Filomat 27:7 (2013), 1259–1268
DOI 10.2298/FIL1307259S
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Multi-valued F-contractions and the solution of
certain functional and integral equations
Margherita Sgroi, Calogero Vetro
a
a
Universit`a degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi, 34, 90123 Palermo, Italy
Abstract. Wardowski [Fixed Point Theory Appl., 2012:94] introduced a new concept of contraction and
proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of
research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued
mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric
spaces or complete ordered metric spaces. An example and two applications, for the solution of certain
functional and integral equations, are given to illustrate the usability of the obtained results.
1. Introduction
It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation
of Banach in 1920, which was published in 1922 [7], is one of the most important theorems in classical
functional analysis. Indeed it is widely considered as the source of metric fixed point theory. Also its
significance lies in its vast applicability in a number of branches of mathematics. Starting from these
considerations, the study of fixed and common fixed points of mappings satisfying a certain metrical
contractive condition attracted many researchers, see for example [3, 4, 6, 11, 14, 15, 21, 23–28]. In [16],
Nadler extended the contraction mapping principle from the single-valued mappings to the multi-valued
mappings. Precisely, Nadler proved the following theorem.
Theorem 1.1. Let (X, d) be a complete metric space and let T : X → CB(X) be a multi-valued mapping satisfying
H(Tx, Ty) ≤ k d(x, y), (1)
for all x, y ∈ X, where k is a constant such that k ∈ [0, 1) and CB(X) denotes the family of non-empty closed bounded
subsets of X. Then T has a fixed point.
The reader can see [1, 2, 13] and references therein, for recent results in this direction. Recently, Wardowski
[29] introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach
contraction principle. Following this direction of research, in this paper, we will present some fixed point
results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive
condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric
spaces. Moreover, an example and two applications, for the solution of certain functional and integral
equations, are given to illustrate the usability of the obtained results.
2010 Mathematics Subject Classification. Primary 34L30 Secondary 47H10; 54H25
Keywords. Closed multi-valued F-contractions, F-contractive condition of Hardy-Rogers-type, fixed points, metric spaces, ordered
metric spaces
Received: 13 December 2012; Accepted: 20 April 2013
Communicated by Vladimir Rako
ˇ
cevi
´
c
Research supported by Universit
`
a degli Studi di Palermo, Local University project R.S. ex 60%.
Email address: calogero.vetro@unipa.it (Calogero Vetro)
M. Sgroi, C. Vetro / Filomat 27:7 (2013), 1259–1268 1260
2. Preliminaries
The aim of this section is to present some notions and results used in the paper. Throughout the article
we denote by R the set of all real numbers, by R
+
the set of all positive real numbers and by N the set of all
positive integers.
Definition 2.1. Let F : R
+
→ R be a mapping satisfying:
(F1) F is strictly increasing;
(F2) for each sequence {α
n
} ⊂ R
+
of positive numbers lim
n→+∞
α
n
= 0 if and only if lim
n→+∞
F(α
n
) = −∞;
(F3) there exists k ∈ (0, 1) such that lim
α→0
+
α
k
F(α) = 0.
We denote with F the family of all functions F that satisfy the conditions (F1)-(F3).
Definition 2.2 ([29]). Let (X, d) be a metric space. A self-mapping T on X is called an F-contraction if there exist
F ∈ F and τ ∈ R
+
such that
τ + F(d(Tx, Ty)) ≤ F(d(x, y)), (2)
for all x, y ∈ X with d(Tx, Ty) > 0.
Definition 2.3 ([12]). Let (X, d) be a metric space. A self-mapping T on X is called an F-contraction of Hardy-
Rogers-type if there exist F ∈ F and τ ∈ R
+
such that
τ + F(d(Tx, Ty)) ≤ F(αd(x, y) + βd(x, Tx) + γd(y, Ty) + δd(x, Ty) + Ld(y, Tx)),
for all x, y ∈ X with d(Tx, Ty) > 0, where α, β, γ, δ, L ≥ 0, α + β + γ + 2δ = 1 and γ , 1.
Now, let (X, d) be a metric space and C(X) be the family of non-empty closed subsets of X. If T : X → C(X)
is a multi-valued mapping, then we put
M(x, y) = max
{
d(x, y), d(x, Tx), d(y, Ty),
d(x, Ty) + d(y, Tx)
2
}
.
