Common Fixed Point Theorem for Almost Weak Contraction Maps with Rational Expression

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 5, 017, no. 1, 43-5 HIKARI Ltd, Common Fixed Point Theorem for Almost Weak Contraction Maps with Rational Expression V. Amarendra Babu Department of Mathematics Acharya Nagarjuna University Guntur-5 510, India O. Bhanu Sekhar 1 Department of Mathematics Potti Sriramulu Chalavadi Mallikarjuna Rao College of Engineering & Technology Vijayawada-50001, India K. Bhanu Chander Department of Mathematics Potti Sriramulu Chalavadi Mallikarjuna Rao College of Engineering & Technology Vijayawada-50001, India Copyright c 016 V. Amarendra Babu, O. Bhanu Sekhar and K. Bhanu Chander. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we prove the existence common fixed points for selfmaps satisfying almost weak contraction condition with rational expression. Also, we prove a fixed point theorem for generalized weak contraction map in a metric space. Mathematics Subject Classification: 47H10, 54H5 1 Corresponding author

2 44 V. Amarendra Babu, O. Bhanu Sekhar and K. Bhanu Chander Keywords: Fixed points, common fixed points, compatible maps, complete metric spaces 1 Introduction The development of fixed point theory is based on the generalization of contraction conditions is one direction or/and generalization of ambient space of the operator under consideration on the other. The Banach contraction mapping is one of the pivotal results of analysis. It is a very popular tool for solving existence problems in many different fields of mathematics Banach contraction principle plays an important role in solving non linear equations, and it is one of the most useful results in fixed point theory. Banach contraction principle has been generalized in various ways either by using contractive conditions or by imposing some additional conditions on the ambiant space. In the direction of generalization of contraction conditions in Berinde [3] in introduced weak contractions as a generalization of contraction maps. Berinde renamed weak contractions as almost contraction in his later work [4] for more works on almost contractions and its generalizations, we refer to Babu, Sandhya and Kameswari [], Abbas, Babu and Alemayehu [1] and the related references cited in this papers. In 008, Suzuki [1] proved a fixed point theorem, which is a new type of generalization of the Banach contraction principle. The following important generalization is due to Suzuki [1]. Theorem 1.1 [1] Define a function θ from [0, 1) on to ( 1, 1] by 1, if 0 r ( 5 1); θ(r) = 1 r, if ( 5 1) r 1 r ; 1, if 1 r < 1. 1+r Let (X, d) be a complete metric space T is a mapping on X. If T satisfy the following θ(r)d(x, T x ) d(x, y) d(t x, T y ) rd(x, y) for all x, y X. Then T has a fixed point. Throughout this paper, we denote R + = [0, ) and Ψ = {ψ/ψ : R + R + is continuous on R +, ψ is nondecreasing and ψ n (t) < } and Φ = {φ : R + R + is lower semi continuous on R + with φ(t) = n=1 0 if and only if t = 0. Here we note that if ψ Ψ then ψ(t) < t for every t > 0 and ψ(0) = 0. The Banach contraction theorem and its several extensions have been generalized using recently developed notion of Weakly contractive maps. The following basic result is due to Rhoades [10].

3 Common fixed point theorem 45 Theorem 1. [10] Let X be a complete metric space and T : X X such that for every x, y X, d(t x, T y ) d(x, y) ϕ(d(x, y)), where ϕ : R + R + is a continuous and non decreasing function with ϕ(0) = 0 and ϕ(t) > 0 for all t > 0. Then T has a unique fixed point. Dutta and Choudhary [7] obtained by the following theorem. Theorem 1.3 [7] Let X be a complete metric space and T : X X such that for every x, y X, ϕ(d(t x, T y )) ψ(d(x, y)) ϕ(d(x, y)), where ψ : [0, ) [0, ) is a continuous and monotone non decreasing function with ψ(t) = 0 if and only if t = 0 and ϕ : R + R + is lower semicontinuous function with ϕ(t) = 0 if and only if t = 0 Recently, Shyam Lal Singh, Rajkamal, M. Delasen and Renu Chugh [11] proved the following theorem. Theorem 1.4 [11]Let X be a complete metric space and T : X X such that for every x, y X, 1d(x, T x) d(x, y) d(x,t y)+d(y,t x) ψ(d(t x, T y) ψ(max{d(x, y), d(x, T x), d(y, T y), d(x,t y)+d(y,t x) ϕ(max{d(x, y), d(x, T x), d(y, T y), } where ψ : R + R + is a continuous and monotone non decreasing function with ψ(t) = 0 if and only if t = 0 and ϕ : R + R + is lower semi-continuous function with ϕ(t) = 0 if and only if t = 0. Then T has a unique fixed point Theorem 1.5 [5] Let (X, d) be a complete metric space and T : X X a mapping such that there exists α, β > 0 with α + β < 1 satisfying d(y, T y)[1 + d(x, T x)] d(t x, T y) αd(x, y) + β 1 + d(x, y) for all x, y X.Then T has a unique fixed point. Dutta and Choudhury [7] proved that every (ψ, ϕ)-weakly contractive map has a unique fixed point in complete metric spaces. On the other hand, Berinde [3] introduced weak contractions as a generalization of contraction maps. Definition 1.6 [4] Let (X, d) be a metric space. A self map T : X X is said to be a weak contraction if there exists δ (0, 1) and L 0 such that for all (x, y) X, d(t x, T y) δd(x, y) + Ld(y, T x). Berinde [4] proved that every weak contraction has a fixed point metric space and provided an example to show that this fixed point need not be unique. In order to obtain the uniqueness of point.berinde [4] used the following condition:there exists θ (0, 1) and L 1 0 such that d(t x, T y) θd(x, y) + L 1 d(x, T x) for all x, y X (1)

