Rao et al. Fixed Point Theory and Applications (2017) 2017:17 DOI 10.1186/s13663-017-0610-3

RESEARCH

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Applications and common coupled fixed point results in ordered partial metric spaces KPR Rao1 , GNV Kishore2 , Kenan Tas3 , S Satyanaraya4* and D Ram Prasad5 *

Correspondence: [email protected] Department of Computing, Adama Science and Technology University, Adama, Ethiopia Full list of author information is available at the end of the article 4

Abstract In this paper, we obtain a unique common coupled fixed point theorem by using (ψ , α , β )-contraction in ordered partial metric spaces. We give an application to integral equations as well as homotopy theory. Also we furnish an example which supports our theorem. Keywords: partial metric; w-compatible maps; coupled fixed point; mixed g-monotone property; ψ -α -β contraction; homotopy theory

1 Introduction The notion of a partial metric space (PMS) was introduced by Matthews [] as a part of the study of denotational semantics of data flow networks. In fact, it is widely recognized that PMSs play an important role in constructing models in the theory of computation and domain theory in computer science (see e.g. [–]). Matthews [, ], Oltra and Valero [] and Altun et al. [] proved some fixed point theorems in PMSs for a single map. For more work on fixed, common fixed point theorems in PMSs, we refer to [, –]. The notion of a coupled fixed point was introduced by Bhaskar and Lakshmikantham [] and they studied some fixed point theorems in partially ordered metric spaces. Later some authors proved coupled fixed and coupled common fixed point theorems (see [, –]). The aim of this paper is to study unique common coupled fixed point theorems of Jungck type maps by using a (ψ, α, β)-contraction condition over partially ordered PMSs. 2 Preliminaries First we recall some basic definitions and lemmas which play a crucial role in the theory of PMSs. Definition . (See [, ]) A partial metric on a non-empty set X is a function p : X ×X → R+ such that, for all x, y, z ∈ X, (p ) x = y ⇔ p(x, x) = p(x, y) = p(y, y), (p ) p(x, x) ≤ p(x, y), p(y, y) ≤ p(x, y), © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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(p ) p(x, y) = p(y, x), (p ) p(x, y) ≤ p(x, z) + p(z, y) – p(z, z). The pair (X, p) is called a PMS. If p is a partial metric on X, then the function dp : X × X → R+ , given by dp (x, y) = p(x, y) – p(x, x) – p(y, y),

()

is a metric on X. Example . (See e.g. [, , ]) Consider X = [, ∞) with p(x, y) = max{x, y}. Then (X, p) is a PMS. It is clear that p is not a (usual) metric. Note that in this case dp (x, y) = |x–y|. Example . (See []) Let X = {[a, b] : a, b ∈ R, a ≤ b} and define p([a, b], [c, d]) = max{b, d} – min{a, c}. Then (X, p) is a PMS. Each partial metric p on X generates a T topology τp on X which has as a base the family of open p-balls {Bp (x, ε), x ∈ X, ε > }, where Bp (x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > . We now state some basic topological notions (such as convergence, completeness, continuity) on PMSs (see e.g. [, , , , , ]). Definition . . A sequence {xn } in the PMS (X, p) converges to the limit x if and only if p(x, x) = lim p(x, xn ). n→∞ . A sequence {xn } in the PMS (X, p) is called a Cauchy sequence if lim p(xn , xm ) n,m→∞

exists and is finite. . A PMS (X, p) is called complete if every Cauchy sequence {xn } in X converges with respect to τp , to a point x ∈ X such that p(x, x) = lim p(xn , xm ). n,m→∞

. A mapping F : X → X is said to be continuous at x ∈ X if, for every  > , there exists δ >  such that F(Bp (x , δ)) ⊆ Bp (Fx , ). The following lemma is one of the basic results as regards PMS [, , , , , ]. Lemma . . A sequence {xn } is a Cauchy sequence in the PMS (X, p) if and only if it is a Cauchy sequence in the metric space (X, dp ). . A PMS (X, p) is complete if and only if the metric space (X, dp ) is complete. Moreover, lim dp (x, xn ) = 

n→∞

⇔

p(x, x) = lim p(x, xn ) = lim p(xn , xm ). n→∞

n,m→∞

()

Next, we give two simple lemmas which will be used in the proofs of our main results. For the proofs we refer []. Lemma . Assume xn → z as n → ∞ in a PMS (X, p) such that p(z, z) = . Then lim p(xn , y) = p(z, y) for every y ∈ X. n→∞

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Lemma . Let (X, p) be a PMS. Then (A) if p(x, y) = , then x = y, (B) if x = y, then p(x, y) > . Remark . If x = y, p(x, y) may not be . Definition . ([]) Let (X, ) be a partially ordered set and F : X × X → X. Then the map F is said to have mixed monotone property if F(x, y) is monotone non-decreasing in x and monotone non-increasing in y; that is, for any x, y ∈ X, x x 

implies

F(x , y) F(x , y)

for all y ∈ X

y y

implies

F(x, y ) F(x, y ) for all x ∈ X.

