Common fixed-point theorems for hybrid generalized $(F, ϕ)$ -contractions under the common limit range property with applications ......................
We consider a relatively new hybrid generalized $F$ -contraction involving a pair of mappings and use this contraction to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized $(F, \varphi)$-contraction condition with the commo...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507666017681408 |
|---|---|
| author | Ahmadullah, M. Imdad, M. Nashine, H. K. Ахмадулла, М. Імдад, М. Нашине, Г. К. |
| author_facet | Ahmadullah, M. Imdad, M. Nashine, H. K. Ахмадулла, М. Імдад, М. Нашине, Г. К. |
| author_sort | Ahmadullah, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:27:02Z |
| description | We consider a relatively new hybrid generalized $F$ -contraction involving a pair of mappings and use this contraction
to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying
generalized $(F, \varphi)$-contraction condition with the common limit range property in complete metric spaces. A similar result
involving a hybrid pair of mappings satisfying the rational-type Hardy – Rogers $(F, \varphi)$-contractive condition is also proved.
We generalize and improve several results available from the existing literature. As applications of our results, we prove
two theorems for the existence of solutions of certain system of functional equations encountered in dynamic programming
and the Volterra integral inclusion. Moreover, we provide an illustrative example. |
| first_indexed | 2026-03-24T02:12:56Z |
| format | Article |
| fulltext |
UDC 517.9
H. K. Nashine (Texas A & M Univ., Amity School Appl. Sci., Amity Univ. Chhattisgarh, India),
M. Imdad, MD Ahmadullah (Aligarh Muslim Univ., India)
COMMON FIXED POINT THEOREMS
FOR HYBRID GENERALIZED (\bfitF , \bfitvarphi )-CONTRACTIONS
UNDER COMMON LIMIT RANGE PROPERTY WITH APPLICATIONS
СПIЛЬНI ТЕОРЕМИ ПРО НЕРУХОМУ ТОЧКУ ДЛЯ ГIБРИДНИХ
УЗАГАЛЬНЕНИХ (\bfitF , \bfitvarphi )-СТИСКАНЬ З ВЛАСТИВIСТЮ СПIЛЬНОГО
ГРАНИЧНОГО ДIАПАЗОНУ З ЗАСТОСУВАННЯМИ
We consider a relatively new hybrid generalized F -contraction involving a pair of mappings and use this contraction
to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying
generalized (F,\varphi )-contraction condition with the common limit range property in complete metric spaces. A similar result
involving a hybrid pair of mappings satisfying the rational-type Hardy – Rogers (F,\varphi )-contractive condition is also proved.
We generalize and improve several results available from the existing literature. As applications of our results, we prove
two theorems for the existence of solutions of certain system of functional equations encountered in dynamic programming
and the Volterra integral inclusion. Moreover, we provide an illustrative example.
Розглянуто вiдносно нове узагальнене гiбридне F -стискання, що включає пару вiдображень. Це стисканя засто-
совано при доведеннi спiльної теормеми про нерухому точку для випадково спiвпадаючих iдемпотентних матриць,
що задовольняють узагальнену умову (F,\varphi )-стискання при влативостi спiльного граничного дiапазону в повних
метричних просторах. Також доведено подiбний результат для гiбридних пар вiдображень, що задовольняють умо-
ву Гардi – Роджерса про (F,\varphi )-стискання рацiонального типу. Узагальнено та покращено деякi вiдомi лiтературнi
результати. Як застосування наших результатiв, доведено двi теореми про iснування розв’язкiв деякої системи
функцiональних рiвнянь, що зустрiчаються в динамiчному програмуваннi, та iнтегрального включення Вольтерра.
Крiм того, наведено iлюстративний приклад.
1. Introduction and preliminaries. Let (X, d) be a metric space. Then, following the Nadler [28],
we adopt the following notations:
CL(X) = \{ A : A is a nonempty closed subset of X}.
CB(X) = \{ A : A is a nonempty closed and bounded subset of X}.
For nonempty closed and bounded subsets A,B of X and x \in X,
d(x,A) = \mathrm{i}\mathrm{n}\mathrm{f}\{ d(x, a) : a \in A\}
and
\scrH (A,B) = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
d(a,B) : a \in A
\bigr\}
, \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
d(b, A) : b \in B
\bigr\} \bigr\}
.
Recall that CB(X) is a metric space with the metric \scrH which is known as the Hausdorff –
Pompeiu metric on CB(X).
In 1969, Nadler [28] proved that every multivalued contraction mapping defined on a complete
metric space has a fixed point. In proving this result, Nadler used the idea of Hausdorff metric to
establish the multivalued version of Banach Contraction Principle which runs as follows:
Theorem 1. Let (X, d) be a complete metric space and \scrT a mapping from X into CB(X)
such that for all x, y \in X,
\scrH (\scrT x, \scrT y) \leq \lambda d(x, y),
where \lambda \in [0, 1). Then \scrT has a fixed point, i.e., there exists a point x \in X such that x \in \scrT x.
c\bigcirc H. K. NASHINE, M. IMDAD, MD AHMADULLAH, 2017
1534 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1535
Hybrid fixed point theory involving pairs of single-valued and multivalued mappings is a relatively
new development in nonlinear analysis (see e.g., [11, 12, 15, 24, 29, 45] and references therein). The
much discussed concepts of commutativity and weak commutativity were extended to hybrid pair of
mappings on metric spaces by Kaneko [20, 21]. In 1989, Singh et al. [40] extended the notion of
compatible mappings and obtained some coincidence and common fixed point theorems for nonlinear
hybrid contractions. It was observed that under compatibility the fixed point results usually require
continuity of one of the underlying mappings. Afterwards, Pathak [30] generalized the concept of
compatibility by defining weak compatibility for hybrid pairs of mappings (including single valued
case as well) and utilized the same to prove common fixed point theorems. Naturally, compatible
mappings are weakly compatible but not conversely.
In 2002, Aamri and El-Moutawakil [1] introduced the property (E.A.) for single-valued mappings.
Later, Kamran [19] extended the notion of (E.A.) property to hybrid pairs of mappings. In 2011,
Sintunavarat and Kumam [44] introduced the notion of common limit range (CLR) property for
single-valued mappings and showed its superiority over the property (E.A.). Motivated by this fact,
Imdad et al. [14] established common limit range property for a hybrid pair of mappings and proved
some fixed point results in symmetric (semimetric) spaces. For more details on hybrid contraction
conditions, one can consult [2, 7, 10, 13, 16, 18, 22, 29, 34, 35, 41 – 43].
The following definitions and results are standard in the theory of hybrid pair of mappings.
Definition 1. Let f : X \rightarrow X and T : X \rightarrow CB(X) be a single-valued and multivalued map-
ping respectively. Then:
A point x \in X is a fixed point of f (resp. T ) if x = fx (resp. x \in Tx). The set of all fixed
points of f (resp. T ) is denoted by F (f) (resp. F (T )).
A point x \in X is a coincidence point of f and T if fx \in Tx. The set of all coincidence points
of f and T is denoted by \scrC (f, T ).
A point x \in X is a common fixed point of f and T if x = fx \in Tx. The set of all common
fixed points of f and T is denoted by F (f, T ).
T is a closed multivalued mapping if the graph of T, i.e., G(T ) = \{ (x, y) : x \in X, y \in Tx\} is
a closed subset of X \times X.
We also recall the following terminology often used in the considerations of a hybrid pairs of
mappings.
