2.1. Multivalued Mappings in GMMS
Two fixed point theorems for multivalued contraction mapping are proved in [
2] by Nadler. The first, a generalization of the contraction mapping principle of Banach, states as a multivalued contraction mapping of a complete metric space into the nonempty closed and bounded subsets of same metric space has a fixed point. The second, a generalization of a result of Edelstein, is a fixed point theorem for compact set-valued local contractions. Nadler’s study is applied through other metric spaces, such as in [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19].
Feng and Liu [
20] gave one of the most important generalization of Nadler’s result without using Pompei-Hausdorff distance. Then many studies focused on those results and applied them in different metric spaces; for example in [
21]. We consider Feng-Liu theorem in GMMS.
Let be a GMMS and sequentially open subset of , while each sequence of has for some , and there exists a point such that most part of the sequence included in B.
Let be a family of all sequentially open subsets of . Any convergent sequence in is convergent in a topological space . When we take as a family of all nonempty closed subsets of and a family of all nonempty subsets A of , we have the following property. Then we threat these two subsets as they are equal. If , then for all , while . If the property is satisfied for any and , then there exists a sequence in B such that . In a topological space , we have such that most part of the sequence included in B, which means , so . As a result . If we have , and a sequence in such that , then no subsequence in A satisfy for any . So is found. We have . The result gives us . In addition, the definition of an open subset is given by using open balls in GMMS as the following. If A is a subset of for any , there exists such that .
In our related paper [
22],
meets properties of usual topology. For example, if we take modular vector spaces as in [
23], the
-ball
, where
and
, is defined by
.
is an open ball and a subset of
A in vector space
. In the example of the topology
for all
-open subsets of
is similar for open subsets of
in a modular space
.
Chistyakov [
24] defined modular open balls and gave their topological structure as: A nonempty set in
X is said to
-open if for every
and
there exists
such that
by using
as a modular metric. Denoted by
for all
-open subsets of
we have a
-topology (modular topology) on
, which is similar to
in a modular metric space.
When we take a JS-metric space and the topology on JS-metric space, as in [
21], we find the usual topology on JS-metric space is, again, equal to
.
Now we can begin with the definition of generalized Hausdorff modular. Next, we interpret some material and produce their relation in the following section.
Let
be a GMMS. For all nonempty
, the generalized Hausdorff modular is defined by:
on
—D-strongly complete version of
is defined in the next section—where
If
, we have:
on
, where
Example 2. If we use the GMMS which is given in the first example, for , we havewhere and . All possible results can be calculated easily. 2.1.1. Fixed Point Results for Multivalued Mappings
Abdou and Khamsi searched the existence of fixed point for contractive-type multivalued map in the setting of modular metric spaces in their study, and they investigated the existence of fixed point of multivalued modular contractive mappings in modular metric spaces in [
25]. They claimed that their results generalize or improve the fixed point result of Nadler in [
2] and Edelstein. Their study inspired us to work on similar ideas and generalize their results for GMMS.
In a primary sense, we define Lipschitzian mapping, fixed point and D-multivalued contraction in GMMS for more generalized form of Lipschitzian maps we take
such as in [
25]. Then we give some essential definitions, such as
D-strongly Cauchy sequence in GMMS. Afterwards, we show relations between these definitions and generalized Hausdorff modular metric. At the end of this section, we give A linked fixed point theorem for
D-multivalued contraction mapping in GMMS.
Definition 3. Let be a GMMS. A mapping is called a multivalued Lipschitzian mapping, if there exists a constant such that for any , for every there exists , such that A point is called a fixed point of f whenever . The set of fixed points of f will be denoted by .
The mapping f is called as D-multivalued contraction, if the constant .
Example 3. If we take the same example as before, a mapping such that for every there exists the inequality is verified in X.
Definition 4. Let be a GMMS and be a sequence of .
- (1)
The sequence in is said to be D-strongly Cauchy if , for some .
- (2)
A subset M of is said to be D-strongly complete if for any D-strongly Cauchy sequence in M such that for some λ, there exists a point such that .
- (3)
D is said to satisfy 1-Fatou property if for any convergent sequence and , such that , we havefor any .
Let
be a GMMS. Let
be a multivalued map and
. Assume that
with
for some constant
and for every
there exists
, such that
where
. If we say
then
.
At this point, we explain that D-multivalued contraction mapping f has a fixed point in particular space .
Theorem 1. Let be a . Assume that is D-strongly complete and D satisfy 1-Fatou property. Let be a D-multivalued contraction mapping. Assume that is finite for some and . Then f has a fixed point.