4 46 V. Amarendra Babu, O. Bhanu Sekhar and K. Bhanu Chander and proved that every weak contraction together with (1)has a unique fixed point in complete metric spaces, and further posed the following problem Find a contractive type condition different from (1), that ensures the uniqueness of fixed point of weak contractions In this context Babu, Sandhya and Kameswari [] answered the above problem by introducing condition(b). Definition 1.7 [] Let(X, d) be a metric space. A map T : X X is said to satisfy condition(b) if there exists 0 < δ < 1 and L 0 such that for all x, y X, d(t x, T y) δd(x, y) + L min{d(x, T x), d(y, T y), d(x, T y), d(y, T x)}. We use the following definitions in our subsequent discussion. Definition 1.8 [8] Let A and B be selfmaps of a metric space (X, d). The pair (A, B) is said to be a compatible pair on X, if lim d(abx n, BAx n ) = 0 n whenever{x n } is a sequence in X such that n lim Ax n = n lim Bx n = t for some t X. Definition 1.9 [9] Let A and B be selfmaps of a metric space (X, d). The pair (A, B) is said to be weakly compatible, if they commute at their coincidence points i.e., ABx = BAx whenever Ax = Bx, x X. Every compatible pair of maps is weakly compatible, but its converse need not true [9]. Recently, Chandok [5] established the following common fixed point result of selfmaps satisfying certain contraction condition involving rational expression. Theorem 1.10 [5] Let M be a subset of a metric space (X, d). Suppose that T, f, g : M M satisfy, T x)d(gy, fy d(t x, T y) α( ) + β(, gy)), gy) +, gy) + d(gy, T x) for all x, y M and for some α, β (0, 1) with α + β < 1. Suppose also that T (M) f(m) g(m) and (g(m), d) is complete. Then T, f and g have a coincidence point in M. Also, if the pairs (g, T ) and (g, f) are weakly comparable, then T, f and g have a unique common fixed point in X.

5 Common fixed point theorem 47 Main Results The following is our main theorem. Theorem.1 Let Mbe a subset of a metric space (X, d). Suppose that T, f, g : M M satisfy,t x)d(gy,fy) d(t x, fy) φ(max{,, gy),gy)+,fy)+d(gy,t x) +L min{d(gy, T x),, fy),, gy), d(t x, fy),, T x)}(.1.1) for all x, y M, L 0 and φ Ψ. Suppose also that T (M) f(m) g(m) and (g(m), d) is complete. Then (i) T, f and g have a coincidence point in M; (ii) If the pairs (T, g) and (f, g) are weakly compatible, then T, f and g have a unique common fixed point. Proof: Let x 0 X. Since T (M) f(m) g(m), we can choose x 1, x M such that gx 1 = T x 0 and gx = fx 1. By induction, we construct a sequence in x n X such that gx n+1 = T x n and gx n+ = fx n+1, n 0. By the inequality (.1.1) n+1, gx n+ ) = d(t x n, fx n+1 ) φ(max{ n,t x n ) n+1,fx n+1 ), n,gx n+1 )+ n,fx n+1 )+ n+1,t x n ) n, gx n+1 ) +L min{ n+1, T x n ), n, fx n+1 ), n, gx n+1 ), d(t x n, fx n+1 ), n, T x n )} = φ(max{ n,gx n+1 ) n+1,gx n+ ), n,gx n+1 )+ n,gx n+ )+ n+1,gx n+1 ) n, gx n+1 ) +L min{ n+1, gx n+1 ), n, gx n+ ), n, gx n+1 ), n+1, gx n+ ), n, gx n+1 )} φ(max{ n,gx n+1 ) n+1,gx n+ ) n+1,gx n+, )) n, gx n+1 ) = φ( n, gx n+1 ) < n, gx n+1 ) n = 0, 1,, 3,... (.1.) Thus { n+1, gx n+ )} is monotone decreasing sequence of non-negative real numbers and hence convergent. Let n lim n+1, gx n+ ) = r 0. On taking as n in (.1.), we get r φ(r) < r, which is a contradiction. Therefore r = 0. i.e., lim n+1, gx n+ ) = 0. n We now prove that{gx n } is Cauchy. It is sufficient to prove that {gx n } is Cauchy. If not there is an ɛ > 0 and there exist sequences {m k }, {n k } with m k > n k > k such that mk, gx nk ) ɛ and mk, gx nk ) < ɛ. (.1.3) We now prove that (i) lim mk, gx nk ) = ɛ. Since ɛ mk, gx nk ) for all k, we have ɛ lim inf m k, gx nk ). Now for each positive integer k, by the triangular inequality, we get mk, gx nk ) mk, gx mk 1)+ mk 1, gx mk )+ mk, gx nk ). On taking limit superior as k, we get lim sup mk, gx nk ) ɛ