and

Definition . ([]) An element (x, y) ∈ X × X is called a coupled fixed point of a mapping F : X × X → X if F(x, y) = x and F(y, x) = y. Definition . ([]) An element (x, y) ∈ X × X is called (g ) a coupled coincident point of mappings F : X × X → X and f : X → X if fx = F(x, y) and fy = F(y, x), (g ) a common coupled fixed point of mappings F : X × X → X and f : X → X if x = fx = F(x, y) and y = fy = F(y, x). Definition . ([]) The mappings F : X × X → X and f : X → X are called wcompatible if f (F(x, y)) = F(fx, fy) and f (F(y, x)) = F(fy, fx) whenever fx = F(x, y) and fy = F(y, x). Inspired by Definition ., Lakshmikantham and Ćirić in [] introduced the concept of a g-mixed monotone mapping. Definition . ([]) Let (X, ) be a partially ordered set, F : X × X → X and g : X → X be mappings. Then the map F is said to have a mixed g-monotone property if F(x, y) is monotone g-non-decreasing in x as well as monotone g-non-increasing in y; that is, for any x, y ∈ X, gx gx

implies

F(x , y) F(x , y) for all y ∈ X

gy gy

implies F(x, y ) F(x, y ) for all x ∈ X.

and

Now we prove our main results.

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3 Results and discussions Definition . Let (X, p) be a PMS, let F : X × X → X and g : X → X be mappings. We say that F satisfies a (ψ, α, β)-contraction with respect to g if there exist ψ, α, β : [, ∞) → [, ∞) satisfying the following: (..) ψ is continuous and monotonically non-decreasing, α is continuous and β is lower semi continuous, (..) ψ(t) =  if and only if t = , α() = β() = , (..) ψ(t) – α(t) + β(t) >  for t > , (..) ψ(p(F(x, y), F(u, v))) ≤ α(M(x, y, u, v)) – β(M(x, y, u, v)), ∀x, y, u, v ∈ X, gx gu, gy  gv and M(x, y, u, v)   p(gx, gu), p(gy, gv), p(gx, F(x, y)), p(gy, F(y, x)), p(gu, F(u, v)), p(gv, F(v, u)), . = max p(gx,F(x,y))p(gy,F(y,x)) p(gu,F(u,v))p(gv,F(v,u)) , +p(gx,gu)+p(gy,gv)+p(F(x,y),F(u,v)) +p(gx,gu)+p(gy,gv)+p(F(x,y),F(u,v)) Theorem . Let (X, ) be a partially ordered set and p be a partial metric such that (X, p) is a PMS. Let F : X × X → X and g : X → X be such that (..) F satisfies a (ψ, α, β)-contraction with respect to g, (..) F(X × X) ⊆ g(X) and g(X) is a complete subspace of X, (..) F has a mixed g-monotone property, (..) (a) if a non-decreasing sequence {xn } → x, then xn x for all n, (b) if a non-increasing sequence {yn } → y, then y yn for all n. If there exist x , y ∈ X such that gx F(x , y ) and gy  F(y , x ), then F and g have a coupled coincidence point in X × X. Proof Let x , y ∈ X be such that gx F(x , y ) and gy  F(y , x ). Since F(X × X) ⊆ g(X), we choose x , y ∈ X such that gx F(x , y ) = gx

and

gy  F(y , x ) = gy

and choose x , y ∈ X such that gx = F(x , y )

and gy = F(y , x ).

Since F has the mixed g-monotone property, we obtain gx gx gx

and

gy  gy  gy .

Continuing this process, we construct the sequences {xn } and {yn } in X such that gxn+ = F(xn , yn ) and

gyn+ = F(yn , xn ),

n = , , , . . .

with gx gx gx · · · and gy  gy  gy  · · · .

 (I)

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Case (a): If gxm = gxm+ and gym = gym+ for some m, then (xm , ym ) is a coupled coincidence point in X × X. Case (b): Assume gxn = gxn+ or gyn = gyn+ for all n. Since gxn gxn+ and gyn  gyn+ , from (..), we obtain      ψ p(gxn , gxn+ ) = ψ p F(xn– , yn– ), F(xn , yn )     ≤ α M(xn– , yn– , xn , yn ) – β M(xn– , yn– , xn , yn ) , ⎧ ⎫ ⎪ p(gxn– , gxn ), p(gyn– , gyn ), p(gxn– , gxn ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p(gyn– , gyn ), p(gxn , gxn+ ), p(gyn , gyn+ ), ⎪ ⎬ . M(xn– , yn– , xn , yn ) = max p(gxn– ,gxn )p(gyn– ,gyn ) ⎪ ⎪ , ⎪ ⎪ +p(gxn– ,gxn )+p(gyn– ,gyn )+p(gxn ,gxn+ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p(gxn ,gxn+ )p(gyn ,gyn+ ) ⎩ ⎭ +p(gxn– ,gxn )+p(gyn– ,gyn )+p(gxn ,gxn+ )

