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VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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Existence theorems for a unified interpolative Kannan contraction with an application on nonlinear matrix equations
Koti N. V V Vara Prasad3, Vinay Mishrab, Stojan Radenovicc
a Guru Ghasidas Vishwavidyalaya, Department of Mathematics,
Bilaspur, Republic of India, e-mail: [email protected],
ORCID iD: ©https://orcid.org/0000-0002-2606-7292 b Guru Ghasidas Vishwavidyalaya, Department of Mathematics,
Bilaspur, Republic of India,
e-mail: [email protected], corresponding author,
ORCID iD: ©https://orcid.org/0000-0002-4223-3924 c University of Belgrade, Faculty of Mechanical Engineering,
Belgrade, Republic of Serbia, e-mail: [email protected],
ORCID iD: ©https://orcid.org/0000-0001-8254-6688
doi https://doi.org/10.5937/vojtehg72-50753
FIELD: mathematics
ARTICLE TYPE: original scientific paper
Abstract:
Introduction/purpose: This paper established a new mathematical framework by uncovering the relationships between Kannan contractions and interpolative Kannan contractions. The concept of unified interpolative Kannan contractions was introduced in the framework of a relational metric space. Additionally, the study aimed to broaden the concept of alpha admissibility by incorporating specific relational metric ideas.
Methods: A detailed exploration of the properties and characteristics of Kannan contractions and interpolative Kannan contractions was conducted. The research introduced the concept of unified interpolative Kan-nan contractions and formulated new fixed point results for these mappings.
Result: The study successfully established fixed point results for unified interpolative Kannan contractions within the framework of relational metric spaces. Additionally, an application of these results to solve a problem concerning nonlinear matrix equations was provided, further emphasizing their significance.
Conclusion: The findings of this study significantly advanced the understanding of Kannan contractions and interpolative Kannan contractions, offering a unified framework for their analysis. The introduction of unified
interpolative Kannan contractions and the expansion of alpha admissibility have broad implications for the field of mathematics.
Key words: unified interpolative Kannan contraction, R-admissible, relational metric space.
Introduction
Kannan made a significant contribution to metric fixed point theory after Banach’s influential fixed point theorem. While mappings satisfying the Banach contraction inequality are necessarily continuous, Kannan introduced a novel class of contractions in 1968, addressing the intriguing question of whether discontinuous mappings defined in a complete metric space and satisfying specific contractive conditions could possess a fixed point.
Kannan stated the following result.
Theorem 1. Let (X, d) be a complete metric space, and S be a self-map defined on X. If S is a Kannan contraction (KC, for brief), meaning that there exists a X in the interval [0, 1) such that,
d(Sv, Sj) < X[d(v, Sv)+ d(j, Sj)], for all v, /л € X, (1)
then, S possesses a unique fixed point y € X, and for each v € X, the sequence of iterates {Snv} converges to y.
Kannan’s fixed-point theorem represents a notable extension of Banach’s remarkable work (Banach, 1922), leading to several generalizations, see (Debnath et al., 2021). Among these, a recent variant introduced by Karapinar, termed as interpolative Kannan-type contraction (or Kannan interpolative contraction), was demonstrated in Karapinar (2018). To guarantee the existence of a fixed point in a complete metric space, this contraction condition allows more flexibility in choosing the constants that control the contraction rate and can also incorporate the distance between points in the contractive condition. Additionally, it is worth mentioning that many classical and advanced contraction concepts have been recently reexamined through interpolation, see (Debnath et al., 2020; Hammad et al., 2023; Jain & Radenovic, 2023; Jain et al., 2022; Karapinar, 2021; Karapinar et al., 2018a,b, 2021).
In his work, Karapinar (2018) presents an example that falls outside the scope of Kannan contractions but aligns with interpolative Kannan contractions. This highlights an additional advantage of interpolative Kannan
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
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contractions over Kannan contractions. Despite existing research on the subject, there is a notable gap in the literature concerning the converse relationship, i.e., whether Kannan contractions imply interpolative Kannan contractions.
