Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 48, No. 1 (2026) pp. 139–145.  
RELATION-THEORETIC NONUNIQUE FIXED  
POINT THEOREMS  
J. B. Dhage1, and B. C. Dhage2,∗  
Abstract. In this paper, we establish a couple of nonunique fixed point results for the  
orbital relationally continuous self maps of a orbital relationally complete metric space  
which include some interesting fixed point results as the special cases. Our abstract results  
are also illustrated with the help of a numerical example.  
Keywords:  
Relational complete metric space; Relational contraction mapping;  
Nonunique fixed point theorem.  
2010 AMS Subject Classification: Primary 54H25; Secondary 47H10.  
1. Introduction  
Ciric [3] proved some nonunique fixed point results for a certain class of contraction  
mappings on a metric space into itself under weaker version of continuity and complete-  
ness of a metric space. Thereafter, several authors extended and generalized the above  
results to other classes of contraction mappings in a metric space which is not necessarily  
complete. See for example, Achari [1], Dhage [4, 5, 6], Pachpatte [12] and references  
therein. Similarly, Alam and Imdad [2] proved a basic relation theoretic fixed point theo-  
rem for relational contraction mappings in relational complete metric spaces. Later, this  
result is generalized to a wider classes of mappings by Dubey et al. [11] and Samet and  
Turinici [13] with fixed point. All the above fixed point results are about the existence of a  
unique fixed point and to the best of our knowledge there is no result for nonunique fixed  
point. In this paper we blend the above two approaches and prove a couple of relation  
theoretic nonunique fixed point theorems for the selfmappings of a metric space under  
weaker suitable conditions. Before going to the main results we give some preliminary  
definitions.  
Let (X, d) be a metric space and let T : X → X be a mapping. Then T -orbit of T at  
a point x ∈ X is a set O(x, T ) in X defined by  
O(x, T ) = {x, T x, T x, · · · , T x, · · · } = {T x}∞n=0  
.
(1)  
2
n
n
The mapping T is called T -orbitally continuous if and only if a sequence {xn} ⊆ O(x, T )  
converges to x implies T xn → T x. X is called T -orbitally complete if every Cauchy  
sequence {xn} in O(x, T ) converges to a point in X. It is known that every continuous  
mapping is T -orbitally continuous and every complete metric space X is T -orbitally  
complete, but the reverse implication need not hold. See Dhage et al. [10] and references  
therein.  
139  
140  
J. B. Dhage, and B. C. Dhage  
A binary relation or simply relation R in X is a subset of the Cartesian product X ×X  
which is full if R = X × X and which is empty if R = ∅. R is called partial relation  
or proper relation if ∅ = R = X × X. We introduce a partial relation R in X and two  
elements x and y in X are said to be relatable if they are related to each other under the  
relation R. In this case we write x R y or (x, y) ∈ R. A metric space X together wth  
a partial relation R is called a relational metric space and it is denoted by (X, d, R). A  
sequence {xn} of points in X is called relatonic or term-wise related if (xn, xn+1) ∈ R  
and inverse relatonic if (xn+1, xn) ∈ R for each n ∈ N. We say the metric space X  
is R-regular or PR-closed or simply closed with respect to the relation R, if {xn} is a  
relatonic (resp. inverse relatonic) sequence of points in X converging to a point x∗, then  
(xn, x∗) ∈ R (resp. (x∗, xn) ∈ R) for all n ∈ N. A relatonic sequence {xn} in (X, d, R)  
is called R-Cauchy if limm,n→∞ d(xm, xn) = 0. Again, (X, d, R) is called R-complete if  
every R-Cauchy sequence converges to a point in X. A mapping T : X → X is called is  
n
called relational R-continuous at a point x ∈ R if and only if xn → x implies T xn → T x  
n
for every sequence {xn} of points relatable to x in R . A few details of above notions  
appear in Anam and Imdad [2] and Dhage [8].  
2. Main Results  
Before going to the main nonunique fixed point theorems we state a couple of mathe-  
matical concepts whih we need in what follows.  
