, we will prove that α n Rθ n for all n. (3.12) Clearly, by the definition of R, (3.12) istrueforn =1,2,...,k. Letthek inequalities α n Rθ n, α n+1 Rθ n+1,...,α n+k 1 Rθ n+k 1 be the induction hypothesis. Then we have α n+k = d(y n+k, y n+k+1 ) = d ( f (x n, x n+1,...,x n+k 1 ), f (x n+1, x n+2,...,x n+k ) ) λφ ( d(gx n, gx n+1 ), d(gx n+1, gx n+2 ),...,d(gx n+k 1, gx n+k ), d ( gx n, f (x n, x n,...,x n ) ), d ( gx n+k, f (x n+k, x n+k,...,x n+k ) )) = λφ(α n, α n+1,...,α n+k 1 ) λφ ( Rθ n, Rθ n+1,...,rθ n+k 1) λφ ( Rθ n, Rθ n,...,rθ n) λrθ n = Rθ n+k. Thus the inductive proof of (3.12)iscomplete.Now,forn, p N,wehave d(y n, y n+p ) sd(y n, y n+1 )+s 2 d(y n+1, y n+2 )+ + s p 1 d(y n+p 1, y n+p ), srθ n + s 2 Rθ n+1 + + s p 1 Rθ n+p 1 srθ n( 1+sθ + s 2 θ 2 + ) = srθ n 1 sθ.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 10 of 17 Hence the sequence y n is a Cauchy sequence in g(x) and since g(x) is complete, there exist v, u X such that lim n y n = v = g(u), d ( gu, f (u, u,...,u) ) s d(gu, y n+k )+d ( y n+k, f (u, u,...,u) ) = s d(gu, y n+k )+d ( f (x n, x n+1,...,x n+k 1 ), f (u, u,...,u) ) = sd(gu, y n+k )+sd ( f (x n, x n+1,...,x n+k 1 ), f (u, u,...,u) ) sd(gu, y n+k )+s 2 d ( f (u, u,...,u), f (u, u,...,x n ) ) + s 3 d ( f (u, u,...,x n ), f (u, u,...,x n, x n+1 ) ) + + s k 1 d ( f (u, x n,...,x n+k 2 ), f (x n, x n+1,...,x n+k 1 ) ) sd(gu, y n+k )+s 2 λφ { d(gu, gu), d(gu, gu),...,d(gu, gx n ) } + s 3 λφ { d(gu, gu), d(gu, gu),...,d(gu, gx n ), d(gx n, gx n+1 ) } + + s k 1 λφ { d(gu, gx n ), d(gx n, gx n+1 ),...,d(gx n+k 2, gx n+k 1 ) } = sd(gu, y n+k )+s 2 λφ ( 0,0,...,d(gu, gx n ) ) + s 3 λφ ( 0,0,...,d(gu, gx n ), d(gx n, gx n+1 ) ) + + s k 1 λφ ( d(gu, gx n ), d(gx n, gx n+1 ),...,d(gx n+k 2, gx n+k 1 ) ). Taking the limit when n tends to infinity, we obtain d(gu, f (u, u,...,u)) 0. Thus gu = f (u, u, u,...,u), i.e., C(g, f ). Thusthereexistv, u X such that lim n y n = v = g(u)= f (u, u, u,...,u). Since g and f are weakly compatible, g(f (u, u,...,u)) = f (gu, gu, gu,...,gu). By (3.10)wehavethat d(ggu, gu)=d ( gf (u, u,...,u), f (u, u,...,u) ) = d ( f (gu, gu, gu,...,gu), f (u, u,...,u) ) < d(ggu, gu) implies d(ggu, gu)=0andsoggu = gu.hencewehavegu = ggu = g(f (u, u,...,u)) = f (gu, gu, gu,...,gu), i.e., gu is a common fixed point of g and f,andlim n y n = g(u). Now suppose that x, y are two fixed points of g and f.then d(x, y)=d ( f (x, x, x,...,x), f (y, y, y,...,y) ) < d(gx, gy) = d(x, y). This implies x = y. Hence the common fixed point is unique. Remark 3.8 Taking s =1,g = I and φ(x 1, x 2,...,x k )=max{x 1, x 2,...,x k } in Theorem 3.7, we obtain Theorem 1.2, i.e., the result of Ciric and Presic 3. 1 Remark 3.9 For λ (0, ), we can drop the condition (3.10)ofTheorem3.7.Infactwe s k+1 have the following.