(α ψ)- Construction with q- function for coupled fixed point

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(α ψ)- Construction with q- function for coupled fixed point

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications30( − ) − Construction with − Function for Coupled FixedPointAnimesh Gupta∗, Sarika Jain∗, Reena M. Patel ∗∗∗, Ramakant Bhardwaj∗∗*Department of Engineering MathematicsSagar Institute of Science, Technology and Research, Ratibad, Bhopal – INDIA***Research Scholar,CMJ University Shilonganimeshgupta10@gmail.com, sarikajain.bpl@gmail.com**Department of Engineering MathematicsTruba Institute of Engineering and Information Technology, Bhopal INDIA.rkbhardwaj100@gmail.com2000 AMS Subject Classification:- 47H10, 54H25, 46J10, 46J15Abstract:- The purpose of this article is to prove coupled fixed point theorem for non linear contractivemappings in partially ordered complete quasi - metric spaces using the concept of monotone mapping with aQ − function q and (α − Ψ) − contractive condition. The presented theorems are generalization and extensionof the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham cite(BL). We also give anexample in support of our theorem.Keywords:- Coupled fixed point, Coupled Coincidence point, Mixed monotone mapping.Introduction and PreliminariesThe Banach contractive mapping principle [8] is an important result of analysis and it has beenapplied widely in a number of branches of mathematics. It has been noted that the Banach contraction principle[8] was defined on complete metric space. Recently, Bhaskar and Lakshmikantham [9] presented some newresults for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [9] noted that theirtheorem can be used to investigate a large class of problems and discussed the existence and uniqueness ofsolution for a periodic boundary value problem.Beside this, Al-Homidan et al [1] introduced the concept of a Q-function defined on a quasi-metricspace which generalizes the notions of a τ −function and a ω −distance and establishes the existence of thesolution of equilibrium problem (see also [2,3,4,5,6]).Our aim of this article is to prove a coupled fixed point theorem in quasi-metric space by using the concept ofQ- function. We also extend and generalized the result of Bhaskar and Lakshmikantham [9].Recall that if (X, ≤) is a partially ordered set and F ∶ X → X such that for each x, y ∈ X, x ≤ yimplies F(x) ≤ F(y), then a mapping F is said to be non decreasing. Similarly, a non increasing mapping isdefined. Bhaskar and Lakshmikantham [9] introduced the following notions of a mixed monotone mapping and acoupled fixed point.Definition 1 :- Let (X, ≤) is a partially ordered set and F ∶ X × X → X. The mapping F is said to have themixed monotone property if F is nondecreasing monotone in first argument and is a nonincreasing monotone inits second argument, that is, for any x, y ∈ Xx1, x2 ∈ X, x1 ≤ x2 → F(x1, y) ≤ F(x2, y)y1, y2 ∈ X, y1 ≤ y2 → F(x, y1) ≥ F(x, y2)Definition 2:- An element (x, y) ∈ X × X is called a coupled fixed point of a mapping F: X × X → X ifF(x, y) = x, F(y, x) = y.Definition 3 :- Let X be a nonempty set. A real valued function d: X × X → R6is said to be quasi metric spaceon X if7(M1)8 d(x, y) ≥ 0 for all x, y ∈ X,7(M2)8d(x, y) = 0 if and only if x = y,7(M<)8d(x, y) ≤ d(x, z) + d(z, y)for all x, y, z ∈ X.The pair (X,d) is called a quasi- metric space.Definition 4:- Let (X,d) be a quasi metric space. A mapping q ∶ X × X → R6is called a Q- function on X if thefollowing conditions are satisfied:7( @)8 for all x, y, z ∈ X,7( A)8Bf x ∈ X and (yC)C D 1 is a sequence in X such that it converges to point y (with respect toquasi metric) and q(x, yC) ≤ M for some M = M(x), then q(x, y) ≤ M;7( F)8for any ϵ > 0 there exists δ > 0 such that q(z, x) ≤ δ and q(z, y) ≤ δ implies thatd(x, y) ≤ ϵ .Remark 5:- If (X, d) is a metric space, and in addition to (Q1) − (Q<), the following condition are also satisfied:
