Some common fixed-point results for mappings satisfying a nonlinear weak contraction condition within the framework of ordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-known from the literature.Отримано дєякі спільні теореми про нерухому точку для відображень, що задовольняють нелінійну слабкостискальну умову в рамках упорядкованих метричних просторів. Отримані результати узагальнюють та розширюють декілька порівняльних результатів, відомих із літературних джерел

Abbas, M.

Chandok, S.

Khan, M.S.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.9S. Chandok ( Khalsa College Engineering and Technology (Punjab Techn. Univ.), India),M. S. Khan (College Sci., Sultan Qaboos Univ. Al-Khod, Sultanate of Oman),M. Abbas (Univ. Pretoria, South Africa)COMMON FIXED POINT THEOREMSFOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGSСПIЛЬНI ТЕОРЕМИ ПРО НЕРУХОМУ ТОЧКУДЛЯ НЕЛIНIЙНИХ СЛАБКОСТИСКАЛЬНИХ ВIДОБРАЖЕНЬSome common fixed point results for mappings satisfying a nonlinear weak contractive condition in the framework ofordered metric spaces are obtained. The accumulated results generalize and extend several comparable results well-knownfrom the literature.Отримано деякi спiльнi теореми про нерухому точку для вiдображень, що задовольняють нелiнiйну слабкостис-кальну умову в рамках упорядкованих метричних просторiв. Отриманi результати узагальнюють та розширюютьдекiлька порiвняльних результатiв, вiдомих iз лiтературних джерел.Introduction and preliminaries. Banach contraction principle is one of the pivotal results of metricfixed point theory. It is a popular tool for solving existence problems in different fields of mathe-matics. There are several generalizations of Banach contraction principle in the related literature onmetric fixed point theory.Ran and Reurings [15] extended Banach contraction principle in partially ordered metric spaceswith some applications to linear and nonlinear matrix equations. While Nieto and Lo´pez [14] extendedthe result of Ran and Reurings and applied their main result to obtain a unique solution for a firstorder ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham[3] introduced a concept of mixed monotone mappings and obtained some coupled fixed point results.Also, they applied their results to a first order differential equation with periodic boundary conditions.Alber and Guerre-Delabriere [1] introduced a concept of weakly contractive mappings and provedthe existence of fixed point of such mappings in Hilbert spaces. Thereafter, in 2001, Rhoades [17]proved the fixed point theorem which is one of the generalizations of Banach’s contraction principle.Weakly contractive mappings are closely related to the mappings of Boyd and Wong [4] and of Reichtypes [16]. Recently, Doric [9] proved a common fixed point theorem for a generalized (ψ, φ)-weaklycontractive mappings. Fixed point problems involving weak contractions and mappings satisfyingweak contractive type inequalities have been studied by many authors (see [1, 5 – 10, 17] and ref-erences cited therein). In this paper, we generalize Chatterjea type contraction mappings to (µ, ψ)-generalized Chatterjea type contraction mappings and derive some common fixed point results forsingle-valued mappings on ordered metric spaces.First, we recall some basic definitions and notations.Let (X, d) be a metric space. A mapping T : X → X is said to be:(a) Kannan type (see [11]) if there exists a k ∈(0,12]such that d(Tx, Ty) ≤ k[d(x, Tx) ++ d(y, Ty)] for all x, y ∈ X;(b) Chatterjea type [7] if there exists a k ∈(0,12]such that d(Tx, Ty) ≤ k[d(x, Ty)+d(y, Tx)]for all x, y ∈ X.c© S. CHANDOK, M. S. KHAN, M. ABBAS, 2014ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4 531532 S. CHANDOK, M. S. KHAN, M. ABBASKhan et al. [12] initiated the