Definition 2.1. A mapping T : X → X is called is called T -orbital R-continuous at a  
point x ∈ X if and only if it is R-continuous on O(x, T ) for each x ∈ X. Again, (X, d, R)  
is called T -orbital R-complete if O(x, T ) is R-complete for each x ∈ X.  
Remark 2.2. It is clear that  
Completeness of X ⇒ T -orbital completeness of X ⇒ T -orbital R-completeness of X  
and  
Continuity of T ⇒ T -orbital continuity of T ⇒ T -orbital R-continuity of T ,  
however the reverse implications may not hold.  
Definition 2.3. A mapping T : X → X is called a relational R-relatone if it preserves  
the binary relation R in X, that is, if x, y ∈ X with (x, y) ∈ R, then (T x, T y) ∈ R.  
We consider the following notations in what follows. We denote  
(i) F (X) = {x ∈ X : T x = x},  
T
(ii) L (X) = {x ∈ X : (x, T x) ∈ R}, and  
T
(iii) U (X) = {x ∈ X : (T x, x) ∈ R}.  
T
Theorem 2.4. Let T be a T -orbital R-continuous and R-relatone self-mapping of a  
R-regular and T -orbital R-complete metric space X. Suppose that there exists a real  
RELATION-THEORETIC NONUNIQUE FIXED POINT THEOREMS  
number b such that  
141  
0 ≤ min{d(T x, T y), d(x, T x), d(x, T x)}  
+ b min{d(x, T y), d(y, T x)} ≤ q d(x, y)  
(2)  
for all relatable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then F (X) = ∅. Moreover, T has a unique relatable fixed point provided  
T
T
b > q.  
Proof. Suppose first that L (S) = ∅ , then there exists an element x0 ∈ S such that  
T
n
(x0, T x0) ∈ R. Define a sequence {xn}∞  
= {T x0}∞  
of iterations of T at x0 ∈ X.  
Since T is relatone, we have (xn, xn+1) ∈nR=0for each n, n = 0, 1, . . . . If xr = xr+1 for some  
r ∈ N, then the conclusion of the theorem follows immediately. Therefore, we assume  
that xn = xn+1 for each n ∈ N. We show that {xn} is a R-Cauchy sequence. Let x = xn−1  
and y = xn in the inequality (2), we obtain  
n=0  
d(xn, xn+1) = d(Txn−1, Txn) ≤ qd(xn−1, xn)  
for all n ∈ N. By repeated application of the above inequality yields  
d(xn, xn+1) ≤ qnd(x0, x1)  
(3)  
(4)  
for each n ∈ N . Next for any positive integer p, by triangle inequality, we set  
d(xn, xn+p) ≤ d(xn, xn+1) + ... + d(xn+p−1, xn+p  
)
≤ (qn + ... + qn+p−1)d(x0, x1)  
ꢀ
ꢁ
1 − qp−1  
= qn  
d(x0, x1)  
1 − q  
ꢀ
ꢁ
qn  
≤
d(x0, x1)  
1 − q  
→ 0 as n → ∞.  
(5)  
This shows that {xn} is a R-Cauchy sequence in O(x, T ). Since X is T -orbital R-  
complete, {xn} converges to a point u in X. From T -orbital R-continuity of T , it follows  
that  
ꢂ
ꢃ
u = lim xn+1 = lim T xn = T  
lim xn = T u.  
n→∞  
n→∞  
n→∞  
Thus u is a fixed point of T . Smilarly, if U (X) = ∅, then following similar arguments,  
T
it can be shown that T has a fixed point. Further assume that b > q and let v(= u)  
be another relatable fixed point of T . Then form condition (2) we get a contradiction  
whence T has a unique relatable fixed point. This completes the proof.  
□
On taking b = 0 in contraction condition (2) we obtain the following fixed point result.  