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications317( J)8 for any sequence (xC)C D 1 in X with limC →Ksup { q(xC, xM): m > N } = 0 and if thereexist a sequence (yC)C D 1 in X such that limC →K q(xC, yC) = 0, then limC →K d(xC, yC) = 0.Then a Q- function is called τ − function, introduced by Lin and Du [16]also in the same paper [16] they showthat every ω − function, introduced and studied by Kada et al. [15], is a τ − function. In fact, if we consider(X, d) as a metric space and replace (Q2) by the following condition:7( Q)8 for any x ∈ X, the function p(x, . ) → R6is lower semi continuous,then a Q- function is called a ω − function on X. Several examples of ω − functions are given in [15]. It is easyto see that if (q(x, . )) is lower semi continuous, then (Q2) holds. Hence, it is obvious that every ω − function isτ − function and every τ − function is Q- function, but the converse assertions do not hold.Example 6:- Let X = R. Define d: X × X → R6byd(x, y) = R0 if x = y|y| otherwiseTand q ∶ X × X → R6byq(x, y) = ∣ y ∣, ∀ x, y ∈ X.Then one can easily see that d is a quasi- metric space and q is a Q- function on X, but q is neither a τ −function nor a ω − function.Example 7:- Define d: X × X → R6byd(x, y) = Ry − x if x = y2(x − y) otherwiseTand q ∶ X × X → R6byq(x, y) = ∣ x − y ∣, ∀ x, y ∈ X.Then one can easily see that d is a quasi- metric space and q is a Q- function on X, but q is neither aτ − function nor a ω − function, because (X, d) is not a metric space.The following lemma lists some properties of a Q- function on X which are similar to that of a ω −function (see [15]).Lemma 8 :- Let q ∶ X × X → R6be a Q - function on X. Let { xC}C∈ X and { yC}C∈ X be sequences in X, andlet { αC}C∈ X and { βC}C∈ X be such that they converges to 0 and x, y, z ∈ X. Then, the following hold:i. if q(xC, y) ≤ αC and q(xC, z) ≤ βC for all n ∈ N, then y = z. In particular, if q(x, y) = 0 andq(x, z) = 0 then y = z;ii. if q(xC, yC) ≤ αC and q(xC, z) ≤ βC for all x ∈ N, then { yC}C∈ X converges to z;iii. if q(xC, xM) ≤ αC for all n, m ∈ N with m > n, then { xC}C∈ X is a Cauchy sequence ;iv. if q(y, xC) ≤ αC for all n ∈ N, then { xC}C∈ X is a Cauchy sequence ;v. if q1, q2, q< . . . . qC are Q- functions on X, then q(x, y) = max { q1(x, y), q2(x, y), . . . . . , qC(x, y) }is also a Q- function on X.Main ResultIn this section we introduced a new concept of coupled fixed point for (α − Ψ) − contractive map in quasiordered metric spaces also we establish some coupled fixed point results by considering maps on quasi metricspaces endowed with partial order.Throughout this article we denote Ψ the family of non decreasing functions Ψ ∶ 70, +∞) → 70, +∞) such thatΣC]1KΨC(t) < ∞ for all t > 0, where ΨCis the n_`iterate of Ψ satisfying,i. Ψa1 ( { 0}) = { 0 },ii. Ψ (t) < b for all t > 0,iii. limc → _d Ψ (t) < b for all t > 0.Lemma 9:- If Ψ ∶ 70, ∞8 → 70, ∞8 is non decreasing and right continuous, the ΨC(t) → 0 as n → ∞ for allt ≥ 0 if and only if Ψ(t) < b efg hii b > 0.Definition 10:- Let F: X × X → X and α ∶ X2× X2→ 70, +∞) be two mappings. Then F is said to be(α) − admissible ifαj(x, y), (u, v)l ≥ 1 → α nj F(x, y), F(y, x)l, jF(u, v), F(v, u)lo ≥ 1,for all x, y, u, v ∈ X.Definition 11:- Let (X, ≤, d) be a partially ordered complete quasi- metric space with a Q- function q on X andF: X × X → X be a mapping. Then a map F is said to be a (α − Ψ) − contractive if there exists two functionsψ ∈ Ψ and α ∶ X2× X2→ 70, +∞) such thatα j (x, y), (u, v)lqjF(x, y), F(u, v)l ≤ ψ nq(r,s)6 q(t,u)2ofor all x ≥ u and y ≤ v.Now we give the main result of this paper, which is as follows.