But

 p(gxn– , gxn )p(gyn– , gyn ) ≤ max p(gxn– , gxn ), p(gxn , gxn+ )  + p(gxn– , gxn ) + p(gyn– , gyn ) + p(gxn , gxn+ ) and p(gxn , gxn+ )p(gyn , gyn+ ) ≤ p(gyn , gyn+ ).  + p(gxn– , gxn ) + p(gyn– , gyn ) + p(gxn , gxn+ ) Therefore 

 p(gxn– , gxn ), p(gyn– , gyn ), . M(xn– , yn– , xn , yn ) = max p(gxn , gxn+ ), p(gyn , gyn+ ) Hence      p(gxn– , gxn ), p(gyn– , gyn ), ψ p(gxn , gxn+ ) ≤ α max p(gxn , gxn+ ), p(gyn , gyn+ )    p(gxn– , gxn ), p(gyn– , gyn ), . – β max p(gxn , gxn+ ), p(gyn , gyn+ ) Similarly      p(gxn– , gxn ), p(gyn– , gyn ), ψ p(gyn , gyn+ ) ≤ α max p(gxn , gxn+ ), p(gyn , gyn+ )    p(gxn– , gxn ), p(gyn– , gyn ), . – β max p(gxn , gxn+ ), p(gyn , gyn+ ) Put Rn = max{p(gxn , gxn+ ), p(gyn , gyn+ )}. Let us suppose that Rn =  for all n ≥ . Let, if possible, for some n, Rn– < Rn .

()

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Now 

 ψ(Rn ) = ψ max p(gxn , gxn+ ), p(gyn , gyn+ )   

 = max ψ p(gxn , gxn+ ) , ψ p(gyn , gyn+ )    p(gxn– , gxn ), p(gyn– , gyn ), ≤ α max p(gxn , gxn+ ), p(gyn , gyn+ )    p(gxn– , gxn ), p(gyn– , gyn ), – β max p(gxn , gxn+ ), p(gyn , gyn+ )     = α max{Rn– , Rn } – β max{Rn– , Rn } = α(Rn ) – β(Rn ). From (..) and (..), it follows that Rn = , a contradiction. Hence Rn ≤ Rn– .

()

Thus {Rn } is a non-increasing sequence of non-negative real numbers and must converge to a real number r ≥ . Also ψ(Rn ) ≤ α(Rn– ) – β(Rn– ). Letting n → ∞, we get ψ(r) ≤ α(r) – β(r). From (..) and (..), we get r = . Thus

 lim max p(gxn , gxn+ ), p(gyn , gyn+ ) = ,

n→∞

lim p(gxn , gxn+ ) =  = lim p(gyn , gyn+ ).

n→∞

n→∞

()

Hence from (p ), we have lim p(gxn , gxn ) =  = lim p(gyn , gyn ).

n→∞

n→∞

()

From () and () and by the definition of dp , we get lim dp (gxn , gxn+ ) =  = lim dp (gyn , gyn+ ).

n→∞

n→∞

Now we prove that {gxn } and {gyn } are Cauchy sequences. To the contrary, suppose that {gxn } or {gyn } is not Cauchy. This implies that dp (gxm , gxn ) →  or dp (gym , gyn ) →  as n, m → ∞.

()

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Consequently 

max dp (gxm , gxn ), dp (gym , gyn ) →  as n, m → ∞. Then there exist an  >  and monotone increasing sequences of natural numbers {mk } and {nk } such that nk > mk > k. We have

 max dp (gxmk , gxnk ), dp (gymk , gynk ) ≥ 

()



max dp (gxmk , gxnk – ), dp (gym , gynk – ) < .

()

and

From () and (), we have 

 ≤ max dp (gxmk , gxnk ), dp (gymk , gynk )

 ≤ max dp (gxmk , gxnk – ), dp (gymk , gynk – ) 

+ max dp (gxnk – , gxnk ), dp (gynk – , gynk )

 <  + max dp (gxnk – , gxnk ), dp (gynk – , gynk ) . Letting k → ∞ and using (), we get

 lim max dp (gxmk , gxnk ), dp (gymk , gynk ) = .

k→∞

()

By the definition of dp and using () we get

  lim max p(gxmk , gxnk ), p(gymk , gynk ) = . 

k→∞

()

From (), we have

  ≤ max dp (gxmk , gxnk ), dp (gymk , gynk )

 ≤ max dp (gxmk , gxmk – ), dp (gymk , gymk – )

 + max dp (gxmk – , gxnk ), dp (gymk – , gynk )

 ≤  max dp (gxmk , gxmk – ), dp (gymk , gymk – )

 + max dp (gxmk , gxnk ), dp (gymk , gynk ) .