6 48 V. Amarendra Babu, O. Bhanu Sekhar and K. Bhanu Chander Therefore lim mk, gx nk ) = ɛ. In the similar way, we prove the following; (ii) lim nk, gx mk +1) = ɛ; (iii) lim mk, gx nk 1) = ɛ; (iv) lim nk 1, gx mk +1) = ɛ; We now consider nk, gx mk +1) = d(t x mk, fx nk 1) ψ(max{ mk,t x mk ) nk 1,fx nk 1), mk,gx nk 1)+ mk,fx nk 1)+ nk 1,T x mk ) mk, gx nk 1) +L min{ nk 1, T x mk ), mk, fx nk 1), mk, gx nk 1), d(t x mk, fx nk 1), mk, T x mk )} = ψ(max{ mk,gx mk +1) nk 1,gx nk ),, mk,gx nk 1)+ mk,gx nk )+ nk 1,gx mk +1) mk, gx nk 1) +L min{ nk 1, gx mk +1), mk, gx nk ), mk, gx nk 1), mk +1, gx nk ), mk, gx mk +1)} On letting k, we get ɛ ψ(max{0, ɛ + L min{ɛ, ɛ, ɛ, ɛ, 0} implies that ɛ ψ(ɛ) < ɛ, a contradiction. Therefore {gx n } is Cauchy. As (g(m), d) is complete, there is t M such that gx n gt as n. We now prove that t is a coincidence point T, f and g. We consider n+1, ft) = d(t x n, ft) φ(max{ n,t x n )d(gt,ft), n,gt)+ n,ft)+d(gt,t x n ) n, gt) +L min{d(gt, T x n ), n, ft), n, gt), d(t x n, ft), n, T x n )} = φ(max{ n,gx n+1 )d(gt,ft), n,gt)+ n,ft)+d(gt,gx n+1 ) n, gt) +L min{d(gt, gx n+1 ), n, ft), n, gt), n+1, ft), n, T x n )} On letting n, we have d(gt, ft) 0 gt = ft. Also, we have d(t t, gt) = d(t t, ft) d(gt,t t)d(gt,ft) φ(max{, d(gt, gt) d(gt,gt)+d(gt,ft)+d(gt,t t) +L min{d(gt, T t), d(gt, ft), d(gt, gt), d(t t, ft), d(gt, T t)} = 0. Therefore d(t t, gt) 0 T t = gt = ft. Thus t is coincidence point of T, f and g. Suppose that the pairs(g, t) and (g, f) are weakly compatible. Let z = ft = gt = T t. Then we have gt t = T gt and fgt = gft, which implies that gz = fz = T z. Suppose gz z. We now consider d(gz,t z)d(gt,ft) d(gz, z) = d(t z, ft) φ(max{, d(gz, gt) d(gz,gt)+d(gz,ft)+d(gt,t z) +L min{d(gt, T z), d(gz, ft), d(gz, gt), d(t z, ft), d(gz, T z)} = φ(max{d(gz, z)+l min{d(z, gz), d(gz, z), d(gz, z), d(gz, z), d(gz, T z)} φ(d(gz, z)) < d(gz, z),

