K. P. R. Rao, K. R. K. Rao UNIQUE COMMON FIXED POINT THEOREMS FOR PAIRS OF HYBRID MAPS UNDER A NEW CONDITION IN PARTIAL METRIC SPACES

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1 DEMONSTRATIO MATHEMATICA Vol. XLVII No K. P. R. Rao, K. R. K. Rao UNIQUE COMMON FIXED POINT THEOREMS FOR PAIRS OF HYBRID MAPS UNDER A NEW CONDITION IN PARTIAL METRIC SPACES Abstract. In this paper, we introduce a new condition namely, condition (W.C.C) and obtain two unique common fixed point theorems for pairs of hybrid mappings on a partial Hausdorff metric space without using any continuity and commutativity of the mappings.. Introduction and preliminaries In 969, Nadler [20] initiated the development of the geometric fixed point theory for multivalued mappings. He used the concept of the Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case. Indeed, the fixed point theorems for multivalued mappings are quite useful in control theory and have been frequently used in solving many problems of economics, game theory, convex optimization and differential equations. Here, we recall that a Hausdorff metric H induced by a metric d on a set X is given by! HpA, Bq max sup xpa ) dpx, Bq, sup dpy, Aq, ypb for every A, B P CBpXq, where dpx, Bq inftdpx, yq : y P Bu and CBpXq is the collection of the closed and bounded subsets of X. Theorem.. [20] Let px, dq be a complete metric space and T : X Ñ CBpXq be a mapping satisfying HpT x, T yq kdpx, yq, where k P r0, q then there exists x P X such that x P T x. 200 Mathematics Subject Classification: 47H0, 54H25. Key words and phrases: partial metric space, multi-valued maps, partial Hausdorff metric, condition (W.C.C). DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

2 Unique common fixed point theorems for pairs of hybrid maps In the last decades, a number of fixed point results (see, for example, [, 2, 8, 4, 5, 7, 8, 9]) have been obtained in attempts to generalize Theorem.. The other basic notion for the development of our work is the concept of the partial metric space, that was introduced by Matthews [2] as a part of the study of denotational semantics of data flow networks. He presented a modified version of the Banach contraction principle, more suitable in this context, see also [3, 6]. In fact, the partial metric spaces constitute a suitable framework to model several distinguished examples of the theory of computation and also to model metric spaces via domain theory, see [4, 5, 7, 0,, 2, 3, 6, 2, 22]. In this direction, Aydi et al. [9] introduced the concept of a partial Hausdorff metric and extended Nadler s fixed point theorem in the setting of partial metric spaces. In view of the above considerations, the aim of this paper is to introduce a new condition namely, condition (W.C.C) and obtain unique common fixed point theorems for pairs of hybrid mappings in a partial Hausdorff metric space without using any continuity and commutativity of the mappings. The presented results extend and unify some recently obtained comparable results for multivalued mappings (see [9] and the references therein). Consistent with [9, 0, 2], the following definitions and results will be needed in the sequel. Definition.2. [2] A partial metric on a nonempty set X is a function p : X ˆ X Ñ R` such that for all x, y, z P X: (p ) x y ô ppx, xq ppx, yq ppy, yq, (p 2 ) ppx, xq ppx, yq, (p 3 ) ppx, yq ppy, xq, (p 4 ) ppx, yq ppx, zq ` ppz, yq ppz, zq. In this case px, pq is called a partial metric space. It is clear that ppx, yq ppy, zq ppx, y, z P X. It is also clear that ppx, yq 0 implies x y from pp q and pp 2 q. But if x y, ppx, yq may not be zero. A basic example of a partial metric space is the pair pr`, pq, where ppx, yq maxtx, yu for all x, y P R`. Each partial metric p on X generates τ 0 topology τ p on X which has as a base the family of open p - balls tb p px, ɛq x P X, ɛ ą 0u for all x P X and ɛ ą 0, where B p px, ɛq ty P X ppx, yq ă ppx, xq ` ɛu for all x P X and ɛ ą 0. If p is a partial metric on X, then the function p s : X ˆ X Ñ R`, given by p s px, yq 2ppx, yq ppx, xq ppy, yq, is a metric on X. Definition.3. [2] Let px, pq be a partial metric space.