Corollary 2.5. Let T be a T -orbital R-continuous and R-relatone self-mapping of a R-  
regular and T -orbital R-complete metric space X. Suppose that T satisfies the contraction  
142  
J. B. Dhage, and B. C. Dhage  
condition  
min{d(T x, T y), d(x, T x), d(x, T x)} ≤ q d(x, y)  
(6)  
for all relatable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then T has a fixed point.  
T
Again taking b = −1 in contraction condition (2), we obtain the following fixed point  
result Ciric [3] type mappings in metric spaces.  
Corollary 2.6. Let T be a T -orbital R-continuous and R-relatone self-mapping of a R-  
regular and T -orbital R-complete metric space X. Suppose that T satisfies the contraction  
condition  
0 ≤ min{d(T x, T y), d(x, T x), d(x, T x)}  
− min{d(x, T y), d(y, T x)} ≤ q d(x, y)  
(7)  
for all relatable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then T has a fixed point.  
T
Remark 2.7. The Ciric’s sufficient contraction condition for the existence of nonunique  
fixed point is  
min{d(T x, T y), d(x, T x), d(x, T x)}  
− min{d(x, T y), d(y, T x)} ≤ q d(x, y)  
for all x, y ∈ X, where 0 ≤ q < 1. Note that the term min{d(x, T y), d(y, T x)} is not  
always equal to 0 which otherwise is assumed in the proof of the fixed point theorem. If we  
assume it never be zero, then the mapping T : R+ → R+ defined by T x = x + 1 satisfies  
the Ciric’s contraction condition with no fixed point in R+. Therefore, we have slightly  
modified it into the form (7) to have greater applicability to nonlinear equations.  
We introduce a partial order ≼ in X which is a reflexive, anti-symmetric and transitive  
relation, so that non-empty set X now becomes a partially ordered metric space  
and it is denoted by (X, d, ≼). A mapping T is called partially continuous at a point  
x ∈ X if and only if xn → x implies T xn → T x for every sequence {xn} of points  
comparable to x in X. T is called partial continuous on X if it is partial continuous at  
each point x ∈ X. A partially ordered complete metric space X is one in which  
every monotonic Cauchy sequence converges to a point in X. The details of these partial  
notions appear in Dhage [7] and Dhage et al. [8] and references therein. We consider the  
following definitions in what follows.  
Definition 2.8. A mapping T : X → X is called is called T -orbital partially continuous  
at a point x ∈ X if and only if it is partial continuous on O(x, T ) for each x ∈ X.  
Again, (X, d, ≼) is called T -orbital partially complete if O(x, T ) is partial complete for  
each x ∈ X.  
Definition 2.9. A mapping T : X → X is called a monotone nondecreasing if it  
preserves the partial order ≼ in X, that is, if x, y ∈ X with x ≼ y, then (T x ≼ T y.  
RELATION-THEORETIC NONUNIQUE FIXED POINT THEOREMS  
143  
Now as a consequence of Theorem 2.4 we obtain the following corollary which is also  
new to the literature.  
Corollary 2.10. Let T be a T -orbital partial continuous and monotone nondecreasing  
self-mapping of a regular and T -orbital partial complete metric space X. Suppose that  
there exists a real number b such that  
0 ≤ min{d(T x, T y), d(x, T x), d(x, T x)}  
+ b min{d(x, T y), d(y, T x)} ≤ q d(x, y)  
(8)  
for all comparable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then F (X) = ∅. Moreover, T has a unique comparable fixed point provided  
T
T
b > q.  
Remark 2.11. We note that Corollary 2.10 includes two more nonunique fixed point  
results for the mapping satisfying the contraction conditions (6) and (7) in the partially  
ordered metric spaces.  
Next, we generalize Corollary 2.5 by the improvement of the contraction condition (6).  
Theorem 2.12. Let T be a T -orbital R-continuous and R-relatone self-mapping of a R-  
regular and T -orbital R-complete metric space X. Suppose that T satisfies the contraction  
condition  
ꢄ
min d(T x, T y),d(x, T x), d(x, T x)}  
ꢅ
(9)  
ꢄ
ꢆ
ꢇꢈ  
≤ q max d(x, y)  
min{d(x, T y), d(y, T x)}  
for all relatable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then T has a fixed point.  