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications32Theorem 12:- Let (X, ≤, d) be a partially ordered complete quasi − metric space with a Q − function q on X.Suppose that F ∶ X × X → X such that F has the mixed monotone property. Assume that ψ ∈ Ψ and α ∶ X2×X2→ 70, +∞) such that for all x, y, u, v ∈ X following holds,α j (x, y), (u, v)lqjF(x, y), F(u, v)l ≤ Ψ nq(r,s)6 q(t,u)2o 2.1for all x ≤ u and y ≥ v. Suppose also that7(v)8F is (α) − admissible7(w)8 there exist xx, yx ∈ X such thatα n(xx, yx), jF(xx, yx), F(yx, xx)lo ≥ 1 and α n(yx, xx), jF(yx, xx), F(xx, yx)lo ≥ 17(y)8 F is continuous.If there exists xx, yx ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.2that is F has a coupled fixed point.Proof:- Let xx, yx ∈ X be such that α n(xx, yx), jF(xx, yx), F(yx, xx)lo ≥ 1 andα n(yx, xx), jF(yx, xx), F(xx, yx)lo ≥ 1 and xx ≤ F(xx, yx) = x_1 and yx ≥ F(yx, xx) = y1. Let x2, y2 ∈X such that F(x1, y1) = x2 and F(y1, x1) = y2. Continuing this process, we can construct two sequences { xC}and { yC} in X as follows,xC61 = F(xC, yC) and yC61 = F(yC, xC)for all n ≥ 0. We will show thatxC ≤ xC61 and yC ≥ yC61 2.3for all n ≥ 0. We will use the mathematical induction. Let n = 0. Since xx ≤ F(xx, yx), and yx ≥F(yx, xx) and as x1 = F(xx, yx), and y1 = F(yx, xx). We have xx ≤ x1 and yx ≥ y1. Thus (2.3) holds forn = 0. Now suppose that (2.3) holds for some n ≥ 0. Then since xC ≤ xC61 and yC ≥ yC61 and by themixed monotone property of F, we havexC62 = F(xC61, yC61) ≥ F(xC, yC61) ≥ F(xC, yC) = xC61andyC62 = F(yC61, xC61) ≤ F(yC, xC61) ≤ F(yC, xC) = yC61From above we conclude thatxC61 ≤ xC62 and yC61 ≥ yC62Thus by the mathematical induction, we conclude that (2.3) holds for n ≥ 0. If for some n we have(xC61, yC61) = (xC, yC), then F(xC, yC) = xC and F(yC, xC) = yC that is, F has a coupled fixed point. Now,we assumed that (xC61, yC61) ≠ (xC, yC) for all n ≥ 0. Since F is (α) − admissible, we haveαj(xx, yx), (x1, y1)l = α n(xx, yx), jF(xx, yx), F(yx, xx)lo ≥ 1which implies α njF(xx, yx), F(yx, xx)l, jF(x1, y1), F(y1, x1)lo = αj(x1, y1), (x2, y2)l ≥ 1Thus, by the mathematical induction, we haveαj(xC, yC), (xC61, yC61)l ≥ 1 2.4and similarly,αj(yC, xC), (yC61, xC61)l ≥ 1 2.5for all n ∈ N. Using (2.1) and (2.4) , we obtainq(xC, xC61) = qjF(xCa1, yCa1), F(xC, yC)l≤ α j(xCa1, yCa1), (xC, yC)lqjF(xCa1, yCa1), F(xC, yC)l≤ Ψ nq(r|}~,r|)6 q(t|}~,t|)2o 2.6Similarly we haveq(yC, yC61) = qjF(yCa1, xCa1), F(yC, xC)lα j(yCa1, xCa1), (yC, xC)lqjF(yCa1, xCa1), F(yC, xC)l ≤ Ψ nq(t|}~,r|)6 q(r|}~,r|)2o 2.7Adding (2.6)and (2.7), we getq(r|,r|d~)6 q(t|,t|d~)2≤ Ψ nq(r|}~,r|)6 q(t|}~,t|)2oRepeating the above process, we getq(r|,r|d~)6 q(t|,t|d~)2≤ ΨCnq(r•,r~)6 q(t•,t~)2ofor all n ∈ N. For ϵ > 0 bℎ•g• exists n(ϵ ) ∈ N such that