Definition 2.4. Let (X, d) be a metric space. A multi-valued mapping T : X → C(X) is called an F-contraction if
there exist F ∈ F and τ ∈ R
+
such that for all x, y ∈ X with y ∈ Tx there exists z ∈ Ty for which
τ + F(d(y, z)) ≤ F
(
M(x, y)
)
(3)
if d(y, z) > 0.
If we choose the mapping F opportunely, then we obtain some classes of contractions known in the literature.
See the following examples.
Example 2.5 ([29]). Let F : R
+
→ R be given by F(x) = ln x. It is clear that F satisfies (F1)-(F3) for any k ∈ (0, 1).
Each mapping T : X → X satisfying (2) is an F-contraction such that
d(Tx, Ty) ≤ e
−τ
d(x, y), for all x, y ∈ X, Tx , Ty.
It is clear that for x, y ∈ X such that Tx = Ty the previous inequality also holds and hence T is a contraction.
Example 2.6. Let F : R
+
→ R be given by F(x) = ln x. For each multi-valued mapping T : X → C(X) satisfying
(3) we have
d(y, z) ≤ e
−τ
M(x, y), for all x, y ∈ X, y , z.
It is clear that for z, y ∈ X such that y = z the previous inequality also holds.
Definition 2.7. Let T : X → C(X) be a multi-valued mapping. The graph of T is the subset {(x, y) : x ∈ X, y ∈ Tx}
of X × X; we denote the graph of T by G(T). Then T is a closed multi-valued mapping if the graph G(T) is a closed
subset of X × X.
M. Sgroi, C. Vetro / Filomat 27:7 (2013), 1259–1268 1261
3. Fixed points for F-contractions in complete metric spaces
In this section, we give some fixed point results for F-contractions in a complete metric space.
Theorem 3.1. Let (X, d) be a complete metric space and let T : X → C(X) be a closed F-contraction. Then T has a
fixed point.
Proof. Let x
0
∈ X be an arbitrary point of X and choose x
1
∈ T x
0
. If x
1
= x
0
, then x
1
is a fixed point of T and
the proof is completed. Assume that x
1
, x
0
. Since T is an F-contraction, then there exists x
2
∈ Tx
1
such that
τ + F(d(x
1
, x
2
)) ≤ F(M(x
0
, x
1
)) and x
2
, x
1
.
Also, we get that there exists x
3
∈ Tx
2
such that
τ + F(d(x
2
, x
3
)) ≤ F(M(x
1
, x
2
)) and x
3
, x
2
.
Repeating this process, we find that there exists a sequence {x
n
} with initial point x
0
such that x
n+1
∈ Tx
n
,
x
n+1
, x
n
and
τ + F(d(x
n
, x
n+1
)) ≤ F(M(x
n−1
, x
n
)), for all n ∈ N.
This implies
F(d(x
n
, x
n+1
)) < F(M(x
n−1
, x
n
)), for all n ∈ N
and consequently, we have
d(x
n
, x
n+1
) < max
{
d(x
n−1
, x
n
), d(x
n−1
, Tx
n−1
), d(x
n
, Tx
n
),
d(x
n−1
, Tx
n
) + d(x
n
, Tx
n−1
)
2
}
= max
{
d(x
n−1
, x
n
), d(x
n
, Tx
n
),
d(x
n−1
, Tx
n
)
2
}
≤ max
{
d(x
n−1
, x
n
), d(x
n
, Tx
n
),
d(x
n−1
, x
n
) + d(x
n
, Tx
n
)
2
}
= max
{
d(x
n−1
, x
n
), d(x
n
, x
n+1
)
}
.
Obviously, if max
{
d(x
n−1
, x
n
), d(x
n
, x
n+1
)
}
= d(x
n
, x
n+1
), we have a contradiction and so
max
{
d(x
n−1
, x
n
), d(x
n
, x
n+1
)
}
= d(x
n−1
, x
n
).
