RESEARC H Open Access Mixed monotone-generalized contractions in partially ordered probabilistic metric spaces Ljubomir Ćirić 1 , Ravi P Agarwal 2* and Bessem Samet 3 * Correspondence: agarwal@fit.edu 2 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA Full list of author information is available at the end of the article Abstract In this article, a new concept of mixed monotone-generalized contraction in partially ordered probabilistic metric spaces is introduced, and some coupled coincidence and coupled fixed point theorems are proved. The theorems Presented are an extension of many existing result s in the literature and include several recent developments. Mathematics Subject Classification: Primary 54H25; Secondary 47H10. Keywords: probabilistic metric spacemixe d monotone property, partially ordered set, coupled coincidence fixed point, coupled fixed point 1 Introduction The Banach contraction principle [1] is one of the most celebrated fixed point theo- rem. Many generalizations of this famous theorem and other important fixed point theorems exist in the literature (cf. [2-37]). Ran and Reurings [3] proved the Banach contraction principle in partially ordered metric spac es. Recently Agarwal et al. [2] presented some new fixed point results for monotone and generalized contractive type mappings in partially ordered metric spaces. Bhaskar and Lakshmikantham [4] initiated and proved some new coupled fixed point results for mixed monotone and contraction mappings in partially ordered metric spaces. The main idea in [2-11] involve combining the ideas of iterative techni- que in the contraction mapping principle with those in the monotone technique. In [3], Ran and Reurings proved the following Banach type principle in partially ordered metric spaces. Theorem 1 (Ran and Reurings [3]). Let (X, ≤) be a partially ordered set such t hat every pair x, y Î X has a lower and an upper bound. Let d be a metric on X such that the metric space (X, d) is complete. Let f : X ® X be a continuous and monotone (that is, either decreasing or increasing) operator. Suppose that the following two assertions hold: (1) there exists k Î (0, 1) such that d(f (x), f (y)) ≤ kd(x, y), for each x, y Î Xwithx≥ y, (2) there exists x 0 Î X such that x 0 ≤ f (x 0 ) or x 0 ≥ f (x 0 ). Thenfhasauniquefixedpointx* Î X, i.e. f(x*) = x*, and for each x Î X, the sequence {f n (x)} of successive approximations of f starting from x converges to x* Î X. Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 ©2011Ćirićć et al; licensee Springer. This is an Open Access article distribu ted under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The results of Ran and Reurings [3] have motivated many authors to undertake further investigation of fixed points in the field of ordered metric spaces: Agarwal et al. [2], Bhas- kar and Lakshmikantham [4], Bhaskar et al. [5], Ćirić and Lakshmikantham [7], Ćirić et al. [8,9], Lakshmikantham and Ćirić [10], Nieto and López [6,11], Samet [12-14], and others. Fixed point theory in probabilistic metric spaces can be considered as a part of prob- abilistic analysis, which is a very dynamic area of mathematical research. The theory of probabilistic metric spaces was introduced in 1942 by Menger [15]. These are generali- zations of metric spaces in which the distances between points are described by prob- ability distributions rather than by numbers. Schweizer and Sklar [ 16,17] studied t his concept and gave some fundamental results on these spaces. In 1972, Sehgal and Bhar- ucha-Reid [ 18] initiated the study of contraction mappings on probabilistic metric