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications33ΣC D C(‚ )ΨCnq(r•,r~)6 q(t•,t~)2o <‚2Let m, n ∈ N be such that m > N > N(ƒ ). Then, by using the triangle inequality, we haveq(r|,r„)6 q(t|,t„)2≤ Σ… ] CMa1nq(r†,r†d~)6 q(t†,t†d~)2o≤ Σ… ] CMa1Ψ…nq(r•,r~)6 q(t•,t~)2o≤ ΣC D C(‚ ) ΨCnq(r•,r~)6 q(t•,t~)2o <‚2This implies that q(xC, xM) + q(yC, yM) < ƒ . Sinced(xC, xM) ≤ q(xC, xM) + q(yC, yM)) < ƒandd(yC, yM) ≤ q(xC, xM) + q(yC, yM) < ƒand hence { xC } and { yC } are Cauchy sequences in X. Since (X, d) is complete quasi metric spaces and hence{ xC} and { yC } are convergent in X. Then there exists x, y ∈ X such thatlimC → KxC = x limC → KyC = y.Since F is continuous and xC61 = F(xC, yC) and yC61 = F(yC, xC), taking limit n → ∞ we getx = limC → KxC = limC → KF(xC, yC) = F(x, y)andy = limC → KyC = limC → K F(yC, xC) = F(y, x)that is, F(x, y) = x and F(y, x) = y and hence F has a coupled fixed point.In the next theorem, we omit the continuity hypothesis of F.Theorem 13:- Let (X, ≤, d) be a partially ordered complete quasi- metric space with a Q − function q on X.Suppose that F ∶ X × X → X such that F has the mixed monotone property. Assume that Ψ ∈ Ψ and α ∶X2× X2→ 70, +∞) such that for all x, y, u, v ∈ X following holds,α ( (x, y), (u, v) )q(F(x, y), F(u, v)) ≤ Ψ nq(r,s)6 q(t,u)2o 2.8for all x ≤ u and y ≥ v. Suppose also that7(v)8F is (α) − admissible7(w)8 there exist xx, yx ∈ X such thatα n(xx, yx), jF(xx, yx), F(yx, xx)lo ≥ 1 and α n(yx, xx), jF(yx, xx), F(xx, yx)lo ≥ 17(y)8 if { xC } and { yC } are sequences in X such thatαj(xC, yC), (xC61, yC61)l ≥ 1 and αj(yC, xC), (yC61, xC61)l ≥ 1for all n and limC → KxC = x ∈ X and limC → KyC = y ∈ X, thenαj(xC, yC), (x, y)l ≥ 1 and αj(yC, xC), (y, x)l ≥ 1.If there exists xx, yx ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x,y ∈ X such thatx = F(x, y), y = F(y, x) 2.9that is F has a coupled fixed point.Proof:- Proceeding along the same line as the above Theorem12, we know that { xC} and { yC } are Cauchysequences in complete quasi metric space X. Then there exists x, y ∈ X such thatlimC → KxC = x and limC → KyC = y. 2.10On the other hand, from (2.4) and hypothesis (c) we obtainαj(xC, yC), (x, y)l ≥ 1 2.11and similarlyαj(yC, xC), (y, x)l ≥ 1 2.12for all n ∈ N. Using the triangle inequality, ref(eq7) and the property of Ψ(t) < b efg all t > 0, we getq(F(x, y), x) ≤ qjF(x, y), F(xC, yC)l + q(xC61, x)≤ αj(xC, yC), (x, y)lqjF(xC, yC), F(x, y)l + q(xC61, x)≤ Ψ nq(r|,r)6 q(t|,t)2o + q(xC61, x)<q(r|,r) 6 q(t|,t)2+ q(xC61, x).Similarly, we obtainq(F(y, x), y) ≤ qjF(y, x), F(yC, xC)l + q(yC61, y)≤ αj(yC, xC), (y, x)lqjF(yC, xC), F(y, x)l + q(yC61, x)≤ Ψ nq(r|,r)6 q(t|,t)2o + q(xC61, x)