()

Letting k → ∞, using (), () and (), we get

 lim max dp (gxmk – , gxnk ), dp (gymk – , gynk ) = .

k→∞

()

Hence, we get

  lim max p(gxmk – , gxnk ), p(gymk – , gynk ) = . 

k→∞

()

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From (), we have

  ≤ max dp (gxmk , gxnk ), dp (gymk , gynk ) 

≤ max dp (gxmk , gxmk – ), dp (gymk , gymk – ) 

+ max dp (gxmk – , gxnk + ), dp (gymk – , gynk + ) 

+ max dp (gxnk + , gxnk ), dp (gynk + , gynk ) 

≤  max dp (gxmk , gxmk – ), dp (gymk , gymk – ) 

+ max dp (gxmk , gxnk ), dp (gymk , gynk ) 

+ max dp (gxnk , gxnk + ), dp (gynk , gynk + ) .

()

Letting k → ∞, using (), () and (), we get

 lim max dp (gxmk – , gxnk + ), dp (gymk – , gynk + ) = .

()

k→∞

Hence, we have

  lim max p(gxmk – , gxnk + ), p(gymk – , gynk + ) = . k→∞ 

()

Now from (), we have 

 ≤ max dp (gxmk , gxnk ), dp (gymk , gynk ) 

≤ max dp (gxmk , gxnk + ), dp (gymk , gynk + ) 

+ max dp (gxnk + , gxnk ), dp (gynk + , gynk ) . Letting k → ∞ and using (), we obtain

  ≤ lim max dp (gxmk , gxnk + ), dp (gymk , gynk + ) +  k→∞



p(gxmk , gxnk + ) – p(gxmk , gxmk ) – p(gxnk + , gxnk + ), ≤ lim max k→∞ p(gymk , gynk + ) – p(gymk , gymk ) – p(gynk + , gynk + )

 =  lim max p(gxmk , gxnk + ), p(gymk , gynk + ) , from ().



k→∞

Thus,

  ≤ lim max p(gxmk , gxnk + ), p(gymk , gynk + ) . k→∞  By the properties of ψ,   

  ≤ lim ψ max p(gxmk , gxnk + ), p(gymk , gynk + ) ψ k→∞ 

    = lim max ψ p(gxmk , gxnk + ) , ψ p(gymk , gynk + ) . k→∞

()

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Now      ψ p(gxmk , gxnk + ) = ψ p F(xmk – , ymk – ), F(xnk , ynk )     ≤ α M(xmk – , ymk – , xnk , ynk ) – β M(xmk – , ymk – , xnk , ynk ) ⎫⎞ ⎧ ⎛ ⎪ p(gxmk – , gxnk ), p(gymk – , gynk ), p(gxmk – , gxmk ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎟ ⎪ ⎪ ⎪ ⎜ ⎪⎟ ⎪ ⎪ p(gy , gy ), p(gx , gx ), p(gy , gy ), mk – mk nk nk + nk nk + ⎪ ⎜ ⎬⎟ ⎨ ⎜ ⎟ = α ⎜max ⎟ p(gxmk – ,gxmk )p(gymk – ,gymk ) ⎜ ⎪ ⎟ ⎪ , ⎪ ⎪ +p(gxmk – ,gxnk ),p(gymk – ,gynk )+p(gxmk ,gxnk + ) ⎜ ⎪ ⎪ ⎪⎟ ⎪ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ p(gxnk ,gxnk + )p(gynk ,gynk + ) ⎪ ⎪ ⎭ ⎩ +p(gxmk – ,gxnk ),p(gymk – ,gynk )+p(gxmk ,gxnk + )

⎫⎞ ⎧ ⎪ p(gxmk – , gxnk ), p(gymk – , gynk ), p(gxmk – , gxmk ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ⎪⎟ ⎜ ⎪ ⎪ p(gy , gy ), p(gx , gx ), p(gy , gy ), mk – mk nk nk + nk nk + ⎪ ⎬⎟ ⎜ ⎨ ⎜ ⎟ – β ⎜max ⎟. p(gxmk – ,gxmk )p(gymk – ,gymk ) ⎪ ⎜ ⎟ ⎪ , ⎪ ⎪ +p(gxmk – ,gxnk ),p(gymk – ,gynk )+p(gxmk ,gxnk + ) ⎪ ⎜ ⎪ ⎪⎟ ⎪ ⎪ ⎪ ⎝ ⎠ ⎪ ⎪ p(gxnk ,gxnk + )p(gynk ,gynk + ) ⎪ ⎪ ⎭ ⎩ ⎛

+p(gxmk – ,gxnk ),p(gymk – ,gynk )+p(gxmk ,gxnk + )

Letting k → ∞, we have         –β . lim ψ p(gxmk , gxnk + ) ≤ α k→∞   Similarly, we obtain         lim ψ p(gymk , gynk + ) ≤ α –β . k→∞   Hence from (), we have          ψ ≤α –β .    From (..) and (..), we get  = , a contradiction. Hence {gxn } and {gyn } are Cauchy sequences in the metric space (X, dp ). Hence we have lim dp (gxn , gxm ) =  = lim dp (gyn , gym ). n,m→∞

n,m→∞

Now from the definition of dp and from (), we have lim p(gxn , gxm ) =  = lim p(gyn , gym ).

n→∞

n→∞

()

Suppose g(X) is a complete subspace of X. Since {gxn } and {gyn } are Cauchy sequences in a complete metric space (g(X), dp ). Then {gxn } and {gyn } converges to some u and v in g(X) respectively. Thus lim dp (gxn , u) = 

n→∞

and lim dp (gyn , v) = 

n→∞

for some u and v in g(X).