spaces. Since th en, several results have been obtained by v arious authors in this direc- tion. For more details, we refer the reader to [19-27]. In [8], Ćirić et al. i ntroduced the concept of monotone generalized contraction i n partially ordered probabilistic metric spaces and proved some fixed and common fixed point theorems on such spaces. In this article, we introduce a new concept of mixed monotone generalized contraction in partially ordered probabilisti c metric spaces and we prove some coupled coincidence and coupled fixed point theorems on such spaces. Presented theorems extend many exist- ing results in the literature, in particular, the results obtained by Bhaskar and Lakshmikan- tham [4], Lakshmikantham and Ćirić [10], and include several recent developments. Throughout this article, the space of all probability distribution functions is denoted by Δ + ={F : ℝ ∪ {-∞,+∞} ® [0,1]: F is left-continuous and non-d ecreasing on ℝ, F(0) =0andF(+∞)=1}andthesubsetD + ⊆ Δ + is the set D + ={F Î Δ + :lim t®+∞ F(t)= 1}. The space Δ + is pa rtially ordered by the usual point-wise ordering of functions, i.e., F ≤ G if and only if F(t) ≤ G(t) for all t in ℝ. The maximal element for Δ + in this order is the distribution function given by ε 0 (t )=  0, if t ≤ 0, 1, if t > 0 . We refer t he reader to [22] for the terminology concerning probabilistic metric spaces (also called Menger spaces). 2 Main results We start by recalling some definitions intro duced by Bhaskar and Laksh mikantham [4] and Lakshmikantham and Ćirić [10]. Definition 2 (Bhaskar and Lakshmikantham [4]). Let X be a non-empty set and A : X × X ® X be a given mapping. An element (x, y) Î X × Xissaidtobeacoupled fixed point of A if A ( x, y ) = xandA ( y, x ) = y . Definition 3 (Lakshmikantham and Ćirić [10]). Let X be a non-empt y set, A : X × X ® X and h : X ® X are given mappings. (1) An element (x, y) Î X × X is said to be a coupled coincidence point of A and h if A ( x, y ) = h ( x ) and A ( y, x ) = h ( y ). Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 2 of 13 (2) An element (x, y) Î X × X is said to be a coupled common fixed point of A and h if A ( x, y ) = h( x ) = xan d A ( y, x ) = h( y ) = y . (3) We say that A and h commute at (x, y) Î X × Xif h ( A ( x, y )) = A ( h ( x ) , h ( y )) and h ( A ( y, x )) = A ( h ( y ) , h ( x )). (4) A and h commute if h ( A ( x, y )) = A ( h ( x ) , h ( y )) , for all ( x, y ) ∈ X × X . Definition 4 (Lakshmikantham and Ćirić [10]). Let (X, ≤) be a partially ordered set, A : X × X ® Xandh: X ® X are given mappings. We say that A has the mixed h- monotone property if for all x, y Î X, we have x 1 , x 2 ∈ X, h(x 1 ) ≤ h(x 2 ) ⇒ A(x 1 , y) ≤ A(x 2 , y) , y 1 , y 2 ∈ X, h ( y 1 ) ≥ h ( y 2 ) ⇒ A ( x, y 1 ) ≤ A ( x, y 2 ) . If h is the identity mapping on X, then A satisfies the mixed monotone property. We need the following lemmas to prove our main results. Lemma 5. Let n ≥ 1. If F Î D + , G 1 , G 2 , , G n : ℝ ® [0,1] are non-decreasing func - tions and, for some k Î (0, 1), F ( kt ) ≥ min{G 1 ( t ) , G 2 ( t ) , ··· , G n ( t ) , F ( t ) }, ∀t > 0 , (1) then F(kt) ≥ min{G 1 (t), G 2 (t), , G n (t)} for all t >0. Proof. The proof is a simple adaptation of that of Lemma 3.3 in [8]. □ Lemma 6. Let (X, F, Δ) be a Menger PM-space and k Î (0, 1). If min{F p , q (kt), F s,v (kt)} = min{F p , q (t ), F s,v (t ) }, for all t > 0 , (2) then p = q and s = v. Proof. From (2) it is easy to show by induction that min{F p , q (k n t), F s,v (k n t) } = min{F p , q (t ), F s,v (t ) },foralln ≥ 1 . (3) Now we shall show that min{F pq (t), F s,v (t)} = 1 for all t > 