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications34<q(t|,t)6 q(r|,r)2+ q(yC61, y).Taking the limit n → ∞ in the above two inequalities, we getq(F(x, y), x) = 0 and q(F(y, x), y) = 0.Hence, F(x, y) = x and F(y, x) = y. Thus, F has a coupled fixed point.In the following theorem, we will prove the uniqueness of the coupled fixed point. If (X, ≤) is a partially orderedset, then the product X × X with the following partial order relation:(x, y) ≤ (u, v) ↔ x ≤ u, y ≥ v,for all (x, y), (u, v) ∈ X × X.Theorem 14:- In addition to the hypothesis of Theorem 12 suppose that for every (x, y), (s, t) ∈ X × X, thereexists (u, v) ∈ X × X such thatα((x, y), u, v)) ≥ 1 and α((s, t), u, v)) ≥ 1and also assume that (u, v) is comparable to (x, y) and (s, t). Then F has a unique coupled fixed point.Proof:- From Theorem 12, the set of coupled fixed point is non empty. Suppose (x, y) and (s, t) are coupledfixed point of the mappings F: X × X → X, that is x = F(x, y), y = F(y, x), s = F(s, t) and t = F(t, s). Byassumption, there exists (u, v) ∈ X × X such that (u, v) is comparable to (x, y) and (s, t). put u = ux and v =vx and choose u1, v1 ∈ X such that u1 = F(ux, vx) and v1 = F(vx, ux). Thus, we can define two sequences{ uC } and { vC } asuC61 = F(uC, vC) and vC61 = F(vC, uC).Since (u, v) is comparable to (x, y), it is easy to show that x ≤ u1 and ≥ v1 . Thus, x ≤ uC and y ≥ vC for alln ≥ 1. Since for every (x, y), (s, t) ∈ X × X, there exists (u, v) ∈ X × X such thatα((x, y), u, v)) ≥ 1 and α((s, t), u, v)) ≥ 1. 2.13Since F is (α) − admissible, so from (2.13), we haveα((x, y), u, v)) ≥ 1 → α((F(x, y), F(y, x)), (F(u, v), F(v, u))) ≥ 1.Since u = ux and v = vx, we getαj(x, y), ux, vxl ≥ 1 → α((F(x, y), F(y, x)), jF(ux, vx), F(vx, ux)l ≥ 1.Thusαj(x, y), (u, v)l ≥ 1 → αj(x, y), (u1, v1)l ≥ 1.Therefore by mathematical induction, we obtainαj(x, y), (uC, vC)l ≥ 1 2.14for all n ∈ N and similarly αj(y, x), (vC, uC)l ≥ 1. From (2.13) and (2.14), we getq(x, u_(n + 1)) = qjF(x, y), F(uC, vC)l≤ αj(x, y), (uC, vC)lqjF(x, y), F(uC, vC)l≤ Ψ nq(r,s|)6 q(t,u|)2o. 2.15Similarly, we getq(y, v_(n + 1)) = qjF(y, v), F(vC, uC)l≤ αj(y, x), (vC, uC)lqjF(y, x), F(vC, xC)l≤ Ψ nq(t,u|)6 q(r,s|)2o. 2.16Adding (2.15) and (2.16), we getq(r,s|d~)6 q(t,u|d~)2≤ Ψ nq(r,s|)6 q(t,u|)2oThusq(r,s|d~)6 q(t,u|d~)2≤ ΨCnq(r,s~)6 q(t,u~)2o 2.17for each n ≥ 1. Letting n → ∞ in 2.17 and using Lemma 8, we getlimC → K7q(x, uC61) + q(y, vC61)8 = 0This implieslimC → Kq(x, uC61) = 0 limC → Kq(y, vC61) = 0. 2.18Similarly we can show thatlimC → Kq(s, uC61) = 0 limC → K q(t, vC61) = 0. 2.19From 2.18 and 2.19, we conclude that x = s and y = t. Hence, F has a unique coupled fixed point.Example 15:- Let X = 70,18, with the usual partial ordered ≤. Defined d: X × X → R6byd(x, y) = Ry − x if x = y2(x − y) otherwiseTand q: X × X → R^ + byq(x, y) = ∣ x − y ∣, ∀ x, y ∈ X. 2.20
