Fixed point results for various contractions in parametric and fuzzy b-metric spaces

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1 Available online at J. Nonlinear Sci. Appl. 8 (05) Research Article Fixed point results for various contractions in parametric fuzzy b-metric spaces Nawab Hussain a Peyman Salimi b Vahid Parvaneh c a Department of Mathematics King Abdulaziz University P.O. Box 8003 Jeddah 589 Saudi Arabia b Young Researchers Elite Club Rasht Branch Islamic Azad University Rasht Iran c Department of Mathematics Gilan-E-Gharb Branch Islamic Azad University Gilan-E-Gharb Iran. Abstract The notion of parametric metric spaces being a natural generalization of metric spaces was recently introduced studied by Hussain et al. [A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces Abstract Applied Analysis Vol. 04 Article ID pp]. In this paper we introduce the concept of parametric b-metric space investigate the existence of fixed points under various contractive conditions in such spaces. As applications we derive some new fixed point results in triangular partially ordered fuzzy b-metric spaces. Moreover some examples are provided here to illustrate the usability of the obtained results. c 05 All rights reserved. Keywords: Fixed point theorem fuzzy b-metric spaces contractions. 00 MSC: 54H5 54A40 54E50.. Introduction preliminaries Fixed point theory has attracted many researchers since 9 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied problems in mathematical sciences engineering. A huge literature on this subject exist this is a very active area of research at present. The concept of metric spaces has been generalized in many directions. The notion of a b-metric space was studied by Czerwik in [7 8] a lot of fixed point results for single multivalued mappings by many authors have been obtained in (ordered) b-metric spaces (see e.g. []-[7]). Khmasi Hussain [] Hussain Shah [9] discussed KKM mappings related results in b-metric cone b-metric spaces. Corresponding author addresses: nhusain@kau.edu.sa (Nawab Hussain) salimipeyman@gmail.com (Peyman Salimi) vahid.parvaneh@kiau.ac.ir (Vahid Parvaneh) Received

2 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) In this paper we introduce a new type of generalized metric space which we call parametric b-metric space as a generalization of both metric b-metric spaces. Then we prove some fixed point theorems under various contractive conditions in parametric b-metric spaces. These contractions include Geraghtytype conditions conditions using comparison functions almost generalized weakly contractive conditions. As applications we derive some new fixed point results in triangular fuzzy b-metric spaces. We illustrate these results by appropriate examples. The notion of a b-metric space was studied by Czerwik in [7 8]. Definition. ([7]). Let X be a (nonempty) set s be a given real number. A function d:x X R + is a b-metric on X if for all x y z X the following conditions hold: (b ) d(x y) = 0 if only if x = y (b ) d(x y) = d(y x) (b 3 ) d(x z) s[d(x y) + d(y z)]. In this case the pair (X d) is called a b-metric space. Note that a b-metric is not always a continuous function of its variables (see e.g. [7 Example ]) whereas an ordinary metric is. Hussain et al. [6] defined studied the concept of parametric metric space. Definition.. Let X be a nonempty set P : X X (0 ) [0 ) be a function. We say P is a parametric metric on X if (i) P(x y t) = 0 for all t > 0 if only if x = y; (ii) P(x y t) = P(y x t) for all t > 0; (iii) P(x y t) P(x z t) + P(z y t) for all x y z X all t > 0. we say the pair (X P) is a parametric metric space. Now we introduce parametric b-metric space as a generalization of parametric metric space. Definition.3. Let X be a non-empty set s be a real number let P : X (0 ) (0 ) be a map satisfying the following conditions: (P b ) P(x y t) = 0 for all t > 0 if only if x = y (P b ) P(x y t) = P(y x t) for all t > 0 (P b 3) P(x z t) s[p(x y t) + P(y z t)] for all t > 0 where s. Then P is called a parametric b-metric on X (X P) is called a parametric b-metric space with parameter s. Obviously for s = parametric b-metric reduces to parametric metric. Definition.4. Let x n be a sequence in a parametric b-metric space (X P).. x n is said to be convergent to x X written as lim x n = x if for all t > 0 lim P(x n x t) = 0.. x n is said to be a Cauchy sequence in X if for all t > 0 lim P(x n x m t) = (X P) is said to be complete if every Cauchy sequence is a convergent sequence. The following are some easy examples of parametric b-metric spaces. Example.5. Let X = [0 + ) P(x y t) = t(x y) p. Then P is a parametric b-metric with constant s = p.

3 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Definition.6. Let (X P b) be a parametric b-metric space T : X X be a mapping. We say T is a continuous mapping at x in X if for any sequence x n in X such that x n x as n then T x n T x as n. In general a parametric b-metric function for s > is not jointly continuous in all its variables. Now we present an example of a discontinuous parametric b-metric. Example.7. Let X = N let P : X (0 ) R be defined by 0 if m = n t P(m n t) = m n if m n are even or mn = 5t if m n are odd m n t otherwise. Then it is easy to see that for all m n p X we have P(m p t) 5 (P(m n t) + P(n p t)). Thus (X P) is a parametric b-metric space with s = 5. Now we show that P is not a continuous function. Take x n = n y n = then we have x n y n. Also On the other h Hence lim P(x n y n t) P(x y t). P(n t) = t n 0 P(y n t) = 0 0. P(x n y n t) = P(x n t) = t P( t) =. So from the above discussion we need the following simple lemma about the convergent sequences in the proof of our main result. Lemma.8. Let (X P s) be a parametric b-metric space suppose that x n y n are convergent to x y respectively. Then we have P(x y t) lim inf s P(x n y n t) lim sup P(x n y n t) s P(x y t) for all t (0 ). In particular if y n = y is constant then for all t (0 ). P(x y t) lim inf s P(x n y t) lim sup P(x n y t) sp(x y t) Proof. Using (P b 3) of Definition.3 in the given parametric b-metric space it is easy to see that P(x y t) sp(x x n t) + sp(x n y t) sp(x x n t) + s P(x n y n t) + s P(y n y t)