Definition 2. Let (X, d) be a metric space with f : X \rightarrow X and T : X \rightarrow CB(X). Then a
hybrid pair of mappings (f, T ) is said to be:
commuting on X [20] if fTx \subseteq Tfx \forall x \in X;
weakly commuting on X [21] if \scrH (fTx, Tfx) \leq d(fx, Tx) \forall x \in X;
compatible [40] if fTx \in CB(X) \forall x \in X and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \scrH (Tfxn, fTxn) = 0, whenever \{ xn\}
is a sequence in X such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn \rightarrow A \in CB(X) \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn \rightarrow t \in A;
noncompatible [22] if exists at least one sequence \{ xn\} in X such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn \rightarrow A \in CB(X) \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn \rightarrow t \in A \mathrm{b}\mathrm{u}\mathrm{t} \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\scrH (Tfxn, fTxn)
is either non-zero or nonexistent;
weakly compatible [17] if Tfx = fTx for each x \in \scrC (f, T );
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1536 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
coincidentally idempotent [13] if for every v \in \scrC (f, T ), ffv = fv, i.e., f is idempotent at the
coincidence points of f and T ;
occasionally coincidentally idempotent [36] if ffv = fv for some v \in \scrC (f, T );
enjoy the property (E.A.) [19] if exists a sequence \{ xn\} in X such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn = t \in A = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn,
for some t \in X and A \in CB(X);
enjoy common limit range property with respect to the mapping f (in short CLRf property)
[14] if exists a sequence \{ xn\} in X such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn = fu \in A = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn,
for some u \in X and A \in CB(X).
The following example demonstrates the interplay of the occasionally coincidentally idempotent
property with other notions described in the preceding definition.
Example 1 ([18], Example 1). Let X = \{ 1, 2, 3\} (with the standard metric),
f :
\biggl(
1 2 3
1 3 2
\biggr)
and T :
\biggl(
1 2 3
\{ 1\} \{ 1, 3\} \{ 1, 3\}
\biggr)
.
Then, it is straight forward to observe the following:
\scrC (f, T ) = \{ 1, 2\} and F (f, T ) = \{ 1\} ,
(f, T ) is not commuting and not weakly commuting,
(f, T ) is not compatible,
(f, T ) is not weakly compatible,
(f, T ) is not coincidentally idempotent since ff2 = f3 = 2 \not = 3 = f2,
(f, T ) is occasionally coincidentally idempotent since ff1 = 1 = f1,
Obviously, in this case (f, T ) is also noncompatible, but simple modifications of this example can
show that the occasionally coincidentally idempotent property is independent of this notion, too.
The following example (taken from [18]) demonstrates the relationship between the property
(E.A.) and common limit range property.
Example 2 ([18], Example 2 and 3). Let X = [0, 2] be a metric space equipped with the usual
metric d(x, y) = | x - y| . Define f, g : X \rightarrow X and T : X \rightarrow CB(X) as follows:
fx=
\left\{ 2 - x, if 0 \leq x < 1,
9
5
, if 1 \leq x \leq 2,
gx=
\left\{ 2 - x, if 0 \leq x \leq 1,
9
5
, if 1 < x \leq 2,
Tx=
\left\{
\biggl[
1
2
,
3
2
\biggr]
, if 0 \leq x \leq 1,\biggl[
1
4
,
1
2
\biggr]
, if 1 < x \leq 2.
One can verify that the pair (f, T ) enjoys the property (E.A.), but not the CLRf property. On
the other hand, the pair (g, T ) satisfies the CLRg property.
Remark 1. If a pair (f, T ) satisfies the property (E.A) along with the closedness of f(X), then
the pair also satisfies the CLRf property.
Throughout this paper, we denote by \BbbR the set of all real numbers, by \BbbR + the set of all positive
real numbers and by \BbbN the set of all positive integers. In what follows, \scrF denote the family of all
functions F : \BbbR + \rightarrow \BbbR that satisfy the following conditions:
(F1) F is continuous and strictly increasing;
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1537
(F2) for each sequence \{ \beta n\} of positive numbers, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta n = 0 \Leftarrow \Rightarrow \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty F (\beta n) = - \infty ;
(F3) there exists k \in (0, 1) such that \mathrm{l}\mathrm{i}\mathrm{m}\beta \rightarrow 0+ \beta kF (\beta ) = 0.
Some examples of functions F \in \scrF are F (t) = \mathrm{l}\mathrm{n} t, F (t) = t + \mathrm{l}\mathrm{n} t, F (t) = - 1/
\surd
t,
F (t) = \mathrm{l}\mathrm{n}(t2 + t), see [47].
Definition 3 [47]. Let (X, d) be a metric space. A self-mapping T on X is called an F -
contraction if there exist F \in \scrF and \tau \in \BbbR + such that
\tau + F (d(Tx, Ty)) \leq F (d(x, y)), (1)
for all x, y \in X with d(Tx, Ty) > 0.
Example 3 [47]. Let F : \BbbR + \rightarrow \BbbR be a mapping given by F (x) = \mathrm{l}\mathrm{n}x. It is clear that F
satisfies (F1) – (F3) for any k \in (0, 1). Under this setting, (1) reduces to
d(Tx, Ty) \leq e - \tau d(x, y) for all x, y \in X, Tx \not = Ty.
Notice that for x, y \in X such that Tx = Ty, the previous inequality also holds and hence T is
a contraction.
In what follows, for a metric space (X, d) and a multivalued mapping T : X \rightarrow CL(X), we
denote
M(x, y) = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
d(x, y), d(x, Tx), d(y, Ty),
1
2
[d(x, Ty) + d(y, Tx)]
\biggr\}
.
Definition 4 [39]. Let (X, d) be a metric space. A multivalued mapping T : X \rightarrow CL(X) is
called an F -contraction if there exist F \in \scrF and \tau \in \BbbR + such that for all x, y \in X with y \in Tx,
exists z \in Ty,
\tau + F (d(y, z)) \leq F (M(x, y)) , \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r} d(y, z) > 0. (2)
Example 4 [39]. Let F : \BbbR + \rightarrow \BbbR be mapping given by F (x) = \mathrm{l}\mathrm{n}x. Then for each multivalued
mapping T : X \rightarrow CL(X) satisfying (2), we have
d(y, z) \leq e - \tau M(x, y) for all x, y \in X, z \in Ty, y \not = z.
It is clear that for z, y \in X such that y = z the previous inequality also holds.
Some fixed point results for single-valued (resp. multivalued) F -contractions were obtained in
[3, 23, 47] (resp. [39]).
Our aim in this paper is to prove a common fixed point theorem for a hybrid pair of occasionally
coincidentally idempotent mappings satisfying generalized (F,\varphi )-contraction condition under CLR
property in complete metric spaces. A similar result for a variant of rational type Hardy – Rogers
generalized (F,\varphi )-contractive condition is also derived. Here, it can be pointed out that Sgroi
and Vetro [39] introduced and studied such conditions for multivalued mappings while the similar
conditions were earlier introduced an studied by Wardowski [47] for single-valued mappings. Our
results generalize and improve several known results of the existing literature. Finally, we utilize our
results to prove the existence of solutions of certain system of functional equations arising in dynamic
programming, as well as Volterra integral inclusion besides providing an illustrative example.