0. Suppose, to the contrary, that there exists some t 0 > 0 such that min{F pq (t 0 ), F s,v (t 0 )} < 1. S ince (X, F)isaMen- ger PM space, then min{F pq ( t), F s,v ( t)} ® 1ast ® ∞. Therefore, there exists t 1 >t 0 such that min{F pq (t 1 ), F s,v (t 1 )} > min{F pq (t 0 ), F s,v (t 0 )} . (4) Since t 0 >0andk Î (0, 1), there exists a positive integer n >1suchthatk n t 1 <t 0 . Then by the monotony of F pq (·) and F s,v (·), it follows that min{F pq (k n t 1 ), F s,v (k n t 1 )} ≤ min{F pq (t 0 ), F s,v (t 0 )}. Hence and from (3) with t = t 1 , we have min{F pq (t 1 ), F s,v (t 1 )} = min{F pq (k n t 1 ), F s,v (k n t 1 )}≤min{F pq (t 0 ), F s,v (t 0 } , a contradiction with (4). Therefore min{F pq (t), F s,v (t)} = 1 for all t > 0, which implies that F pq (t) = 1 and F s,v (t) = 1 for all t > 0. Hence p = q and s = v. □ Now, we state and prove our first result. Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 3 of 13 Theorem 7. Let (X, ≤) be a partiall y ordered set and (X, F , Δ) be a complete Menger PM-space under a T-n orm Δ of H-type (Hadžić-type). Suppose A : X × X ® X and h : X ® X are two mappings such th at A has the h-mixed monotone property on X and, for some k Î (0, 1), F A(x,y),A(u,v) (kt) ≥ min{F h(x),h(u) (t ), F h(y),h(v) (t ), F h(x),A(x,y) (t ) , F h ( u ) ,A ( u,v ) (t ), F h ( y ) ,A ( y,x ) (t ), F h ( v ) ,A ( v,u ) (t ) } (5) for all x , y Î X for which h(x) ≤ h(u) and h(y) ≥ h(v) and all t >0.Suppose also that A(X × X) ⊆ h(X), h(X) is closed and if {h(x n )}⊂Xisanon- decreasing sequence with h(x n ) → h(z) in h(X ) then h ( x n ) ≤ h ( z ) for all n hold, (6) if {h(x n )}⊂Xisanon- decreasing sequence with h(x n ) → h(z) in h(X ) then h ( z ) ≤ h ( x n ) for all n hold. (7) If there exist x 0 , y 0 Î X such that h ( x 0 ) ≤ A ( x 0 , y 0 ) and h ( y 0 ) ≥ A ( y 0 , x 0 ), then A and h have a coupled coincidence point, that is, there exist p, q Î X such that A(p, q) =h(p) and A(q, p)=h(q). Proof. By hypothesis, there exist (x 0 , y 0 ) Î X × X such that h(x 0 ) ≤ A( x 0 , y 0 )andh (y 0 ) ≥ A(y 0 , x 0 ). Since A(X × X) ⊆ h (X), we can choose x 1 , y 1 Î X such that h(x 1 )=A (x 0 , y 0 ) and h (y 1 )=A(y 0 , x 0 ). Now A(x 1 , y 1 ) and A(y 1 , x 1 ) are well defined. Again, from A(X × X) ⊆ h(X), we can choose x 2 , y 2 Î X such that h(x 2 )=A(x 1 , y 1 )andh(y 2 )=A (y 1 , x 1 ). Continuing this process, we can construct sequences {x n }and{y n }inX such that h ( x n+1 ) = A ( x n , y n ) and h ( y n+1 ) = A ( y n , x n ) ,foralln ∈ N . (8) We shall show that h ( x n ) ≤ h ( x n+1 ) ,foralln ∈ N (9) and h ( y n ) ≥ h ( y n+1 ) ,foralln ∈ N . (10) We shall use the mathematical induction. Let n = 0. Since h(x 0 ) ≤ A(x 0 , y 0 ) and h(y 0 ) ≥ A(y 0 , x 0 ), and as h(x 1 )=A(x 0 , y 0 )andh(y 1 )=A(y 0 , x 0 ), we have h(x 0 ) ≤ h(x 1 )andh (y 0 ) ≥ h(y 1 ). Thus (9) and (10) hold for n = 0. Suppose now that (9) and (10) hold for some fixed n Î N. Then, since h(x n ) ≤ h(x n+1 ) and h(y n+1 ) ≤ h(y n ), and as A has the h- mixed monotone property, from (8), h ( x n+1 ) = A ( x n , y n ) ≤ A ( x n+1 , y n ) and A ( y n+1 , x n ) ≤ A ( y n , x n ) = h ( y n+1 ), (11) and from (8), h ( x n+2 ) = A ( x n+1 , y n+1 ) ≥ A ( x n+1 , y n ) and A ( y n+1 , x n ) ≥ A ( y n+1 , x n+1 ) = h ( y n+2 ). (12) Now from (11) and (12), we get h ( x n+1 ) ≤ h ( x n+2 ) Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 4 of 13 and h ( y n+1 ) ≥ h ( y n+2 ). Thus by the mathematical induction we conclude t hat (9) and (10) hold for a ll n Î N. Therefore, h ( x 0 ) ≤ h ( x 1 ) ≤ h ( x 2 ) ≤ h ( x 3 ) ≤···≤ h ( x n ) ≤ h ( x n+1 ) ≤·· · (13) and h ( y 0 ) ≥ h ( y 1 ) ≥ h ( y 2 ) ≥ h ( y 3 ) ≥···≥ h ( y n ) ≥ h ( y n+1 ) ≥ ··· . (14) Now, from (13) and (14), we can apply (5) for (x, y)=(x n , y n )and(u, v)=(x n+1 , y n +1 ). Thus, for all t > 0, we have F A(x n ,y n ),A(x n+1 ,y n+1 ) (kt) ≥ min{F h(x n ),h(x n+1 ) (t ), F h(y n ),h(y n+1 ) (t ), F h(x n ),A(x n ,y n ) (t ), F h(x n +1 ),A(x n+1 ,y n+1 ) (t ), F h(y n ),A(y n ,x n ) (t ), F h(y n +1 ),A(y n +1 ,x n +1 ) (t ) } . Using (8), we obtain F h(x n+1 ),h(x n+2 ) (kt) ≥ min{F h(x n ),h(x n+1 ) (t ), F h(y n ),h(y n+1 ) (t ), F h(x n+1 ),h(x n+2 ) (t ) , F h ( y n+1 ) ,h ( y n+2 ) (t ) }. (15) Similarly, from (13) and (14), we can apply (5) for (x, y)=(y n+1 , x n+1 )and(u, v)= (y n , x n ). Thus, by using (8), for all t > 0 we get F h(y n+2 ),h(y n+1 ) (kt) ≥ min{F h(y n+1 ),h(y n ) (t ), F h(x n+1 ),h(x n ) (t ), F h(y n+1 ),h(y n+2 ) (t ) , F h ( x n+1 ) ,h ( x n+2 ) (t ) }. (16) From (15) and (16), we have min{F h(x n+1 ),h(x n+2 ) (kt), F h(y n+1 ),h(y n+2 ) (kt)} ≥ min{F h(x n ),h(x n+1 ) (t ), F h(y n ),h(y n+1 ) (t ), F h(x n+1 ),h(x n+2 ) (t ), F h( y n+1 ),h(y n+2 ) (t ) } = min{F h ( x n ) ,h ( x n+1 ) (t ), F h ( y n ) ,h ( y n+1 ) (t ), min{F h ( x n+1 ) ,h ( x n+2 ) (t ), F h ( y n+1 ) ,h ( y n+2 ) (t ) }} . Now, from Lemma 5, we have min{F h ( x n+1 ) ,h ( x n+2 ) (kt), F h ( y n+1 ) ,h ( y n+2 ) (kt)}≥min{F h ( x n ) ,h ( x n+1 ) (t ), F h ( y n ) ,h ( y n+1 ) (t ) } (17) for all t > 0. From (17) it follows that min{F h(x n+1 ),h(x n+2 ) (t), F h(y n+1 ),h(y n+2 ) (t)}≥min{F h(x n ),h(x n+1 ) (t  k), F h(y n ),h(y n+1 ) (t  k) } (18) for all t > 0. Repeating the inequality (18), for all t > 0 we get min{F h(x n+1 ),h(x n+2 ) (t), F h(y n+1 ),h(y n+2 ) (t)}≥min{F h(x n ),h(x n+1 ) (t  k), F h(y n ),h(y n+1 ) (t  k)} ≥ min{F h(x n−1 ),h(x n ) (t  k 2 ), F h(y n−1 ),h(y n ) (t  k 2 )} ≥ ··· ≥ min{F h(x 0 ),h(x 1 ) (t  k n+1 ), F h(y 0 ),h(y 1 ) (t  k n+1 )} . Thus min{F h(x n+1 ),h(x n+2 ) (t), F h(y n+1 ),h(y n+2 ) (t)}≥min{F h(x 0 ),h(x 1 ) (t  k n+1 ), F h(y 0 ),h(y 1 ) (t  k n+1 )} , (19) Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 5 of 13 for all t > 0 and n Î N. Letting n ® +∞ in (19), we obtain lim n →∞ F h(x n ),h(x n+1 ) (t )=1, f or all t > 0 , (20) and lim n →∞ F h(y n ),h(y n+1 ) (t )=1, forallt > 0 . (21) We now prove that {h(x n )} and {h(y n )} ar e Cauchy se quences in X. We need to show that for each δ > 0 and 0 <ε < 1, there exists a positive integer n 0 = n 0 (δ, ε) such that F h ( x n ) ,h ( x m ) (δ) > 1 − ε,forallm > n ≥ n 0 (δ, ε ) and F h ( y n ) ,h ( y m ) (δ) > 1 − ε, f or all m > n ≥ n 0 (δ, ε) , that is, min{F h ( x n ) ,h ( x m ) (δ), F h ( y n ) ,h ( y m ) (δ)} > 1 − ε,forallm > n ≥ n 0 (δ, ε) . (22) Now we shall prove that for each r >0, min{F h(x n ),h(x m ) (ρ), F h(y n ),h(y m ) (ρ)} ≥  m−n (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)} ) (23) for all m ≥ n + 1. We prove (23) by the mathematical induction. Let m = n +1. Then from monotony of F, for m = n +1 we have F h(x n ),h(x n+1 ) (ρ) ≥ F h(x n ),h(x n+1 ) (ρ − kρ) = (F h(x n ),h(x n+1 ) (ρ − kρ), 1) ≥ (F h(x n ),h(x n+1 ) (ρ − kρ), F h(x n ),h(x n+1 ) (ρ − kρ)) =  1 (F h(x n ),h(x n+1 ) (ρ − kδ)) ≥  1 (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}) . Similarly, F h(y n ),h(y n+1 ) (ρ) ≥ F h(y n ),h(y n+1 ) (ρ − kρ) = (F h(y n ),h(y n+1 ) (ρ − kρ), 1) ≥ (F h(y n ),h(y n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)) =  1 (F h(y n ),h(y n+1 ) (ρ − kδ)) ≥  1 (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}) . Then min{F h ( x n ) ,h ( x n+1 ) (ρ), F h ( y n ) ,h ( y n+1 ) (ρ)}≥ 1 (min{F h ( x n ) ,h ( x n+1 ) (ρ−kρ), F