3 76 K. P. R. Rao, K. R. K. Rao (i) A sequence tx n u in px, pq is said to converge to a point x P X if and only if ppx, xq lim ppx, x n q. (ii) A sequence tx n u in px, pq is said to be Cauchy sequence if lim ppx n, x m q exists and is finite. n,mñ8 (iii) px, pq is said to be complete if every Cauchy sequence tx n u in X converges, with respect to τ p, to a point x P X such that ppx, xq lim ppx n, x m q. n,mñ8 Lemma.4. [2] Let px, pq be a partial metric space. (a) tx n u is a Cauchy sequence in px, pq if and only if it is a Cauchy sequence in the metric space px, p s q. (b) px, pq is complete iff the metric space px, p s q is complete. Further more, lim ps px n, xq 0 if and only if ppx, xq lim ppx n, xq lim ppx n, x m q. n,mñ8 Lemma.5. [0] Let px, pq be a partial metric space and A any nonempty set in px, pq, then a P A if and only if ppa, Aq ppa, aq, where A denotes the closure of A with respect to the partial metric p. Consistent with [9], let px, pq be a partial metric space and let CB p pxq be the family of all non-empty, closed and bounded subsets of the partial metric space px, pq, induced by the partial metric p. Note that the closedness is taken from px, τ p q pτ p is the topology induced by p) and the boundedness is given as follows: A is a bounded subset in px, pq if there exist x 0 P X and M 0 such that for all a P A, we have a P B p px 0, Mq, that is, ppx 0, aq ă ppx 0, x 0 q ` M. For A, B P CB p pxq, x P X, δ p : CB p pxq ˆ CB p pxq Ñ R` define ppx, Aq inf tppx, aq, a P Au, δ p pa, Bq sup tppa, Bq : a P Au, δ p pb, Aq sup tppb, Aq : b P Bu, H p pa, Bq max tδ p pa, Bq, δ p pb, Aqu. The mapping H p : CB p pxq ˆ CB p pxq Ñ R` is called the partial Hausdorff metric induced by partial metric p. Every Hausdorff metric is a partial Hausdorff metric but the converse is not true, see Example 2.6 in [9]. Lemma.6. [9] Let px, pq be a partial metric space. For any A, B, C P CB p pxq, we have (i) δ p pa, Aq = sup tppa, aq : a P Au, (ii) δ p pa, Aq δ p pa, Bq, (iii) δ p pa, Bq = 0 implies that A Ď B, (iv) δ p pa, Bq δ p pa, Cq ` δ p pc, Bq inf ppc, cq. cpc

4 Unique common fixed point theorems for pairs of hybrid maps Lemma.7. [9] Let px, pq be a partial metric space. For any A, B, C P CB p pxq, we have (i) H p pa, Aq H p pa, Bq, (ii) H p pa, Bq H p pb, Aq, (iii) H p pa, Bq H p pa, Cq ` H p pc, Bq - inf ppc, cq. cpc Lemma.8. [9] Let px, pq be a partial metric space. For any A, B P CB p pxq the following holds: H p pa, Bq 0 implies that A B. In [9], they also show that by an example, H p pa, Aq need not be zero. Lemma.9. [9] Let px, pq be a partial metric space, A, B P CB p pxq and h ą. For any a P A, there exists b P B such that ppa, bq hh p pa, Bq. Theorem.0. [9] Let px, pq be a complete partial metric space and T : X Ñ CB p pxq is a multi-valued mapping such that for all x, y P X, we have H p pt x, T yq k ppx, yq, where k P p0, q then T has a fixed point. We state and prove our main results. 2. Main results Lemma 2.. Let x n Ñ x as n Ñ 8 in a partial metric space px, pq such that ppx, xq 0 then lim ppx n, Bq ppx, Bq for any B P CB p pxq. Proof. Since x n Ñ x, we have lim ppx n, xq ppx, xq 0. By using triangular inequality for x n P X and y P B, we have which implies that ppx n, yq ppx n, xq ` ppx, yq ppx, xq, ppx n, Bq ppx n, xq ` ppx, Bq. Therefore, we get lim ppx n, Bq ppx, Bq...(i). On the other hand, we have Thus ppx, yq ppx, x n q ` ppx n, yq ppx n, x n q. ppx, yq ppx, x n q ` ppx n, yq. By taking infinimum over y P B, we get ppx, Bq ppx, x n q ` ppx n, Bq. Therefore, we get ppx, Bq lim ppx n, Bq...(ii). From (i) and (ii), we have lim ppx n, Bq ppx, Bq. Now we introduce the following new condition, namely, the condition (W.C.C) on mappings which are not necessarily continuous and commutative.