Recent studies Nazam et al. (2023a,b) have suggested that Kannan contractions do indeed imply interpolative Kannan contractions. However, this paper diverges from this perspective and, through illustrative examples, establishes that not every Kannan contraction implies an interpolative Kannan contraction. Consequently, this paper asserts that these two classes of contractions are independent from each other. This comprehensive understanding emphasizes the significance of both contraction types, providing valuable insights into their practical applications.
Karapinar (2018) introduced the concept of an interpolative Kannan contraction as follows:
Definition 1. A self-mapping S defined on a metric space (X,d) is considered as an interpolative Kannan type contraction (IKC, for brief) if there exists a pair of constant a, A e [0,1) with a = 0, satisfying
d(Sv, S ц) < A[d(v, Sv)a ■ д(ц, S ц)1-а], for all v,ц e X, and v = Sv. (2)
By employing the interpolative Kannan contraction, Karapinar (2018) established a unique fixed point theorem. Subsequently, Karapinar et al. (2018a) identified a limitation in the aforementioned result, highlighting that fixed points obtained from the contractive condition (2) may not necessarily be unique. To illustrate that not every Kannan contraction implies an interpolative Kannan contraction, these authors examine the following example.
Example 1. Let X = [0,1] and consider the mapping S : X ^ X defined by Sv = 5. Let d denote the usual metric.
Then, one can observe that, d(Sv, Sц) = 5\v - ц\,д(v, Sv) = 4V, and д(ц, Sц) = ^.
For A = 5 e [o, ±), one can verify that:
This confirms that S fulfills condition (1). Now, the next task is to demonstrate that S does not satisfy (2). Suppose if possible S satisfies (2),
д(Sv, Sц)
1
5
2 4
\v - ц\< 5 ■ 5(v + ц) = A ■ [d(v, Sv) + д(ц, Sц)\.
then, two points are chosen, and -00, from the interval [0,1]. Clearly,
-00 = S (тш) and -00 = S (1L^).
Case I: When v = and i = -00, there have,
49 \ .qq--«
250 = d(Sv, Si) < A [d(v, Sv)a • d(i, Sц)1-а] =
125
Case II: When v = -00 and ц = , there exists,
49 A•99a
250 = 9(Sv Si) < A [d(v,Sv)a • d(i,Si)l-a] = 125 .
(3)
(4)
Since S satisfies (2) for all v,i e X\F(S), from (3) and (4), there exists a pair of constants A, a e [0,1) with a = 0, such that
49
2 < A • min{99--“, 99“}.
(5)
Now, if A = 0, a contradiction of (5) is obtained. Therefore, A e (0,1), and in such a case, there is
49
< 2 • min{99--“, 99“}.
A
However, this again leads to a contradiction, as expressed by the following inequality
49
inf > 2 • sup min{99--“, 99“} . ле(0,-) A ae(o,-)
Therefore, there does not exist any a e (0, 1) and A e (0, 1) for which equation (5) holds true for all v,i e X\F(S). Thus, the initial assumption is incorrect, and S does not satisfy condition (2).
Therefore, based on Example 1 and Example 2.3 of Karapinar et al. (2018a), it can be inferred that conditions (1) and (2) are independent. In the current study, these authors endeavor to establish connections between these conditions by extending them to a more generalized contraction condition in a relational metric space.
It is noteworthy that in relational metric spaces, one often considers weaker properties such as ^-continuous (not necessarily continuous), R-complete (not necessarily complete), etc. In this setting, additional flexibility is beneficial in that the contraction condition need not be applied to every element but rather to related elements only. Importantly, these contraction conditions revert to their conventional counterparts when the universal relation is taken into account.