4 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) P(x n y n t) sp(x n x t) + sp(x y n t) sp(x n x t) + s P(x y t) + s P(y y n t) for all t > 0. Taking the lower limit as n in the first inequality the upper limit as n in the second inequality we obtain the desired result. If y n = y then P(x y t) sp(x x n t) + sp(x n y t) for all t > 0. P(x n y t) sp(x n x t) + sp(x y t). Main results.. Results under Geraghty-type conditions Fixed point theorems for monotone operators in ordered metric spaces are widely investigated have found various applications in differential integral equations (see [ 5 0 4] references therein). In 973 M. Geraghty [] proved a fixed point result generalizing Banach contraction principle. Several authors proved later various results using Geraghty-type conditions. Fixed point results of this kind in b-metric spaces were obtained by -Dukić et al. in [0]. Following [0] for a real number s let F s denote the class of all functions β : [0 ) [0 s ) satisfying the following condition: β(t n ) s as n implies t n 0 as n. Theorem.. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that for all t > 0 for all comparable elements x y X where M(x y t) = max P(x y t) If f is continuous then f has a fixed point. sp(fx fy t) β(p(x y t))m(x y t) () P(x fx t)p(y fy t) + P(fx fy t) P(x fx t)p(y fy t). + P(x y t) Proof. Starting with the given x 0 put x n = f n x 0. Since x 0 fx 0 f is an increasing function we obtain by induction that x 0 fx 0 f x 0 f n x 0 f n+ x 0. Step I: We will show that lim P(x n x n+ t) = 0. Since x n x n+ for each n N then by () we have sp(x n x n+ t) = sp(fx n fx n t) β(p(x n x n t))m(x n x n t) < s P(x n x n t) P(x n x n t) ()

5 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) because M(x n x n t) = max P(x n x n t) P(x n fx n t)p(x n fx n t) P(x n fx n t)p(x n fx n t) + P(fx n fx n t) + P(x n x n t) = max P(x n x n t) P(x n x n t)p(x n x n+ t) P(x n x n t)p(x n x n+ t) + P(x n x n+ t) + P(x n x n t) maxp(x n x n t) P(x n x n+ t). If maxp(x n x n t) P(x n x n+ t) = P(x n x n+ t) then from () we have P(x n x n+ t) β(p(x n x n t))p(x n x n+ t) < s P(x n x n+ t) P(x n x n+ t) which is a contradiction. Hence maxp(x n x n t) P(x n x n+ t) = P(x n x n t) so from (3) (3) P(x n x n+ t) β(p(x n x n t))p(x n x n t) P(x n x n t). (4) Therefore the sequence P(x n x n+ t) is decreasing so there exists r 0 such that lim P(x n x n+ t) = r. Suppose that r > 0. Now letting n from (4) we have s r r lim β(p(x n x n t))r r. So we have lim β(p(x n x n t)) s since β F s we deduce that lim P(x n x n t) = 0 which is a contradiction. Hence r = 0 that is lim P(x n x n+ t) = 0. (5) Step II: Now we prove that the sequence x n is a Cauchy sequence. Using the triangle inequality by () we have P(x n x m t) sp(x n x n+ t) + s P(x n+ x m+ t) + s P(x m+ x m t) sp(x n x n+ t) + s P(x m x m+ t) + sβ(p(x n x m t))m(x n x m t). Letting m n in the above inequality applying (5) we have Here lim P(x n x m t) s lim β(p(x n x m t)) lim M(x n x m t). (6) m m m P(x n x m t) M(x n x m t) = max P(x n x m t) P(x n fx n t)p(x m fx m t) + P(fx n fx m t) = max Letting m n in the above inequality we get P(x n fx n t)p(x m fx m t) + P(x n x m t) P(x n x m t) P(x n x n+ t)p(x m x m+ t) P(x n x n+ t)p(x m x m+ t) + P(x n+ x m+ t) + P(x n x m t) lim M(x n x m t) = lim P(x n x m t). (7) m m.

6 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) From (6) (7) we obtain lim P(x n x m t) s lim β(p(x n x m t)) lim P(x n x m t). (8) m m m Now we claim that lim m P(x n x m t) = 0. On the contrary if lim m P(x n x m t) 0 then we get Since β F s we deduce that s lim β(p(x n x m t)). m lim P(x n x m t) = 0. (9) m which is a contradiction. Consequently x n is a b-parametric Cauchy sequence in X. Since (X P) is complete the sequence x n converges to some z X that is lim P(x n z t) = 0. Step III: Now we show that z is a fixed point of f. Using the triangle inequality we get P(fz z t) sp(fz fx n t) + sp(fx n z t). Letting n using the continuity of f we have fz = z. Thus z is a fixed point of f. Example.. Let X = [0 ) be endowed with the parametric b-metric t(x + y) if x y P(x y t) = 0 if x = y for all x y X all t > 0. Define T : X X by 8 x if x [0 ) T x = 8x if x [ ) 4 if x [ ) Also define β : [0 ) [0 ) by β(t) = 4. Clearly (X P ) is a complete parametric b-metric space T is a continuous mapping β F. Now we consider the following cases: Let x y [0 ) with x y then Let x y [ ) with x y then P(T x T y t) = t( 8 x + 8 y ) = 3 t(x + y ) 4 t(x + y) = 4P(x y t) 4M(x y t) = β(p(x y t))m(x y t) P(T x T y t) = t( 8 x + 8 y) = 3t(x + y) 4 t(x + y) = 4P(x y t) 4M(x y t) = β(p(x y t))m(x y t)