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 11 of 17 Theorem 3.10 Let (X, d) be a b-metric space with s 2. For any positive integer k, let f : X k Xandg: X Xbemappingssatisfyingconditions(3.7), (3.8) and (3.9) with 1 λ (0, ). Then all conclusions of Theorem 3.7 hold. s k+1 Proof As proved in Theorem 3.7, thereexistv, u X such that lim n y n = v = g(u) = f (u, u,...,u), i.e., C(g, f ). Sinceg and f are weakly compatible, g(f (u, u,...,u)) = f (gu, gu, gu,...,gu). By (3.8)wehave d(ggu, gu) =d ( gf (u, u,...,u), f (u, u,...,u) ) = d ( f (gu, gu, gu,...,gu), f (u, u,...,u) ) sd ( f (gu, gu, gu,...,gu), f (gu, gu,...,gu, u) ) + s 2 d ( f (gu, gu,...,gu, u), f (gu, gu,...,u, u) ) + + s k 1 d ( f (gu, gu,...,u, u), f (u, u,...,u) ) + s k 1 d ( f (gu, u,...,u, u), f (u, u,...,u) ) sλφ ( d(ggu, ggu),...,d(ggu, ggu), d(ggu, gu) ) + s 2 λφ ( d(ggu, ggu),...,d(ggu, gu), d(gu, gu) ) + + s k 1 λφ ( d(ggu, gu),...,d(gu, gu), d(gu, gu) ) = sλφ ( 0,0,0,...,d(ggu, gu) ) + s 2 λφ ( 0,0,...,0,d(ggu, gu), 0 ) + + s k 1 λφ ( d(ggu, gu),0,0,...,0 ) = sλ 1+s + s 2 + s 3 + + s k 2 + s k 2 d(ggu, gu) sλ 1+s + s 2 + s 3 + + s k 2 + s k 1 d(ggu, gu) = sλ sk 1 d(ggu, gu). s 1 sλ sk 1 < 1 implies d(ggu, gu) =0andsoggu = gu. Hencewehavegu = ggu = g(f (u, u, s 1...,u)) = f (gu, gu, gu,...,gu), i.e., gu is a common fixed point of g and f,andlim n y n = g(u). Now suppose that x, y are two fixed points of g and f.then d(x, y) =d ( f (x, x, x,...,x), f (y, y, y,...,y) ) sd ( f (x, x,...,x), f (x, x,...,x, y) ) + s 2 d ( f (x, x,...,x, y), f (x, x, x,...,x, y, y) ) + + s k 1 d ( f (x, x, y,...,y), f (y, y,...,y) ) + s k 1 d ( f (x, y, y,...,y), f (y, y,...,y) ) sλφ { d(fx, fx), d(fx, fx),...,d(fx, fy) } + s 2 λφ { d(fx, fx), d(fx, fx),...,d(fx, fy), d(fy, fy) } + + s k 1 λφ { d(fx, fy), d(fy, fy),...,d(fy, fy) } = sλφ ( 0,0,...,d(fx, fy) ) + s 2 λφ ( 0,0,...,d(fx, fy), 0 ) + + s k 1 λφ ( d(fx, fy),0,0,0,...,0 )
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 12 of 17 = λ s + s 2 + s 3 + + s k 1 + s k 1 d(fx, fy) = sλ 1+s + s 2 + s 3 + + s k 2 + s k 2 d(fx, fy) sλ 1+s + s 2 + s 3 + + s k 2 + s k 1 d(fx, fy) = sλ sk 1 d(fx, fy). s 1 = sλ sk 1 d(x, y). s 1 This implies x = y. Hence the common fixed point is unique. Example 3.11 Let X = R and d : X X X such that d(x, y)= x y 3.Thend is a b-metric on X with s =4.Letf : X 2 X and g : X X be defined as follows: f (x, y)= x2 + y 2 + 18 13 13 gx = x 2 2 ifx R. if (x, y) R, We will prove that f and g satisfy condition (3.8): d ( f (x, y), f (y, z) ) = f (x, y) f (y, z) 3 = x 2 z 2 3 13 = x 2 y 2 + y 2 z 2 13 ( x 2 y 2 3 4 13 + y 2 z 2 3) 13 = 4 x 2 y 2 3 + y 2 z 2 3 13 3 = 8 1 x 2 y 2 3 + y 2 z 2 3 13 3 2 8 13 max{ x 2 y 2 3, y 2 z 2 3 } 3 = 8 13 3 max{ d(gx, gy), d(gy, gz) }. Thus, f and g satisfy condition (3.8) withλ = 8 1 (0, ). Clearly C(f, g) =2,f and g 13 3 4 3 commute at 2. Finally, 2 is the unique common fixed point of f and g. Butf and g do not satisfy condition (3.10) asatx = 1andy =1,d(f (x, x), f (y, y)) = d(f ( 1, 1), f (1, 1)) = d( 2 13 + 18 13, 2 13 + 18 )=0=d( 1, 1) = d(g( 1), g(1)) = d(gx, gy). 13 4 Application to matrix equation InthissectionwehaveappliedTheorem3.7 to study the existence of solutions of the nonlinear matrix equation 3 X = Q + m A i X δ i A i, 0< δ i <1, (4.1) where Q is an n n positive semidefinite matrix and A i s are nonsingular n n matrices, or Q is an n n positive definite matrix and A i s are arbitrary n n matrices, and a positive