use of a control function that alters distance between two points ina metric space. So they called it an altering distance function.A function µ : [0,∞)→ [0,∞) is called an altering distance function if the following propertiesare satisfied:(i) µ is monotone increasing and continuous;(ii) µ(t) = 0 if and only if t = 0.Using the control function, we generalize the Chatterjea type contraction mappings as follows:Suppose that T and f are self-mappings defined on a metric space X. A pair of mappings (T, f)is said to satisfy (µ, ψ)-generalized Chatterjea type contractive condition if for all x, y ∈ X,µ(d(Tx, fy)) ≤ µ(12[d(x, fy) + d(y, Tx)])− ψ(d(x, fy), d(y, Tx)), (1)holds, where µ : [0,∞)→ [0,∞) is an altering distance function and ψ : [0,∞)2 → [0,∞) is a lowersemicontinuous mapping such that ψ(x, y) = 0 if and only if x = y = 0.Let M be a nonempty subset of a metric space X, a point x ∈M is a common fixed (coincidence)point of f and T if x = fx = Tx (fx = Tx). The set of fixed points (respectively, coincidencepoints) of f and T is denoted by F (f, T ) (respectively, C(f, T )).Definition 1. Let (X,≤) be a partially ordered set. Two mappings f, g : X → X are said to beweakly increasing if fx ≤ gfx and gx ≤ fgx for all x ∈ X.The following example shows that there exist discontinuous not nondecreasing mappings whichare weakly increasing.Example 1. Let X = (0,∞), endowed with usual ordering. Let f, g : X → X be defined byfx =3x+ 2 if 0 < x < 1,2x+ 1 if 1 ≤ x <∞andgx =4x+ 1 if 0 < x < 1,3x if 1 ≤ x <∞.For 0 < x < 1, fx = 3x+2 ≤ 3(3x+2) = gfx and gx = 4x+1 ≤ 4x+3 = 2(2x+1)+1 = fgxand for 1 ≤ x < ∞, fx = 2x + 1 ≤ 3(2x + 1) = gfx and gx = 3x ≤ 2(3x) + 1 = fgx. Thus fand g are weakly increasing maps but not nondecreasing.Common fixed point theorem in ordered metric spaces. Suppose that (X,) is a partiallyordered set. A mapping T : X → X is said to be monotone increasing if for all x, y ∈ X,x  y if and only if Tx  Ty. (2)A subset W of a partially ordered set X is said to be well ordered if every two elements of Ware comparable.Theorem 1. Let (X,) be a partially ordered set such that there exists a complete metric don X. Suppose that T and f are weakly increasing self mappings on X, and satisfy (1) for allcomparable elements x, y ∈ X.Also suppose that eitherISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 533(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn  z, for every n ∈ N,or(ii) T or f is continuous.Then T and f have a common fixed point. Moreover, the set of common fixed points of f and Tis well ordered if and only if f and T have one and only one common fixed point.Proof. Let x0 ∈ X. We can choose x1, x2 ∈ X such that x1 = Tx0 and x2 = fx1. By induction,we construct a sequence {xn} in X such that x2n+1 = Tx2n and x2n+2 = fx2n+1, for every n ≥ 0.As T and f are weakly increasing mappings, so we obtainx1 = Tx0  fx1 = x2  Tx2 = x3.By induction on n, we conclude thatx1  x2  . . .  x2n+1  x2n+2  . . . .Since x2n+1 and x2n+2 are comparable, by inequality (1) we haveµ(d(x2n+1, x2n+2)) = µ(d(Tx2n, fx2n+1)) ≤≤ µ(12[d(x2n, fx2n+1) + d(x2n+1, Tx2n)])− ψ(d(x2n, fx2n+1), d(x2n+1, Tx2n)) == µ(12d(x2n, x2n+2))− ψ(d(x2n, x2n+2), 0) ≤≤ µ(12d(x2n, x2n+2)).Since µ is a monotone increasing function, for all n = 1, 2, . . . , we getd(x2n+1, x2n+2) ≤ 12d(x2n, x2n+2) ≤ 12[d(x2n, x2n+1) + d(x2n+1, x2n+2)].This implies that d(x2n+1, x2n+2) ≤ d(x2n, x2n+1). Following the similar arguments, we obtaind(x2n+2, x2n+3) ≤ d(x2n+1, x2n+2). Hence, d(xn, xn+1) ≤ d(xn−1, xn). Thus {d(xn, xn+1)} is amonotone decreasing sequence of nonnegative real numbers. Hence there exists r ≥ 0 such thatd(xn, xn+1)→ r. Asd(x2n+1, x2n+2) ≤ 12d(x2n, x2n+2) ≤ 12[d(x2n, x2n+1) + d(x2n+1, x2n+2)].Taking limit as n → ∞, we have r ≤ lim 12d(x2n, x2n+2) ≤ 12r +12r. Therefore limn→∞ d(x2n,x2n+2) = 2r. Using the continuity of µ and lower semicontinuity of ψ, we have µ(r) ≤ µ(r) −− ψ(2r, 0). This implies that ψ(2r, 0) = 0 and hence r = 0. Thus d(xn+1, xn)→ 0.Now, we prove that {xn} is a Cauchy sequence. It is sufficient to show that {x2n} is a Cauchysequence. On contrary, suppose that {x2n} is not a Cauchy sequence. Then there exists  > 0 forwhich we can find subsequences {x2m(k)} and {x2n(k)} of {x2n} such that n(k) is the smallest indexfor which n(k) > m(k) > k, d(x2m(k), x2n(k)) ≥ . This means that d(x2m(k), x2n(k)−2) < . So,we haveISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4534 S. CHANDOK, M. S. KHAN, M. ABBAS ≤ d(x2m(k), x2n(k)) ≤≤ d(x2m(k), x2n(k)−2) + d(x2n(k)−2, x2n(k)−1) + d(x2n(k)−1, x2n(k)) << + d(x2n(k)−2, x2n(k)−1) + d(x2n(k)−1, x2n(k)).Taking limit as k →∞, we getlimn→∞ d(x2m(k), x2n(k)) = . (3)Also, ≤ d(x2m(k), x2n(k)) ≤ d(x2m(k), x2m(k)−1) + d(x2m(k)−1, x2n(k)) ≤≤ 2d(x2m(k), x2m(k)−1) + d(x2m(k), x2n(k)).On letting k →∞, we obtainlimn→∞ d(x2m(k)−1, x2n(k)) = . (4)On the other hand, we haved(x2m(k), x2n(k)) ≤ d(x2m(k), x2n(k)+1) + d(x2n(k)+1, x2n(k)) ≤≤ d(x2m(k), x2n(k)) + 2d(x2n(k)+1, x2n(k)).On taking limit as k →∞, we getlimn→∞ d(x2m(k), x2n(k)+1) = .Also,d(x2m(k)−1, x2n(k)) ≤ d(x2m(k)−1, x2n(k)+1) + d(x2n(k)+1, x2n(k)) ≤≤ d(x2m(k)−1, x2n(k)) + 2d(x2n(k)+1, x2n(k)).On taking limit as k →∞, we obtainlimn→∞ d(x2m(k)−1, x2n(k)+1) = .Considerµ() ≤ µ(d(x2m(k), x2n(k))) = µ(d(Tx2m(k)−1, fx2n(k)−1)) ≤≤ µ(12[d(x2m(k)−1, fx2n(k)−1) + d(x2n(k)−1, Tx2m(k)−1)])−−ψ(d(x2m(k)−1, fx2n(k)−1), d(x2n(k)−1, Tx2m(k)−1)) == µ(12[d(x2m(k)−1, x2n(k)) + d(x2n(k)−1, x2m(k))])−ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 535−ψ(d(x2m(k)−1, x2n(k)), d(x2n(k)−1, x2m(k))).Taking limit as k → ∞, and using the continuity of µ and lower semicontinuity of ψ, we haveµ() ≤ µ(12[+ ])− ψ(, ) and consequently ψ(, ) ≤ 0, a contradiction as  > 0. Thus {x2n}is a Cauchy sequence and hence {xn} is a Cauchy sequence. As X is a complete metric space, thereexists t ∈ X such that limn→∞ xn = t. Since {xn} is a nondecreasing sequence, by (i), we havexn  t. Considerµ(d(x2n+1, ft)) = µ(d(Tx2n, ft)) ≤≤ µ(12[d(x2n, ft) + d(t, Tx2n)])− ψ(d(x2n, ft), d(t, Tx2n)) == µ(12[d(x2n, ft) + d(t, x2n+1)])− ψ(d(x2n, ft), d(t, x2n+1)).Taking limit as n → ∞, we have µ(d(t, ft)) ≤ µ(12d(t, ft))− ψ(d(t, ft), 0)) ≤ µ(12d(t, ft)).This implies that d(t, ft) = 0 and hence t = ft.Again, considerµ(d(Tt, t)) = µ(d(Tt, ft)) ≤ µ(12[d(t, ft) + d(t, T t)])− ψ(d(t, ft), d(t, T t)) == µ(12d(t, T t))− ψ(0, d(t, T t)) ≤ µ(12d(t, T t)).This implies that d(Tt, t) = 0, T t = t. Therefore, t = Tt = ft, i.e., t is a common fixed point of Tand f.If condition (ii) holds: Assume that T is continuous. Then t = limn→∞ Txn = x2n+1 = Tt.Nowµ(d(t, ft)) = µ(d(Tt, ft)) ≤ µ(12[d(t, ft) + d(t, T t)])− ψ(d(t, ft), d(t, T t)) == µ(12d(t, ft))− ψ(d(t, ft), 0) ≤ µ(12d(t, ft))implies that d(t, ft) = 0, ft = t. Therefore, t = Tt = ft, i.e., t is a common fixed point of T and f.If f is continuous, then following arguments similar to those given above, the result follows.Now suppose that the set of common fixed points of T and f is well ordered. Now, we claimthe uniqueness of common fixed points of T and f. Assume on contrary that Tu = fu = u andTv = fv = v but u 6= v. Considerµ(d(u, v)) = µ(d(Tu, fv)) ≤≤ µ(12[d(u, fv) + d(v, Tu)])− ψ(d(u, fv), d(v, Tu)) =ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4536 S. CHANDOK, M. S. KHAN, M. ABBAS= µ(12[d(u, v) + d(v, u)])− ψ(d(u, v), d(v, u)) == µ(d(u, v))− ψ(d(u, v), d(u, v)).This implies that d(u, v) = 0, by the property of ψ. Hence u = v. Conversely, if T and f haveonly one common fixed point then the set of common fixed point of f and T being singleton is wellordered.Theorem 1 is proved.If T = f, we have the following result.Corollary 1. Let (X,) be a partially ordered set such that there exists a complete metric d onX. Suppose that T is a monotone nondecreasing self mapping on X such thatµ(d(Tx, Ty)) ≤ µ(12[d(x, Ty) + d(y, Tx)])− ψ(d(x, Ty), d(y, Tx)),is satisfied for all x, y ∈ X with x and y comparable.Also suppose that either(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn  z, for every n ∈ N,or(ii) T is continuous.Then T has a fixed point.If µ(t) = t, we have the following result.Corollary 2 (see [5, 10]). Let (X,) be a partially ordered set such that there exists a completemetric d on X. Suppose that T is a monotone nondecreasing self mapping on X such thatµ(d(Tx, Ty)) ≤ µ(12[d(x, Ty) + d(y, Tx)])− ψ(d(x, Ty), d(y, Tx)),is satisfied for all comparable elements x, y ∈ X.Also suppose that either(i) if {xn} ⊂ X is a nondecreasing sequence with xn → z in X, then xn  z, for every n ∈ N,or(ii) T is continuous.Then T has a fixed point.Example 2. Let M = [0, 1] be endowed with partial order x  y if and only if x ≥ y. Let d bedefined by d(x, y) = |x− y|. We set Tx = 0 and fx = x28for all x ∈M. It is easy to see that f andg are weakly increasing maps. We define µ : [0,∞)→ [0,∞) and ψ : [0,∞)× [0,∞)→ [0,∞) byµ(t) =t2and ψ(t, s) =t+ s16.Then for x, y ∈M, we haveµ(d(Tx, fy)) = µ(d(0,y28))= µ(y28)=y216,andISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4COMMON FIXED POINT THEOREMS FOR NONLINEAR WEAKLY CONTRACTIVE MAPPINGS 537µ(12[d(x, fy) + d(y, Tx)])− ψ(d(x, fy), d(y, Tx)) == µ(12[d(x,y28)+ d(y, 0)])− ψ(d(x,y28), d(y, 0))== µ(12[ ∣∣∣∣x− y28∣∣∣∣+ y])− ψ(∣∣∣∣x− y28∣∣∣∣, y) ==14[ ∣∣∣∣x− y28∣∣∣∣+ y]−∣∣∣∣x− y28∣∣∣∣+ y16=316[ ∣∣∣∣x− y28∣∣∣∣+ y] ≥ 3y16 ≥ y216 .Henceµ(d(Tx, fy)) ≤ µ(12[d(x, fy) + d(y, Tx)])− ψ(d(x, fy), d(y, Tx)).Thus all conditions of Theorem 1 are satisfied. Moreover, T and f have a unique common fixedpoint 0.1. Alber Ya. I., Guerre-Delabriere S. Principles of weakly contractive maps in Hilbert spaces // New Results in OperatorTheory, Adv. Appl. / Eds I. Gohberg and Yu. Lyubich. – Basel: Birkha¨user, 1997. – 8. – P. 7 – 22.2. Altun I., Damjanovic´ B., Djoric´ D. Fixed point and common fixed point theorems on ordered cone metric spaces //Appl. Math. Lett. – 2010. – 23. – P. 310 – 316.3. Bhaskar T. G., Lakshmikantham V. 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Fixed point theorems for weakly C-contractive mapping in ordered metricspaces // Comput. and Math. with Appl. – 2011. – 61, № 4. – P. 790 – 796.11. Kannan R. Some results on fixed points-II // Amer. Math. Monthly. – 1969. – 76. – P. 405 – 408.12. Khan M. S., Swaleh M., Sessa S. Fixed point theorems by altering distances between the points // Bull. Aust. Math.Soc. – 1984. – 30. – P. 1 – 9.13. Nadler S. B. Multivalued contraction mappings // Pacif. J. Math. – 1969. – 30. – P. 475 – 488.14. Nieto J. J., Lo´pez R. R. Contractive mapping theorems in partially ordered sets and applications to ordinary differentialequations // Order. – 2005. – 22. – P. 223 – 239.15. Ran A. C. M., Reurings M. C. B. A fixed point theorem in partially ordered sets and some applications to matrixequations // Proc. Amer. Math. Soc. – 2004. – 132, № 5. – P. 1435 – 1443.16. Reich S. Some fixed point problems // Atti Acad. naz. Lincei. Rend. Cl. sci. fis., mat. e natur. – 1975. – 57. –P. 194 – 198.17. Rhoades B. E. Some theorems on weakly contractive maps // Nonlinear Anal. – 2001. – 47. – P. 2683 – 2693.Received 11.06.12,after revision — 04.04.13ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4

Common fixed-point theorems for nonlinear weakly contractive mappings

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Common fixed-point theorems for nonlinear weakly contractive mappings

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)