2. The Main Results. This section is divided into two parts. In the first subsection, we prove
a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings
satisfying a generalized (F,\varphi )-contractions condition via CLR property in complete metric spaces,
while in the second one we obtain results for hybrid pairs which satisfy a rational Hardy – Rogers
type (F,\varphi )-contractive condition.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1538 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
Definition 5. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Then hy-
brid pair (f, T ) is said to be a generalized (F,\varphi )-contraction, if there exist an increasing, upper
semicontinuous mapping from the right-hand side
\Phi =
\bigl\{
\varphi : [0,\infty ) \rightarrow [0,\infty ) | \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
s\rightarrow t+
\varphi (s) < \varphi (t), \varphi (t) < t\forall t > 0
\bigr\}
,
F \in \scrF and \tau \in \BbbR + such that
\tau + F (\scrH p(Tx, Ty)) \leq
\leq F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fx, Tx), dp(fy, Ty), dp(fy, fx),
1
2
[dp(fx, Ty) + dp(fy, Tx)] ,
dp(fx, Tx)dp(fy, Ty)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(Tx, Ty)
\right\}
\right)
\right) (3)
for all x, y \in X, p \geq 1 with \scrH (Tx, Ty) > 0.
Definition 6. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Then hybrid pair
(f, T ) is said to be a rational Hardy – Rogers type (F,\varphi )-contraction, if there exist an increasing,
upper semicontinuous mapping from the right-hand side
\Phi = \{ \varphi : [0,\infty ) \rightarrow [0,\infty ) | \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
s\rightarrow t+
\varphi (s) < \varphi (t), \varphi (t) < t,\forall t > 0\} ,
F \in \scrF and \tau \in \BbbR + such that
\tau + F (\scrH p(Tx, Ty)) \leq
\leq F
\left( \varphi
\left( \alpha dp(fx, fy) + \beta [1 + dp(fx, Tx)] dp(fy, Ty)
1 + dp(fx, fy)
+ \gamma [dp(fx, Tx) + dp(fy, Ty)] +
+\delta [dp(fx, Ty) + dp(fy, Tx)]
\right) \right) (4)
for all x, y \in X with Tx \not = Ty, where p \geq 1, \alpha , \beta , \gamma , \delta \geq 0, \alpha + \beta + 2\gamma + 2\delta \leq 1.
Now we propose our first main result as follows:
Theorem 2. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). If the hybrid pair
(f, T ) satisfies generalized (F,\varphi )-contraction condition (3), and also enjoys the CLRf property,
then the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Proof. Since the pair (f, T ) enjoys the CLRf property, there exists a sequence \{ xn\} in X
such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn = fu \in A = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn,
for some u \in X and A \in CB(X). We assert that fu \in Tu. If not, then using condition (3), we have
\tau + F (\scrH p(Txn, Tu)) \leq
\leq F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fxn, Txn), d
p(fu, Tu), dp(fu, fxn),
1
2
[dp(fxn, Tu) + dp(fu, Txn)] ,
dp(fxn, Txn)d
p(fu, Tu)
1 + dp(fu, fxn)
,
dp(fxn, Tu)d
p(fu, Txn)
1 + dp(fu, fxn)
,
dp(fxn, Tu)d
p(fu, Txn)
1 + dp(Txn, Tu)
\right\}
\right)
\right) .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1539
Passing to the limit as n \rightarrow \infty , we get
\tau + F (\scrH p(A, Tu)) \leq
\leq F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fu,A), dp(fu, Tu), 0,
1
2
[dp(fu, Tu) + dp(fu,A)] ,
dp(fu,A)dp(fu, Tu)
1 + dp(fu, fu)
,
dp(fu, Tu)dp(fu,A)
1 + dp(fu, fu)
,
dp(fu, Tu)dp(fu,A)
1 + dp(A, Tu)
\right\}
\right)
\right)
= F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fu,A), dp(fu, Tu), 0,
1
2
[dp(fu, Tu) + dp(fu,A)] ,
dp(fu,A)dp(fu, Tu),
dp(fu,A)dp(fu, Tu)
1 + dp(A, Tu)
\right\}
\right)
\right) .
Using fu \in A, \tau > 0, (F1) and property of \Phi , we obtain
\scrH p(A, Tu) \leq \varphi
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
0, dp(fu, Tu), 0,
1
2
[dp(fu, Tu) + 0], 0, 0
\biggr\} \biggr)
=
= \varphi (dp(fu, Tu)) < dp(fu, Tu).
Since fu \in A the above inequality implies
d(fu, Tu) \leq \scrH (A, Tu) < d(fu, Tu),
a contradiction. Hence fu \in Tu which shows that the pair (f, T ) has a coincidence point (i.e.,
\scrC (f, T ) \not = \varnothing ).
Now, assume that the hybrid pair (f, T ) is occasionally coincidentally idempotent. Then for some
v \in \scrC (f, T ),\mathrm{w}\mathrm{e} \mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e} ffv = fv \in Tv. Our claim is that Tv = Tfv. If not, then using condition
(3), we get
\tau + F (\scrH p(Tfv, Tv)) \leq
\leq F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(ffv, Tfv), dp(fv, Tv), dp(fv, ffv),
1
2
[dp(fv, Tfv) + dp(ffv, Tv)] ,
dp(ffv, Tfv)dp(fv, Tv)
1 + dp(fv, ffv)
,
dp(fv, Tfv)dp(ffv, Tv)
1 + dp(fv, ffv)
,
dp(fv, Tfv)dp(ffv, Tv)
1 + dp(Tfv, Tv)
\right\}
\right)
\right)
= F
\left( \varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fv, Tfv), dp(fv, Tv), 0,
1
2
[dp(fv, Tfv) + dp(fv, Tv)] ,
dp(fv, Tfv)dp(fv, Tv), dp(fv, Tfv)dp(fv, Tv),
dp(fv, Tfv)dp(fv, Tv)
1 + dp(Tfv, Tv)
\right\}
\right)
\right) .
Since fv \in Tv, the above inequality implies
\tau + F (\scrH p(Tfv, Tv)) \leq F
\biggl(
\varphi
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
dp(fv, Tfv), 0, 0,
1
2
dp(fv, Tfv), 0, 0, 0
\biggr\} \biggr) \biggr)
=
= F (\varphi (dp(Tfv, fv))) .
Using (F1) and property of \Phi , we obtain
dp(Tfv, fv) < dp(Tfv, fv),
which is a contradiction. Thus we have fv = ffv \in Tv = Tfv which shows that fv is a common
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1540 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
fixed point of the mappings f and T.
Theorem 2 is proved.
In view of Remark 1, we have the following natural result:
Corollary 1. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). If the hybrid pair
(f, T ) satisfies generalized (F,\varphi )-contraction condition (3), and enjoys the property (E.A.) along
with the closedness of f(X), then the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Notice that, a noncompatible hybrid pair always satisfies the property (E.A.). Hence, we get the
following corollary:
Corollary 2. Let f be a self mapping on a metric space (X, d), T a mapping from X into
CB(X) satisfying generalized (F,\varphi )-contraction condition (3). If the hybrid pair (f, T ) is non-
compatible and f(X) a closed subset of X, then the mappings f and T have a coincidence point.
Moreover, if the pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T ) has a
common fixed point.
If F : \BbbR + \rightarrow \BbbR is defoned by F (t) = \mathrm{l}\mathrm{n} t and denoting e - \tau = k, then we have the following
corollary:
Corollary 3. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Assume that
there exist k \in (0, 1), \varphi \in \Phi such that
\scrH p(Tx, Ty)\leq k\varphi
\left( \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fx, Tx), dp(fy, Ty), dp(fy, fx),
1
2
[dp(fx, Ty) + dp(fy, Tx)] ,
dp(fx, Tx)dp(fy, Ty)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(Tx, Ty)
\right\}
\right)
for all x, y \in X with \scrH (Tx, Ty) > 0, p \geq 1, and the hybrid pair (f, T ) enjoys the CLRf . Then
the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Since every members of \scrF and \Phi are increasing, we can deduce the following far more natural
results from Theorem 2:
Corollary 4. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Assume that
there exist F \in \scrF , \varphi \in \Phi and \tau \in \BbbR + such that
\tau + F (\scrH p(Tx, Ty)) \leq F
\biggl(
\varphi
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
dp(fx, Tx), dp(fy, Ty), dp(fy, fx),
1
2
[dp(fx, Ty) + dp(fy, Tx)]
\biggr\} \biggr) \biggr)
for all x, y \in X with \scrH (Tx, Ty) > 0, p \geq 1, and the hybrid pair (f, T ) enjoys the CLRf . Then
the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Remark 2. Corollary 4 is an improved version of Theorem 11 due to Kadelburg et al. [18].