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Page 10 of 20

Since u, v ∈ g(X), there exist x, y ∈ X such that u = gx and v = gy. Since {gxn } and {gyn } are Cauchy sequences, gxn → u, gyn → v, gxn+ → u and gyn+ → v. From Lemma .() and (), we obtain p(u, u) = lim p(gxn , u) = p(v, v) = lim p(gyn , v) = . n→∞

()

n→∞

Now we prove that lim p(F(x, y), gxn ) = p(F(x, y), u). n→∞ By definition of dp ,       dp F(x, y), gxn = p F(x, y), gxn – p F(x, y), F(x, y) – p(gxn , gxn ). Letting n → ∞, we have       dp F(x, y), u =  lim p F(x, y), gxn – p F(x, y), F(x, y) – , from (). n→∞

By definition of dp and (), we have     lim p F(x, y), gxn = p F(x, y), u .

n→∞

Similarly, lim p(F(y, x), gyn ) = p(F(y, x), v). n→∞ From (p ), we have     p u, F(x, y) ≤ p(u, gxn+ ) + p gxn+ , F(x, y) – p(gxn+ , gxn+ )   = p(u, gxn+ ) + p gxn+ , F(x, y) . Letting n → ∞, we have     p u, F(x, y) ≤  + lim p F(xn , yn ), F(x, y) . n→∞

Also from (..), we get gxn gx and gyn  gy. Since ψ is a continuous and nondecreasing function, we get       ψ p u, F(x, y) ≤ lim ψ p F(xn , yn ), F(x, y) n→∞

     ≤ lim α M(xn , yn , x, y) – β M(xn , yn , x, y) , n→∞

⎧ ⎫ ⎪ p(gxn , u), p(gyn , v), p(gxn , gxn+ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p(gyn , gyn+ ), p(u, F(x, y)), p(v, F(y, x)), ⎪ ⎬

M(xn , yn , x, y) = max

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

p(gxn ,gxn+ )p(gyn ,gyn+ ) , +p(gxn ,u)+p(gyn ,v)+p(gxn+ ,F(x,y)) p(u,F(x,y))p(v,F(y,x)) +p(gxn ,u)+p(gyn ,v)+p(gxn+ ,F(x,y))

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

    → max p u, F(x, y) , p v, F(y, x) as n → ∞. Therefore          p(u, F(x, y)), p(u, F(x, y)), ψ p u, F(x, y) ≤ α max – β max . p(v, F(y, x)) p(v, F(y, x))

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Page 11 of 20

Similarly,          p(u, F(x, y)), p(u, F(x, y)), ψ p v, F(y, x) ≤ α max – β max . p(v, F(y, x)) p(v, F(y, x)) Hence 

    ψ max p u, F(x, y) , p v, F(y, x)    

  = max ψ p u, F(x, y) , ψ p v, F(y, x)       p(u, F(x, y)), p(u, F(x, y)), ≤ α max – β max . p(v, F(y, x)) p(v, F(y, x)) It follows that max{p(u, F(x, y)), p(v, F(y, x))} = . So F(x, y) = u and F(y, x) = v. Hence F(x, y) = gx = u and F(y, x) = gy = v. Hence F and g have a coincidence point in X × X.



Theorem . In addition to the hypothesis of Theorem ., we suppose that for every (x, y), (x , y ) ∈ X × X there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable to (F(x, y), F(y, x)) and (F(x , y ), F(y , x )). If (x, y) and (x , y ) are coupled coincidence points of F and g, then   F(x, y) = gx = gx = F x , y and   T(y, x) = gy = gy = F y , x . Moreover, if (F, g) is w-compatible, then F and g have a unique common coupled fixed point in X × X. Proof The proof follows from Theorem . and the definition of comparability.



Theorem . Let (X, ) be a partially ordered set and p be a partial metric such that (X, p) is a complete PMS. Let F : X × X → X be such that    

 

 ψ p F(x, y), F(u, v) ≤ α max p(x, u), p(y, v) – β max p(x, u), p(y, v) , (..) ∀x, y, u, v ∈ X, x u and y  v, where ψ, α and β are defined in Definition . and (..) (a) If a non-decreasing sequence {xn } → x, then xn x for all n, and (b) if a non-increasing sequence {yn } → y, then y yn for all n. If there exist x , y ∈ X such that x F(x , y ) and y  F(y , x ), then F has a unique coupled fixed point in X × X. Example . Let X = [, ], let be partially ordered on X by x y

⇔

x ≥ y. 



x +y and p : X × X → [, ∞) by p(x, y) = The mapping F : X × X → X defined by F(x, y) = (x+y+) max{x, y} is a complete partial metric on X. Define ψ, α, β : [, ∞) → [, ∞) by ψ(t) = t,

Rao et al. Fixed Point Theory and Applications (2017) 2017:17

t 

α(t) =

Page 12 of 20

and β(t) = t . We have

      u + v x +y , p F(x, y), F(u, v) = max (x + y + ) (u + v + )      u v  x y max , + max , =  x+y+ u+v+ x+y+ u+v+        u v  x y max , + max , ≤  x+ u+ y+ v+      u v  x y max , + max , ≤  x+ u+ y+ v+    ≤ max{x, u} + max{y, v}    = p(x, u) + p(y, v) 

  ≤ max p(x, u), p(y, v)  

 

 = α max p(x, u), p(y, v) – β max p(x, u), p(y, v) . Hence all conditions of Theorem . hold. From Theorem ., (, ) is a unique coupled fixed point of F in X × X.

