Liu et al. Journal of Inequalities and Applications 2014, 2014:394
http://www.journalofinequalitiesandapplications.com/content/2014/1/394
RESEARCH
Open Access
Common fixed points for a pair of mappings
satisfying contractive conditions of integral
type
Zeqing Liu1 , Xiaochen Zou1 , Shin Min Kang2* and Jeong Sheok Ume3
Dedicated to Professor Shih-sen Chang on his 80th birthday.
*
Correspondence:
[email protected]
Department of Mathematics and
the RINS, Gyeongsang National
University, Jinju, 660-701, Korea
Full list of author information is
available at the end of the article
2
Abstract
Four common fixed point theorems for a pair of weakly compatible mappings
satisfying contractive conditions of integral type in metric spaces are proved. The
existence result of bounded solutions for a system of functional equations arising in
dynamic programming is discussed by using one of the common fixed point
theorems obtained in this paper. An example is given to illustrate that our results
extend properly two fixed point theorems due to Branciari and Rhoades.
MSC: 54H25
Keywords: contractive mappings of integral type; weakly compatible mappings;
common fixed point; system of functional equations; dynamic programming
1 Introduction and preliminaries
In , Branciari [] proved an interesting fixed point theorem for a single-valued contractive mapping of integral type satisfying an analog of the Banach contraction principle
in metric spaces. Afterwards many researchers [–] extended the result of Branciari and
obtained a lot of fixed point and common fixed point theorems for various single-valued
and multi-valued mappings involving a large amount of general contractive conditions of
integral type in metric spaces, modular spaces, symmetric spaces, fuzzy metric spaces,
cone metric spaces, uniform spaces, and Hausdorff topological spaces etc.
Liu et al. [] and Rhoades [] got the existence, uniqueness, and iterative approximations of fixed points for general contractive mappings of integral type, Djoudi and
Merghadi [] and Vijayaraju et al. [] showed several common fixed point theorems for
a pair of weakly compatible mappings satisfying certain contractive mappings of integral
type, Altun and Türkoǧlu [], Altun et al. [] and Djoudi and Aliouche [] discussed a few
common fixed point theorems for two pairs of weakly compatible mappings satisfying
an implicit relation and contractive conditions of integral type, respectively, Suzuki []
proved that Meir-Keeler contractions of integral type are still Meir-Keeler contractions,
Jachymski [] discussed that most contractive conditions of integral type given recently
by many authors coincide with classical ones and got a new contractive condition of integral type which is independent of classical ones, and Sintunavarat and Kumam [, ]
© 2014 Liu et al.; licensee Springer. This is an Open Access article distributed 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.
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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proved Gregus-type common fixed point theorems for tangential multi-valued mappings
satisfying strict general contractive conditions of integral type in metric spaces.
Beygmohammadi and Razani [], Hussain and Salimi [], and Mongkolkeha and Kumam
[] presented more general fixed point and common fixed point theorems for some
integral-type contractions in modular spaces. Murthy et al. [] proved common fixed
point theorems for different variant of compatible mappings, satisfying a contractive condition of integral type in fuzzy metric spaces. De la Sen [] investigated the existence
of fixed points and best proximity points of p-cyclic self-mappings in a set of subsets of
a certain uniform space under integral-type contractive conditions. Khojasteh et al. []
obtained a fixed point theorem of an integral-type contraction in complete cone metric
spaces. Aliouche [] proved a common fixed point theorem for two pairs of weakly compatible mappings satisfying a contractive condition of integral type in symmetric spaces.
Altun and Türkoǧlu [] established two fixed point and common fixed point theorems
for mappings satisfying contractive conditions of integral type in Hausdorff d-complete
topological spaces.
However, to the best of our knowledge, no one studied the existence and uniqueness
problems of common fixed points for a pair of contractive mappings of integral type satisfying (.), (.), (.), and (.), respectively.
The aim of this paper is to show the existence and uniqueness of common fixed points
for the four kinds of weakly compatible mappings (.), (.), (.), and (.) in metric
spaces under weaker conditions. As an application, we use Theorem . to study solvability of system of functional equations (.). Our results extend, improve, and unify Theorem . of Branciari [], Theorems .-. of Liu et al. [], Theorem  of Rhoades [],
Theorem  of Bhakta and Mitra [], Theorem . of Liu and Kang [], and Theorem .
of Liu et al. []. A nontrivial example with uncountably many points is included.
Throughout this paper we assume that R+ = [, +∞), N = {} ∪ N, where N denotes the
set of all positive integers, and
= ϕ: ϕ : R+ → R+ satisfies the requirement that ϕ is Lebesgue integrable,
ε
summable on each compact subset of R and
+
= ψ: ψ : R+ → R+ is upper semi-continuous,
ψ() =  and ψ(t) >  for each t >  .
ϕ(t) dt >  for each ε >  ,

Recall that a pair of self-mappings f and g in a metric space (X, d) are said to be weakly
compatible if they commute at their coincidence points.
The following lemma plays an important role in this paper.
Lemma . ([]) Let ϕ ∈ and {rn }n∈N be a nonnegative sequence with limn→∞ rn = a.
Then
lim
n→∞ 
rn
ϕ(t) dt =
a
ϕ(t) dt.

Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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Lemma . ([]) Let E be a set, p and q : E → R be mappings. If opty∈E p(y) and opty∈E q(y)
are bounded, then
opt p(y) – opt q(y) ≤ supp(y) – q(y).
y∈E
y∈E
y∈E
2 Common fixed point theorems
Now we show the existence and uniqueness of common fixed points for four classes of
weakly compatible mappings satisfying contractive conditions of integral type.
Theorem . Let (X, d) be a metric space and let f and g be weakly compatible selfmappings on X satisfying
d(fx,fy)
M (x,y)
ϕ(t) dt ≤ ψ

ϕ(t) dt ,
∀x, y ∈ X,
(.)

where (ϕ, ψ) ∈ × and
d(fx, gy) + d(fy, gx)
M (x, y) = max d(gx, gy), d(fx, gx), d(fy, gy),
,

d(fx, gy)d(fy, gx) d(fx, gx)d(fy, gx) d(fy, gy)d(fx, gy)
,
,
,
 + d(gx, gy)
[ + d(gx, gy)] [ + d(gx, gy)]
∀x, y ∈ X.
(.)
If f (X) ⊆ g(X) and g(X) is complete, then f and g have a unique common fixed point in X.
Proof Firstly we prove that f and g have at most one common fixed point in X. Suppose
that f and g possess two common fixed points a, b ∈ X and a = b. It follows from (.),
(.), and (ϕ, ψ) ∈ × that
d(fa, gb) + d(fb, ga)
M (a, b) = max d(ga, gb), d(fa, ga), d(fb, gb),
,

d(fa, gb)d(fb, ga) d(fa, ga)d(fb, ga) d(fb, gb)d(fa, gb)
,
,
 + d(ga, gb)
[ + d(ga, gb)] [ + d(ga, gb)]
d (a, b)
, , 
= max d(a, b), , , d(a, b),
 + d(a, b)
= d(a, b)
and
d(a,b)
d(fa,fb)
ϕ(t) dt =

ϕ(t) dt ≤ ψ

=ψ
<
d(a,b)
M (a,b)
ϕ(t) dt

ϕ(t) dt

d(a,b)
ϕ(t) dt,

which is impossible.
Secondly we show that f and g have a common fixed point in X. Let x be an arbitrary
point in X. Since f (X) ⊆ g(X), it follows that there exists a sequence {xn }n∈N in X satisfying
fxn = gxn+ for each n ∈ N . Put dn = d(fxn , fxn+ ) for all n ∈ N .
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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Assume that dn =  for some n ∈ N . It follows that
fxn = fxn + = gxn + .
(.)
Because f and g are weakly compatible, by (.) we get
f  xn + = fgxn + = gfxn + = g  xn + .
(.)
Now we assert that fxn + = f  xn + . Otherwise we infer that in view of (.)-(.) and
(ϕ, ψ) ∈ × M (fxn + , xn + )
= max d(gfxn + , gxn + ), d f  xn + , gfxn + , d(fxn + , gxn + ),
d(f  xn + , gxn + ) + d(fxn + , gfxn + ) d(f  xn + , gxn + )d(fxn + , gfxn + )
,
,

 + d(gfxn + , gxn + )
d(f  xn + , gfxn + )d(fxn + , gfxn + ) d(fxn + , gxn + )d(f  xn + , gxn + )
,
[ + d(gfxn + , gxn + )]
[ + d(gfxn + , gxn + )]
= max d f  xn + , fxn + , , , d f  xn + , fxn + ,
d(f  xn + , fxn + )d(fxn + , f  xn + )
, , 
 + d(f  xn + , fxn + )
= d f  xn + , fxn +
and
d(f  xn + ,fxn + )
M (fxn + ,xn + )
ϕ(t) dt ≤ ψ

ϕ(t) dt

d(f  xn + ,fxn + )
=ψ
ϕ(t) dt

d(f  x
<
n + ,fxn + )
ϕ(t) dt,

which is absurd. Therefore fxn + = f  xn + , which together with (.) means that fxn + is
a common fixed point of f and g in X.
Assume that dn =  for all n ∈ N . Observe that
dn– d(fxn+ , fxn– ) d(fxn+ , fxn– ) dn– + dn
≤
≤
≤ max{dn– , dn },
( + dn– )


It follows from (.) and (.) that
M (xn , xn+ )
= max d(gxn , gxn+ ), d(fxn , gxn ), d(fxn+ , gxn+ ),
∀n ∈ N.
(.)
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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d(fxn , gxn+ ) + d(fxn+ , gxn ) d(fxn , gxn+ )d(fxn+ , gxn )
,
,

 + d(gxn , gxn+ )
d(fxn , gxn )d(fxn+ , gxn ) d(fxn+ , gxn+ )d(fxn , gxn+ )
,
[ + d(gxn , gxn+ )]
[ + d(gxn , gxn+ )]
= max d(fxn– , fxn ), d(fxn , fxn– ), d(fxn+ , fxn ),
d(fxn , fxn ) + d(fxn+ , fxn– ) d(fxn , fxn )d(fxn+ , fxn– )
,
,