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 13 of 17 definite solution X is sought. Here A i denotes the conjugate transpose of the matrix A i. The existence and uniqueness of positive definite solutions and numerical methods for finding a solution of (4.1) have recently been studied by many authors (see 25 30). The Thompson metric on the open convex cone P(N)(N 2), the set of all N N Hermitian positive definite matrices, is defined by d(a, B)=max { log M(A/B), log M(B/A) }, (4.2) where M(A/B) =inf{λ >0:A λb} = λ max (B 1/2 AB 1/2 ), the maximal eigenvalue of B 1/2 AB 1/2.HereX Y means that Y X is positive semidefinite and X < Y means that Y X is positive definite. Thompson 31 has proved that P(N) is a complete metric space with respect to the Thompson metric d and d(a, B)= log(a 1/2 BA 1/2 ),where stands for the spectral norm. The Thompson metric exists on any open normal convex cone of real Banach spaces 31, 32; in particular, the open convex cone of positive definite operators of a Hilbert space. It is invariant under the matrix inversion and congruence transformations: d(a, B)=d ( A 1, B 1) = d ( MAM, MBM ) (4.3) for any nonsingular matrix M. One remarkable and useful result is the nonpositive curvature property of the Thompson metric: d ( X r, Y r) rd(x, Y ), r 0, 1. (4.4) By the invariant properties of the metric, we then have d ( MX r M, MY r M ) = r d(x, Y ), r 1, 1 (4.5) for any X, Y P(N) and a nonsingular matrix M. Proceeding as in 30 weprovethefollowing lemma. Lemma 4.1 For any A 1, A 2,...,A k P(N), B 1, B 2,...,B k P(N), d(a 1 + A 2 + + A k, B 1 + B 2 + + B k ) max{d(a 1, B 1 ), d(a 2, B 2 ),...,d(a k, B k )}. Proof Without loss of generality we can assume that d(a 1, B 1 ) d(a 2, B 2 ) d(a k, B k )=log r. ThenB 1 ra 1, B 2 ra 2,...,B k ra k and A 1 rb 1, A 2 rb 2,...,A k rb k,andthusb 1 + A 1 r(a 1 + B 1 ), B 2 + A 2 r(a 2 + B 2 ),...,B k + A k r(a k + B k ). Hence A 1 + A 2 + + A k rb 1 + B 2 + + B k andb 1 + B 2 + + B k ra 1 + A 2 + + A k. Hence d(a 1 + A 2 + + A k, B 1 + B 2 + + B k ) log r = d(a k, B k )= max{d(a 1, B 1 ), d(a 2, B 2 ),...,d(a k, B k )}. For arbitrarily chosen positive definite matrices X n r, X n (r 1),...,X n, consider the iterative sequence of matrices, given by X n+1 = Q + A 1 Xα 1 n r A 1 + A 2 Xα 2 n (r 1) A 2 + + A r+1 Xα r+1 n A r+1, (4.6) α 1, α 2,...,α r+1 are real numbers.