3.1 Application to integral equations In this section, we study the existence of a unique solution to an initial value problem, as an application to Theorem .. Consider the initial value problem   x (t) = f t, x(t), x(t) ,

t ∈ I = [, ],

x() = x ,

()

where f : I × [ x , ∞) × [ x , ∞) → [ x , ∞) and x ∈ R. Theorem . Consider the initial value problem () with f ∈ C(I × [ x , ∞) × [ x , ∞)) and 

t





f s, x(s), y(s) ds ≤ max





 t f (s, x(s), x(s)) ds –    t f (s, y(s), y(s)) ds –  

x ,  x 

 .

Then there exists a unique solution in C(I, [ x , ∞)) for the initial value problem (). Proof The integral equation corresponding to initial value problem () is 

t

x(t) = x + 

  f s, x(s), x(s) ds.

()

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Let X = C(I, [ x , ∞)) and p(x, y) = max{x – x , y – x } for x, y ∈ X. Define ψ, α, β : [, ∞) → [, ∞) by ψ(t) = t, α(t) =  t and β(t) =  t. Define F : X × X → X by  F(x, y)(t) = x +

t

  f s, x(s), y(s) ds.



Now   p F(x, y)(t), F(u, v)(t)   x x = max F(x, y) – , F(u, v) –      t  t     x x = max + + f s, x(s), y(s) ds, f s, u(s), v(s) ds      ⎫  t ⎧ x ⎪ ⎪  f (s, x(s), x(s)) ds –  ,  x ⎪  ⎪ ⎪  + max ,⎪ ⎪ ⎪ t ⎪ ⎪ x   ⎬ ⎨ f (s, y(s), y(s)) ds –    ≤ max   ⎪  t ⎪ ⎪ f (s, u(s), u(s)) ds – x , ⎪ ⎪ ⎪   x ⎪ ⎪ ⎪ ⎪ + max ⎭ ⎩  x  t f (s, v(s), v(s)) ds –    " ⎫ ⎧ ! ⎨ max x(t) – x , y(t) – x , ⎬    ! " = max ⎩ max u(t) – x , v(t) – x ⎭          x x x x  , max y(t) – , v(t) – = max max x(t) – , u(t) –     

  = max p(x, u), p(y, v)  

 

 = α max p(x, u), p(y, v) – β max p(x, u), p(y, v) . Thus F satisfies the condition (..) of Theorem .. From Theorem ., we conclude that F has a unique coupled fixed point (x, y) with x = y. In particular x(t) is the unique solution of the integral equation (). 

3.2 Application to homotopy In this section, we study the existence of a unique solution to homotopy theory. Theorem . Let (X, p) be a complete PMS, U be an open subset of X and U be a closed subset of X such that U ⊆ U. Suppose H : U × U × [, ] → X is an operator such that the following conditions are satisfied: (i) x = H(x, y, λ) and y = H(y, x, λ) for each x, y ∈ ∂U and λ ∈ [, ] (here ∂U denotes the boundary of U in X), (ii) ψ(p(H(x, y, λ), H(u, v, λ))) ≤ α(max{p(x, y), p(u, v)}) – β(max{p(x, y), p(u, v)}) ∀x, y ∈ U and λ ∈ [, ], where ψ, α : [, ∞) → [, ∞) is continuous and non-decreasing and β : [, ∞) → [, ∞) is lower semi continuous with ψ(t) – α(t) + β(t) >  for t > , (iii) there exists M ≥  such that   p H(x, y, λ), H(x, y, μ) ≤ M|λ – μ|

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Page 14 of 20

for every x ∈ U and λ, μ ∈ [, ]. Then H(·, ) has a coupled fixed point if and only if H(·, ) has a coupled fixed point. Proof Consider the set