h ( y n ) ,h ( y n+1 ) (ρ−kρ)}) , and (23) holds for m = n +1. Suppose now that (23) hold s for some m ≥ n +1.Sincer - kr > 0, from the prob- abilistic triangle inequality, we have F h(x n ),h(x m+1 ) (ρ)=F h(x n ),h(x m+1 ) ((ρ − kρ)+kρ) ≥ (F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( x n+1 ) ,h ( x m+1 ) (kρ)) . (24) Similarly, F h ( y n ) ,h ( y m+1 ) (ρ) ≥ (F h ( y n ) ,h ( y n+1 ) (ρ − kρ), F h ( y n+1 ) ,h ( y m+1 ) (kρ)) . (25) Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 6 of 13 From (24) and (25), we get min{F h(x n ),h(x m+1 ) (ρ), F h(y n ),h(y m+1 ) (ρ)} ≥ (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}, min{F h ( x n+1 ) ,h ( x m+1 ) (kρ), F h ( y n+1 ) ,h ( y m+1 ) (kρ)}) . (26) Now we shall consider min{F h ( x n+1 ) ,h ( x m+1 ) (kρ), F h ( y n+1 ) ,h ( y m+1 ) (kρ) } .From(5)andthe hypothesis (23), we have min{F h(x n+1 ),h(x m+1 ) (kρ), F h(y n+1 ),h(y m+1 ) (kρ)} = min{F A(x n ,y n ),A(x m ,y m ) (kρ), F A(y n ,x n ),A(y m ,x m ) (kρ)} ≥ min{F h(x n ),h(x m ) (ρ), F h(y n ),h(y m ) (ρ), F h(x n ),h(x n+1 ) (ρ), F h(x m ),h(x m+1 ) (ρ), F h(y n ),h(y n+1 ) (ρ), F h(y m ),h(y m+1 ) (ρ)} = min{min{F h(x n ),h(x m ) (ρ), F h(y n ),h(y m ) (ρ)}, F h(x n ),h(x n+1 ) (ρ), F h(x m ),h(x m+1 ) (ρ), F h(y n ),h(y n+1 ) (ρ), F h(y m ),h(y m+1 ) (ρ)} ≥ min{ m−n (min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}), F h ( x n ) ,h ( x n+1 ) (ρ), F h ( x m ) ,h ( x m+1 ) (ρ), F h ( y n ) ,h ( y n+1 ) (ρ), F h ( y m ) ,h ( y m+1 ) (ρ)} . (27) Note that from (18), for every positive integer m ≥ n, we have min{F h(x m ),h(x m+1 ) (ρ), F h(y m ),h(y m+1 ) (ρ)} ≥ min{F h(x n ),h(x n+1 ) (ρ  k m−n ), F h(y n ),h(y n+1 ) (ρ  k m−n )} ≥ min{F h ( x n ) ,h ( x n+1 ) (ρ), F h ( y n ) ,h ( y n+1 ) (ρ)} for all n ∈ N . (28) Therefore, from (27) and (28), we get min{F h(x n+1 ),h(x m+1 ) (kρ), F h(y n+1 ),h(y m+1 ) (kρ)} ≥ min{ m−n (min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}) , min{F h ( x n ) ,h ( x n+1 ) (ρ), F h ( y n ) ,h ( y n+1 ) (ρ)}}. Since r ≥ r - kr, using the monotony of F, we have min{F h ( x n ) ,h ( x n+1 ) (ρ), F h ( y n ) ,h ( y n+1 ) (ρ)}≥min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ −kρ)} . Then, we have min{F h(x n+1 ),h(x m+1 ) (kρ), F h(y n+1 ),h(y m+1 ) (kρ)} ≥ min{ m−n (min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}) , min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}}. Since {Δ i (t)} i≥0 is a decreasing sequence for all t > 0, we have min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)} ≥  m−n (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}) . Then, we get min{F h(x n+1 ),h(x m+1 ) (kρ), F h(y n+1 ),h(y m+1 ) (kρ)} ≥  m−n (min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}) . (29) Now, from (26) and (29), we obtain min{F h(x n ),h(x m+1 ) (ρ), F h(y n ),h(y m+1 ) (ρ)} ≥ ( m−n (min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}) , min{F h(x n ),h(x n+1 ) (ρ − kρ), F h(y n ),h(y n+1 ) (ρ − kρ)}) =  m−n+1 (min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)}). Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 7 of 13 Hence and by the induction we conclude that (23) holds for all m ≥ n +1. Now we show that {h(x n )} and {h(y n )} a re Cauchy sequences, that is, for each δ >0 and 0 <ε < 1, there exists a positive integer n 0 = n 0 (δ, ε)suchthat(22)holds.SinceΔ is of H-type, then {Δ n : n Î ≁} is equicontinuous at 1, that is, ∀ε ∈ ( 0, 1 ) ∃r ∈ ( 0, 1 ) | s > 1 − r ⇒  n ( s ) > 1 − ε ( for all n ∈ N ). Since δ - kδ > 0, from (20) and (21) it follows that for any 0 <r <1thereexistsa positive integer n 1 = n 1 ((δ - kδ), r) such that