Consequently, we get
τ + F(d(x
n
, x
n+1
)) ≤ F(d(x
n−1
, x
n
)), for all n ∈ N. (4)
Now, let d
n
= d(x
n
, x
n+1
) > 0 for all n ∈ N ∪ {0}. By (4), we have
F(d
n
) ≤ F(d
n−1
) − τ ≤ · · · ≤ F(d
0
) − nτ, for all n ∈ N (5)
and hence lim
n→+∞
F(d
n
) = −∞. By the property (F2), we get that d
n
→ 0 as n → +∞. Now, let k ∈ (0, 1)
such that lim
n→+∞
d
k
n
F(d
n
) = 0. By (5), the following holds for all n ∈ N:
d
k
n
F(d
n
) − d
k
n
F(d
0
) ≤ d
k
n
(F(d
0
) − nτ) − d
k
n
F(d
0
) = −nτ d
k
n
≤ 0. (6)
Letting n → +∞ in (6), we deduce lim
n→+∞
n d
k
n
= 0 and hence lim
n→+∞
n
1/k
d
n
= 0. Clearly, lim
n→+∞
n
1/k
d
n
=
0 ensures that the series
∑
+∞
n=1
d
n
is convergent. This implies that {x
n
} is a Cauchy sequence. Since X is a
complete metric space, there exists u ∈ X such that x
n
→ u as n → +∞. Now, we prove that u is a fixed
point of T. Since T is a closed multi-valued mapping and (x
n
, x
n+1
) → (u, u), we get u ∈ Tu and hence u is a
fixed point of T.
M. Sgroi, C. Vetro / Filomat 27:7 (2013), 1259–1268 1262
We note that in a metric space every upper semicontinuous multi-valued mapping is closed. Precisely,
we recall that T is upper semicontinuous if and only if for each closed set B ∈ C(X), we have that T
−1
(B) =
{x : T(X) ∩ B , ∅} is closed. Then, from Theorem 3.1, we obtain the following corollary.
Corollary 3.2. Let (X, d) be a complete metric space and let T : X → C(X) be an upper semicontinuous F-contraction.
Then T has a fixed point.
Example 3.3. Consider the complete metric space (X, d), where X = {0, 1, 2, . . .} and d : X × X → [0, +∞) is given
by
d(x, y) =
0 if x = y,
x + y if x , y,
Let T : X → C(X) be defined by
Tx =
{0} if x ∈ {0, 1},
{0, . . . , x − 1} if x ≥ 2.
Clearly the multi-valued mapping T is closed.
Now, we show that T satisfies (3), where τ = 1 and F(x) = ln x + x for each x ∈ R
+
. To this aim, for all x, y ∈ X
with y ∈ Tx, we choose z = 0 ∈ Ty. First, we note that d(y, z) > 0 if and only if x ≥ 2 and y > 0. If this holds true,
then d(y, z) = y < x + y = d(x, y) and hence
d(y, z) − M(x, y) ≤ d(y, z) − d(x, y) ≤ −2.
It follows easily that
d(y, z)
M(x, y)
e
d(x,y)−M(x,y)
≤ e
−1
and hence
1 + F(d(y, z)) ≤ F(M(x, y)),
for all x, y ∈ X with d( y, z) > 0. Then, by Theorem 3.1, T has a fixed point.
On the other hand, it is easy to show that Theorem 1.1 is not applicable in this case. Indeed, assume there exists
k ∈ [0, 1) such that (1) holds true, then H(Tx, T0) = x − 1 ≤ kx, for all x ≥ 1. This implies that
x − 1
x
≤ k and, for
x → +∞, we get 1 ≤ k, a contradiction.
Next, we give a fixed point result for multi-valued F-contractions of Hardy-Rogers-type in a complete
metric space.
Theorem 3.4. Let (X, d) be a complete metric space and let T : X → CB(X). Assume that there exist F ∈ F and
τ ∈ R
+
such that
2τ + F(H(Tx, Ty)) ≤ F(αd(x, y) + βd(x, Tx) + γd(y, Ty ) + δd(x, Ty) + Ld(y, Tx)), (7)
for all x, y ∈ X, with Tx , Ty, where α, β, γ, δ, L ≥ 0, α + β + γ + 2δ = 1 and γ , 1. Then T has a fixed point.
Proof. Let x
0
∈ X be an arbitrary point of X and choose x
1
∈ Tx
0
. If x
1
∈ Tx
1
, then x
1
is a fixed point of T and
the proof is completed. Assume that x
1
< Tx
1
, then Tx
0
, Tx
1
. Since F is continuous from the right, there
exists a real number h > 1 such that
F(h H (Tx
0
, Tx
1
)) < F(H(Tx
0
, Tx
1
)) + τ.