7 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Let x y [ ) with x y then P(T x T y t) = t( ) = t t = 4t( + ) 4 t(x + y) = 4P(x y t) Let x [0 ) y [ ) (clearly with x y) then 4M(x y t) = β(p(x y t))m(x y t) P(T x T y t) = t( 8 x + 8 y) t( 8 x + 8 y) = 3 t(x + y ) 4 t(x + y) = 4P(x y t) 4M(x y t) = β(p(x y t))m(x y t) Let x [0 ) y [ ) (clearly with x y) then P(T x T y t) = t( 8 x + 4 ) t( 8 x + 8 y) = 3t(x + y) 4 t(x + y) = 4P(x y t) Let x [ ) y [ ) (clearly with x y) then Therefore 4M(x y t) = β(p(x y t))m(x y t) P(T x T y t) = t( 8 x + 4 ) t( 8 x + 8 y) = 3t(x + y) 4 t(x + y) = 4P(x y t) 4M(x y t) = β(p(x y t))m(x y t) P(T x T y t) β(p(x y t))m(x y t) for all x y X with x y all t > 0. Hence all conditions of Theorem. holds T has a unique fixed point. Note that the continuity of f in Theorem. is not necessary can be dropped. Theorem.3. Under the hypotheses of Theorem. without the continuity assumption on f assume that whenever x n is a nondecreasing sequence in X such that x n u one has x n u for all n N. Then f has a fixed point. Proof. Repeating the proof of Theorem. we construct an increasing sequence x n in X such that x n z X. Using the assumption on X we have x n z. Now we show that z = fz. By () Lemma.8 [ ] s P(z fz t) s s lim sup P(x n+ fz t) lim sup β(p(x n z t)) lim sup M(x n z t)

8 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) where lim M(x n z t) = lim max P(x n z t) P(x n fx n t)p(z fz t) n + P(fx n fz t) = lim max n P(x n fx n t)p(z fz t) + P(x n z t) P(x n z t) P(x n x n+ t)p(z fz t) P(x n x n+ t)p(z fz t) + P(x n+ fz t) + P(x n z t) Therefore we deduce that P(z fz t) 0. As t is arbitrary hence we have z = fz. = 0 (see (5)). If in the above theorems we take β(t) = r where 0 r < s then we have the following corollary. Corollary.4. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that for some r with 0 r < s sp(fx fy t) rm(x y t) holds for each t > 0 all comparable elements x y X where M(x y t) = max P(x y t) P(x fx t)p(y fy t) + P(fx fy t) P(x fx t)p(y fy t). + P(x y t) If f is continuous or for any nondecreasing sequence x n in X such that x n u X one has x n u for all n N then f has a fixed point. Corollary.5. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that P(x fx t)p(y fy t) P(x fx t)p(y fy t) P(fx fy t) αp(x y t) + β + γ + P(fx fy t) + P(x y t) for each t > 0 all comparable elements x y X where α β γ 0 α + β + γ s. If f is continuous or for any nondecreasing sequence x n in X such that x n u X one has x n u for all n N then f has a fixed point... Results using comparison functions Let Ψ denote the family of all nondecreasing functions ψ : [0 ) [0 ) such that lim n ψ n (t) = 0 for all t > 0 where ψ n denotes the n-th iterate of ψ. It is easy to show that for each ψ Ψ the following is satisfied: (a) ψ(t) < t for all t > 0; (b) ψ(0) = 0. Theorem.6. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that where sp(fx fy t) ψ(n(x y t)) (0) P(x fx t)d(x fy t) + P(y fy t)p(y fx t) N(x y t) = max P(x y t) + s[p(x fx t) + P(y fy t)] P(x fx t)p(x fy t) + P(y fy t)p(y fx t) + P(x fy t) + P(y fx t) for some ψ Ψ for all comparable elements x y X all t > 0. If f is continuous then f has a fixed point.

9 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Proof. Since x 0 fx 0 f is an increasing function we obtain by induction that Putting x n = f n x 0 we have x 0 fx 0 f x 0 f n x 0 f n+ x 0. x 0 x x x n x n+. If there exists n 0 N such that x n0 = x n0 + then x n0 = fx n0 so we have nothing for prove. Hence we assume that x n x n+ for all n N. Step I. We will prove that lim P(x n x n+ t) = 0. Using condition (39) we obtain Here P(x n x n+ t) sp(x n x n+ t) = sp(fx n fx n t) ψ(n(x n x n t)). N(x n x n t) = maxp(x n x n t) P(x n fx n t)p(x n fx n t) + P(x n fx n t)p(x n fx n t) + s[p(x n fx n t) + P(x n fx n t)] P(x n fx n t)p(x n fx n t) + P(x n fx n t)p(x n fx n t) + P(x n fx n t) + P(x n fx n t) = maxp(x n x n t) P(x n x n t)p(x n x n+ t) + P(x n x n+ t)p(x n x n t) + s[p(x n x n t) + P(x n x n+ t)] P(x n x n t)p(x n x n+ t) + P(x n x n+ t)p(x n x n t) + P(x n x n+ t) + P(x n x n t) = P(x n x n t). Hence By induction we get that P(x n x n+ t) sp(x n x n+ t) ψ(p(x n x n t)). P(x n x n+ t) ψ(p(x n x n t)) ψ (P(x n x n t)) ψ n (P(x 0 x t)). As ψ Ψ we conclude that lim P(x n x n+ t) = 0. () Step II. We will prove that x n is a parametric Cauchy sequence. Suppose the contrary. Then there exist t > 0 ε > 0 for them we can find two subsequences x mi x ni of x n such that n i is the smallest index for which n i > m i > i P(x mi x ni t) ε. () This means that From () using the triangle inequality we get Taking the upper limit as i we get P(x mi x ni t) < ε. (3) ε P(x mi x ni t) sp(x mi x mi + t) + sp(x mi + x ni t). ε s lim sup P(x mi + x ni t). (4)