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications35Then d is a quasi metric and q is a Q − function on X. Thus, (X, d, ≤) is a partially ordered complete quasi metricspace with Q- function q on X.Consider a mapping α ∶ X2× X2→ 70, +∞) be such thatαj(x, y), (u, v)l = ‰1 if x ≥ y, u ≥ v0 otherwiseTLet Ψ (t) =_2, for t > 0. Defined F: X × X → X by F(x, y) =1Šxy for all x, y ∈ X.Since ∣ xy − uv ∣ ≤ ∣ x − u ∣ + ∣ y − v ∣ holds for all x, y, u, v ∈ X. Therefore, we haveqjF(x, y), F(u, v)l = ∣rtŠ–suŠ∣≤1Šj ∣ x – u ∣ + ∣ y – v ∣l=1Šjq(x, u) + q(y, v)lIt follows thatα ( (x, y), (u, v) )q(F(x, y), F(u, v)) ≤1Š(q(x, u) + q(y, v))Thus ref(eq1) holds for Ψ(t) =_2for all t > 0 and we also see that all the hypothesis of Theorem 12arefulfilled. Then there exists a coupled fixed point of F. In this case (0,0) is coupled fixed point of F.Example 16:- Let X = 70,18, with the usual partial ordered ≤. Defined d: X × X → R6byd(x, y) = Ry − x if x = y2(x − y) otherwiseTand q: X × X → R6byq(x, y) = ∣ x − y ∣, ∀ x, y ∈ X 2.21Then d is a quasi metric and q is a Q − function on X. Thus, (X, q, ≤) is a partially ordered complete quasi metricspace with Q- function q on X.Consider a mapping α ∶ X2× X2→ 70, +∞) be such thatαj(x, y), (u, v)l = ‰1 if x ≥ y, u ≥ v0 otherwiseTLet (t) = 2 t, for t > 0. Defined F: X × X → X by F(x, y) = sin x + sin y for all x, y ∈ X.Since ∣ sin x − sin y ∣ ≤ ∣ x − y ∣ holds for all x, y ∈ X. Therefore, we haveqjF(x, y), F(u, v)l = ∣ sin x + sin y − sin u − sin v ∣≤ ∣ sin x − sin u ∣ + ∣ sin y − sin v ∣≤ ∣ x − u ∣ + ∣ y − v ∣≤ Ψ njq(r,s)6 q(t,u)l2o.Then there exists a coupled fixed point of F. In this case (0,0) is coupled fixed point of F.Corollary 17:- Let (X, ≤, d) be a partially ordered complete quasi- metric space with a Q − function q on X.Suppose that F ∶ X × X → X such that F is continuous and has the mixed monotone property. Assume thatΨ ∈ Ψ and such that for all x, y, u, v ∈ X following holds,q(F(x, y), F(u, v)) ≤ Ψ nq(r,s)6 q(t,u)2o 2.22for all x ≤ u and y ≥ v.If there exists x_0, y_0 ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.23that is F has a coupled fixed point.Proof:- It is easily to see that if we take α ( (x, y), (u, v) ) = 1 in Theorem 12 then we get Corollary 17.Corollary 18:- Let (X, ≤, d) be a partially ordered complete quasi- metric space with a Q- function q on X.Suppose that F ∶ X × X → X such that F is continuous and has the mixed monotone property. Assume thatΨ ∈ Ψ and such that for all x, y, u, v ∈ X following holds,q(F(x, y), F(u, v)) ≤…27q(x, u) + q(y, v)8 2.24for k ∈ 70,1) and for all x ≤ u and y ≥ v.If there exists x_0, y_0 ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.25that is F has a coupled fixed point.