7 Common fixed point theorem 49 which is a contradiction. Therefore gz = fz = T z = z. Hence z is common fixed point. Uniqueness follows from the inequality (.1.1). Corollary. Let M be a subset of a Metric space (X, d). Suppose that,t x)d(gy,fy) T, f, g : M M satisfy d(t x, fy) φ(max{,, gy),gy)+,fy)+d(gy,t x) for all x, y M, and φ Ψ. Suppose also that T (M) f(m) g(m) and (g(m), d) is complete. Then (i) T, f and g have a coincidence point in M; (ii) If the pairs (T, g) and (f, g) are weakly compatible, then T, f and g have a unique common fixed point. Proof: Take L = 0 in Theorem.1, then the proof of Corollary follows. Theorem.3 Let X be a complete metric space and T : X X such that for every x, y X, 1d(x, T x) d(x, y) ψ(d(t x, T y)) ψ(m(x, y)) φ(m(x, y))+ln(x, y), (.3.1) where ψ Ψ, φ Φ and L 0. Then T has a unique fixed point. Proof: Let x 0 X. construct a sequence {x n } in X such that x n+1 = T x n, n = 0, 1,, 3,... For any n we have 1 d(x n 1, x n ) d(x n 1, x n ) and using (.3.1), we get ψ(d(t x n 1, T x n )) ψ(m(x n 1, x n )) φ(m(x n 1, x n )) + LN(x n 1, x n ) which implies that ψ(d(x n, x n+1 )) ψ(max{d(x n 1, x n ), d(x n 1, T x n 1 ), d(x n, T x n ), d(x n 1,T x n)+d(x n,t x n 1 ) ϕ(max{d(x n 1, x n ), d(x n 1, T x n 1 ), d(x n, T x n ), d(x n 1,T x n)+d(x n,t x n 1 ) +L min{d(x n 1, x n ), d(x n 1, T x n 1 ), d(x n, T x n ), d(x n 1, T x n ), d(x n, T x n 1 )} = ψ(max{d(x n 1, x n ), d(x n 1, x n ), d(x n, x n+1 ), d(x n 1,x n+1 )+d(x n,x n) ϕ(max{d(x n 1, x n ), d(x n 1, x n ), d(x n, x n+1 ), d(x n 1,x n+1 )+d(x n,x n) +L min{d(x n 1, x n ), d(x n 1, x n ), d(x n, x n+1 ), d(x n 1, x n+1 ), d(x n, x n )} = ψ(max{d(x n 1, x n ), d(x n, x n+1 ϕ(max{d(x n 1, x n ), d(x n, x n+1 +L min{d(x n 1, x n ), d(x n 1, x n ), d(x n, x n+1 ), d(x n 1, x n+1 ), 0} = ψ(max{d(x n 1, x n ), d(x n, x n+1 ϕ(max{d(x n 1, x n ), d(x n, x n+1. If max{d(x n 1, x n ), d(x n, x n+1 } = d(x n, x n+1 ), then ψ(d(x n, x n+1 )) ψ(d(x n, x n+1 )) ϕ(d(x n, x n+1 )) < ψ(d(x n, x n+1 )), which is a contradiction. Therefore ψ(d(x n, x n+1 )) ψ(d(x n 1, x n )) ϕ(d(x n 1, x n )) < ψ(d(x n 1, x n )) (.3.) By the property of ψ, we get d(x n, x n+1 ) d(x n 1, x n ), for all n Hence the sequence {d(x n, x n+1 )} is monotone decreasing sequence of nonnegative reals and bounded bellow by 0. So, there exists r 0 such that lim d(x n, x n+1 ) = r (.3.3) n If r > 0, then taking upper limits as n in (.3.) and using the property