 + d(fxn– , fxn )
d(fxn , fxn– )d(fxn+ , fxn– ) d(fxn+ , fxn )d(fxn , fxn )
,
[ + d(fxn– , fxn )]
[ + d(fxn– , fxn )]
d(fxn+ , fxn– )
dn– d(fxn+ , fxn– )
, ,
,
= max dn– , dn– , dn ,

( + dn– )
= max{dn– , dn },
∀n ∈ N.
(.)
If dn > dn– for some n ∈ N, using (.), (.), and (ϕ, ψ) ∈ × , we conclude that
dn
d(fxn ,fxn+ )
ϕ(t) dt =
ϕ(t) dt ≤ ψ


=ψ
ϕ(t) dt

dn
M (xn ,xn+ )
ϕ(t) dt

dn
<
ϕ(t) dt,

which is a contradiction. Hence dn ≤ dn– for each n ∈ N. Consequently, the sequence
{dn }n∈N is nondecreasing and bounded, which imply that there exists a constant W with
limn→∞ dn = W ≥ .
Next we show that W = . Otherwise W > . Taking the upper limit in (.) and using
(.), (ϕ, ψ) ∈ × , and Lemma ., we infer that
W
n→∞

dn
ϕ(t) dt = lim sup
n→∞
n→∞
M (xn ,xn+ )
ϕ(t) dt

ϕ(t) dt = lim sup ψ

≤ ψ lim sup
<
n→∞

≤ lim sup ψ
d(fxn ,fxn+ )
ϕ(t) dt = lim sup
dn–
n→∞
W
ϕ(t) dt = ψ

dn–
ϕ(t) dt

ϕ(t) dt

W
ϕ(t) dt,

which is impossible. Therefore, W = , that is,
lim dn = .
n→∞
(.)
Now we prove that {fxn }n∈N is a Cauchy sequence. Suppose that {fxn }n∈N is not a
Cauchy sequence, which means that there exist a constant ε >  and two sequences
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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{m(p)}n∈N and {n(p)}n∈N in N such that m(p) < n(p) < m(p + ) and
d(fxm(p) , fxn(p) ) ≥ ε,
∀p ∈ N.
d(fxm(p) , fxn(p)– ) < ε,
(.)
Note that
d(fxm(p) , fxn(p) ) ≤ d(fxn(p)– , fxm(p) ) + dn(p)– , ∀p ∈ N;
d(fxm(p) , fxn(p) ) – d(fxm(p) , fxn(p)– ) ≤ dn(p)– , ∀p ∈ N;
d(fxm(p) , fxn(p) ) – d(fxm(p)– , fxn(p) ) ≤ dm(p)– , ∀p ∈ N;
d(fxm(p)– , fxn(p)– ) – d(fxm(p)– , fxn(p) ) ≤ dn(p)– , ∀p ∈ N.
(.)
By virtue of (.)-(.), we deduce that
ε = lim d(fxn(p) , fxm(p) ) = lim d(fxm(p) , fxn(p)– )
p→∞
p→∞
= lim d(fxm(p)– , fxn(p) ) = lim d(fxm(p)– , fxn(p)– ).
p→∞
(.)
p→∞
In light of (.), (.), (.), (.), (ϕ, ψ) ∈ × , and Lemma ., we conclude that
lim M (xm(p) , xn(p) )
p→∞
= lim max d(gxm(p) , gxn(p) ), d(fxm(p) , gxm(p) ), d(fxn(p) , gxn(p) ),
p→∞
d(fxm(p) , gxn(p) ) + d(fxn(p) , gxm(p) ) d(fxm(p) , gxn(p) )d(fxn(p) , gxm(p) )
,
,

 + d(gxm(p) , gxn(p) )
d(fxm(p) , gxm(p) )d(fxn(p) , gxm(p) ) d(fxn(p) , gxn(p) )d(fxm(p) , gxn(p) )
,
[ + d(gxm(p) , gxn(p) )]
[ + d(gxm(p) , gxn(p) )]
= lim max d(fxm(p)– , fxn(p)– ), d(fxm(p) , fxm(p)– ), d(fxn(p) , fxn(p)– ),
p→∞
d(fxm(p) , fxn(p)– ) + d(fxn(p) , fxm(p)– ) d(fxm(p) , fxn(p)– )d(fxn(p) , fxm(p)– )
,
,

 + d(fxm(p)– , fxn(p)– )
d(fxm(p) , fxm(p)– )d(fxn(p) , fxm(p)– ) d(fxn(p) , fxn(p)– )d(fxm(p) , fxn(p)– )
,
[ + d(fxm(p)– , fxn(p)– )]
[ + d(fxm(p)– , fxn(p)– )]
ε
= max ε, , , ε,
, , 
+ε
=ε
and
ε
d(fxm(p) ,fxn(p) )
ϕ(t) dt = lim sup

p→∞
p→∞
ε
ϕ(t) dt,


M (xm(p) ,xn(p) )
M (xm(p) ,xn(p) )
ϕ(t) dt
p→∞

≤ ψ lim sup
<
ϕ(t) dt ≤ lim sup ψ
ϕ(t) dt = ψ

ε
ϕ(t) dt

Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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which is a contradiction. Hence {fxn }n∈N is a Cauchy sequence. Since g(X) is complete,
there exist a, z ∈ X with
lim fxn = lim gxn = a = gz.
n→∞
(.)
n→∞
Suppose that fz = a. Making use of (.), (.), (.), and (ϕ, ψ) ∈ × and Lemma .,
we arrive at
lim M (z, xn ) = lim max d(gz, gxn ), d(fz, gz), d(fxn , gxn ),
n→∞
n→∞
d(fz, gxn ) + d(fxn , gz) d(fz, gxn )d(fxn , gz)
,
,