F h ( x n ) ,h ( x n+1 ) (δ − kδ) > 1 − r and F h ( y n ) ,h ( y n+1 ) (δ − kδ) > 1 − r,foralln ≥ n 1 . Then by (23), with min{F h ( x n ) ,h ( x n+1 ) (ρ − kρ), F h ( y n ) ,h ( y n+1 ) (ρ − kρ)} = s ,weconclude that (22) holds for n 0 (δ, ε)=n 1 ((δ - kδ), r). Thus we proved that {h(x n )} and {h(y n )} are Cauchy sequences in X. Since h(X) is complete, there is some p, q Î X such that lim n → ∞ h(x n )=h(p) and lim n → ∞ h(y n )=h(q) , that is, for all t >0, lim n → ∞ F h(x n ),h(p) (t ) = 1 and lim n → ∞ F h(y n ),h(q) (t )=1 . (30) Now we show that (p, q) is a coupled coincidence point of A and h. Since {h(x n )} is a non-decreasing sequence, from (30) and (6), we have h ( x n ) ≤ h ( p ). (31) Since {h(y n )} is a non-increasing sequence, from (30) and (7), we have h ( q ) ≤ h ( y n ). (32) For all t > 0 and a Î (0, 1), we have F h ( p ) ,A ( p,q ) (kt) ≥ (F h ( p ) ,h ( x n+1 ) (kt − αkt), F h ( x n+1 ) ,A ( p,q ) (kαt) ) and F h ( q ) ,A ( q,p ) (kt) ≥ (F h ( q ) ,h ( y n+1 ) (kt − αkt), F h ( y n+1 ) ,A ( q,p ) (kαt)) . Then min{F h ( p ) ,A ( p,q ) (kt), F h ( q ) ,A ( q,p ) (kt)}≥(A n , min{F h ( x n+1 ) ,A ( p,q ) (kαt), F h ( y n+1 ) ,A ( q,p ) (kαt)}) , (33) where A n = min{F h ( p ) ,h ( x n+1 ) (kt − αkt), F h ( q ) ,h ( y n+1 ) (kt − αkt)} . (34) Now, using (31), (32) and (5), we have F h(x n+1 ),A(p,q) (kαt)=F A(x n ,y n ),A(p,q) (kαt) ≥ min{F h(x n ),h(p) (αt), F h(y n ),h(q) (αt), F h(x n ),h(x n+1 ) (αt) , F h(p),A(p,q) (αt), F h(y n ),h(y n+1 ) (αt), F h(q),A(q,p) (αt)} := B n ( αt ) = B n . (35) Similarly, we get F h ( y n+1 ) ,A ( q,p ) (kαt) ≥ B n . (36) Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 8 of 13 Combining (35) and (36), we obtain min{F h ( x n+1 ) ,A ( p,q ) (kαt), F h ( y n+1 ) ,A ( q,p ) (kαt) }≥B n . (37) Therefore, from (37) and (33), we have min{F h ( p ) ,A ( p,q ) (kt), F h ( q ) ,A ( q,p ) (kt)}≥(A n , B n ) . (38) Now, letting n ® +∞ in (38), using the continuity of the T-norm Δ, (30), (20), (21) and the property Δ(1, a)=a for all a Î [0, 1], we get min{F h ( p ) ,A ( p,q ) (kt), F h ( q ) ,A ( q,p ) (kt)}≥min{F h ( p ) ,A ( p,q ) (αt), F h ( q ) ,A ( q,p ) (αt)} . Now, letting a ® 1 - in the above inequality, using the left-continuity of F and the monotony of F, we get min{F h(p),A(p,q) (t ), F h(q),A(q,p) (t ) }≥min{F h(p),A(p,q) (kt), F h(q),A(q,p) (kt) } ≥ min{F h ( p ) ,A ( p,q ) (t ), F h ( q ) ,A ( q,p ) (t ) }. Hence, for all t > 0, we have min{F h ( p ) ,A ( p,q ) (t ), F h ( q ) ,A ( q,p ) (t ) } = min{F h ( p ) ,A ( p,q ) (kt), F h ( q ) ,A ( q,p ) (kt)} . Now, applying Lemma 6, we get A ( p, q ) = h ( p ) and A ( q, p ) = h ( q ), that is, (p, q) is a coupled coincidence point of A and h. This makes end to the proof. □ The following result is an immediate consequence of Theorem 7. Corollary 8 . Let (X, ≤) be a partially ordered set and (X, F, Δ) be a complete Menger PM-space under a T-norm Δ of H-type. Let A : X × X ® X be mapping satisfying the mixed monotone property, for which there exists k Î (0, 1) such that F A ( x,y ) ,A ( u,v ) (kt) ≥ min{F x,u (t), F y,v (t), F x,A ( x,y ) (t), F u,A ( u,v ) (t), F y,A ( y,x ) (t), F v,A ( v,u ) (t) } for all x, y Î X for which x ≤ u and y ≥ v and all t >0.Suppose also that i f {x n }⊂Xisanon- decreasing sequence with x n → zinXthenx n ≤ z f or all n hold , i f {x n }⊂Xisanon- increasin g se q uence with x n → zinXthenz≤ x n f or all n hold. If there exist x 0 , y 0 Î X such that x 0 ≤ A ( x 0 , y 0 ) and y 0 ≥ A ( y 0 , x 0 ), then A has a coupled fixed point, that is, there exist p, q Î X such that A(p, q)=p and A(q, p)=q. Now, we prove the following