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 14 of 17 Theorem 4.2 Suppose that λ = max{ α 1, α 2,..., α r+1 } (0, 1). (i) Equation (4.6) has a unique equilibrium point in P(N), that is, there exists unique U P(N) such that U = Q + A 1 Uα 1 A 1 + A 2 Uα 2 A 2 + + A r+1 Uα r+1 A r+1. (4.7) (ii) The iterative sequence {X n } defined by (4.6) converges to a unique solution of (4.1). Proof Define the mapping f : P(N) P(N) P(N) P(N) P(N)by f (X 1, X 2, X n (r 2),...,X k )=Q + A 1 Xα 1 1 A 1 + A 2 Xα 2 2 A 2 + + A r+1 Xα r+1 k A r+1, (4.8) where X 1, X 2,...,X k P(N). For all X n r, X n (r 1), X n (r 2),...,X n+1 P(N), we have d ( f (X n r, X n (r 1), X n (r 2),...,X n ), f (X n (r 1), X n (r 2), X n (r 2),...,X n+1 ) ) = d ( Q + A 1 Xα 1 n r A 1 + A 2 Xα 2 n (r 1) A 2 + + A r+1 Xα r+1 n A r+1, Q + A 2 Xα 1 n (r 1) A 2 + A 3 Xα 3 n (r 2) A 3 + + A r+2 Xα r+2 n+1 A r+2 d ( A 1 Xα 1 n r A 1 + A 2 Xα 2 n (r 1) A 2 + + A r+1 Xα r+1 n A r+1, A 2 Xα 1 n (r 1) A 2 + A 3 Xα 3 n (r 2) A 3 + + A r+2 Xα r+2 n+1 A r+2 max { d ( A 1 Xα 1 n r A 1, A 2 Xα 1 n (r 1) A ) ( 2, d A 2 X α 2 n (r 1) A 2, A 3 Xα 3 n (r 2) A 3),...,d ( A r+1 Xα r+1 n A r+1, A r+2 Xα r+2 n+1 A r+2 max { α 1 d(x n r, X n (r 1) ), α 2 d(x n (r 1), X n (r 2) ),..., α r+1 d(x n, X n+1 ) } max { α 1, α 2,..., α r+1 } max { d(x n r, X n (r 1) ), d(x n (r 1), X n (r 2) ),...,d(x n, X n+1 ) } )} λ max { d(x n r, X n (r 1) ), d(x n (r 1), X n (r 2) ),...,d(x n, X n+1 ) } (4.9) ) ) for all X n r, X n (r 1), X n (r 2),...,X n+1 P(N). X, Y P(N), we have d ( f (X, X,...,X), f (Y, Y,...,Y ) ) = d ( Q + A 1 Xα 1 A 1 + A 2 Xα 2 A 2 + + A r+1 Xα r+1 A r+1, Q + A 2 Y α 1 A 2 + A 3 Y α 3 A 3 + + A r+2 Y α r+2 A r+2 ) d ( A 1 Xα 1 A 1 + A 2 Xα 2 A 2 + + A r+1 Xα r+1 A r+1, A 2 Y α 1 A 2 + A 3 Y α 3 A 3 + + A r+2 Y α r+2 A r+2 ) max { d ( A 1 Xα 1 A 1, A 2 Y α ) ( 1 A 2, d A 2 X α 2 A 2, A 3 Y α ) 3 A 3,...,d ( A r+1 Xα r+1 A r+1, A r+2 Y α )} r+2 A r+2 max { α 1 d(x, Y ), α 2 d(x, Y ),..., α r+1 d(x, Y ) }
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 15 of 17 max { α 1, α 2,..., α r+1 } max { d(x, Y ), d(x, Y ),...,d(x, Y ) } λ max { d(x, Y ), d(x, Y ),...,d(x, Y ) } < d(x, Y ). Since λ (0, 1), (i) and (ii) follow immediately from Theorem 3.7 with s =1andg = I. Numerical experiment illustrating the above convergence algorithm Consider the nonlinear matrix equation X = Q + A X 1 2 A + B X 1 3 B + C X 1 4 C, (4.10) where 14/3 1/3 1/4 2/5 3/2 4/6 A = 2/15 1/12 1/23, B = 10/4 6/13 7/46, 3/10 9/20 11/4 5/2 4/7 6/13 1/3 19/24 22/55 1 2 3 C = 17/10 27/15 45/17, Q = 2 6 4. 13/8 1/3 1/4 1 2 7 We define the iterative sequence {X n } by X n+1 = Q + A X 1 2 n 2 A + B X 1 3 n 1 B + C X 1 4 n C. (4.11) Let R m (m 2) be the residual error at the iteration m,thatis,r m = X m+1 (Q+A 2 X 1 m+1 A+ B 3 X 1 m+1 B + C 4 X 1 m+1 C),where is the spectral norm. For initial values 1 0 0 1 1 0 X 0 = 0 1 0, X 1 = 1 1 0, 0 0 1 1 0 1 1 1 1 X 2 = 1 1 1, 1 1 1 we computed the successive iterations and the error R m using MATLAB and found that after thirty five iterations the sequence given by (4.11)convergesto 639.1810 54.1681 107.3574 U = X 35 = 54.1285 44.7768 44.1469, 104.3977 42.1095 112.5509 which is clearly a solution of (4.10). The convergence history of algorithm (4.11) isgiven in Figure 1.