 + d(gz, gxn )
d(fz, gz)d(fxn , gz) d(fxn , gxn )d(fz, gxn )
,
[ + d(gz, gxn )]
[ + d(gz, gxn )]
d(fz, a) + d(a, a)
,
= max d(a, a), d(fz, a), d(a, a),

d(fz, a)d(a, a) d(fz, a)d(a, a) d(a, a)d(fz, a)
,
,
 + d(a, a) [ + d(a, a)] [ + d(a, a)]
d(fz, a)
, , , 
= max , d(fz, a), ,

= d(fz, a)
and
d(fz,a)
d(fz,fxn )
ϕ(t) dt = lim sup

n→∞
<
ϕ(t) dt ≤ lim sup ψ
n→∞

≤ ψ lim sup
n→∞
M (z,xn )
ϕ(t) dt
ϕ(t) dt = ψ

M (z,xn )

d(fz,a)
ϕ(t) dt

d(fz,a)
ϕ(t) dt,

which is a contradiction. Thus a = fz = gz. Since f and g are weakly compatible, it follows
that
fa = f  z = fgz = gfz = g  z = ga.
(.)
Suppose that a = fa. In view of (.), (.) (.), and (ϕ, ψ) ∈ × , we infer that
d(fz, ggz) + d(fgz, gz)
,
M (z, gz) = max d(gz, ggz), d(fz, gz), d(fgz, ggz),

d(fz, ggz)d(fgz, gz) d(fz, gz)d(fgz, gz) d(fgz, ggz)d(fz, ggz)
,
,
 + d(gz, ggz)
[ + d(gz, ggz)]
[ + d(gz, ggz)]
d(a, fa) + d(fa, a) d (a, fa)
,
, , 
= max d(a, fa), , ,

 + d(a, fa)
= d(a, fa)
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and
d(a,fa)
d(fz,fgz)
ϕ(t) dt =

ϕ(t) dt ≤ ψ

=ψ
ϕ(t) dt

d(a,fa)
M (z,gz)
ϕ(t) dt

d(a,fa)
<
ϕ(t) dt,

which is absurd. Hence a = fa. It follows from (.) that f and g have a common fixed
point a ∈ X. This completes the proof.
Theorem . Let (X, d) be a metric space and let f and g be weakly compatible selfmappings on X satisfying
d(fx,fy)
M (x,y)
ϕ(t) dt ≤ ψ

ϕ(t) dt ,
∀x, y ∈ X,
(.)

where (ϕ, ψ) ∈ × and
d(fy, gx) + d(fx, gy)
M (x, y) = max d(gx, gy), d(fx, gx), d(fy, gy),
,

d(fx, gy)d(fy, gx) d(fx, gy)d(fx, gx) d(fy, gx)d(fy, gy)
,
,
,
 + d(fx, fy)
[ + d(fx, fy)] [ + d(fx, fy)]
∀x, y ∈ X.
(.)
If f (X) ⊆ g(X) and g(X) is complete, then f and g have a unique common fixed point in X.
Proof Firstly we prove that f and g have at most one common fixed point in X. Suppose
that f and g possess two common fixed points a, b ∈ X and a = b. It follows from (.),
(.), and (ϕ, ψ) ∈ × that
d(fb, ga) + d(fa, gb)
M (a, b) = max d(ga, gb), d(ga, fa), d(fb, gb),
,

d(fa, gb)d(fb, ga) d(fa, gb)d(fa, ga) d(fb, ga)d(fb, gb)
,
,
 + d(fa, fb)
[ + d(fa, fb)]
[ + d(fa, fb)]
d (a, b)
, , 
= max d(a, b), , , d(a, b),
 + d(a, b)
= d(a, b)
and
d(a,b)
d(fa,fb)
ϕ(t) dt =

ϕ(t) dt ≤ ψ

<
d(a,b)
=ψ
ϕ(t) dt

d(a,b)
ϕ(t) dt,

which is absurd.
M (a,b)
ϕ(t) dt

Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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Secondly we show that f and g have a common fixed point in X. Let x be an arbitrary
point in X. Since f (X) ⊆ g(X), it follows that there exists a sequence {xn }n∈N in X satisfying
fxn = gxn+ for each n ∈ N . Put dn = d(fxn , fxn+ ) for all n ∈ N .
Assume that dn =  for some n ∈ N . Now we assert that fxn + = f  xn + . Otherwise we
infer that in light of (.), (.), (.), (.), and (ϕ, ψ) ∈ × M (fxn + , xn + )
= max d(gfxn + , gxn + ), d f  xn + , gfxn + , d(fxn + , gxn + ),
d(fxn + , gfxn + ) + d(f  xn + , gxn + ) d(f  xn + , gxn + )d(fxn + , gfxn + )
,
,