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COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1541
Corollary 5. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Assume that
there exist F \in \scrF , \varphi \in \Phi and \tau \in \BbbR + such that
\tau + F (\scrH p(Tx, Ty)) \leq F
\biggl(
\varphi
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
dp(fx, Tx)dp(fy, Ty)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(Tx, Ty)
\biggr\} \biggr) \biggr)
for all x, y \in X with \scrH (Tx, Ty) > 0, p \geq 1, and the hybrid pair (f, T ) enjoys the CLRf . Then
the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Now, we present our second main result as follows:
Theorem 3. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). If the hybrid
pair (f, T ) satisfies a rational Hardy – Rogers (F,\varphi )-contraction condition (4) and also enjoys the
CLRf property, then the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
Proof. As the pair (f, T ) shares the CLRf property, there exists a sequence \{ xn\} in X such
that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn = fu \in A = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Txn,
for some u \in X and A \in CB(X). We assert that fu \in Tu. If not, then using condition (4), we have
\tau +F (\scrH p(Txn, Tu))\leq F
\left( \varphi
\left( \alpha dp(fxn, fu) +
\beta [1 + dp(fxn, Txn)]d
p(fu, Tu)
1 + dp(fxn, fu)
+
+\gamma [dp(fxn, Txn)+dp(fu, Tu)]+\delta [dp(fxn, Tu)+dp(fu, Txn)]
\right) \right) .
Passing to the limit as n \rightarrow \infty in the above inequality, we obtain
\tau + F (\scrH p(A, Tu)) \leq F (\varphi (\beta + \gamma + \delta )dp(fu, Tu)).
Using \tau > 0 and (F1) and property of \Phi , it follows that
dp(fu, Tu) \leq dp(A, Tu) < (\beta + \gamma + \delta )dp(fu, Tu),
a contradiction, as \beta + \gamma + \delta \leq 1. Hence, fu \in Tu which shows that the hybrid pair (f, T ) has a
coincidence point (i.e., \scrC (f, T ) \not = \varnothing ).
Now, if the mappings f and T are occasionally coincidentally idempotent, then there exists
v \in \scrC (f, T ) such that ffv = fv \in Tv. Our claim is that fu is the common fixed point of f and T.
It is sufficient to show that Tv = Tfv. If not, then using condition (4), we get
\tau +F (\scrH p(Tfv, Tv))\leq
\leq F
\left( \varphi
\left( \alpha dp(ffv, fv) +
\beta [1 + dp(ffv, Tfv)]dp(fv, Tv)
1 + dp(ffv, fv)
+
+\gamma [dp(ffv, Tfv)+dp(fv, Tv)]+\delta [dp(ffv, Tv)+dp(fv, Tfv)]
\right) \right) =
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1542 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
= F
\Biggl(
\varphi
\Biggl(
\beta [1 + dp(fv, Tfv)]dp(fv, Tv) + \gamma [dp(fv, Tfv) + dp(fv, Tv)]+
+\delta [dp(fv, Tv) + dp(fv, Tfv)]
\Biggr) \Biggr)
.
Since fv \in Tv, the above inequality implies
\tau + F (dp(Tfv, Tv)) \leq F (\varphi (\gamma + \delta )dp(fv, Tfv)) .
Using (F1) and property of \Phi , we can have
dp(Tfv, fv) < (\gamma + \delta )dp(fv, Tfv),
a contradiction, as \gamma + \delta \leq 1. Thus, fv = ffv \in Tv = Tfv which shows that fv is a common
fixed point of the mappings f and T.
Theorem 3 is proved.
If F : \BbbR + \rightarrow \BbbR is given by F (t) = \mathrm{l}\mathrm{n} t and denoting e - \tau = k, then we have the following
corollary:
Corollary 6. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Suppose that
there exist k \in (0, 1), \varphi \in \Phi such that
\scrH p(Tx, Ty)\leq k\varphi
\left( \alpha dp(fx, fy)+
\beta [1 + dp(fx, Tx)]dp(fy, Ty)
1 + dp(fx, fy)
+\gamma [dp(fx, Tx) + dp(fy, Ty)]+
+\delta [dp(fx, Ty) + dp(fy, Tx)]
\right)
for all x, y \in X with Tx \not = Ty, where p \geq 1, \alpha , \beta , \gamma , \delta \geq 0, \alpha + \beta + 2\gamma + 2\delta \leq 1, and the hybrid
pair (f, T ) enjoys the CLRf . Then the mappings f and T have a coincidence point.
Moreover, if the pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T ) has a
common fixed point.
In view of Remark 1 and increasingness of the members of \scrF and \Phi , we have the following
natural corollary:
Corollary 7. Let (X, d) be a metric space, f : X \rightarrow X and T : X \rightarrow CB(X). Suppose there
exist F \in \scrF , \varphi \in \Phi and \tau \in \BbbR + such that
\tau + F (\scrH p(Tx, Ty)) \leq
\leq F (\varphi (\alpha dp(fx, fy) + \beta [dp(fx, Tx) + dp(fy, Ty)] + \gamma [dp(fx, Ty) + dp(fy, Tx)]))
for all x, y \in X with Tx \not = Ty, where p \geq 1, \alpha , \beta , \gamma \geq 0, \alpha +2\beta +2\gamma \leq 1, and enjoys the property
(E.A.) along with the closedness of f(X). Then the mappings f and T have a coincidence point.
Moreover, if the hybrid pair (f, T ) is occasionally coincidentally idempotent, then the pair (f, T )
has a common fixed point.
3. Illustrative example. In this section, we provide an example to establish the genuineness of
our extension.
Example 5. Let X = [0, 3] be a metric space equipped with the metric d(x, y) = | x - y| . Define
f : X \rightarrow X and T : X \rightarrow CB(X) as follows:
fx =
\left\{ 3 - x, if x \in [0, 2],
3, if x \in (2, 3],
Tx =
\left\{
[1, 2] , if x \in [0, 2],\biggl[
0,
1
2
\biggr]
, if x \in (2, 3].
Let F : \BbbR + \rightarrow \BbbR such that F (t) = t + ln(t), \varphi : \BbbR + \rightarrow \BbbR + such that \varphi (t) =
9
10
t and \tau =
1
5
> 0
and p \geq 1, then the condition (3) takes the form
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COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1543
\scrH p(Tx, Ty) \leq 9
10
\Theta (x, y)e
9
10\Theta (x,y) - \scrH p(Tx,Ty) - 1
5 , (5)
where
\Theta (x, y) = \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
dp(fx, Tx), dp(fy, Ty), dp(fy, fx),
1
2
[dp(fx, Ty) + dp(fy, Tx)] ,
dp(fx, Tx)dp(fy, Ty)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(fy, fx)
,
dp(fx, Ty)dp(fy, Tx)
1 + dp(Tx, Ty)
\right\} .