8 50 V. Amarendra Babu, O. Bhanu Sekhar and K. Bhanu Chander of ϕ, we get ψ(r) ψ(r) ϕ(r) < ψ(r), a contradiction. Hence, n lim d(x n, x n+1 ) = 0 (.3.4) We know prove that {x n } is a Cauchy sequence. On the contrary suppose that {x n } is not Cauchy. Then there is an ɛ > 0 and there exists integers m k and n k with m k > n k > k such that d(x mk, x nk ) ɛ and d(x mk 1, x nk ) < ɛ. (.3.5) (i) We prove lim d(x mk, x nk ) = ɛ From (.3.5), we have ɛ d(x mk, x nk ). Taking limit infimum as k, we get ɛ lim inf d(x m k, x nk ). (.3.6) We consider d(x mk, x nk ) d(x mk, x mk 1) + d(x mk 1, x nk ) < d(x mk, x mk 1) + ɛ On letting limit superior as k, we get lim sup d(x mk, x nk ) ɛ. (.3.7) From (.3.6) and (.3.7), we get lim d(x m k, x nk ) = ɛ In the similar way we prove the following; (ii) lim d(x mk, x nk +) = ɛ; (iii) lim d(x mk +1, x nk ) = ɛ. (iv) lim d(x mk +1, x nk +1) = ɛ; (v) lim d(x mk +1, x nk +) = ɛ. Now, 1d(x m k, T x mk ) = 1d(x m k, x mk +1) d(x mk, x mk +1) d(x mk, x nk +1) as m k n k implies m k+1 n k+1. Therefore by (.3.1), ψ(d(t x mk, T x nk +1)) ψ(m(x mk, x nk +1)) ϕ(m(x mk, x nk +1))+LN(x mk, x nk +1) which implies that ψ(d(x mk +1, x nk +)) ψ(max{d(x mk, x nk +1), d(x mk, T x mk ), d(x nk +1, T x nk +1), d(x mk,t x nk +1)+d(x nk +1,T x mk ϕ(max{d(x mk, x nk +1), d(x mk, T x mk ), d(x nk +1, T x nk +1), d(x mk,t x nk +1)+d(x nk +1,T x mk + L min{d(x mk, x nk +1), d(x mk, T x mk ), d(x nk +1, T x nk +1), d(x mk, T x nk +1), d(x nk +1, T x mk )} = ψ(max{d(x mk, x nk +1), d(x mk, x mk +1), d(x nk +1, x nk +), d(x mk,x nk +)+d(x nk +1,x mk +1) ϕ(max{d(x mk, x nk +1), d(x mk, x mk +1), d(x nk +1, x nk +), d(x mk,x nk +)+d(x nk +1,x mk +1) + L min{d(x mk, x nk +1), d(x mk, x mk +1), d(x nk +1, x nk +), d(x mk, x nk +), d(x nk +1, x mk +1) By taking limits k, it follows that ψ(ɛ) ψ(ɛ) ϕ(ɛ) < ψ(ɛ), which is contradiction with ɛ > 0.

9 Common fixed point theorem 51 Therefore {x n } is a Cauchy sequence in X. Since X is complete, let {x n } converges to some point z X. We now show that z is a fixed point of T. We claim that 1 d(x n, T x n ) d(x n, z) or 1 d(x n+1, T x n+1 ) d(x n+1, z) Otherwise, we have d(x n, x n+1 ) d(x n, z) + d(z, x n+1 ) < 1 d(x n, T x n ) + 1 d(x n+1, T x n+1 ) = 1 [d(x n, T x n ) + d(x n+1, T x n+1 ] = 1 [d(x n, x n+1 ) + d(x n+1, x n+ ] 1 [d(x n, x n+1 ) + d(x n, x n+1 )] = d(x n, x n+1 ), which is a contradiction. Therefore, there exists a sub sequence {n k } of {n} such that 1 d(x n k, x nk +1) d(x nk, z). By (.3.1), we have ψ(d(t x nk, T z) ψ(m(x nk, z)) ϕ(m(x nk, z)) + LN(x nk, z) = ψ(max{d(x nk, z), d(x nk, T x nk ), d(z, T z), d(xn k,t z)+d(z,t xn k ) ϕ(max{d(x nk, z), d(x nk, T x nk ), d(z, T z), d(xn k,t z),d(z,t xn k ) +L min{d(x nk, z)d(x nk, T x nk ), d(z, T z), d(x nk, T z), d(z, T x nk )} = ψ(max{d(x nk, z), d(x nk, x nk +1, d(z, T z), d(xn k,t z)+d(z,x n k +1 ϕ(max{d(x nk, z), d(x nk, x nk +1), d(z, T z), d(xn k,t z)+d(z,x n k +1 +L min{d(x nk, z), d(x nk, x nk +1, d(z, T z), d(x nk, T z), d(z, x nk +1)}. On letting k, we get ψ(d(z, T z)) ψ(d(z, T z)) ϕ(d(z, T z)) which implies that z = T z. Therefore z is fixed point of T. Uniqueness trivially follows from the inequality(.3.1) Acknowledgements. The authors are sincerely thankful to the anonymous referee for his/her valuable suggestions which helped us to improve the quality of the paper. References [1] M. Abbas, G. V. R. Babu and G.N Alemayehu, On common fixed points of weakly compatible mappings satisfying Generalized condition (B), Filomat, 5 (011), [] G. V. R. Babu, M. L. Sandhya and M. V. R. Kameswari, A note on a fixed point theorem of Berinde on Weak contractions, Carpath. J. Math., 4 (008), 8-1.

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