10 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) From the definition of M(x y t) we have M(x mi x ni t) = maxp(x mi x ni t) P(x m i fx mi t)p(x mi fx ni t) + P(x ni fx ni t)p(x ni fx mi t) + s[p(x mi fx mi t) + P(x ni fx ni t)] P(x mi fx mi t)p(x mi fx ni t) + P(x ni fx ni t)p(x ni fx mi t) + P(x mi fx ni t) + P(x ni fx mi t) = maxp(x mi x ni t) P(x m i x mi + t)p(x mi x ni t) + P(x ni x ni t)p(x ni x mi + t) + s[p(x mi x mi + t) + P(x ni x ni t)] P(x mi x mi + t)p(x mi x ni t) + P(x ni x ni t)p(x ni x mi + t) + P(x mi x ni t) + P(x ni x mi + t) if i by () (3) we have Now from (39) we have Again if i by (4) we obtain lim sup M(x mi x ni t) ε. sp(x mi + x ni t) = sp(fx mi fx ni t) ψ(m(x mi x ni t)). ε = s ε s s lim sup P(x mi + x ni a) ψ(ε) < ε which is a contradiction. Consequently x n is a Cauchy sequence in X. Therefore the sequence x n converges to some z X that is lim n P(x n z t) = 0 for all t > 0. Step III. Now we show that z is a fixed point of f. Using the triangle inequality we get Letting n using the continuity of f we get Hence we have fz = z. Thus z is a fixed point of f. P(z fz t) sp(z fx n t) + sp(fx n fz t). P(z fz t) 0. Theorem.7. Under the hypotheses of Theorem.6 without the continuity assumption on f assume that whenever x n is a nondecreasing sequence in X such that x n u X one has x n u for all n N. Then f has a fixed point. Proof. Following the proof of Theorem.6 we construct an increasing sequence x n in X such that x n z X. Using the given assumption on X we have x n z. Now we show that z = fz. By (39) we have sp(fz x n t) = sp(fz fx n t) ψ(m(z x n t)) (5) where M(z x n t) = maxp(x n z t) P(x n fx n t)p(x n fz t) + P(z fz t)p(z fx n t) + s[p(x n fx n t) + P(z fz t)] P(x n fx n t)p(x n fz t) + P(z fz t)p(z fx n t) + P(x n fz t) + P(z fx n t) = maxp(x n z t) P(x n x n t)p(x n fz t) + P(z fz t)p(z x n t) + s[p(x n x n t) + P(z fz t)] P(x n x n t)p(x n fz t) + P(z fz t)p(z x n t). + P(x n fz t) + P(z x n t)

11 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Letting n in the above relation we get lim sup M(z x n a) = 0. (6) Again taking the upper limit as n in (5) using Lemma.8 (6) we get [ ] s P(z fz t) s lim sup P(x n fz t) s lim sup ψ(m(z x n t)) = 0. So we get P(z fz t) = 0 i.e. fz = z. Corollary.8. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that where 0 r < N(x y t) = max sp(fx fy t) rm(x y t) P(x fx t)d(x fy t) + P(y fy t)p(y fx t) P(x y t) + s[p(x fx t) + P(y fy t)] P(x fx t)p(x fy t) + P(y fy t)p(y fx t) + P(x fy t) + P(y fx t) for all comparable elements x y X all t > 0. If f is continuous or whenever x n is a nondecreasing sequence in X such that x n u X one has x n u for all n N then f has a fixed point..3. Results for almost generalized weakly contractive mappings Berinde in [5] studied the concept of almost contractions obtained certain fixed point theorems. Results with similar conditions were obtained e.g. in [4] [5]. In this section we define the notion of almost generalized (ψ ϕ) st -contractive mapping prove our new results. In particular we extend Theorems...3 of Ćirić et al. in [6] to the setting of b-parametric metric spaces. Recall that Khan et al. introduced in [] the concept of an altering distance function as follows. Definition.9. A function ϕ : [0 + ) [0 + ) is called an altering distance function if the following properties hold:. ϕ is continuous non-decreasing.. ϕ(t) = 0 if only if t = 0. Let (X P) be a parametric b-metric space let f : X X be a mapping. For x y X for all t > 0 set P(x fy t) + P(y fx t) M t (x y) = max P(x y t) P(x fx t) P(y fy t) s N t (x y) = minp(x fx t) P(x fy t) P(y fx t) P(y fy t). Definition.0. Let (X P) be a parametric b-metric space. We say that a mapping f : X X is an almost generalized (ψ ϕ) st -contractive mapping if there exist L 0 two altering distance functions ψ ϕ such that ψ(sp(fx fy t)) ψ(m t (x y)) ϕ(m t (x y)) + Lψ(N t (x y)) (7) for all x y X for all t > 0.