5 78 K. P. R. Rao, K. R. K. Rao Definition 2.2. Let px, pq be a partial metric space. Let f, g : X Ñ X and S : X Ñ CB p pxq be mappings. Then (i) the triplet pf, g; Sq is said to satisfy the condition pw.c.cq if ppfx, gyq ppy, x, y P X, (ii) the pair pf; Sq is said to satisfy the condition pw.c.cq if ppfx, fyq ppy, x, y P X. The following example illustrates the condition (W.C.C). Example 2.3. Let X r0, s and ppx, yq maxtx, x, y P X. Let f, g : X Ñ X and S : X Ñ CB p pxq be defined by fx x, y P X, 0, if x P r0, gx 2 s, x 32, if x P p 2, s, and Sx r0, x, y P X. We consider the following two cases. Case (a): x P X and y P r0, 2s. Then ppfx, gyq 0 ppy, Sxq. Case (b): x P X and y P p 2, s. Then ppfx, gyq y 32 ă y ppy, Sxq. Thus pf, g; Sq satisfies the condition pw.c.cq. The following example shows that the triplet pf, g; Sq, satisfying the condition (W.C.C), need not be continuous even when S is a single-valued mapping. Example 2.4. Let X r0, s and ppx, yq maxtx, y P X. Let f, g, S : X Ñ X be defined by x fx 2, if x, x 24, if x, x 6 if x, 2 if x, and x Sx 2, if x, 4, if x. Clearly, all the mappings f, g and S are discontinuous. Now, we distinguish the following cases to show that pf, g; Sq satisfies the condition pw.c.cq. Case (i): x and y. " x ppfx, gyq max 2, y * 6 " * x2 6 max, y ppy, Sxq. 6 Case (ii): x and y. " x ppfx, gyq max 2, * ă 6 " 2 max, x * ppy, Sxq. 2 6

6 Unique common fixed point theorems for pairs of hybrid maps Case (iii): x and y. " ppfx, gyq max 24, y * 6 Case (iv): x and y. " ppfx, gyq max 24, * 2 Thus, pf, g; Sq satisfies the condition (W.C.C). ppy, Sxq. 6 " 2 ă max, * ppy, Sxq. 4 The following example shows that the triplet pf, g; Sq satisfying the condition (W.C.C), need not be commuting even when S is a single-valued mapping. Example 2.5. Let a and b be non-negative real numbers such that b ă a. Let X ta, bu and ppx, yq maxtx, y P X. Let f, g, S : X Ñ X be defined by fa fb b, ga b, gb a and Sa Sb a. Clearly the triplet pf, g; Sq satisfies the condition pw.c.cq and the pairs pf, Sq, pg, Sq and pf, gq are not commuting. Now, we state and prove our main results. Theorem 2.6. Let px, pq be a complete partial metric space and let S, T : X Ñ CB p pxq and f, g : X Ñ X be mappings satisfying ppfx, gyq, p2.6.q HppSx, T yq α max 2rppfx, Sxq ` ppgy, T yqs, 2rppfx, T yq ` ppgy, Sxqs for Ť all x, y P X and α P p0, q, p2.6.2q Sx Ď gpxq and Ť T x Ď fpxq, xpx xpx p2.6.3q the triplet pf, g; Sq or the triplet pf, g; T q satisfies the condition pw.c.cq. Then f, g, S and T have a unique common fixed point in X. Proof. Let x 0 P X. From p2.6.2q, there exist x, y P X such that y gx P Sx 0. From p2.6.2q and Lemma.9 with h? α, there exist x 2, y 2 P X such that y 2 fx 2 P T x and ppy, y 2 q? H p psx 0, T x q. α Again from p2.6.2q and Lemma.9, there exist x 3, y 3 P X such that y 3 gx 3 P Sx 2 and ppy 2, y 3 q? H p psx 2, T x q. α