 + d(f  xn + , fxn + )
d(f  xn + , gxn + )d(f  xn + , gfxn + ) d(fxn + , gfxn + )d(fxn + , gxn + )
,
[ + d(f  xn + , fxn + )]
[ + d(f  xn + , fxn + )]
= max d f  xn + , fxn + , , , d f  xn + , fxn + ,
d(f  xn + , fxn + )d(fxn + , f  xn + )
, , 
 + d(f  xn + , fxn + )
= d f  xn + , fxn +
and
d(f  xn + ,fxn + )
M (fxn + ,xn + )
ϕ(t) dt ≤ ψ

ϕ(t) dt

d(f  xn + ,fxn + )
=ψ
ϕ(t) dt

<
d(f  xn + ,fxn + )
ϕ(t) dt,

which is absurd. Therefore fxn + = f  xn + , which together which (.) means that fxn +
is a common fixed point of f and g in X.
Suppose that dn =  for all n ∈ N . Using (.), (.), and (.), we deduce that
M (xn , xn+ )
= max d(gxn , gxn+ ), d(fxn , gxn ), d(fxn+ , gxn+ ),
d(fxn+ , gxn ) + d(fxn , gxn+ ) d(fxn , gxn+ )d(fxn+ , gxn )
,
,

 + d(fxn , fxn+ )
d(fxn , gxn+ )d(fxn , gxn ) d(fxn+ , gxn )d(fxn+ , gxn+ )
,
[ + d(fxn , xn+ )]
[ + d(fxn , fxn+ )]
= max d(fxn– , fxn ), d(fxn , fxn– ), d(fxn+ , fxn ),
d(fxn+ , fxn– ) + d(fxn , fxn ) d(fxn , fxn )d(fxn+ , fxn– )
,
,

 + d(fxn , fxn+ )
d(fxn , fxn )d(fxn , fxn– ) d(fxn+ , fxn– )d(fxn+ , fxn )
,
[ + d(fxn , fxn+ )]
[ + d(fxn , fxn+ )]
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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d(fxn+ , fxn– )dn
d(fxn+ , fxn– )
, , ,
= max dn– , dn– , dn ,

( + dn )
= max{dn– , dn }.
(.)
If dn > dn– for some n ∈ N, making use of (.), (.), and (ϕ, ψ) ∈ × , we obtain
dn
d(fxn ,fxn+ )
ϕ(t) dt =
ϕ(t) dt ≤ ψ


=ψ
ϕ(t) dt

dn
M (xn ,xn+ )
ϕ(t) dt

dn
<
ϕ(t) dt,

which is impossible. Thus dn ≤ dn– for each n ∈ N. Hence the sequence {dn }n∈N is nondecreasing and bounded, which imply that there exists a constant Q with limn→∞ dn = Q ≥ .
Next we show that Q = . Otherwise Q > . Taking the upper limit in (.) and using
(.), (ϕ, ψ) ∈ × , and Lemma ., we infer that
Q
n→∞

dn
ϕ(t) dt = lim sup
n→∞

≤ lim sup ψ
n→∞
n→∞
M (xn ,xn+ )
ϕ(t) dt

ϕ(t) dt = lim sup ψ

≤ ψ lim sup
d(fxn ,fxn+ )
ϕ(t) dt = lim sup
dn–
ϕ(t) dt = ψ

n→∞
Q
dn–
ϕ(t) dt

ϕ(t) dt

Q
<
ϕ(t) dt,

which is absurd. Therefore, Q = , that is,
lim dn = ,
n→∞
∀n ∈ N.
(.)
Now we claim that {fxn }n∈N is a Cauchy sequence. Suppose that {fxn }n∈N is not a
Cauchy sequence. According to (.), (.), (.), (.), and (ϕ, ψ) ∈ × and Lemma ., we have
lim M (xm(p) , xn(p) )
p→∞
= lim max d(gxm(p) , gxn(p) ), d(fxm(p) , gxm(p) ), d(fxn(p) , gxn(p) ),
p→∞
d(fxn(p) , gxm(p) ) + d(fxm(p) , gxn(p) ) d(fxm(p) , gxn(p) )d(fxn(p) , gxm(p) )
,

 + d(fxm(p) , fxn(p) )
d(fxm(p) , gxn(p) )d(fxm(p) , gxm(p) ) d(fxn(p) , gxm(p) )d(fxn(p) , gxn(p) )
,
[ + d(fxm(p) , fxn(p) )]
[ + d(fxm(p) , fxn(p) )]
= lim max d(fxm(p)– , fxn(p)– ), d(fxm(p) , fxm(p)– ), d(fxn(p) , fxn(p)– ),
p→∞
d(fxn(p) , fxm(p)– ) + d(fxm(p) , fxn(p)– ) d(fxm(p) , fxn(p)– )d(fxn(p) , fxm(p)– )
,

 + d(fxm(p) , fxn(p) )
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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d(fxm(p) , fxn(p)– )d(fxm(p) , fxm(p)– ) d(fxn(p) , fxm(p)– )d(fxn(p) , fxn(p)– )
,
[ + d(fxm(p) , fxn(p) )]
[ + d(fxm(p) , fxn(p) )]
ε
= max ε, , , ε,
, , 
+ε
=ε
(.)
and
ε
ϕ(t) dt ≤ lim sup
p→∞

d(fxm(p) ,fxn(p) )
p→∞
ϕ(t) dt
p→∞
M (xm(p) ,xn(p) )
M (xm(p) ,xn(p) )
ϕ(t) dt ≤ lim sup ψ