Then it is easy to verify that
F \in \scrF ; \varphi \in \Phi ; f(X) = [1, 3] \cup \{ 3\} , a closed set in X; \scrC (f, F ) = [1, 2];
the hybrid pair (f, T ) satisfies CLRf property, as for the sequence
\biggl\{
1 +
1
n
\biggr\}
n\in \BbbN
,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
fxn = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\biggl(
2 - 1
n
\biggr)
= 2 = f1 \in [1, 2] = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
T
\biggl(
1 +
1
n
\biggr)
;
(f, T ) is not coincidentally idempotent because ff1 = f2 = 1 \not = 2 = f1;
(f, T ) is occasionally coincidentally idempotent, because ff
3
2
= f
3
2
=
3
2
.
Now, in order to verify condition (5), we distinguish two cases:
Case I. If x \in [0, 2] \mathrm{a}\mathrm{n}\mathrm{d} y \in (2, 3], then
\scrH (Tx, Ty) = \scrH
\biggl(
[1, 2],
\biggl[
0,
1
2
\biggr] \biggr)
=
= \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
d
\biggl(
[1, 2],
\biggl[
0,
1
2
\biggr] \biggr)
, d
\biggl( \biggl[
0,
1
2
\biggr]
, [1, 2]
\biggr) \biggr\}
= \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
3
2
, 1
\biggr\}
=
3
2
,
and d(fy, Ty) = d
\biggl(
3,
\biggl[
0,
1
2
\biggr] \biggr)
=
5
2
. Therefore, (5) reduces to
\biggl(
3
2
\biggr) p
<
9
10
\biggl(
5
2
\biggr) p
e
9
10
\Bigl(
5
2
\Bigr) p
-
\Bigl(
3
2
\Bigr) p
- 1
5
which is true for all p \geq 1.
Case II. If x \in (2, 3] \mathrm{a}\mathrm{n}\mathrm{d} y \in [1, 2], then
\scrH (Tx, Ty) = \scrH
\biggl( \biggl[
0,
1
2
\biggr]
, [1, 2]
\biggr)
=
3
2
and d(fx, Tx) = d
\biggl(
3,
\biggl[
0,
1
2
\biggr] \biggr)
=
5
2
.
Therefore, (5) reduces to \biggl(
3
2
\biggr) p
<
9
10
\biggl(
5
2
\biggr) p
e
9
10
\Bigl(
5
2
\Bigr) p
-
\Bigl(
3
2
\Bigr) p
- 1
5
which is true for all p \geq 1.
Notice that for x, y \in [1, 2] (or x, y \in (2, 3]) \scrH (Tx, Ty) = 0 and so (5) is true.
Thus, all the hypotheses of Theorem 2 are satisfied and the hybrid pair (f, T ) has the common
fixed point
\Bigl(
namely
3
2
\Bigr)
.
With a view to establish genuineness of our extension, notice that for x = 1, y = 3, we have
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1544 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
\scrH (Tx, Ty) =
3
2
; d(fx, fy) = d(2, 3) = 1;
1
2
[d(fx, Tx) + d(fy, Ty)] =
1
2
\biggl[
d(2, [1, 2]) + d
\biggl(
3,
\biggl[
0,
1
2
\biggr] \biggr) \biggr]
=
1
2
\biggl(
0 +
5
2
\biggr)
=
5
4
and
1
2
[d(fx, Ty) + d(fy, Tx)] =
1
2
\biggl[
d
\biggl(
2,
\biggl[
0,
1
2
\biggr] \biggr)
+ d(3, [1, 2])
\biggr]
=
1
2
\biggl(
3
2
+ 1
\biggr)
=
5
4
,
which shows that the contractive condition of Theorem 11 (due to Kadelburg et al. [18]) is not
satisfied. Thus, in all our results (Corollary 4 as well as Theorem 2) are applicable to the present
example while Theorem 11 of Kadelburg et al. [18] is not which substantiates the utility of Theorem 2.
4. Applications. As applications of our main results, we prove an existence theorem on bounded
solutions of a system of functional equations. Also, an existence theorem on the solution of integral
inclusion is proved.
4.1. Application to dynamic programming. In 1978, Bellman and Lee [5] first studied the
existence of solutions for functional equations wherein authors notice that the basic form of functional
equations in dynamic programming can be described as follows:
q(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in D
\{ G(x, y, q(\tau (x, y)))\} , x \in W,
where \tau : W \times D \rightarrow W, G : W \times D\times \BbbR \rightarrow \BbbR are mappings, while W \subseteq U is a state space, D \subseteq V
is a decision space, and U, V are Banach spaces.
In 1984, Bhakta and Mitra [6] obtained some existence theorems for the following functional
equation which arises in multistage decision process related to dynamic programming:
q(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in D
\bigl\{
g(x, y) +G(x, y, q(\tau (x, y)))
\bigr\}
, x \in W,
where \tau : W \times D \rightarrow W, g : W \times D \rightarrow \BbbR , G : W \times D \times \BbbR \rightarrow \BbbR are mappings, while W \subseteq U is a
state space, D \subseteq V is a decision space, and U, V are Banach spaces.
In recent years, a lot of work have been done in this direction wherein a multitude of existence
and uniqueness results have been obtained for solutions and common solutions of some functional
equations, including systems of functional equations in dynamic programming using suitable fixed
point results. For more details one can consults [26, 27, 31 – 33, 37] and the references therein.
Consider now a multistage process, reduced to the system of functional equations
qi(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in D
\bigl\{
g(x, y) +Gi(x, y, qi(\tau (x, y)))
\bigr\}
, x \in W, i \in \{ 1, 2\} , (6)
where \tau : W \times D \rightarrow W, g : W \times D \rightarrow \BbbR , Gi : W \times D\times \BbbR \rightarrow \BbbR are given mappings, while W \subseteq U
is a state space, D \subseteq V is a decision space, and U, V are Banach spaces. The purpose of this section
is to prove the existence of solutions for a system of functional equations (6) using Theorem 2.
Let B(W ) be the set of all bounded real-valued functions on W. For an arbitrary h \in B(W )
define | | h| | = \mathrm{s}\mathrm{u}\mathrm{p}x\in W | h(x)| , with respective metric d. Also, (B(W ), | | \cdot | | ) is a Banach space
wherein convergence is uniform. Therefore, if we consider a Cauchy sequence \{ hn\} in B(W ), then
the sequence \{ hn\} converges uniformly to a function, say h\ast , so that h\ast \in B(W ).
We consider the operators Ti : B(W ) \rightarrow B(W ) given by
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COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1545
Tihi(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in D
\Bigl\{
g(x, y) +Gi(x, y, hi(\tau (x, y)))
\Bigr\}
, (7)
for hi \in B(W ), x \in W, for i = 1, 2; these mappings are well-defined if the functions g and Gi are
bounded. Also, denote
\Theta (h, k)=\mathrm{m}\mathrm{a}\mathrm{x}
\left\{
d(T2h, T2k), d(T2h, T1h), d(T2k, T1k),
d(T1h, T2k) + d(T1k, T2h)
2
,
d(T1h, T2h)d(T1k, T2k)
1 + d(T2k, T2h)
,
d(T1h, T2k)d(T1k, T2h)
1 + d(T2k, T2h)
,
d(T1h, T2k)d(T1k, T2h)
1 + d(T1h, T1k)
\right\} (8)
for h, k \in B(W ).
Theorem 4. Let Ti : B(W ) \rightarrow B(W ) be given by (7), for i = 1, 2. Suppose that the following
hypotheses hold:
(i) there exist \tau \in \BbbR + and \varphi \in \Phi such that
| G1(x, y, h(x)) - G2(x, y, k(x))| \leq
\varphi (\Theta (h, k)(x))
(1 + \tau
\sqrt{}
\mathrm{s}\mathrm{u}\mathrm{p}x\in W \varphi (\Theta (h, k)(x)))2
for all x \in W, y \in D;
(ii) g : W \times D \rightarrow \BbbR and Gi : W \times D \times \BbbR \rightarrow \BbbR are bounded functions, for i = 1, 2;
(iii) there exists a sequence \{ hn\} in B(W ) and a function h\ast \in B(W ) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
T1hn = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
T2hn = T1h
\ast ;
(iv) T1T1h = T1h, whenever T1h = T2h, for some h \in B(W ).