result. Theorem 9. Let (X, ≤) be a partiall y ordered set and (X, F , Δ) be a complete Menger PM-space under a T-norm Δ of H-type. Suppose A : X × X ® Xandh: X ® Xare two continuous mappings such that A(X × X) ⊆ h(X), A has the h-mixed monotone property on X and h commutes with A. Suppose that for some k Î (0, 1), F A(x,y),A(u,v) (kt) ≥ min{F h(x),h(u) (t ), F h(y),h(v) (t ), F h(x),A(x,y) (t ) , F h ( u ) ,A ( u,v ) (t ), F h ( y ) ,A ( y,x ) (t ), F h ( v ) ,A ( v,u ) (t ) } Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 9 of 13 for all x, y Î X for which h(x) ≤ h(u) and h(y) ≥ h(v) and all t >0.If there exist x 0 , y 0 Î X such that h ( x 0 ) ≤ A ( x 0 , y 0 ) and h ( y 0 ) ≥ A ( y 0 , x 0 ), then A and h have a coupled coincidence point. Proof. Following the proof of Theorem 7, {h(x n )} and {h(y n )} are Cauchy sequences in the complete Menger PM-space (X, F, Δ). Then, there is some p, q Î X such that lim n → ∞ h(x n )=p and lim n → ∞ h(y n )=q . (39) Since h is continuous, we have lim n → ∞ h(h(x n )) = h(p) and lim n → ∞ h(h(y n )) = h(q) . (40) From (8) and the commutativity of A and h, we have h ( h ( x n+1 )) = h ( A ( x n , y n )) = A ( h ( x n ) , h ( y n )) (41) and h ( h ( y n+1 )) = h ( A ( y n , x n )) = A ( h ( y n ) , h ( x n )). (42) We now show that h(p)=A(p, q) and h(q)=A (q, p). Taking the limit as n ® +∞ in (41) and (42), by (39), (40) and the continuity of A, we get h(p) = lim n →∞ h(h(x n+1 )) = lim n →∞ A(h(x n ), h(y n )) = A( lim n →∞ h(x n ), lim n →∞ h(y n )) = A(p, q ) and h(q) = lim n → ∞ h(h(y n+1 )) = lim n → ∞ A(h(y n ), h(x n )) = A( lim n → ∞ h(y n ), lim n → ∞ h(x n )) = A(q, p) . Thus we proved that h(p)=A(p, q)andh(q)=A(q, p), that is, (p, q)isacoupled coincidence point of A and h. This makes end to the proof. □ The following result is an immediate consequence of Theorem 9. Corol lary 10. Let (X, ≤) be a partially ordered set a nd (X, F, Δ) be a complete Men- ger PM-space under a T-norm Δ of H-type. Let A : X × X ® X be a continuous map- ping having the mixed monontone property, for which there exists k Î (0, 1) such that F A ( x,y ) ,A ( u,v ) (kt) ≥ min{F x,u (t), F y,v (t), F x,A ( x,y ) (t), F u,A ( u,v ) (t), F y,A ( y,x ) (t), F v,A ( v,u ) (t) } for all x, y Î X for which x ≤ u and y ≥ v and all t >0.If there exist x 0 , y 0 Î X such that x 0 ≤ A ( x 0 , y 0 ) and y 0 ≥ A ( y 0 , x 0 ), then A has a coupled fixed point. Now, we end the article with two examples to illustrate our obtained results. Example 11. Let (X, d) be a metric space defined by d(x, y )=|x - y|, where X =[0,1] and ( X, F, Δ) be the induced Menger space with F x,y (t )= t t + d ( x, y ) for all t >0and x, y Î X. We endow X with the natural ordering of real numbers. Let h : X ® X be defined as h ( x ) = x 4 , for all x ∈ X . Let A : X × X ® X be defined as A(x, y)=  x 4 −y 4 4 , if x ≥ y 0, if x < y . Ćirić et al. Fixed Point Theory and Applications 2011, 2011:56 http://www.fixedpointtheoryandapplications.com/content/2011/1/56 Page 10 of 13 [...]... contractions in partially ordered probabilistic metric spaces Topol Appl 156(17), 2838–2844 (2009) doi:10.1016/j.topol.2009.08.029 Ćirić, LB, Cakić, N, Rajović, M, Ume, JS: Monotone generalized nonlinear contractions in partially ordered metric spaces Fixed Point Theory Appl Article ID 131294 (2008) Lakshmikantham, V, Ćirić, LB: Coupled fixed point theorems for nonlinear contractions in partially ordered. .. contraction in partially ordered metric spaces Nonlinear Anal 72(12), 4508–4517 (2010) doi:10.1016/j.na.2010.02.026 Samet, B, Vetro, C: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces Nonlinear Anal 74, 4260–4268 (2011) doi:10.1016/j.na.2011.04.007 Samet, B, Yazidi, H: Coupled Fixed point theorems in partially ordered ε-chainable metric spaces J... D: Generalized contractions in partially ordered metric spaces Appl Anal 87, 1–8 (2008) doi:10.1080/00036810701714164 3 Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations Proc Am Math Soc 132, 1435–1443 (2004) doi:10.1090/S0002-9939-03-07220-4 4 Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces... doi:10.1016/j.aml.2007.09.006 Nashine, HK, Samet, B: Fixed point results for mappings satisfying (ψ, φ) weakly contractive condition in partially ordered metric spaces Nonlinear Anal 74, 2201–2209 (2011) doi:10.1016/j.na.2010.11.024 Samet, B: Ćirić’s fixed point theorem in a cone metric space J Nonlinear Sci Appl 3(4), 302–308 (2010) Singh, SL, Mishra, SN: On a Ljubomir Ćirić’s fixed point theorem for nonexpansive... in probabilistic metric spaces (II) Int J Math Math Sci 5, 729–736 (2005) Miheţ, D: Fixed point theorems in probabilistic metric spaces Chaos Solitons Fractals 41(2), 1014–1019 (2009) doi:10.1016/j.chaos.2008.04.030 Miheţ, D: Altering distances in probabilistic Menger spaces Nonlinear Anal 71, 2734–2738 (2009) doi:10.1016/j na.2009.01.107 O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic. .. and anticipation Nonlinear Anal 66(10), 2237–2242 (2007) doi:10.1016/j.na.2006.03.013 Nieto, JJ, López, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations Order 22, 223–239 (2005) doi:10.1007/s11083-005-9018-5 Ćirić, LB, Lakshmikantham, V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces Stoch... point theorems under strict contractive conditions in Menger spaces Nonlinear Anal 70, 184–193 (2009) doi:10.1016/j.na.2007.11.045 Hadžić, O, Pap, E: Fixed Point Theory in PM Spaces Kluwer, Boston (2001) Khamsi, MA, Kreinovich, VY: Fixed point theorems for dissipative mappings in complete probabilistic metric spaces Math Jap 44, 513–520 (1996) Miheţ, D: A generalization of a contraction principle in. .. ordered metric spaces Nonlinear Anal 70(12), 4341–4349 (2009) doi:10.1016/j.na.2008.09.020 Nieto, JJ, López, RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations Acta Math Sinica, English Series 23, 2205–2212 (2007) doi:10.1007/s10114-005-0769-0 Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially. .. Mishra, SN: On a Ljubomir Ćirić’s fixed point theorem for nonexpansive type maps with applications Indian J Pure Appl Math 33, 531–542 (2002) doi:10.1186/1687-1812-2011-56 Cite this article as: Ćirić et al.: Mixed monotone-generalized contractions in partially ordered probabilistic metric spaces Fixed Point Theory and Applications 2011 2011:56 Page 13 of 13 ... doi:10.1007/BF01706080 Ćirić, LB: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces Chaos Solitons Fractals 42, 146–154 (2009) doi:10.1016/j.chaos.2008.11.010 Ćirić, LB, Ješić, SN, Ume, JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces Chaos Solitons Fractals 37(3), 781–791 (2008) doi:10.1016/j.chaos.2006.09.093 . for mixed monotone and contraction mappings in partially ordered metric spaces. The main idea in [2-11] involve combining the ideas of iterative techni- que in the contraction mapping principle. those in the monotone technique. In [3], Ran and Reurings proved the following Banach type principle in partially ordered metric spaces. Theorem 1 (Ran and Reurings [3]). Let (X, ≤) be a partially. mappings in partially ordered metric spaces. Nonlinear Anal. 74, 4260–4268 (2011). doi:10.1016/j.na.2011.04.007 14. Samet, B, Yazidi, H: Coupled Fixed point theorems in partially ordered ε-chainable