≤ ψ lim sup

ε
ϕ(t) dt ≤ ψ
ϕ(t) dt


ε
<
ϕ(t) dt,

which is a contradiction. Hence {fxn }n∈N is a Cauchy sequence. Since g(X) is complete,
there exist a, z ∈ X such that
lim fxn = lim gxn = a = gz.
n→∞
(.)
n→∞
Suppose that fz = a. By means of (.), (.), (.), (ϕ, ψ) ∈ × and Lemma ., we
arrive at
lim M (z, xn ) = lim max d(gz, gxn ), d(fz, gz), d(fxn , gxn ),
n→∞
n→∞
d(fxn , gz) + d(fz, gxn ) d(fz, gxn )d(fxn , gz)
,
,

 + d(fz, fxn )
d(fz, gxn )d(fz, gz) d(fxn , gz)d(fxn , gxn )
,
[ + d(fz, fxn )]
[ + d(fz, fxn )]
d(a, a) + d(fz, a)
= max d(a, a), d(fz, a), d(a, a),
,

d(fz, a)d(a, a) d(fz, a)d(fz, a) d(a, a)d(a, a)
,
,
 + d(a, a) [ + d(fz, a)] [ + d(fz, a)]
d (fz, a)
d(fz, a)
, ,
,
= max , d(fz, a), ,

[ + d(fz, a)]
= d(fz, a)
and
d(fz,a)
d(fz,fxn )
ϕ(t) dt = lim sup

n→∞
<
n→∞
M (z,xn )

d(fz,a)
M (z,xn )
ϕ(t) dt
n→∞
ϕ(t) dt,

ϕ(t) dt ≤ lim sup ψ

≤ ψ lim sup
ϕ(t) dt = ψ


d(fz,a)
ϕ(t) dt
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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which is a contradiction. Therefore, a = fz = gz. Because f and g are weakly compatible, it
follows that
fa = f  z = fgz = gfz = g  z = ga.
(.)
Suppose that a = fa. In view of (.), (.) (.), and (ϕ, ψ) ∈ × , we acquire
d(fgz, gz) + d(fz, g  z)
,
M (z, gz) = max d gz, g  z , d(fz, gz), d fgz, g  z ,

d(fz, g  z)d(fgz, gz) d(fz, g  z)d(fz, gz) d(fgz, gz)d(fgz, g  z)
,
,
 + d(fz, fgz)
[ + d(fz, fgz)]
[ + d(fz, fgz)]

d(fa, a) + d(a, fa) d (a, fa)
,
, , 
= max d(a, fa), , ,

 + d(a, fa)
= d(a, fa)
and
d(a,fa)
d(fz,fgz)
ϕ(t) dt =

ϕ(t) dt ≤ ψ

=ψ
d(a,fa)
M (z,gz)
ϕ(t) dt

ϕ(t) dt

d(a,fa)
<
ϕ(t) dt,

which is absurd. Hence a = fa. It follows from (.) that f and g have a common fixed
point a ∈ X. This completes the proof.
Similar to the arguments of Theorems . and ., we conclude the following results and
omit their proofs.
Theorem . Let (X, d) be a metric space and let f and g be weakly compatible selfmappings on X satisfying
d(fx,fy)
ϕ(t) dt ≤ ψ

M (x,y)
ϕ(t) dt ,
∀x, y ∈ X,
(.)

where (ϕ, ψ) ∈ × and
M (x, y) = max d(gx, gy), d(fx, gx), d(fy, gy),
d(fy, gx) + d(fx, gy) d(fx, gy)d(fy, gx)
,
,

 + d(gx, gy)
d(fx, gx)d(fy, gx) d(fy, gy)d(fx, gy)
,
,
min
 + d(gx, gy)
 + d(gx, gy)
∀x, y ∈ X.
(.)
If f (X) ⊆ g(X) and g(X) is complete, then f and g have a unique common fixed point in X.
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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Theorem . Let (X, d) be a metric space and let f and g be weakly compatible selfmappings on X satisfying
d(fx,fy)
ϕ(t) dt ≤ ψ

M (x,y)
ϕ(t) dt ,
∀x, y ∈ X,
(.)

where (ϕ, ψ) ∈ × and
M (x, y) = max d(gx, gy), d(fx, gx), d(fy, gy),
d(fy, gx) + d(fx, gy) d(fx, gy)d(fx, gx)
,
,

 + d(fy, gx)
d(fx, gy)d(fx, gx) d(fy, gx)d(fy, gy)
,
,
min
 + d(fx, fy)
 + d(fx, fy)
∀x, y ∈ X.
(.)
If f (X) ⊆ g(X) and g(X) is complete, then f and g have a unique common fixed point in X.
Remark . In case f = g and ψ(t) = ct for each t ∈ R+ , where c is a constant in (, ),
then Theorems .-. reduce to four results which include Theorem . in [] and Theorem  in [] as special cases. Example . below shows that Theorems .-. extend
substantially Theorem . in [] and Theorem  in [].
Example . Let X = R+ be endowed with the Euclidean metric d(x, y) = |x – y| for all
x, y ∈ X. Let f : X → X be defined by
fx =
⎧
⎨,
∀x ∈ X – {},
⎩, x = .
Now we prove that Theorem . in [] and Theorem  in [] cannot be used to prove
the existence of fixed points of the mapping f in X. Suppose that there exist ϕ ∈ and
c ∈ [, ) satisfying the condition of Theorem  in [], that is,
d(fx,fy)
m(x,y)
ϕ(t) dt ≤ c