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications36Proof:- It is easily to see that if we take Ψ(t) = kt in Corollary 17 then we get Corollary 18.Corollary 19:- Let (X, ≤, d) be a partially ordered complete quasi metric space with a Q-function q on X.Assume that the function Ψ ∶ 70, +∞) → 70, +∞) is such that Ψ (t) < b for each t > 0. Further suppose thatF: X × X → X is such that F has the mixed monotone property andq(F(x, y), F(u, v)) ≤ Ψ nq(r,s)6 q(t,u)2o 2.29for all x, y, u, v ∈ X for which x ≤ u and y ≤ v. Suppose that F satisfies following,[(a)]F is continuous or[(b)] X has the following property:[(i)] if a non decreasing sequence { x_n } → x then x_n ≤ x for all n,[(ii)] if a non increasing sequence { y_n } → y then y_n ≥ y for all n.If there exists xx, yx ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx) 2.30then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.31that is F has a coupled fixed point.Proof:- It is easily to see that if we take α j (x, y), (u, v)l = 1 and from the property in Theorem 12 then weget Corollary 19.Corollary20 :- Let (X, ≤, d) be a partially ordered complete quasi metric space with a Q-function q on X.Assume that the function Ψ ∶ 70, +∞) → 70, +∞) is such that Ψ (t) < b for each t > 0. Further suppose thatF: X × X → X is such that F has the mixed monotone property andqjF(x, y), F(u, v)l ≤…27q(x, u) + q(y, v)8 2.32for all k ∈ 70,1), x, y, u, v ∈ X for which x ≤ u and y ≤ v. Suppose that F satisfies following,7(a)8F is continuous or7(b)8 X has the following property:7(i)8 if a non decreasing sequence { xC } → x then xC ≤ x for all n,7(ii)8 if a non increasing sequence { yC } → y then yC ≥ y for all n.If there exists xx, y 0 ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.33that is F has a coupled fixed point.Proof:- It is easily to see that if we take Ψ (t) = kt in Theorem 12 then we get Corollary 20.Now our next result show that (α) − admissible function is work like as a control function, but converges maynot be true in general. We also give an example in support of this fact.Theorem 21:- Let (X, ≤, d) be a partially ordered complete quasi- metric space with a Q- function q on X.Suppose that F ∶ X × X → X such that F has the mixed monotone property. Assume that α: X2× X2→70, +∞) such that for all x, y, u, v ∈ X following holds,α ( (x, y), (u, v) )q(F(x, y), F(u, v)) ≤…27 q(x, u) + q(y, v)8 2.34for k ∈ 70,1) and for all x ≤ u and y ≥ v. Suppose also that7(a)8F is (α) − admissible7(b)8 there exist xx , yx ∈ X such thatα n(xx, yx), jF(xx, yx), F(yx, xx)lo ≥ 1andα n(yx, xx), jF(yx, xx), F(xx, yx)lo ≥ 17(c)8 F is continuous.If there exists xx, yx ∈ X such thatxx ≤ F(xx, yx), yx ≥ F(yx, xx)then there exist x, y ∈ X such thatx = F(x, y), y = F(y, x) 2.35that is F has a coupled fixed point.Proof:- If we take Ψ (t) = kt in Theorem 12 then the remaining prove of the above Theorem 21 is similar tothe prove of Theorem 12.Example 22:- Let X = 70, ∞), with the usual partial ordered ≤. Defined d: X × X → R6byd(x, y) = Ry − x if x = y2(x − y) otherwiseT
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications37and q: X × X → R6byq(x, y) = ∣ x − y ∣, ∀ x, y ∈ X.Then d is a quasi metric and q is a Q- function on X. Thus, (X, d, ≤) is a partially ordered complete quasi metricspace with Q- function q on X.Consider a mapping α ∶ X2× X2→ 70, +∞) be such thatαj(x, y), (u, v)l = ‰1 if x ≥ y, u ≥ v0 otherwiseTDefined F: X × X → X byF(x, y) = Žr a t2if x ≤ y0 otherwiseTThen there is no any k ∈ 70,1) for which satisfying all conditions of Theorem ref(thm4).If we take α ∶ X2× X2→ 70, +∞) as follows,αj(x, y), (u, v)l = ‰2 if x ≥ y, u ≥ v0 otherwiseTThen there is k =12∈ 70,1) such that all conditions of Theorem 21 are satisfies and (0,0) is a coupled fixedpoint of F.Acknowledgements:- The authors would like to express their sincere thanks to the editor and the anonymousreferees for their valuable comments and useful suggestions in improving the article.References1. S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “Some generalizations of Ekeland-type variationalprinciple with applications to equilibrium problems and fixed point theory,” Nonlinear Analysis:Theory, Methods and Applications, vol. 69, no. 1, pp. 126--139, 2008.2. Q. H. Ansari, “Vectorial form of Ekeland-type variational principle with applications to vectorquilibrium problems and fixed point theory,” Journal of Mathematical Analysis and Applications, vol.334, no. 1, pp. 561--575, 2007.3. Q. H. Ansari, I. V. Konnov, and J. C. Yao, “On generalized vector equilibrium problems,” NonlinearAnalysis: Theory, Methods and Applications, vol. 47, no. 1, pp. 543--554, 2001.4. Q. H. Ansari, A. H. Siddiqi, and S. Y. Wu, “Existence and duality of generalized vector equilibriumproblems,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 115--126, 2001.5. Q. H. Ansari and J.