12 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Now let us prove our result. Theorem.. Let (X ) be a partially ordered set suppose that there exists a parametric b-metric P on X such that (X P) is a complete parametric metric space. Let f : X X be a continuous non-decreasing mapping with respect to. Suppose that f satisfies condition (7) for all comparable elements x y X. If there exists x 0 X such that x 0 fx 0 then f has a fixed point. Proof. Starting with the given x 0 define a sequence x n in X such that x n+ = fx n for all n 0. Since x 0 fx 0 = x f is non-decreasing we have x = fx 0 x = fx by induction x 0 x x n x n+. If x n = x n+ for some n N then x n = fx n hence x n is a fixed point of f. So we may assume that x n x n+ for all n N. By (7) we have ψ(p(x n x n+ t)) ψ(sp(x n x n+ t)) = ψ(sp(fx n fx n t)) ψ(m t (x n x n )) ϕ(m t (x n x n )) + Lψ(N t (x n x n )) (8) where M t (x n x n ) = max P(x n x n t) P(x n fx n t) P(x n fx n t) P(x n fx n t) + P(x n fx n t) s = max P(x n x n t) P(x n x n+ t) P(x n x n+ t) s max P(x n x n t) P(x n x n+ t) P(x n x n t) + P(x n x n+ t) (9) N t (x n x n ) = min P(x n fx n t) P(x n fx n t) P(x n fx n t) P(x n fx n t) = min P(x n x n t) P(x n x n+ t) 0 P(x n x n+ t) = 0. (0) From (8) (0) the properties of ψ ϕ we get ( ) ψ(p(x n x n+ t)) ψ max P(x n x n t) P(x n x n+ t) ϕ ( max P(x n x n t) P(x n x n+ t) P(x n x n+ t) s ). () If max P(x n x n t) P(x n x n+ t) = P(x n x n+ t) then by () we have ( ψ(p(x n x n+ t)) ψ(p(x n x n+ t)) ϕ max P(x n x n t) P(x n x n+ t) P(x ) n x n+ t) s which gives that x n = x n+ a contradiction.

13 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Thus P(x n x n+ t) : n N 0 is a non-increasing sequence of positive numbers. Hence there exists r 0 such that lim P(x n x n+ t) = r. Letting n in () we get ( ) P(x n x n+ t) ψ(r) ψ(r) ϕ max r r lim ψ(r). n s Therefore ( ϕ max r r lim ) P(x n x n+ t) = 0 s hence r = 0. Thus we have lim P(x n x n+ t) = 0 () for each t > 0. Next we show that x n is a Cauchy sequence in X. Suppose the contrary that is x n is not a Cauchy sequence. Then there exist t > 0 ε > 0 for them we can find two subsequences x mi x ni of x n such that n i is the smallest index for which n i > m i > i P(x mi x ni t) ε. (3) This means that P(x mi x ni t) < ε. (4) Using () taking the upper limit as i we get On the other h we have lim sup P(x mi x ni t) ε. (5) P(x mi x ni t) sp(x mi x mi + t) + sp(x mi + x ni t). Using () (4) taking the upper limit as i we get Again using the triangular inequality we have ε s lim sup P(x mi + x ni t). P(x mi + x ni t) sp(x mi + x mi t) + sp(x mi x ni t) P(x mi x ni t) sp(x mi x ni t) + sp(x ni x ni t). Taking the upper limit as i in the first inequality above using () (5) we get lim sup P(x mi + x ni t) εs. (6) i0 Similarly taking the upper limit as i in the second inequality above using () (4) we get From (7) we have ψ(sp(x mi + x ni t)) = ψ(sp(fx mi fx ni t)) lim sup P(x mi x ni t) εs. (7) ψ(m t (x mi x ni )) ϕ(m t (x mi x ni )) + Lψ(N t (x mi x ni )) (8)

14 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) where M t (x mi x ni ) = max P(x mi x ni t) P(x mi fx mi t) P(x ni fx ni t) P(x m i fx ni t) + P(fx mi x ni t) s = max P(x mi x ni t) P(x mi x mi + t) P(x ni x ni t) P(x m i x ni t) + P(x mi + x ni t) (9) s N t (x mi x ni ) = min P(x mi fx mi t) P(x mi fx ni t) P(x ni fx mi t) P(x ni fx ni t) = min P(x mi x mi + t) P(x mi x ni t) P(x ni x mi + t) P(x ni x ni t). (30) Taking the upper limit as i in (9) (30) using () (5) (6) (7) we get lim sup M t (x mi x ni ) = max lim sup P(x mi x ni t) 0 0 lim sup P(x mi x ni t) + lim sup P(x mi + x ni t) s εs + εs max ε = ε. (3) s So we have lim sup M t (x mi x ni ) ε (3) lim sup N t (x mi x ni ) = 0. (33) Now taking the upper limit as i in (8) using (3) (3) (33) we have ( ψ s ε ) ψ ( s lim sup P(x mi + x ni t) ) s which further implies that ψ ( lim sup M t (x mi x ni t) ) lim inf ψ(ε) ϕ ( lim inf M t(x mi x ni ) ) ϕ ( lim inf M t(x mi x ni ) ) = 0 ϕ(m t(x mi x ni )) so lim inf M t(x mi x ni ) = 0 a contradiction to (3). Thus x n+ = fx n is a Cauchy sequence in X. As X is a complete space there exists u X such that x n u as n that is lim x n+ = lim fx n = u. Now suppose that f is continuous. Using the triangular inequality we get Letting n we get So we have fu = u. Thus u is a fixed point of f. P(u fu t) sp(u fx n t) + sp(fx n fu t). P(u fu t) s lim P(u fx n t) + s lim P(fx n fu t).