7 720 K. P. R. Rao, K. R. K. Rao Continuing in this way, we get the sequences tx n u and ty n u in X such that y 2n` gx 2n` P Sx 2n, y 2n`2 fx 2n`2 P T x 2n`, n 0,, 2, 3,... and ppy 2n`, y 2n q? H p psx 2n, T x 2n q, n, 2, 3,... α ppy 2n`, y 2n`2 q? α H p psx 2n, T x 2n` q, n 0,, 2, 3,... Now from p2.6.q, we have ppy 2n`, y 2n`2 q? H p psx 2n, T x 2n` q, α? ppfx 2n, gx 2n` q, α max 2 rppfx 2n, Sx 2n q ` ppgx 2n`, T x 2n` qs, 2 rppfx 2n, T x 2n` q ` ppgx 2n`, Sx 2n qs? ppy 2n, y 2n` q, α max 2 rppy 2n, y 2n` q ` ppy 2n`, y 2n`2 qs, 2 rppy 2n, y 2n`2 q ` ppy 2n`, y 2n` qs? " α max ppy 2n, y 2n` q, * 2 rppy 2n, y 2n` q ` ppy 2n`, y 2n`2 qs Thus, we have () ppy 2n`, y 2n`2 q βppy 2n, y 2n` q,!?α,? ) where β max α{2 ă.?α{2 Similarly, we can show that (2) ppy 2n`, y 2n q βppy 2n, y 2n q. From pq and p2q, we have (3) ppy n`, y n q βppy n, y n q, for all n, 2, 3,... By continuing in this way, we get (4) ppy n`, y n q β n ppy, y 0 q. Since β ă, which in turn yields that (5) ppy n`, y n q Ñ 0 as n Ñ 8. For m ą n, we have (6) ppy n, y m q ppy n, y n` q ` ppy n`, y n`2 q ` ` ppy m, y m q, `β n ` β n` ` ` β m ppy, y 0 q from p2q βn β ppy, y 0 q Ñ 0 as n Ñ 8. from pp 4 q.

8 Unique common fixed point theorems for pairs of hybrid maps Thus ty n u is a Cauchy sequence in X. Hence from Lemma.4, ty n u is a Cauchy sequence in px, p s q. Since px, pq is complete and from Lemma.4, it follows that px, p s q is complete. So ty n u converges to some z P X. That is lim ps py n, zq 0. Now from Lemma.4 and (6), we have (7) ppz, zq lim ppy n, zq lim ppy n, y m q 0. Suppose the triplet pf, g; Sq satisfies the condition (W.C.C) then (8) ppfx, gyq ppy, Sxq for all x, y P X. Let x x 2n and y z in (8), we have ppfx 2n, gzq ppz, Sx 2n q ppz, gx 2n` q. Letting n Ñ 8, using Lemma 2. and (7), we can obtain Now by using p2.6.q, we have ppz, gzq 0 so that gz z. ppgx 2n`, T zq H p psx 2n, T zq, ppfx 2n, gzq, α max 2 rppfx 2n, Sx 2n q ` ppgz, T zqs, 2 rppfx 2n, T zq ` ppgz, Sx 2n qs α max ppfx 2n, zq, 2 rppfx 2n, gx 2n` q ` ppz, T zqs, 2 rppfx. 2n, T zq ` ppz, gx 2n` qs Letting n Ñ 8, using Lemma 2., (7) and (5), we get ppz, T zq α ppz, T zq. 2 Hence ppz, T zq 0, which in turn yields from Lemma.5 and (7) that z P T z T z. Thus (9) gz z P T z. Now from (8), we have (0) ppfz, zq ppfz, gzq ppz, Szq.

9 722 K. P. R. Rao, K. R. K. Rao Using p2.6.q, we have ppz, Szq H p psz, T zq ppfz, gzq, α max 2rppfz, Szq ` ppgz, T zqs, 2rppfz, T zq ` ppgz, Szqs α max tppz, Szq, ppz, Szq, ppz, Szqu from pp 4 q, p0q, p9q αppz, Szq, which in turn yields that ppz, Szq 0. From Lemma.5 and (7), we have z P Sz Sz. Now from (8), we get ppfz, zq 0 so that fz z. Thus () fz z P Sz. From (9) and (), it follows that z is a common fixed point of f, g, S and T. Suppose z is another common fixed point of f, g, S and T. From (8), we have (2) ppz, z q ppfz, gz q ppz, Szq H p psz, T z q. Now H p psz, T z q α max α max ppfz, gz q, 2 rppfz, Szq ` ppgz, T z qs, 2 rppfz, T z q ` ppgz, Szqs H p psz, T z q, 2 rh ppsz, Szq ` H p pt z, T z qs, 2 rh ppsz, T z q ` H p pt z, Szqs αh p psz, T z q from Lemma.7 (i). from (2) Thus H p psz, T z q 0 so that from (2), z z. Hence z is the unique common fixed point of f, g, S and T. Similarly we can prove the theorem when pf, g; T q satisfies the condition (W.C.C). Proceeding as in Theorem 2.6, one can easily prove the following. Theorem 2.7. Let px, pq be a complete partial metric space and let S, T : X Ñ CB p pxq and f : X Ñ X be mappings satisfying ppfx, fyq, ppfx, Sxq, ppfy, T yq, p2.7.q HppSx, T yq α max for all 2rppfx, T yq ` ppfy, Sxqs x, Ť y P X where 0 α ă, p2.7.2q Sx Ď fpxq and Ť T x Ď fpxq, xpx xpx p2.7.3q the pair pf; Sq or the triplet pf; T q satisfies the condition pw.c.cq.

