point results
*1* 2 3 2
1 Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan.
2 Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan.
Department of Mathematics, King Abdul Aziz University, Jeddah 21589, Saudi Arabia.
- dominated set-valued mappings sequence contained in a closed ball in a complete dislocated b
- dominated mappings in ordered complete dislocated b-metric space. Fixed point results with graphic contractions on a closed ball for graph dominated single-valued mappings were also given. Examples were given to demonstrate the variety of obtained results. The obtained results extended and generalised several comparable results in the existing literature.
- dominated set-valued mapping, closed ball, b-metric space, graph dominated mapping, partial order.
Let M : W P(W w W
M if w Mw. Fixed point theory plays a fundamental role in functional analysis and it has a large number of applications (Leibovic,
, et al.
et al.
conditions on a subset of domains instead of a whole et al., et al. et al., 2015).
Hussain et al. (2013) introduced the concept of dislocated b-metric space as a generalisation of dislocated metric space (Hitzler & Seda, 2000) and b
et al. et al.
et al.
et al. et al., 2017). They established dislocated b-metric space.
et al. (2012) -admissible set-valued mapping and multifunction in ordered metric space. For further results et al. (2015) et al. (2017) -dominated multivalued multi -dominated mappings in an ordered complete dislocated quasi b-metric space. For further results of
et al.
-dominated set-valued mappings satisfying a generalised contraction in complete dislocated b-metric point of multi -dominated mappings in an ordered complete dislocated b-metric space is established. The notion of multi-graph dominated mapping is introduced and the results with graphic contractions on a closed
Abdullah Shoaib et al.
and results for single-valued mappings are also given.
compelling results in metric space, partial metric space and dislocated metric space can be obtained as applications of theorems, which are still not available in literature.
the unique solution of a problem. These constraints are results, the contractive condition holds partially but the problem has a solution. In many cases contractive conditions do not satisfy the entire space X but the solution of such problems, we established an iterative sequence {TS(xn)} based on two multivalued mappings S,T : X P(X).
to understand the method.
(Hussain et al., 2013) Let H be a nonempty set and let db : H × H , ) be a function called a dislocated b-metric (or simply db -metric), if for any x,y,z H; and 1 the following conditions hold:
(i) If db(z, y) = 0, then z = y (ii) db (z, y) = db(y, z
(iii) db(z,y b db(z, x) + db(x, y)].
The pair (H, db) is called a dislocated b-metric space.
It should be noted that the class of (H, db) metric spaces is effectively larger than that of dislocated metric spaces (H, dl ), since every dl metric is a db metric when b = 1.
It is clear that if db(z, y) = 0, then from (i), z = y.
However, if z = y, db(z, y) may not be 0. For z H and is a closed ball in (H, db).
Example 1.2. (Hussain et al., 2013) If ,
then b-metric db on
H with b = 2.
(Hussain et al., 2013) Let (H, db) be a dislocated b-metric space.
in (H, db
if given , there corresponds such that for all s, 0 we have or
is called a dislocated b-converges (for short db -converges) to z if . In this case z is called a db-limit of .
(iii) (H,db H
converges to a point z H.
Let K be a nonempty subset of dislocated b-metric space (H,db) and let z H. y0 K is called a best approximation in K if,
If each z H has at least one best approximation in K, then K is called a proximinal set.
We denote P(H) to be the set of all closed proximinal subsets of z: Let , where 1, denote the family of all nondecreasing functions
such that and for
all , where is the kth iterate of . Further, .
et al., 2017) Let (H,db) be a dislocated b-metric space, S : (H) be a multivalued
mapping and . Let , thereby
S -dominated on A, whenever for all
z A, where . If A =
H, then S is -dominated. et al.,
2018) that there are mappings which are -dominated but not et al., 2012). We denote the
set .
The function ,
is called a dislocated Hausdorff b-metric on P(H).
Lemma 1.7. Let (H,db) be a dislocated metric space. Let (P(H);Hdb) be a dislocated Hausdorff b-metric space on P(H): Then, for all and for each
there exists then
.
Let be a dislocated
b-metric space, and be
the multifunctions on H. Let be an element
such that Let be
such that Let be
such that .
process, we construct a sequence of points in H
such that and , where
. Further
. We denote this iterative sequence by and that is a sequence in H generated by z0.
*
Theorem 2.1. Let be a complete dislocated b-metric space. Suppose there exist a function
. Let, :
be two -dominated mappings on and be a sequence in H generated by . that, for some , the following holds:
...(1)
whenever and
Further, for all
and
...(2) Then is a sequence in
for all and .
and for all , then S and T
in and
. From equation (2), we get
.
It follows that, .
Let for some ,
where .
Since be a - dominated mapping
on so 1 and
this implies
Further, therefore
Now by using Lemma 1.7, we obtain,
then
This is the contradiction to the fact that for all Therefore,
. Hence, we obtain,
...(3)
and ,
then . Now, by using Lemma 1.7, we
have
If then
This is a contradiction to the fact that for all Hence, we obtain
...(4) is nondecreasing,
.
...(5)
Now, if , where , by using
equation (4) and similar procedure as above, we have, ...
for all ...(7)
Abdullah Shoaib et al.
Now, by using triangle inequality and by equation (7), we have
Thus Hence for all
and therefore is a sequence in
are -dominated on , 1
and This implies
for all written as
...(8)
then the series converges for each . so
for all
Fix then there exists such that
Let then, we obtain
Thus we proved that in
dislocated b-metric space is complete, there exists,
such that that is
...(9)
Now,
(by Lemma 1.7)
Since we obtain
Letting , and using the inequality in equation (9), we obtain
. Similarly, by using,
we can show that . Hence, S and T have a
in . Now,
This implies that, .