M. Sgroi, C. Vetro / Filomat 27:7 (2013), 1259–1268 1263
Now, from d(x
1
, Tx
1
) < h H(Tx
0
, Tx
1
), we deduce that there exists x
2
∈ Tx
1
such that d(x
1
, x
2
) ≤ h H(Tx
0
, Tx
1
).
Consequently, we get
F(d(x
1
, x
2
)) ≤ F(h H(Tx
0
, Tx
1
)) < F(H(Tx
0
, Tx
1
)) + τ,
which implies
2τ + F(d(x
1
, x
2
)) ≤ 2τ + F(H(Tx
0
, Tx
1
)) + τ
≤ F(αd(x
0
, x
1
) + βd(x
0
, Tx
0
) + γd(x
1
, Tx
1
) + δd(x
0
, Tx
1
) + Ld(x
1
, Tx
0
)) + τ
≤ F(αd(x
0
, x
1
) + βd(x
0
, x
1
) + γd(x
1
, x
2
) + δd(x
0
, x
2
)) + τ
≤ F((α + β + δ)d(x
0
, x
1
) + (γ + δ)d(x
1
, x
2
)) + τ.
Since F is strictly increasing, we deduce
d(x
1
, x
2
) ≤ (α + β + δ)d(x
0
, x
1
) + (γ + δ)d(x
1
, x
2
)
and hence
(1 − γ − δ)d(x
1
, x
2
) < (α + β + δ)d(x
0
, x
1
).
From α + β + γ + 2δ = 1 and γ , 1, we deduce that 1 − γ − δ > 0 and so
d(x
1
, x
2
) <
α + β + δ
1 − γ − δ
d(x
0
, x
1
) = d(x
0
, x
1
).
Consequently,
τ + F(d(x
1
, x
2
)) ≤ F(d(x
0
, x
1
)).
Continuing in this manner, we can define a sequence {x
n
} ⊂ X such that x
n
< Tx
n
, x
n+1
∈ Tx
n
and
τ + F(d(x
n+1
, x
n+2
)) ≤ F(d(x
n
, x
n+1
)), for all n ∈ N ∪ {0}.
Proceeding as in the proof of Theorem 3.1, we obtain that {x
n
} is a Cauchy sequence. Since X is a complete
metric space, there exists u ∈ X such that x
n
→ u as n → +∞. Now, we prove that u is a fixed point of T. If
there exists an increasing sequence {n
k
} ⊂ N such that x
n
k
∈ Tu for all k ∈ N, since Tu is closed and x
n
k
→ u,
we get u ∈ Tu and the proof is completed. So we can assume that there exists n
0
∈ N such that x
n
< Tu for
all n ∈ N with n ≥ n
0
. This implies that Tx
n−1
, Tu for all n ≥ n
0
. Now, using (7) with x = x
n
and y = u, we
obtain
2τ + F(H(Tx
n
, Tu)) ≤ F(αd(x
n
, u) + βd(x
n
, Tx
n
) + γd(u, Tu) + δd(x
n
, Tu) + Ld(u, Tx
n
)),
which implies
2τ + F(d(x
n+1
, Tu)) ≤ 2τ + F(H(Tx
n
, Tu))
≤ F(αd(x
n
, u) + βd(x
n
, Tx
n
) + γd(u, Tu) + δd(x
n
, Tu) + Ld(u, Tx
n
))
≤ F(αd(x
n
, u) + βd(x
n
, x
n+1
) + γd(u, Tu) + δd(x
n
, Tu) + Ld(u, x
n+1
)).
Since F is strictly increasing, we have
d(x
n+1
, Tu) < αd(x
n
, u) + βd(x
n
, x
n+1
) + γd(u, Tu) + δd(x
n
, Tu) + Ld(u, x
n+1
).
Letting n → +∞ in the previous inequality, as γ + δ < 1 we get
d(u, Tu) ≤ (γ + δ)d(u, Tu ) < d(u, Tu),
which implies d(u, Tu) = 0. Since Tu is closed we obtain that u ∈ Tu, that is, u is a fixed point of T.