Then the system of functional equations (6) has a bounded solution.
Proof. By hypothesis (iii), the pair (T1, T2) shares the common limit range property with respect
to T1. Now, let \lambda be an arbitrary positive number, x \in W and h1, h2 \in B(W ). Then there exist
y1, y2 \in D such that
T1h1(x) < g(x, y1) +G1(x, y1, h1(\tau (x, y1))) + \lambda , (9)
T2h2(x) < g(x, y2) +G2(x, y2, h2(\tau (x, y2))) + \lambda , (10)
T1h1(x) \geq g(x, y2) +G1(x, y2, h1(\tau (x, y2))), (11)
T2h2(x) \geq g(x, y1) +G2(x, y1, h2(\tau (x, y1))). (12)
Next, by using (9) and (12), we obtain
T1h1(x) - T2h2(x) < G1(x, y1, h1(\tau (x, y1))) - G2(x, y1, h2(\tau (x, y1))) + \lambda \leq
\leq | G1(x, y1, h1(\tau (x, y1))) - G2(x, y1, h2(\tau (x, y1)))| + \lambda \leq
\leq \varphi (\Theta (h, k)(x))
(1 + \tau
\sqrt{}
\mathrm{s}\mathrm{u}\mathrm{p}x\in W \varphi (\Theta (h, k)(x)))2
+ \lambda
and so we have
T1h1(x) - T2h2(x) <
\varphi (\Theta (h, k)(x))
(1 + \tau
\sqrt{}
\mathrm{s}\mathrm{u}\mathrm{p}x\in W \varphi (\Theta (h, k)(x)))2
+ \lambda . (13)
Analogously, by using (10) and (11), we get
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1546 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
T2h2(x) - T1h1(x) <
\varphi (\Theta (h, k)(x))
(1 + \tau
\sqrt{}
\mathrm{s}\mathrm{u}\mathrm{p}x\in W \varphi (\Theta (h, k)(x)))2
+ \lambda . (14)
Combining (13) and (14), we obtain
| T1h1(x) - T2h2(x)| <
\varphi (\Theta (h, k)(x))
(1 + \tau
\sqrt{}
\mathrm{s}\mathrm{u}\mathrm{p}x\in W \varphi (\Theta (h, k)(x)))2
+ \lambda ,
implying thereby
d(T1h1, T2h2) \leq
\varphi (\Theta (h, k))
(1 + \tau
\sqrt{}
\varphi (\Theta (h, k)))2
+ \lambda .
Notice that, the last inequality does not depend on x \in W and \lambda > 0 is taken arbitrarily, therefore
we have
d(T1h1, T2h2) \leq
\varphi (\Theta (h, k))
(1 + \tau
\sqrt{}
\varphi (\Theta (h, k)))2
.
If we consider F \in \scrF defined by F (t) =
- 1\surd
t
, for each t \in (0,+\infty ), and put f = T1, T = T2, then
we get condition
\tau + F (d(fh1, Th2)) \leq F (\varphi (\Theta (h, k)))
where \Theta (h, k) is given in (8). Thus all the hypotheses of Theorem 2 are satisfied for the pair (f, T )
and p = 1. Moreover, in view of the hypotheses (iv), the pair (T1, T2) is occasionally coincidentally
idempotent, so by using Theorem 2, the mapping T1 and T2 have a common fixed point, that is, the
system of functional equations (6) has a bounded solution.
4.2. Application to Volterra integral inclusions. Here, we present yet another application of
Theorem 3. This application is essentially inspired by [46].
We establish new results on the existence of solutions of integral inclusion of the type
x(t) \in q(t) +
\sigma (t)\int
0
k(t, s)F (s, x(s)) ds (15)
for t \in J = [0, 1] \subset \BbbR , where \sigma : J \rightarrow J, q : J \rightarrow E, k : J \times J \rightarrow \BbbR are continuous and F :
J \times E \rightarrow C(E), where E is a Banach space with norm \| \cdot \| E and C(E) denotes the class of all
nonempty closed subsets of E.
Let C(J,E) be the space of all continuous E -valued functions on J. Define a norm \| \cdot \| on
C(J,E) by
\| x\| = \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\| x(t)\| E .
Definition 7. A continuous function a \in C(J,E) is called a lower solution of the integral
inclusion (15), if it satisfies
a(t) \leq q(t) +
\sigma (t)\int
0
k(t, s)v1(s)ds \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} v1 \in B(J,E)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1547
such that v1(t) \in F (t, a(t)) \mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t} \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} (a.e.) \mathrm{f}\mathrm{o}\mathrm{r} t \in J, where B(J,E) is the space of all
E -valued Bochner-integrable functions on J. Similarly, a continuous function b \in C(J,E) is called
an upper solution of the integral inclusion (15), if it satisfies
b(t) \geq q(t) +
\sigma (t)\int
0
k(t, s)v2(s)ds, for all v2 \in B(J,E)
such that v2(t) \in F (t, b(t)) a.e. for t \in J.
Notice that, all the solution lies between lower solution `a’ as well upper solution `b’. We can
denote the solution set as an interval [a, b].
Definition 8. A continuous function x : J \rightarrow E is said to be a solution of the integral inclusion
(15), if
x(t) = q(t) +
\sigma (t)\int
0
k(t, s)v(s) ds
for some v \in B(J,E) satisfying v(t) \in F (t, x(t)) for all t \in J.
In what follows, we also need the following definitions:
Definition 9. A multivalued mapping F : J \rightarrow 2E is said to be measurable if for any y \in E,
the function t \mapsto \rightarrow d(y, F (t)) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \| y - x\| : x \in F (t)\} is measurable.
Definition 10. A multivalued mapping \beta : J \times E \rightarrow 2E is called Carathéodory if
(i) t \mapsto \rightarrow (t, x) is measurable for each x \in E, and
(ii) x \mapsto \rightarrow (t, x) is upper semicontinuous almost everywhere for t \in J.
Denote
\| F (t, x)\| = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| u\| E : u \in F (t, x)\} .
Definition 11. A Carathéodory multimapping F (t, x) is called L1-Carathéodory if for every
real number r > 0, there exists a function hr \in L1(J,\BbbR ) such that
\| F (t, x)\| \leq hr(t) for almost every t \in J
and for all x \in E with \| x\| E \leq r.
Denote
S1
F (x) =
\bigl\{
v \in B(J,E) : v(t) \in F (t, x(t)) a.e. t \in J
\bigr\}
.
Lemma 1 [25]. If \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m} (E) < \infty and F : J \times E \rightarrow 2E is L1-Carathéodory, then S1
F (x) \not = \varnothing
for each x \in C(J,E).
Lemma 2 [46]. Let E be a Banach space, F a Carathéodory multimappping with S1
F \not = \varnothing and
\scrL : L1(J,E) \rightarrow C(J,E) a continuous linear mapping. Then the operator
\scrL \circ S1
F : C(J,E) \rightarrow 2C(J,E)
is a closed graph operator on C(J,E)\times C(J,E).