We have the following result without closed ball and -dominated mappings for one multivalued mapping.
Theorem 2.2. Let be a complete dislocated b-metric space. Suppose be a multivalued
mapping: , the following
hold:
for all . Then and S
in H:
et al., 2017) Let H be a nonempty set, is a partial order on and . We say that whenever for all
mapping is said to be multi -dominated on A if for each . If then is said to be multi -dominated. We denote the set
We have the following result for multi -dominated mappings on in an ordered complete dislocated b-metric space.
Theorem 2.4. Let be an ordered complete
be two multi -dominated mappings on and be a sequence in H generated
by , the following hold.
...(10)
whenever and
...(11) Then is a sequence in
and .
equation (10) holds for and for all . Then S and T
in and .
Let
by for all and and
for all other elements . S and T are the dominated mappings on , so and for all . This implies that for
all and for all . So, for
all and for all .
This implies that and
. Hence
for all So,
are the -dominated mapping on Moreover, inequality in equation (10) can be written as,
whenever and
2.1, we have is a sequence in
and . Now,
and implies that for all
Therefore all the conditions of Theorem S and T have a in
We have the following result without closed ball in an ordered complete dislocated b-metric space. The result is only for one multivalued mapping.
Theorem 2.5. Let be an ordered complete dislocated b-metric space. Let be a multi
- dominated mapping on H.
, the following holds:
whenever and Then,
and Further, if
the inequality in equation (12) holds for and for all . Then S
point and .
Example 2.6. Let and let ) be the complete dislocated b-metric on H by
with parameter b = 2.
S,T : (H) by,
, and
. then . Now db(z0;Sz0) =
Therefore we obtain a sequence in H generated by . Let
then
Then be the -dominated mappings
on Now take and then, we
have,
Therefore, the contractive condition does not hold the whole space H. Now for all
with we have
240 Abdullah Shoaib et al.
Therefore, the contractive condition holds on . Further, for all we have
is a sequence in
and .
Further, for all Hence, all the S and T.
satisfying graphic contraction. Let be a metric space and represents the diagonal of the cartesian
product G is a directed graph.
The set having vertices along with H, and the set E(G) denoted the edges of H included all loops, i.e., If G has no parallel edges, then we can unify
G with pair et al.,
Let H be a nonempty set and be a graph such that
is said to be multi graph dominated on A if for all and
. We denote the set
or .
Theorem 3.2. Let be a complete dislocated b-metric space endowed with a graph G. Let,
and be a sequence in H generated by
following holds:
(i) S and T are multi graph dominated on
(ii) there exists such that
...(13)
whenever and
(iii) for all
and
Then, is a sequence in
and .
inequality in equation (13) holds for and
for all then S and T
point in and
( ) by
otherwise.
S and T are graph dominated on then
for for all and
for all So, for all and for all This implies that
and
Hence for all
Therefore are the -dominated
mappings on Moreover, inequality in equation (13) can be written as
whenever and
Further, (iii) in Theorem 3.2 holds. Then, by Theorem 2.1, is a sequence in and
Now, and
implies that Therefore, all the conditions S
and T in and
We have the following result without closed ball in complete dislocated b-metric space for multi graph dominated mapping. Therefore we write the result only for one multivalued mapping and for
Theorem 3.3. Let be a complete dislocated b-metric space endowed with a graph G. Let,
and be a sequence in H generated by
following holds:
(i) S is a multi graph dominated on (ii) there exists such that
...(14) whenever
Then, and
Further, if the inequality in equation (14) holds for
and for all then S and T have
in H and
et al. (2017) have given the following
Let be a dislocated b-metric space, be a self mapping, and
) be a function. We say that,
(i) T is -dominated mapping on A, if for all
(ii) is -regular on A if for any sequence in A such that 1 for all and
we have for all
Theorem 4.2. Let be a complete dislocated b-metric space. Suppose there exists a function
). Let and
be two -dominated mappings on , the following holds:
whenever and Further,
If is -regular on , then there of S and T in
et al., 2013) if be a f on H is called dominated if for each in H. Two elements
are called comparable if or holds.
Theorem 4.3. Let be an ordered complete dislocated b-metric space, be dominated maps and be an arbitrary point in H. Suppose that for some and for , we have,
for all comparable elements in . Further,
If for a nonincreasing sequence in implies that or Then,
there exists such that and
et al.
2013) as a corollary of Theorem 4.3.
et al. 2013) Let be an ordered complete dislocated metric space,
be dominated maps and be an arbitrary point in H.
Suppose that for , we have,
for all comparable elements in and
If for a non-increasing sequence in implies that Then there
exists such that and
Let H be a nonempty set
and be a graph such that
is said to be graph dominated on A if for all . Theorem 4.6. Let be a complete dislocated b-metric space endowed with a graph G. Let,
and the following holds:
(i) S and T are graph dominated on (ii) there exists , such that
for all and
(iii) for all
and
Then, there exists a sequence in such
that and Further,
if for all then S and T have
in and
Now, we present only one new result in metric space.
Many other results can be derived as corollaries of our previous results.
Theorem 4.7 Let be a complete metric space endowed with a graph G and be a mapping.
(i) S is a graph dominated on H (ii) there exists such that
for all and
242 Abdullah Shoaib et al.
Then, there exist a sequence such that and Further, if
for all , then S in H.
generalised contraction on an intersection of a closed ball and a sequence for a more general class of -dominated mappings rather than
with graphic contractions on a closed ball for graph dominated mappings are established, and Theorems for ordered multi -dominated mappings are obtained.
Moreover, we investigated the obtained results as a new framework of dislocated b-metric space. New results in partial b-metric space, dislocated metric space, partial metric space, b-metric space and metric space can be obtained as corollaries of the obtained results.
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