Proof. Fix
such that
for some
then there exists
such that
where
.
where
. By induction, we build a sequence
there is
, for every
, then there exists
, since
f is a
D-multivalued contraction:
where
, for every
. Since
,
is convergent, i.e.,
is
D-strongly Cauchy. Since
is
D-strogly complete, there exists a point
such that
. Since there is
, for every
,
and
has 1-Fatou property,
we conclude that
, then
x is fixed point of
f. □
2.1.2. From Caristi-Type to Feng-Liu-Type Fixed Point Results for Multivalued Mappings
Caristi proved a general fixed point theorem and applied it to derive a generalization of the Contraction Mapping Principle in a complete metric space, then gave an application together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces [
26]. Following that, many authors expanded his approach through different metric spaces; for example in [
27]. In addition, there exist an application of Caristi-type mappings in [
28]. We examine Caristi-type mappings and state Feng-Liu-type results in GMMS in this section.
Theorem 2. Let be a D-complete GMMS and be a nonexpansive mapping such that for each and we havefor , while is D-closed and bounded subsets of and the function is lower semicontinuous with its first variable. Then , so f has a fixed point. Proof. Let
and
. If
, then proof is completed. Let
. Using the above inequality of the theorem, then
for
. When we continue the process, we have
while
, then we have
for
. We have
nonincreasing sequence and converges to
. If we take limit for the above inequality, we have
for
. It is the same way to show
is
D-Cauchy sequence. Then we assume
is a fixed point of
f:
for the last equality we pass the limit, and then we have
Then is a fixed point of f. □
Next, it is available to generalize even more as below.
Theorem 3. Let be a D-complete GMMS and be a multivalued mappingfor all and is a lower semicontunious map defined as for and satisfied that is nondecreasing. Then , so f has a fixed point. Feng and Liu [
20] gave the following theorem without using Hausdorff distance. To state their result, we use the following notation for a multivalued mapping
f on
, let and we define
The function f is called D-lower semicontinuous, and for any sequence is convergent to , if
Example 4. If we take the same example as before, a mapping such that , , we can show for any calculation of , where , it is satisfied. Then f is called D-lower semicontinuous for any sequence is convergent to , if .
Theorem 4. Let be a complete GMMS and f be D-multivalued mapping on . Suppose there exists a constant such that for any there is satisfying If there exists such that . Assume there exists a sequence in such that and ; while and for any .
The sequence is D-strongly Cauchy, and if we assume is D-lower semicontinuous, then f has a fixed point.
Proof. Since
for all
, then
is nonempty. Let us start choosing
such that
. From
, there exists
such that,
Since
, then
and
Choosing
such that
. From
there exists
such that,
Since
, then
and
By choosing
such that
. From
, there exists
such that,
Since
, then
and
Then, we have,
which give us, while
Then, we have
for
for any
,
while
and
is
D-strongly Cauchy and
is
D-strongly complete; then
is
D-lower semicontinuous,
since
, then we have
□
2.1.3. An Application
When we mention applications of multivalued mappings, one of them is given by Khamsi et al. in [
23] for modular vector spaces. They pointed out a fixed point theorem for uniformly Lipschitzian mappings in modular vector spaces which has the uniform normal structure property in the modular sense. They expanded their results in the variable exponent space. Another application of them is given by Borisut et al. in [
29]. They proved some fixed point theorems in generalized metric spaces by using the generalized contraction and they applied the fixed point theorems to show the existence and uniqueness of solution to the ordinary difference equation (ODE), partial difference equation (PDEs) and fractional boundary value problem.
For a non-homogeneous linear parabolic partial differential equation, initial value problem is given in [
6], such as
for same valued
, where
S is continuous and
assumed to be continuously differentiable such that
and
are bounded. By a solution of this problem, a function
defined on
, where
I satisfying the following conditions:
- (i)
while it denotes the space of all continuous real valued functions,
- (ii)
are bounded ,
- (iii)
,
- (iv)
for all ,
The differential equation problem below, is equivalent to the following integral equation problem:
for all
and
where
This problem admits a solution if and only if the corresponding problem just below has a solution. Let
where
Now, we take a function
as
is a GMM on
B. Obviously, the GMMS
is a
complete and independent of generators.
While is a GMMS, lower semicontinous it is easy to proof for Feng-Liu-type.
Theorem 5. Let the problemand assume the following: - (i)
For with and the function is uniformly Hölder continuous in x and t for each compact subset of ,
- (ii)
There exists a constant , where such thatfor all with and , - (iii)
S is bounded for bounded s and p;
Then the problem has a solution.
Proof. Let choose is a solution of the problem below, if and only if is a solution integral equivalent.
When we take the graph G with and . is partially ordered and satisfy property (A).
The mapping
defined as
for all
and when we solve the problem, the solution gives us the existence of fixed point of
f.
Since
and from the definition of
f and
When we take the solutions together:
From Feng-Liu’s perspective, we have
since we have
, while
. Then, there exists a
such that
, which is the solution of the problem. □