 A = λ ∈ [, ] : (x, y) = H(x, y, λ) for some x, y ∈ U . Since H(·, ) has a coupled fixed point in U, we have  ∈ A, so that A is a non-empty set. We will show that A is both open and closed in [, ] so by the connectedness we have A = [, ]. As a result, H(·, ) has a fixed point in U. First we show that A is closed in [, ]. To see this let {λn }∞ n= ⊆ A with λn → λ ∈ [, ] as n → ∞. We must show that λ ∈ A. Since λn ∈ A for n = , , , . . . , there exist xn , yn ∈ U with un = (xn , yn ) = H(xn , yn λn ). Consider   p(xn , xn+ ) = p H(xn , yn , λn ), H(xn+ , yn+ , λn+ )   ≤ p H(xn , yn , λn ), H(xn+ , yn+ λn )   + p H(xn+ , yn+ , λn ), H(xn+ , yn+ , λn+ )   – p H(xn+ , yn+ , λn ), H(xn+ , yn+ , λn )   ≤ p H(xn , yn , λn ), H(xn+ , yn+ , λn ) + M|λn – λn+ |. Letting n → ∞, we get   lim p(xn , xn+ ) ≤ lim p H(xn , yn , λn ), H(xn+ , yn+ , λn ) + .

n→∞

n→∞

Since ψ is continuous and non-decreasing we obtain   lim ψ p(xn , xn+ )

n→∞

   ≤ lim ψ p H(xn , yn , λn ), H(xn+ , yn+ , λn ) n→∞

 

 

 ≤ lim α max p(xn , xn+ ), p(yn , yn+ ) – β max p(xn , xn+ ), p(yn , yn+ ) . n→∞

Similarly   lim ψ p(yn , yn+ )

n→∞

 

 

 ≤ lim α max p(xn , xn+ ), p(yn , yn+ ) – β max p(xn , xn+ ), p(yn , yn+ ) . n→∞

It follows that lim p(xn , xn+ ) =  = lim p(yn , yn+ ).

n→∞

n→∞

()

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Page 15 of 20

From (p ), lim p(xn , xn ) =  = lim p(yn , yn ).

n→∞

n→∞

()

By the definition of dp , we obtain lim dp (xn , xn+ ) =  = lim dp (yn , yn+ ).

n→∞

n→∞

()

Now we prove that {xn } and {yn } are Cauchy sequences in (X, dp ). Contrary to this hypothesis, suppose that {xn } or {sn } is not Cauchy. There exists an  >  and a monotone increasing sequence of natural numbers {mk } and {nk } such that nk > mk ,

 max dp (xmk , xnk ), dp (ymk , ynk ) ≥ 

()

 max dp (xmk , xnk – ), dp (ymk , ynk – ) < .

()

and

From () and (), we obtain

  ≤ max dp (xmk , xnk ), dp (ymk , ynk ) 

≤ max dp (xmk , xnk – ), dp (ymk , ynk – ) 

+ max dp (xnk – , xnk ), dp (ynk – , ynk ) 

<  + max dp (xnk – , xnk ), dp (ynk – , ynk ) . Letting k → ∞ and then using (), we get

 lim max dp (xmk , xnk ), dp (ymk , ynk ) = .

k→∞

()

Hence from the definition of dp and from (), we get

  lim max p(xmk , xnk ), p(ymk , ynk ) = . k→∞ 

()

Letting k → ∞ and then using () and () in # # #dp (xm , xn + ) – dp (xm , xn )# ≤ dp (xn + , xn ), k k k k k k we get lim dp (xnk + , xmk ) = .

k→∞

()

Hence, we have  lim p(xnk + , xmk ) = . 

k→∞

()

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Page 16 of 20

Similarly  lim p(ynk + , ymk ) = . 

()

k→∞

Consider   p(xmk , xnk + ) = p H(xmk , ymk , λmk ), H(xnk + , ynk + , λnk + )   ≤ p H(xmk , ymk , λmk ), H(xmk , ymk , λnk + )   + p H(xmk , ymk , λnk + ), H(xnk + , ynk + , λnk + )   – p H(xmk , ymk , λnk + ), H(xmk , ymk , λnk + )   ≤ M|λmk – λnk + | + p H(xmk , ymk , λnk + ), H(xnk + , ynk + , λnk + ) . Since {λn } is Cauchy, letting k → ∞ in the above, we get    ≤ lim p H(xmk , ymk , λnk + ), H(xnk + , ynk + , λnk + ) .  k→∞ Since ψ is continuous and non-decreasing we obtain ψ

      ≤ lim ψ p H(xmk , ymk , λnk + ), H(xnk + , ynk + , λnk + ) k→∞   

 ≤ lim α max p(xmk , xnk + ), p(ymk , ynk + ) k→∞

 

– β max p(xmk , xnk + ), p(ymk , ynk + )       =α –β .   It follows that  ≤ , which is a contradiction. Hence {xn } and {yn } are Cauchy sequences in (X, dp ) and lim dp (xn , xm ) =  = lim dp (yn , ym ).

n,m→∞

n,m→∞

By the definition of dp and (), we get lim p(xn , xm ) =  = lim p(yn , ym ). n,m→∞

n,m→∞

From Lemma ., we conclude (a) {xn } and {yn } are Cauchy sequences in (X, p). Since (X, p) is complete, from Lemma .(b), we conclude there exist u, v ∈ U with p(u, u) = lim p(xn , u) = lim p(xn+ , u) = lim p(xn , xm ) = ,

()

p(v, v) = lim p(xn , v) = lim p(xn+ , v) = lim p(yn , ym ) = .