10 Unique common fixed point theorems for pairs of hybrid maps Then f, S and T have a unique common fixed point in X. Finally, we give the following. Theorem 2.8. Let px, pq be a complete partial metric space and let S, T : X Ñ CB p pxq be mappings satisfying ppx, yq, ppx, Sxq, ppy, T yq, p2.8.q HppSx, T yq α max for all x, y P X, 2rppx, T yq ` ppy, Sxqs where 0 α ă. Then S and T have a common fixed point in X. Further, if we assume that ppx, yq ppy, Sxq or ppx, yq ppy, T xq for all x, y P X then S and T have a unique common fixed point in X. References [] B. Damjanovic, B. Samet, C. Vetro, Common fixed point theorems for multi-valued maps, Acta Math. Sci. (English Ed.) 32 (202), [2] B. D. Rouhani, S. Moradi, Common fixed point of multivalued generalized φ-weak contractive mappings, Fixed Point Theory Appl. vol. 200, Artical ID , 3 pages. [3] C. Di Bari, P. Vetro, Fixed points for weak φ-contractions on partial metric spaces, Int. J. Contemp. Math. Sci. () (20), 5 3. [4] C. Di Bari, Z. Kadelburg, H. K. Nashine, S. Radenovic, Common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces, Fixed Point Theory Appl. vol. 202, 202:3, 3 pages. [5] C. Di Bari, M. Milojevic, S. Radenovic, P. Vetro, Common fixed points for selfmappings on partial metric spaces, Fixed Point Theory Appl. vol. 202, 202:40, 0 pages. [6] D. Paesano, P. Vetro, Suzki s type characterization of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl. 59(3) (202), [7] F. Vetro, S. Radenovic, Nonlinear ψ-quasi-contractions of Ciric-type in partial metric spaces, Appl. Math. Comput. 29 (202), [8] H. Covitz, S. B. Nadler Jr., Multi-valued contraction mappings in generalized metric spaces, Israel J. Math. 8 (970), 5. [9] H. Aydi, M. Abbas, C. Vetro, Partial Hausdrouff metric and Nadler s fixed point theorem on partial metric space, Topology Appl. 59(4) (202), [0] I. Altun, H. Simsek, Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud. (2008), 8. [] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 57 (200), [2] K. P. R. Rao, G. N. V. Kishore, A unique common fixed point theorem for four maps under pψ Φq contractive condition in partial metric spaces, Bull. Math. Anal. Appl. 3(3) (20),

11 724 K. P. R. Rao, K. R. K. Rao [3] K. P. R. Rao, G. N. V. Kishore, K. A. S. N. V. Prasad, A unique common fixed point theorem for two maps under pψ Φq contractive condition in partial metric spaces, Math. Sci. (Springer open Journal), 6:9, 202, 4 pages. [4] Lj. Ciric, Fixed points for generalized multi-valued contractions, Mat. Vesnik 9 (972), [5] Lj. Ciric, Multi-valued nonlinear contraction mappings, Nonlinear Anal. 7 (2009), [6] Lj. Ciric, B. Samet, H. Aydi, C. Vetro, Common fixed points of generalized contractions on partial metric spaces and an application, Appl. Math. Comput. 28 (20), [7] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (2008), [8] N. Mizoguchi, W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 4 (989), [9] P. Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 92(2) (995), [20] S. B. Nadler, Mutivalued contraction mappings, Pacific. J. Math. 30 (969), [2] S. G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. vol. 728, 994, [22] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. vol. 200, Article ID , 6 pages. K. P. R. Rao DEPARTMENT OF MATHEMATICS ACHARYA NAGARJUNA UNIVERSITY NAGARJUNA NAGAR , A.P., INDIA kprrao2004@yahoo.com K. R. K. Rao DEPARTMENT OF MATHEMATICS GITAM UNIVERSITY RUDRARAM(V), PATANCHERU(M), HYDERABAD , A.P., INDIA krkr08@gmail.com Received March, 203, revised version October 28, 203. Communicated by E. Weber.