ϕ(t) dt,
∀x, y ∈ X,
(.)

where

m(x, y) = max d(x, y), d(x, fx), d(y, fy), d(y, fx) + d(x, fy) ,

∀x, y ∈ X.
Taking (x , y ) = (, ) and using (.), (.), ϕ ∈ , and c ∈ [, ), we get

m(x , y ) = max d(x , y ), d(x , fx ), d(y , fy ), d(y , fx ) + d(x , fy )


= max d(, ), d(, ), d(, ), d(, ) + d(, )

=
(.)
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and

d(fx ,fy )
ϕ(t) dt =


m(x ,y )
ϕ(t) dt ≤ c

ϕ(t) dt = c

ϕ(t) dt


<
ϕ(t) dt,

which is a contradiction. Note that Theorem  in [] is a generalization of Theorem .
in []. Therefore, Theorem . in [] is also futile in proving the existence of fixed points
for the mapping f in X.
Define three mappings g : X → X and ϕ, ψ : R+ → R+ by
⎧
⎪
⎪
⎨,
gx =
⎪
⎪
⎩
ϕ(t) =
∀x ∈ X \ {, },
, x = ,
,
x = ,
ln( + t)
,
+t
∀t ∈ R+
and
⎧
⎨ sin t, ∀t ∈ [, ],
ψ(t) =
⎩t  ,
∀t ∈ (, +∞).
It is clear that (ϕ, ψ) ∈ × , f and g are weakly compatible in X, f (X) = {, } ⊆
{, , } = g(X) and g(X) is complete. Let x, y ∈ X with x < y. In order to verify (.), (.),
(.), and (.) hold, we have to consider the following four possible cases:
Case . x, y ∈ X \ {}. It follows that
d(fx,fy)
d(,)
ϕ(t) dt =

Mi (x,y)
ϕ(t) dt =  ≤ ψ

ϕ(t) dt ,
∀i ∈ {, , , }.

Case . x =  and y = . Observe that ψ is strictly increasing in (, +∞) and
Mi (x, y) ≥ d(gx, gy) = d(, ) = ,
∀i ∈ {, , , }.
It is easy to verify that
d(fx,fy)

 

ln  < < 






 
< ln  = ψ
ln  = ψ
ϕ(t) dt



Mi (x,y)
ϕ(t) dt , ∀i ∈ {, , , }.
≤ψ
ϕ(t) dt =

d(,)
ϕ(t) dt =
ϕ(t) dt =

Case . x =  and y ∈ (, ) ∪ (, +∞). Note that ψ is strictly increasing in (, +∞) and
Mi (x, y) ≥ d(gx, gy) = d(, ) = ,
∀i ∈ {, , , }.
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It is clear that
d(fx,fy)
d(,)

 

ln  < < 






 
< ln  = ψ
ln  = ψ
ϕ(t) dt



Mi (x,y)
≤ψ
ϕ(t) dt , ∀i ∈ {, , , }.
ϕ(t) dt =

ϕ(t) dt =
ϕ(t) dt =

Case . x <  and y = . Note that ψ is strictly increasing in (, +∞) and
Mi (x, y) ≥ d(gx, gy) = d(, ) = ,
∀i ∈ {, , , }.
Clearly we have
d(fx,fy)

d(,)

 

ln  < < 




 
< ln  = ψ
ln  = ψ
ϕ(t) dt



Mi (x,y)
≤ψ
ϕ(t) dt , ∀i ∈ {, , , }.
ϕ(t) dt =
ϕ(t) dt =

ϕ(t) dt =


That is, (.), (.), (.), and (.) hold. Hence each of Theorems .-. guarantees
that f and g possess a unique common fixed point in X.
Finally, we use Theorem . to discuss solvability of the following system of functional
equations arising in dynamic programming:
f (x) = opt u(x, y) + H x, y, f a(x, y) ,
∀x ∈ S,
y∈D
g(x) = opt v(x, y) + G x, y, g b(x, y) ,
∀x ∈ S,
(.)
y∈D
where opt stands for sup or inf, Z and Y are Banach spaces, S ⊆ Z is the state space, D ⊆ Y
is the decision space, x and y signify the state and decision vectors, respectively, a and
b represent the transformations of the process, f (x) and g(x) denote the optimal return
functions with the initial state x, B(S) denotes the Banach space of all bounded real-valued
functions on S with norm
w = sup w(x) : x ∈ S for any w ∈ B(S).
Example . Let u, v : S × D → R, a, b : S × D → S, H, G : S × D × R → R and (ϕ, ψ) ∈
× satisfy the following conditions:
u, v, H and G are bounded;
(.)
fgh = gfh for each h ∈ B(S) with fh = gh;
f B(S) ⊆ g B(S) and g B(S) are complete
(.)
(.)
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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and
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|
M∗ (h,w)
ϕ(t) dt ≤ ψ