()

n→∞

n→∞

n→∞

n→∞

n,m→∞

n,m→∞

From Lemma ., we get lim p(xn , H(u, v, λ)) = p(u, H(u, v, λ)). n→∞ Now,     p xn , H(u, v, λ) = p H(xn , yn , λn ), H(u, v, λ)     ≤ p H(xn , yn , λn ), H(xn , yn , λ) + p H(xn , yn , λ), H(u, v, λ)

Rao et al. Fixed Point Theory and Applications (2017) 2017:17

  – p H(xn , yn , λ), H(xn , yn , λ)   ≤ M|λn – λ| + p H(xn , yn , λ), H(u, v, λ) . Letting n → ∞, we obtain     p u, H(u, v, λ) ≤ lim p H(xn , yn , λ), H(u, v, λ) . n→∞

Since ψ is continuous and non-decreasing, we obtain       ψ p u, H(u, v, λ) ≤ lim ψ p H(xn , yn , λ), H(u, v, λ) n→∞

 

 

 ≤ lim α max p(xn , u), p(yn , v) – β max p(xn , u), p(yn , v) n→∞

= . It follows that p(u, H(u, v, λ)) = . Thus u = H(u, v, λ). Similarly v = H(v, u, λ). Thus λ ∈ A. Hence A is closed in [, ]. Let λ ∈ A. Then there exist x , y ∈ U with x = H(x , y , λ ). Since U is open, there exists r >  such that Bp (x , r) ⊆ U. Choose λ ∈ (λ – , λ + ) such that |λ – λ | ≤ Mn < . Then x ∈ Bp (x , r) = {x ∈ X/p(x, x ) ≤ r + p(x , x )}. We have     p H(x, y, λ), x = p H(x, y, λ), H(x , x , λ )     ≤ p H(x, y, λ), H(x, y, λ ) + p H(x, y, λ ), H(x , y , λ )   – p H(x, y, λ ), H(x, y, λ )   ≤ M|λ – λ | + p H(x, y, λ ), H(x , y , λ ) ≤

   + p H(x, y, λ ), H(x , y , λ ) . n– M

Letting n → ∞, we obtain     p H(x, y, λ), x ≤ p H(x, y, λ ), H(x , y , λ ) . Since ψ is continuous and non-decreasing, we have       ψ p H(x, y, λ), x ≤ ψ p H(x, y, λ ), H(x , y , λ )  

≤ α max p(x, x ), p(y, y ) 

 – φ max p(x, x ), p(y, y ) . Similarly    ψ p H(y, x, λ), y 

 

 ≤ α max p(x, x ), p(y, y ) – φ max p(x, x ), p(y, y ) .

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Thus    

 ψ max p H(x, y, λ), x , p H(y, x, λ), y  

 

≤ α max p(x, x ), p(y, y ) – φ max p(x, x ), p(y, y )  

≤ ψ max p(x, x ), p(y, y ) . Since ψ is non-decreasing, we have   



 max p H(x, y, λ), x , p H(y, x, λ), y ≤ max p(x, x ), p(y, y ) 

≤ max r + p(x , x ), r + p(y , y ) . Thus for each fixed λ ∈ (λ – , λ + ), H(·, λ) : Bp (x , r) → Bp (x , r). Since also (ii) holds and ψ and α are continuous and non-decreasing and β is continuous with ψ(t) – α(t) + β(t) >  for t > , all conditions of Theorem . are satisfied. Thus we deduce that H(·, λ) has a coupled fixed point in U. But this coupled fixed point must be in U since (i) holds. Thus λ ∈ A for any λ ∈ (λ – , λ + ). Hence (λ – , λ + ) ⊆ A and therefore A is open in [, ]. For the reverse implication, we use the same strategy.  Corollary . Let (X, p) be a complete PMS, U be an open subset of X and H : U × U × [, ] → X with the following properties: () x = H(x, y, t) and y = H(y, x, t) for each x, y ∈ ∂U and each λ ∈ [, ] (here ∂U denotes the boundary of U in X), () there exist x, y ∈ U and λ ∈ [, ], L ∈ [, ), such that  

 p H(x, y, λ), H(u, v, μ) ≤ L max p(x, u), p(y, v) , () there exists M ≥ , such that   p H(x, λ), H(x, μ) ≤ M · |λ – μ| for all x ∈ U and λ, μ ∈ [, ]. If H(·, ) has a fixed point in U, then H(·, ) has a fixed point in U. Proof The proof follows by taking ψ(x) = x, φ(x) = x – Lx with L ∈ [, ) in Theorem .. 

4 Conclusions In this paper we conclude some applications on homotopy theory and integral equations by using coupled fixed point theorems in ordered PMSs. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Author details 1 Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur, Andhra Pradesh 522 510, India. 2 Department of Mathematics, K L University, Vaddeswaram, Guntur, Andhra Pradesh 522 502, India. 3 Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey. 4 Department of Computing, Adama Science and Technology University, Adama, Ethiopia. 5 Department of Mathematics, Nallamallareddy Engineering College, Divya Nagar, Hyderabad, Telangana 500088, India.

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