Pathak et al. Fixed Point Theory and Applications (2015) 2015:101 Page 16 of 17 Figure 1 Convergence history for Equation (4.11). Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally in this work. All authors read and approved the final manuscript. Author details 1 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, India. 2 Department of Mathematics, College of Science, Salman bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia. 3 Department of Mathematics and Computer Science, St. Thomas College, Bhilai, Chhattisgarh, India. 4 Department of Engineering Basic Sciences, Faculty of Engineering, Menofia University, Menofia, Egypt. 5 Department of Mathematics, Faculty of Commerce, Alazhar University, Menofia, Egypt. 6 Department of Mathematics, Government VYT PG Autonomous College, Durg, Chhattisgarh, India. 7 Department of Mathematics, Rungta College of Engineering and Technology, Bhilai, Chhattisgarh, India. Acknowledgements The authors are thankful to the learned referees for their valuable comments which helped in bringing this paper to its present form. The second, third and fourth authors would like to thank the Deanship of Scientific Research, Salman bin Abdulaziz University, Al Kharj, Kingdom of Saudi Arabia for the financial assistance provided. The research of second, third and fourth authors is supported by the Deanship of Scientific Research, Salman bin Abdulaziz University, Alkharj, Kingdom of Saudi Arabia, Research Grant No. 2014/01/2046. Received: 30 December 2014 Accepted: 29 May 2015 References 1. Presic, SB: Sur la convergence des suites. C. R. Acad. Sci. Paris 260, 3828-3830 (1965) 2. Presic, SB: Sur une classe d inequations aux differences finite et sur la convergence de certaines suites. Publ. Inst. Math. (Belgr.) 5(19),75-78 (1965) 3. Ciric, LB, Presic, SB: On Presic type generalisation of Banach contraction principle. Acta Math. Univ. Comen. LXXVI(2), 143-147 (2007) 4. Pacurar, M: Approximating common fixed points of Presic-Kannan type operators by a multi-step iterative method. An. Ştiinţ. Univ. Ovidius Constanţa 17(1), 153-168 (2009) 5. Pacurar, M: Fixed points of almost Presic operators by a k-step iterative method. An. Ştiinţ. Univ. Al.I. Cuza Iaşi, Mat. LVII, 199-210 (2011) 6. Rao, KPR, Mustaq Ali, M, Fisher, B: Some Presic type generalizations of the Banach contraction principle. Math. Morav. 15(1), 41-47 (2011) 7. Pacurar, M: Common fixed points for almost Presic type operators. Carpath. J. Math. 28(1),117-126 (2012) 8. Khan, MS, Berzig, M, Samet, B: Some convergence results for iterative sequences of Presic type and applications. Adv. Differ. Equ. (2012). doi:10.1186/1687-1847-2012-38 9. George, R, Reshma, KP, Rajagopalan, R: A generalised fixed point theorem of Presic type in cone metric spaces and application to Markov process. Fixed Point Theory Appl. (2011). doi:10.1186/1687-1812-2011-85 10. Shukla, S: Presic type results in 2-Banach spaces. Afr. Math. (2013). doi:10.1007/s13370-013-0174-2 11. Shukla, S, Fisher, B: A generalization of Presic type mappings in metric-like spaces. J. Oper. 2013, Article ID368501 (2013)
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