Let us list the following set of conditions:
(H0) the function k(t, s) is continuous and nonnegative on J \times J with
\mathrm{s}\mathrm{u}\mathrm{p}
t,s\in J
k(t, s) \leq 1;
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1548 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
(H1) the multivalued mapping F (t, x) is Carathéodory;
(H2) the multivalued mapping F (t, x) is increasing in x almost everywhere for t \in J ;
(H3) There exist \tau \in \BbbR + and \varphi \in \Phi such that
| F (s, x(s)) - F (s, y(s))| \leq
\sqrt{}
[e - \tau
\biggl[
(\varphi (\Delta (x, y)))2 + \varphi (\Delta (x, y))] +
1
4
\biggr]
- 1
2
for all s \in J, x \in E, where
\Delta (x, y) = \alpha | fx - fy| + \beta [1 + | fx - Tx| ] | fy - Ty|
1 + | fx - fy|
+ \gamma [| fx - Tx| + | fy - Ty| ] +
+\delta [| fx - Ty| + | fy - Tx| ]
with \alpha , \beta , \gamma , \delta \geq 0, \alpha + \beta + 2\gamma + 2\delta \leq 1;
(H4) S1
F (x) \not = \varnothing for each x \in C(J,E).
Theorem 5. Suppose that the conditions (H0) – (H4) hold. Then the integral inclusion (15) has
a solution in [a, b] defined on J.
Proof. Let X = C(J,E). Define a multivalued mapping T : [a, b] \subset X \rightarrow 2X given by
Tx =
\left\{ u \in [a, b] : u(t) = q(t) +
\sigma (t)\int
0
k(t, s)v(s) ds; v \in S1
F (x), for every t \in [0, 1]
\right\} .
Observe that T is well-defined, as owing to (H4), S
1
F (x) \not = \varnothing . To show that T satisfies all hypotheses
of Theorem 3 defined on [a, b].
For all \vargamma , \mu \in 2X on t \in J and making use of (H0) and (H3), we have
\bigl(
for v1, v2 \in S1
F (x)
\bigr)
\| \vargamma (t) - \mu (t)\| E =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\sigma (t)\int
0
k(t, s)v1(s) ds -
\sigma (t)\int
0
k(t, s)v2(s) ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
E
\leq
\leq
\sigma (t)\int
0
k(t, s) ds\| v1(s) - v2(s)\| E \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
t,s\in J
k(t, s)
\sqrt{}
[e - \tau
\biggl[
(\varphi (\Delta (v1, v2)))2 + \varphi (\Delta (v1, v2))] +
1
4
\biggr]
- 1
2
.
This implies that
\| \vargamma (t) - \mu (t)\| E \leq
\sqrt{}
[e - \tau
\biggl[
(\varphi (\Delta (v1, v2)))2 + \varphi (\Delta (v1, v2))] +
1
4
\biggr]
- 1
2
,
for each t \in J.
On considering F \in \scrF defined by F (t) = ln(t2 + t), for each t \in (0,+\infty ), then we have
condition
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
COMMON FIXED POINT THEOREMS FOR HYBRID GENERALIZED (F,\varphi )-CONTRACTIONS . . . 1549
\tau + F (\| \vargamma (t) - \mu (t)\| E) \leq F (\varphi (\Delta (v1, v2))).
Thus we deduce that the operator T satisfy condition (4) where f is an identity mapping and p = 1.
Also T is a closed mapping, using Theorem 3, we conclude that the given integral inclusion has a
solution in [a, b].
Theorem 5 is proved.
Acknowledgement. The first author is thankful to the United State-India Education Foun-
dation, New Delhi, India and IIE/CIES, Washington, DC, USA for Fulbright-Nehru PDF Award
(No. 2052/FNPDR/2015).
References
1. Aamri M., El Moutawakil D. Some new common fixed point theorems under strict contractive conditions // J. Math.
Anal. and Appl. – 2002. – 270, № 1. – P. 181 – 188.
2. Ali J., Imdad M. Common fixed points of nonlinear hybrid mappings under strict contractions in semi-metric spaces //
Nonlinear Anal. Hybrid Syst. – 2010. – 4, № 4. – P. 830 – 837.
3. Alsulami H. H., Karapinar E., Piri H. Fixed points of modified F -contractive mappings in complete metric-like
spaces // J. Funct. Spaces. – 2014. – Article ID 270971.
4. Aubin J. P., Cellina A. Differential Inclusions. – Berlin: Springer, 1984.
5. Bellman R., Lee E. S. Functional equations in dynamic programming // Aequat. Math. – 1978. – 17. – P. 1 – 18.
6. Bhakta T. C., Mitra S. Some existence theorems for functional equations arising in dynamic programming // J. Math.
Anal. and Appl. – 1984. – 98. – P. 348 – 362.
7. Chauhan S., Khan M. A., Kadelburg Z., Imdad M. Unified common fixed point theorem for a hybrid pair of mappings
via an implicit relation involving altering distance function // Abstr. and Appl. Anal. – Vol. 2014. – Article ID 718040. –
8 p.
8. Deimling K. Multivalued differential equations. – Berlin: Walter de Gruyter, 1992.
9. Dhage B. C. A functional integral inclusion involving Carathéodories // Electron. J. Qual. Theory Different. Equat. –
2003. – 14. – P. 1 – 18.
10. Dhompongsa S., Yingtaweesittikul H. Fixed points for multivalued mappings and the metric completeness // Fixed
Point Theory and Appl. – 2009. – Article ID 972395.
11. Fisher B. Common fixed point theorems for mappings and set-valued mappings // Rostock. Math. Kolloq. – 1981. –
18. – P. 69 – 77.
12. Hu T. Fixed point theorems for multivalued mappings // Can. Math. Bull. – 1980. – 23. – P. 193 – 197.
13. Imdad M., Ahmad A., Kumar S. On nonlinear nonself hybrid contractions // Rad. Mat. – 2001. – 10, № 2. –
P. 233 – 244.
14. Imdad M., Chauhan S., Soliman A. H., Ahmed M. A. Hybrid fixed point theorems in symmetric spaces via common
limit range property // Demonstr. Math. – 2014. – 47, № 4. – P. 949 – 962.
15. Imdad M., Khan M. S., Sessa S. On some weak conditions of commutativity in common fixed point theorems // Int.
J. Math. and Math. Sci. – 1988. – 11, № 2. – P. 289 – 296.
16. Imdad M., Soliman A. H. Some common fixed point theorems for a pair of tangential mappings in symmetric spaces
// Appl. Math. Lett. – 2010. – 23, № 4. – P. 351 – 355.
17. Jungck G., Rhoades B. E. Fixed points for set valued functions without continuity // Indian J. Pure Appl. and Math. –
1998. – 29, № 3. – P. 227 – 248.
18. Kadelburg Z., Chauhan S., Imdad M. A hybrid common fixed point theorem under certain recent properties // Sci.
World J. – 2014. – Article ID 860436. – 6 p.
19. Kamran T. Coincidence and fixed points for hybrid strict contractions // J. Math. Anal. and Appl. – 2004. – 299, № 1. –
P. 235 – 241.
20. Kaneko H. Single-valued and multi-valued f -contractions // Boll. Unione Mat. Ital. – 1985. – 4. – P. 29 – 33.
21. Kaneko H. A common fixed point of weakly commuting multi-valued mappings // Math. Jap. – 1988. – 33, № 5. –
P. 741 – 744.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1550 H. K. NASHINE, M. IMDAD, MD AHMADULLAH
22. Kaneko H., Sessa S. Fixed point theorems for compatible multi-valued and single-valued mappings // Int. J. Math.
and Math. Sci. – 1989. – 12, № 2. – P. 257 – 262.
23. Karapinar E., Piri H., Al-Sulami H. Fixed points of generalized F -Suzuki type contraction in complete b-metric
spaces // Discrete Dyn. Nat. Sci. – 2015. – Article ID 969726. – 8 p.