-C. Yao, “An existence result for the generalized vector equilibrium problem,”Applied Mathematics Letters, vol. 12, no. 8, pp. 53 --56, 1999.6. Q. H. Ansari and J.-C. Yao, “A fixed point theorem and its applications to a system of variationalinequalities,” Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp. 433--442, 1999.7. A. D. Arvanitakis, A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc.131(12, 3647--3656 (2003).8. S. Banach, Sur les op(erations dans les ensembles abstraits et leurs applications aux (equationsi(egrales, Fund. Math. 3 (1922 133-181.).9. T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces andapplications. Nonlinear Anal. TMA 65, 1379--1393 (2006).10. D. W. Boyd, J. S. W. Wong, On nonlinear contractions. Proc. Am. Math. Soc. 20, 458--464 (1969).11. Dz. Burgic, S. Kalabusic, M.R.S. Kulenovic, Global attractivity results for mixed monotone mappingsin partially ordered complete metric spaces, Fixed Point Theory Appl. (2009 Article ID 762478.)12. Lj. B. Ciric, N. Cakid, M. Rajovic, J.S, Ume, Monotone generalized nonlinear contractions in partiallyordered metric spaces, Fixed Point Theory Appl./(2008 Article ID 131294.13. J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applicationsto ordinary differntial equations, Nonlinear Anal. 72, 1188--1197 (2010.).14. D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications. NonlinearAnal., Theory Methods Appl. 11 (1987 )623--632.15. O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theoremsin complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.16. L.-J. Lin and W.-S. Du, “Ekeland’s variational principle, minimax theorems and existence ofnonconvex equilibria in complete metric spaces,” Journal of Mathematical Analysis and Applications,vol. 323, no. 1, pp. 360-370, 2006.17. S.G. Matthews, Partial metric topology, Ann. New York Acad. Sci. vol. 728 183-197 , 1994.18. J.J. Nieto, R. R. Lopez, Contractive mappin theorems in partially ordered sets and applications toordinary differential equations, Order, 22, 223--239 (2005.)
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications3819. J.J. Nieto, R. R. Lopez, Existence and uniqueness of fixed point in partially ordered sets andapplications to ordinary differential equations, Acta Math.Sin. (Engl. Set. 23, 2205 -- 2212 (2007.)20. S. Romaguera, M. Schellekens, Partial metric monoids and semivaluation spaces, Topol. Appl. vol.153, 948-962, 2005.)21. S. Romaguera, M.P. Schellekens, O. Valero, Complexity spaces as quantitative domains ofcomputation, Topology Appl. vol. 158, 853-860, 2011.)22. S. Romaguera, P. Tirado, O. Valero, Complete partial metric spaces have partially metrizablecomputational models, Int. J. Comput. Math. 89, 284-290, 2012.)23. S. Romaguera, O. Valero, A quantitative computational model for complete partial metric spaces viaformal balls, Math. Struc. Comp. Sci. vol. 19, 541-563, 2009.)24. S. Romaguera, O. Valero, Domain theoretic characterizations of quasimetric completeness in terms offormal balls, Math. Struc. Comp. Sci. vol. 20, 453-472, 2010).25. W. Sintunavarat, Y.J. Cho, P. Kumam, Coupled fixed point theorems for weak contraction mappingunder F-invariant set. Abstr. Appl. Anal. 2012, 15 (Article ID 324874 (2012.)26. W. Sintunavarat, P. Kumam,: Gregus type fixed points for a tangential multi-valued mappingssatisfying contractive conditions of integral type. J. Inequal. Appl. 2011, 3 (2011.)27. W. Sintunavarat, Y.J. Cho, Kumam, P: Common fixed point theorems for c-distance in ordered conemetric spaces. Comput. Math. Appl. 62, 1969--1978 (2011.)28. W. Sintunavarat, P. Kumam,: Common fixed point theorems for hybrid generalized multivaluedcontraction mappings. Appl. Math. Lett. 25, 52--57 (2012).29. W. Sintunavarat, P. Kumam,: Common fixed point theorems for generalized operator classes andinvariant approximations. J. Inequal. Appl. 2011, 67 (2011).30. W. Sintunavarat, P. Kumam,: Generalized common fixed point theorems in complex valued metricspaces and applications. J. Inequal. Appl. 2012, 84 (2012).31. T. Suzuki,: A generalized Banach contraction principle that characterizes metric completeness. Proc.Am. Math. Soc. 136(5, 1861--1869 (2008).
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