ϕ(t) dt ,

∀(x, y, h, w) ∈ S × D × B(S) × B(S),
(.)
where the mappings f and g are defined by
fh(x) = opt u(x, y) + H x, y, h a(x, y) ,
∀(x, h) ∈ S × B(S),
y∈D
gh(x) = opt v(x, y) + G x, y, h b(x, y) ,
∀(x, h) ∈ S × B(S)
(.)
y∈D
and
M∗ (h, w) = max gh – gw, fh – gh, fw – gw,
fh – gw + fw – gh fh – gwfw – gh
,
,

 + gh – gw
fh – ghfw – gh fw – gwfh – gw
,
, ∀h, w ∈ B(S).
( + gh – gw)
( + gh – gw)
(.)
Then the system of functional equations (.) has a unique common solution w ∈ B(S).
Proof It follows from (.) that there exists M >  satisfying
sup u(x, y), v(x, y), H(x, y, t), G(x, y, t) : (x, y, t) ∈ S × D × R ≤ M.
(.)
It is easy to see that f and g are self-mappings in B(S) by (.), (.), and Lemma .. It
is clear that Theorem . in [] and ϕ ∈ yield, for each ε > : there exists δ ∈ (, M)
satisfying
ϕ(t) dt < ε, ∀C ⊆ [, M] with m(C) ≤ δ,
(.)
C
where m(C) denotes the Lebesgue measure of C.
Let (x, h, w) ∈ S × B(S) × B(S). Suppose that opty∈D = infy∈D . Clearly (.) implies that
there exist y, z ∈ D satisfying
fh(x) > u(x, y) + H x, y, h a(x, y) – δ;
fw(x) > u(x, z) + H x, z, w a(x, z) – δ;
fh(x) ≤ u(x, z) + H x, z, h a(x, z) ;
fw(x) ≤ u(x, y) + H x, y, w a(x, y) ,
which means that
fh(x) – fw(x) > H x, y, h a(x, y) – H x, y, w a(x, y) – δ
≥ – max H x, y, h a(x, y) – H x, y, w a(x, y) ,
H x, z, h a(x, z) – H x, z, w a(x, z) – δ
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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and
fh(x) – fw(x) < H x, z, h a(x, z) – H x, z, w a(x, z) + δ
≤ max H x, y, h a(x, y) – H x, y, w a(x, y) ,
H x, z, h a(x, z) – H x, z, w a(x, z) + δ,
which yield
fh(x) – fw(x) < max H x, y, h a(x, y) – H x, y, w a(x, y) ,
H x, z, h a(x, z) – H x, z, w a(x, z) + δ
= max H x, y, h a(x, y) – H x, y, w a(x, y) + δ,
H x, z, h a(x, z) – H x, z, w a(x, z) + δ .
(.)
Similarly we infer that (.) holds also for opty∈D = supy∈D . Combining (.), (.), and
(.), we arrive at
|fh(x)–fw(x)|
ϕ(t) dt ≤ max

|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|+δ
ϕ(t) dt,

|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|+δ
ϕ(t) dt

= max
+
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|
ϕ(t) dt

|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|+δ
ϕ(t) dt,
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|
|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|
ϕ(t) dt

+
|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|+δ
|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|
≤ max
ϕ(t) dt
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|
ϕ(t) dt,

|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|
ϕ(t) dt

+ max
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|+δ
|H(x,y,h(a(x,y)))–H(x,y,w(a(x,y)))|
|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|+δ
|H(x,z,h(a(x,z)))–H(x,z,w(a(x,z)))|
≤ψ
M∗ (h,w)
ϕ(t) dt + ε,
ϕ(t) dt,
ϕ(t) dt
∀(x, h, w) ∈ S × B(S) × B(S),

which means that

fh–fw
M∗ (h,w)
ϕ(t) dt ≤ ψ

ϕ(t) dt + ε,
∀h, w ∈ B(S),
Liu et al. Journal of Inequalities and Applications 2014, 2014:394
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letting ε → + in the above inequality, we deduce that

fh–fw
M∗ (h,w)
ϕ(t) dt ≤ ψ
ϕ(t) dt ,
∀h, w ∈ B(S).

Thus Theorem . ensures that the mappings f and g have a unique common fixed point
w ∈ B(S), which is a unique common solution of the system of functional equations (.).
This completes the proof.
Remark . The conclusion of Example . generalizes and improves Theorems .-.
in [], Theorem  in [], Theorem . in [], and Theorem . in [].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, People’s Republic of China.
2
Department of Mathematics and the RINS, Gyeongsang National University, Jinju, 660-701, Korea. 3 Department of
Applied Mathematics, Changwon National University, Changwon, 641-773, Korea.
Acknowledgements
The authors thank the referees for useful comments and suggestions. This research was supported by the Science
Research Foundation of Educational Department of Liaoning Province (L2012380) and Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning
(2013R1A1A2057665).
Received: 18 March 2014 Accepted: 18 September 2014 Published: 16 October 2014
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Cite this article as: Liu et al.: Common fixed points for a pair of mappings satisfying contractive conditions of
integral type. Journal of Inequalities and Applications 2014 2014:394.
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