15 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Note that the continuity of f in Theorem. is not necessary can be dropped. Theorem.. Under the hypotheses of Theorem. without the continuity assumption on f assume that whenever x n is a non-decreasing sequence in X such that x n x X one has x n x for all n N. Then f has a fixed point in X. Proof. Following similar arguments to those given in the proof of Theorem. we construct an increasing sequence x n in X such that x n u for some u X. Using the assumption on X we have that x n u for all n N. Now we show that fu = u. By (7) we have ψ(sp(x n+ fu t)) = ψ(sp(fx n fu t)) ψ(m t (x n u)) ϕ(m t (x n u)) + Lψ(N t (x n u)) (34) where M t (x n u) = max P(x n u t) P(x n fx n t) P(u fu t) P(x n fu t) + P(fx n u t) s = max P(x n u t) P(x n x n+ t) P(u fu t) P(x n fu t) + P(x n+ u t) s N t (x n u) = min P(x n fx n t) P(x n fu t) P(u fx n t) P(u fu t) (35) = min P(x n x n+ t) P(x n fu t) P(u x n+ t) P(u fu t). (36) Letting n in (35) (36) using Lemma.8 we get sp(u fu t) lim inf s M t(x n u) lim sup M t (x n u) max P(u fu t) N t (x n u) 0. Again taking the upper limit as i in (34) using Lemma.8 (37) we get ψ(p(u fu t) = ψ(s s P(u fu t)) ψ( s lim sup P(x n+ fu t) ) ψ ( lim sup M t (x n u) ) lim inf ϕ(m t(x n u)) ψ(p(u fu t)) ϕ ( lim inf M t(x n u) ). sp(u fu t) = P(u fu t) (37) s Therefore ϕ ( lim inf M t(x n u) ) 0 equivalently lim inf M t(x n u) = 0. Thus from (37) we get u = fu hence u is a fixed point of f. Corollary.3. Let (X ) be a partially ordered set suppose that there exists a b-parametric metric P on X such that (X P) is a complete parametric b-metric space. Let f : X X be a non-decreasing continuous mapping with respect to. Suppose that there exist k [0 ) L 0 such that P(fx fy t) k s max P(x y t) P(x fx t) P(y fy t) P(x fy t) + P(y fx t) s + L s minp(x fx t) P(y fx t) for all comparable elements x y X all t > 0. If there exists x 0 X such that x 0 fx 0 then f has a fixed point.

16 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) Proof. Follows from Theorem. by taking ψ(t) = t ϕ(t) = ( k)t for all t [0 + ). Corollary.4. Under the hypotheses of Corollary.3 without the continuity assumption of f let for any non-decreasing sequence x n in X such that x n x X we have x n x for all n N. Then f has a fixed point in X. 3. Fuzzy b-metric spaces In 988 Grabiec [4] defined contractive mappings on a fuzzy metric space extended fixed point theorems of Banach Edelstein in such spaces. Successively George Veeramani [] slightly modified the notion of a fuzzy metric space introduced by Kramosil Michálek then defined a Hausdorff first countable topology on it. Since then the notion of a complete fuzzy metric space presented by George Veeramani has emerged as another characterization of completeness many fixed point theorems have also been proved (see for more details [ ] the references therein). In this section we develop an important relation between parametric b-metric fuzzy b-metric deduce certain new fixed point results in triangular partially ordered fuzzy b-metric space. Definition 3.. (Schweizer Sklar [6]) A binary operation : [0 ] [0 ] [0 ] is called a continuous t-norm if it satisfies the following assertions: (T) is commutative associative; (T) is continuous; (T3) a = a for all a [0 ]; (T4) a b c d when a c b d with a b c d [0 ]. Definition 3.. A 3-tuple (X M ) is said to be a fuzzy metric space if X is an arbitrary set is a continuous t-norm M is fuzzy set on X (0 ) satisfying the following conditions for all x y z X t s > 0 (i) M(x y t) > 0; (ii) M(x y t) = for all t > 0 if only if x = y; (iii) M(x y t) = M(y x t); (iv) M(x y t) M(y z s) M(x z t + s); (v) M(x y.) : (0 ) [0 ] is continuous; The function M(x y t) denotes the degree of nearness between x y with respect to t. Definition 3.3. A fuzzy b-metric space is an ordered triple (X B ) such that X is a nonempty set is a continuous t-norm B is a fuzzy set on X X (0 + ) satisfying the following conditions for all x y z X t s > 0: (F) B(x y t) > 0; (F) B(x y t) = if only if x = y; (F3) B(x y t) = B(y x t); (F4) B(x y t) B(y z s) B(x z b(t + s)) where b ;