24. Kubiak T. Fixed point theorems for contractive type multivalued mappings // Math. Jap. – 1985. – 30, № 1. –
P. 89 – 101.
25. Lasota A., Opial Z. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations //
Bull. acad. pol. sci. Ser. sci. math., astron. et phys. – 1965. – 13. – P. 781 – 786.
26. Liu Z., Ume J. S. On properties of solutions for a class of functional equations arising in dynamic programming // J.
Optim. Theory and Appl. – 2003. – 117. – 533.
27. Liu Z., Wong L., Kug Kim H., Kang S. M. Common fixed point theorems for contactive mappings and their applications
in dynamic programming // Bull. Korean Math. Soc. – 2008. – 45, № 3. – P. 573 – 585.
28. Nadler Jr. S. B. Multivalued contraction mappings // Pacif. J. Math. – 1969. – 20, № 2. – P. 457 – 488.
29. Naimpally S. A., Singh S. L., Whitfield J. H. M. Coincidence theorems for hybrid contractions // Math. Nachr. – 1986. –
127. – P. 177 – 180.
30. Pathak H. K. Fixed point theorems for weak compatible multi-valued and single-valued mappings // Acta Math.
Hung. – 1995. – 67, № 1-2. – P. 69 – 78.
31. Pathak H. K., Cho Y. J., Kang S. M., Lee B. S. Fixed point theorems for compatible mappings of type (P ) and
applications to dynamic programming // Matematiche. – 1995. – 50. – P. 15 – 33.
32. Pathak H. K., Deepmala Some existing theorems for solvability of certain functional equations arising in dynamic
programming // Bull. Calcutta Math. Soc. – 2012. – 104, № 3. – P. 237 – 244.
33. Pathak H. K., Fisher B. Common fixed point theorems with applications in dynamic programming // Glas. Mat. –
1996. – 31. – P. 321 – 328.
34. Pathak H. K., Kang S. M., Cho Y. J. Coincidence and fixed point theorems for nonlinear hybrid generalized
contractions // Math. J. – 1998. – 48(123). – P. 341 – 357.
35. Pathak H. K., Khan M. S. Fixed and coincidence points of hybrid mappings // Arch. Math. – 2002. – 38. – P. 201 – 208.
36. Pathak H. K., Rodrı́guez – López R. Noncommutativity of mappings in hybrid fixed point results // Bound. Value
Probl. – 2013. – 145. – 2013. – 21 p.
37. Pathak H. K., Tiwari R. Common fixed points for weakly compatible mappings and applications in dynamic
programming // Ital. J. Pure and Appl. Math. – 2013. – 30. – P. 253 – 268.
38. O’Regan D. Integral inclusions of upper semi-continuous or lower semi-continuous type // Proc. Amer. Math. Soc. –
1996. – 124. – P. 2391 – 2399.
39. Sgroi M., Vetro C. Multi-valued F-contractions and the solution of certain functional and integral equations // Filomat. –
2013. – 27, № 7. – P. 1259 – 1268.
40. Singh S. L., Ha K. S., Cho Y. J. Coincidence and fixed points of nonlinear hybrid contractions // Int. J. Math. and
Math. Sci. – 1989. – 12, № 2. – P. 247 – 256.
41. Singh S. L., Hashim A. M. New coincidence and fixed point theorems for strictly contractive hybrid maps // Aust. J.
Math. Anal. and Appl. – 2005. – 2, № 1. – Article 12. – 7 p.
42. Singh S. L., Mishra S. N. Coincidence and fixed points of non-self hybrid contractions // J. Math. Anal. and Appl. –
2001. – 256. – P. 486 – 497.
43. Singh S. L., Mishra S. N. Coincidence theorems for certain classes of hybrid contractions // Fixed Point Theory and
Appl. – 2010. – Article ID 898109.
44. Sintunavarat W., Kumam P. Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric
spaces // J. Appl. Math. – 2011. – Article ID 637958. – 14 p.
45. Sessa S., Khan M. S., Imdad M. Common fixed point theorem with a weak commutativity condition // Glas. Mat. –
1986. – 21(41). – P. 225 – 235.
46. Türkoǧlu D., Altun I. A fixed point theorem for multi-valued mappings and its applications to integral inclusions //
Appl. Math. Lett. – 2007. – 20. – P. 563 – 570.
47. Wardowski D. Fixed points of a new type of contractive mappings in complete metric spaces // Fixed Point Theory
and Appl. – 2012. – 2012, № 94.
Received 09.02.16
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|
| id | umjimathkievua-article-1801 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:56Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/07/bd5fbc05a2a380dc467ffb7490b83407.pdf |
| spelling | umjimathkievua-article-18012019-12-05T09:27:02Z Common fixed-point theorems for hybrid generalized $(F, ϕ)$ -contractions under the common limit range property with applications ...................... Спiльнi теореми про нерухому точку для гiбридних узагальнених $(F, ϕ)$-стискань з властивiстю спiльного граничного дiапазону з застосуваннями Ahmadullah, M. Imdad, M. Nashine, H. K. Ахмадулла, М. Імдад, М. Нашине, Г. К. We consider a relatively new hybrid generalized $F$ -contraction involving a pair of mappings and use this contraction to prove a common fixed-point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized $(F, \varphi)$-contraction condition with the common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying the rational-type Hardy – Rogers $(F, \varphi)$-contractive condition is also proved. We generalize and improve several results available from the existing literature. As applications of our results, we prove two theorems for the existence of solutions of certain system of functional equations encountered in dynamic programming and the Volterra integral inclusion. Moreover, we provide an illustrative example. Розглянуто вiдносно нове узагальнене гiбридне $F$ -стискання, що включає пару вiдображень. Це стисканя застосовано при доведеннi спiльної теормеми про нерухому точку для випадково спiвпадаючих iдемпотентних матриць, що задовольняють узагальнену умову $(F, \varphi)$-стискання при влативостi спiльного граничного дiапазону в повних метричних просторах. Також доведено подiбний результат для гiбридних пар вiдображень, що задовольняють умо- ву Гардi – Роджерса про $(F, \varphi)$-стискання рацiонального типу. Узагальнено та покращено деякi вiдомi лiтературнi результати. Як застосування наших результатiв, доведено двi теореми про iснування розв’язкiв деякої системи функцiональних рiвнянь, що зустрiчаються в динамiчному програмуваннi, та iнтегрального включення Вольтерра. Крiм того, наведено iлюстративний приклад. Institute of Mathematics, NAS of Ukraine 2017-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1801 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 11 (2017); 1534-1550 Український математичний журнал; Том 69 № 11 (2017); 1534-1550 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1801/783 Copyright (c) 2017 Ahmadullah M.; Imdad M.; Nashine H. K. |
| spellingShingle | Ahmadullah, M. Imdad, M. Nashine, H. K. Ахмадулла, М. Імдад, М. Нашине, Г. К. Common fixed-point theorems for hybrid generalized $(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title | Common fixed-point theorems for hybrid generalized
$(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title_alt | Спiльнi теореми про нерухому точку для гiбридних
узагальнених $(F, ϕ)$-стискань з властивiстю спiльного
граничного дiапазону з застосуваннями |
| title_full | Common fixed-point theorems for hybrid generalized
$(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title_fullStr | Common fixed-point theorems for hybrid generalized
$(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title_full_unstemmed | Common fixed-point theorems for hybrid generalized
$(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title_short | Common fixed-point theorems for hybrid generalized
$(F, ϕ)$ -contractions under the common limit range property with applications ...................... |
| title_sort | common fixed-point theorems for hybrid generalized
$(f, ϕ)$ -contractions under the common limit range property with applications ...................... |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1801 |
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