T
Proof. Suppose first that L (S) = ∅ , then there exists an element x0 ∈ S such that  
T
n
(x0, T x0) ∈ R. Define a sequence {xn}∞  
= {T x0}∞  
of iterations of T at x0 ∈ S.  
Since T is relatone, we have (xn, xn+1) ∈nR=0for each n, n = 0, 1, . . . . If xr = xr+1 for some  
r ∈ N, then the conclusion of the theorem follows immediately. Therefore, we assume  
that xn = xn+1 for each n ∈ N. We show that {xn} is a R-Cauchy sequence. Let x = xn−1  
and y = xn in the inequality (2), we obtain  
n=0  
ꢄ
min d(T xn−1, T xn),d(xn−1, T xn−1), d(xn, T xn)}  
ꢅ
ꢄ
ꢆ
ꢇꢈ  
≤ q max d(xn−1, xn)  
min{d(xn−1, T xn), d(xn, T xn−1)} ,  
ꢄ
i.e.  
min d(xn, xn+),d(xn−1, xn), d(xn, xn+1)}  
ꢅ
ꢄ
ꢆ
ꢇꢈ  
≤ q max d(xn−1, xn)  
min{d(xn−1, xn+1), d(xn, xn)} ,  
which implies  
ꢄ
min d(xn, xn+), d(xn−1, xn)} ≤ q d(xn−1, xn).  
Since d(xn−1, xn)} ≤ q d(xn−1, xn) is not possible because q < 1, we have that  
d(xn, xn+1) ≤ q d(xn−1, xn)  
144  
J. B. Dhage, and B. C. Dhage  
for all n ∈ N. The rest of the proof is similar to the proof of Theorem 2.4. Hence we omit  
the details.  
□
As a consequence of Theorem 2.12, we obtain the following result which is also new the  
metric fixed point theory.  
Corollary 2.13. Let T be a T -orbital partial continuous and monotone nondecreasing  
self-mapping of a regular and T -orbital partial complete metric space X. Suppose that T  
satisfies the contraction condition  
ꢄ
min d(T x, T y),d(x, T x), d(x, T x)}  
ꢅ
(10)  
ꢄ
ꢆ
ꢇꢈ  
≤ q max d(x, y)  
min{d(x, T y), d(y, T x)}  
for all comparable elements x, y ∈ X, where 0 ≤ q < 1. Furthermore, if L (X) = ∅ or  
T
U (X) = ∅, then T has a fixed point.  
T
Example 2.14. Let X = [0, 1) and let ≤ be the partial order defined in [0, 1). Then the  
ꢉ
triplet  
[0, 1), | · |, ≤) becomes a partially ordered metric space. Clearly, [0, 1) is not a  
complete metric space. Define a mapping T : [0, 1) → [0, 1) by  
ꢊ
ꢆ
ꢎ
ꢎ
x
3
4
ꢋ
if x ∈ 0,  
,
ꢌ
2
1
T x =  
ꢆ
ꢋ
ꢍ
if x ∈ 34 , 1 .  
2
Note that T is not continuous on [0, 1). However, T is T -orbital partial continuous on  
[0, 1) and [0, 1) is T -orbital partially complete metric space. Moreover the condition (8) is  
1
4
satisfied with b = 0 and q = . Also we have U (X) = ∅ . Therefore, by Corollary 2.10,  
T
T has a fixed point u = 0.  
Acknowledgment. The authors are thankful to the referee for giving some useful sug-  
gestions for the improvement of this paper.  
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(Received, July 07, 2025)  
(Revised, October 09, 2025)  
1,2Kasubai, Gurukul Colony,  
Thodga Road, Ahmedpur - 413515,  
Dist. Latur, Maharashtra,  
India5004.  
Email1: jbdhage@gmail.com  
Email2: bcdhage@gmail.com  
∗
Corresponding author