17 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) (F5) B(x y ) : (0 + ) (0 ] is left continuous. Definition 3.4. Let (X B ) be a fuzzy b-metric space. Then (i) a sequence x n converges to x X if only if lim n + B(x n x t) = for all t > 0; (ii) a sequence x n in X is a Cauchy sequence if only if for all ɛ (0 ) t > 0 there exists n 0 such that B(x n x m t) > ɛ for all m n n 0 ; (iii) the fuzzy b-metric space is called complete if every Cauchy sequence converges to some x X. Definition 3.5. Let (X B b) be a fuzzy b-metric space. The fuzzy b-metric B is called triangular whenever B(x y t) b[ B(x z t) + B(z y t) )] for all x y z X all t > 0. t t+d(xy). Example 3.6. Let (X d s) be a b-metric space. Define B : X X (0 ) [0 ) by B(x y t) = Also suppose a b = mina b. Then (X B ) is a fuzzy b-metric spaces with constant b = s. Further B is a triangular fuzzy B-metric. Remark 3.7. Notice that P(x y t) = b-metric. B(xyt) is a parametric b-metric whenever B is a triangular fuzzy As an applications of Remark 3.7 the results established in section we can deduce the following results in ordered fuzzy b-metric spaces. Theorem 3.8. Let (X ) be a partially ordered set suppose that there exists a triangular fuzzy b-metric B on X such that (X B b) is a complete fuzzy b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that b[ B(fx fy t)) ] β( )M(x y t) (38) B(x y t) for all t > 0 for all comparable elements x y X where M(x y t) = max B(x y t) [ If f is continuous then f has a fixed point. B(xfxt) ][ B(yfyt) ] B(fxfyt) [ B(xfxt) ][ B(xyt) B(yfyt) ] Theorem 3.9. Under the hypotheses of Theorem 3.8 without the continuity assumption on f assume that whenever x n is a nondecreasing sequence in X such that x n u one has x n u for all n N. Then f has a fixed point. Theorem 3.0. Let (X ) be a partially ordered set suppose that there exists a triangular fuzzy b- metric B on X such that (X B b) is a complete fuzzy b-metric space. Let f : X X be a continuous non-decreasing mapping with respect to. Also suppose that there exist L 0 two altering distance functions ψ ϕ such that ψ(b[ B(fx fy t)) ]) ψ(m t(x y)) ϕ(m t (x y)) + Lψ(N t (x y)) for all comparable elements x y X where M t (x y) = max B(x y t) B(x fx t) B(y fy t) b [ B(x fy t) + B(y fx t) ]

18 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) N t (x y) = min B(x fx t) B(y fy t) B(y fx t) B(x fy t). If there exists x 0 X such that x 0 fx 0 then f has a fixed point. Theorem 3.. Under the hypotheses of Theorem 3.0 without the continuity assumption on f assume that whenever x n is a nondecreasing sequence in X such that x n u X one has x n u for all n N. Then f has a fixed point. Theorem 3.. Let (X ) be a partially ordered set suppose that there exists a triangular fuzzy b- metric B on X such that (X B b) is a complete fuzzy b-metric space. Let f : X X be an increasing mapping with respect to such that there exists an element x 0 X with x 0 fx 0. Suppose that where b[ ] ψ(n (x y t)) (39) B(fx fy t)) N (x y t) = max B(x y t)) [ B(xfxt) ][ B(xfyt) ] + [ B(yfyt) ][ B(yfxt) ] + b[ B(xfxt) + B(yfyt) ] [ B(xfxt) ][ B(xfyt) ] + [ B(yfyt) ][ B(yfxt) ] B(xfyt) + B(yfxt) for some ψ Ψ for all comparable elements x y X all t > 0. If f is continuous then f has a fixed point. 4. Application to existence of solutions of integral equations Let X = C([0 T ] R) be the set of real continuous functions defined on [0 T ] P : X X (0 ) [0 + ) be defined by P(x y α) = sup t [0T ] e αt x(t) y(t) for all x y X all t > 0. Then (X P ) is a complete parametric b metric space. Let be the partial order on X defined by x y if only if x(t) y(t) for all t [0 T ]. Then (X d α ) is a complete partially ordered metric space. Consider the following integral equation T x(t) = p(t) + S(t s)f(s x(s))ds (40) 0 where (A) f : [0 T ] R R is continuous (B) p : [0 T ] R is continuous (C) S : [0 T ] [0 T ] [0 + ) is continuous sup e αt ( t [0T ] (D) there exist k [0 ) L 0 such that ( ke αs 0 f(s y(s)) f(s x(s)) T 0 max S(t s)ds) x(s) Hy(s) + y(s) Hx(s) 4 + Le αs x(s) y(s) x(s) Hx(s) y(s) Hy(s) min x(s) Hx(s) y(s) Hx(s) )

19 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) for all x y X with x y s [0 T ] α > 0 where Hx(t) = p(t) + (E) there exist x 0 X such that T 0 S(t s)f(s x(s))ds t [0 T ] for all x X. T x 0 (t) p(t) + S(t s)f(s x 0 (s))ds. 0 We have the following result of existence of solutions for integral equations. Theorem 4.. Under assumptions (A) (E) the integral equation (40) has a unique solution in X = C([0 T ] R). Proof. Let H : X X be defined by Hx(t) = p(t) + T 0 S(t s)f(s x(s))ds t [0 T ] for all x X. First we will prove that H is a non-decreasing mapping with respect to. Let x y then by (D) we have 0 f(s y(s)) f(s x(s)) for all s [0 T ]. On the other h by definition of H we have Hy Hx = T 0 S(t s)[f(s y(s)) f(s x(s))]ds 0 for all t [0 T ]. Then Hx Hy that is H is a non-decreasing mapping with respect to. Now suppose that x y X with x y. Then by (C) (D) the definition of H we get P(Hx Hy α) = sup e αt Hx(t) Hy(t) t [0T ] = sup e αt t [0T ] T 0 S(t s)[f(s x(s)) f(s y(s))]ds sup e αt( T S(t s) f(s x(s)) f(s y(s)) ds t [0T ] 0 ( ke αs sup e αt( T S(t s) t [0T ] 0 ) max x(s) y(s) x(s) Hy(s) + y(s) Hx(s) x(s) Hx(s) y(s) Hy(s) 4 + Le αs min x(s) Hx(s) y(s) Hx(s) ( k max sup e αt( T S(t s) t [0T ] 0 sup s [0T ] ) ds ) e αs x(s) y(s) sup e αs x(s) Hx(s) s [0T ] sup e αs y(s) Hy(s) sup s [0T ] e αs x(s) Hy(s) + sup s [0T ] e αs y(s) Hx(s) s [0T ] 4 + L min sup e αs x(s) Hx(s) sup e αs y(s) Hx(s) s [0T ] s [0T ] = sup e αt( T ( k S(t s) t [0T ] 0 max P(x Hy α) + P(y Hx α) 4 ) ) ds P(x y α) P(x Hx α) P(y Hy α)

20 N. Hussain P. Salimi V. Parvaneh J. Nonlinear Sci. Appl. 8 (05) = + L minp(x Hx α) P(y Hx α) ) ( sup t [0T ] T e αt ( S(t s)ds) )( k 0 max P(y Hy α) ) ds P(x Hy α) + P(y Hx α) 4 k max P(x y α) P(x Hx α) P(y Hy α) P(x y α) P(x Hx α) + L minp(x Hx α) P(y Hx α) ) P(x Hy α) + P(y Hx α) 4 + L minp(x Hx α) P(y Hx α) Now by (E) there exists x 0 X such that x 0 Hx 0. Then the conditions of Corollary.3 are satisfied hence the integral equation (40) has a unique solution in X = C([0 T ] R). Acknowledgement This article was funded by the Deanship of Scientific Research (DSR) King Abdulaziz University Jeddah. The authors acknowledge with thanks DSR KAU for financial support. References [] R. P. Agarwal N. Hussain M. A. Taoudi Fixed point theorems in ordered Banach spaces applications to nonlinear integral equations Abstr. Appl. Anal. 0 (0) 5 pages.. [] M. A. Alghamdi N. Hussain P. Salimi Fixed point coupled fixed point theorems on b-metric-like spaces J. Inequal. Appl. 03 (03) 5 pages. [3] I. Altun D. Turkoglu Some fixed point theorems on fuzzy metric spaces with implicit relations Commun. Korean Math. Soc. 3 (008) 4. 3 [4] G. V. R. Babu M. L. Shya M. V. R. Kameswari A note on a fixed point theorem of Berinde on weak contractions Carpathian J. Math. 4 (008) 8..3 [5] V. Berinde General contrcactive fixed point theorems for Ćirić-type almost contraction in metric spaces Carpathian J. Math. 4 (008) [6] L. Ćirić M. Abbas R. Saadati N. Hussain Common fixed points of almost generalized contractive mappings in ordered metric spaces Appl. Math. Comput. 7 (0) [7] S. Czerwik Contraction mappings in b-metric spaces Acta Math. Inf. Univ. Ostravensis (993) 5.. [8] S. Czerwik Nonlinear set-valued contraction mappings in b-metric spaces Atti Sem. Mat. Fis. Univ. Modena 46 (998) [9] C. Di Bari C. Vetro A fixed point theorem for a family of mappings in a fuzzy metric space Rend. Circ. Math. Palermo 5 (003) [0] D. -Dukić Z. Kadelburg S. Radenović Fixed points of Geraghty-type mappings in various generalized metric spaces Abstr. Appl. Anal. 0 (0) 3 pages.. [] A. George P. Veeramani On some results in fuzzy metric spaces Fuzzy Sets Systems 64 (994) [] M. A. Geraghty On contractive mappings Proc. Amer. Math. Soc. 40 (973) [3] D. Gopal M. Imdad C. Vetro M. Hasan Fixed point theory for cyclic weak φ-contraction in fuzzy metric space J. Nonlinear Anal. Appl. 0 (0) pages. 3 [4] M. Grabiec Fixed points in fuzzy metric spaces Fuzzy Sets Systems 7 (988) [5] N. Hussain S. Al-Mezel P. Salimi Fixed points for ψ-graphic contractions with application to integral equations Abstr. Appl. Anal. 03 (03) pages.. [6] N. Hussain S. Khaleghizadeh P. Salimi A. A. N. Abdou A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces Abstr. Appl. Anal. 04 (04) 6 pages. 3 [7] N. Hussain V. Parvaneh J. R. Roshan Z. Kadelburg Fixed points of cyclic weakly (ψ ϕ L A B)-contractive mappings in ordered b-metric spaces with applications Fixed Point Theory Appl. 03 (03) 8pages. [8] N. Hussain P. Salimi Implicit contractive mappings in Modular metric Fuzzy Metric Spaces The Sci. World J. 04 (04) 3 pages.3 [9] N. Hussain M. H. Shah KKM mappings in cone b-metric spaces Comput. Math. Appl. 6 (0) [0] N. Hussain M. A. Taoudi Krasnosel skii-type fixed point theorems with applications to Volterra integral equations Fixed Point Theory Appl. 03 (03) 6 pages.. [] M. A. Khamsi N. Hussain KKM mappings in metric type spaces Nonlinear Anal. 73 (00)

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