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
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Preliminaries
Before presenting the main results of this paper, it is important to introduce formal notations that will be used throughout. Let X be a non-empty set, with a binary relation R. In this context, the pair (X, R) is acknowledged as a relational set. Similarly, within the framework of a metric space (X,d), one designates the triplet (X,d, R) which constitutes a relational metric space (RMS, for brevity). The collection of fixed points of the selfmapping S is indicated by F(S), and let XR denote the set defined by, XR = {(v,x) £ X2 : (v,x) £ R and v,x £ F(S)}. Furthermore, X(S, R)
is a subset of X, containing elements v such that (v, Sv) € R. These formalized notations ensure precision and consistency throughout the subsequent analyses and discussions.
Definition 2. (Alam & Imdad, 2015) Let S be self-map on X, and (X, R) be a relational set,
(i) any two elements v,x £ X are considered R-comparative if (v, x) £ R or (x,v) £ R. This relationship is symbolically represented as
[v, x] £ R,
(ii) a sequence {vk} с X satisfies the condition (vk,vk+i) £ R for all k £ No, is referred to as an R-preserving sequence.
(iii) R is designated as S-closed when it satisfies the condition that if (v, x) belongs to R, then (Sv, Sx) also belongs to R, for any v,x £ X.
(iv) R is referred to as d-self-closed under the condition that whenever
there exists a R-preserving sequence {vk} such that vk v, there
can always be found a subsequence {vkn} of {vk} such that [vkn ,v] belongs to R for all n £ N0.
Definition 3. (Alam & Imdad, 2017) (X, d, R) is considered R-complete if every sequence in X, which is both R-preserving and Cauchy, converges.
Definition 4. (Alam & Imdad, 2017) A self-map S defined on X is termed R-continuous at v £ X, if any R-preserving sequence vk —+ v, implies Svk —+ Sv. Furthermore, if S exhibits this behavior at every point in X, it is simply categorized as R-continuous.
Definition 5. (Alam & Imdad, 2018) Consider a self-mapping S defined on X. If for every R-preserving sequence {vn} c S (X), with a range denoted as E = {vn : n e N}, R\e is transitive, then S is designated as locally S-transitive.
Samet et al. (2012) introduced the concept of a-admissible mappings, which has been applied by various authors in numerous fixed-point theorems.
Definition 6. (Samet et al., 2012) Suppose S is a self-map on X, and a : X x X ^ R+ is a function. Then, S is considered a-admissible if
a(v, ц) > 1 ^ a(Sv, Sц) > 1 for all v,y e X.
In the following definition, this concept is generalized by incorporating certain relational metrical notions.
Definition 7. Let (X, R) be a relational set. A self-map S defined on X is termed R-admissible if there exists a function § : X x X ^ [0, +rc>), satisfying the following conditions:
(ri) §(v,^) > 1 for all (и,ц) e R,
(r2) R is S-closed.
Remark 1. From the above two definitions, it can be observed that if S is a-admissible, it also holds that S is R-admissible when considering R = {(v,y) e X2 : §(v, ц) > 1}. However, it should be noted that the converse is not necessarily true, as illustrated in the following example.
Example 2. Let X = {0,1,2,3}, § : X x X ^ R+ by
(2, (v,v) e {(0,1), (1, 2), (2, 3)}
§(v, ц) = h, (v, n) e {(0, 2), (1,1), (2,1), (2, 2)}
{ , otherwise.
and S : X ^ X is defined by S0 = 0, S1 = 2, S2 = 1, and S3 = 3.
In this example, it is evident that §(2,3) > 1, but §(S2, S3) = §(1,3) ^ 1,
indicating that S is not §-admissible. Now, let us consider the binary relation R defined as,
R = {(0,1), (0, 2), (1, 2), (2,1), (1,1), (2, 2)}.
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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It is straightforward to observe that R is S-closed, and for all v,/ e X with (v,/) e R, §(v,/) > 1. Therefore, S is R-admissible.
Let ф, ф : [0, +rc>) ^ [0, +rc>) be two functions. Then the following conditions are considered:
(Ci) ф is u.s.c. such that ф(0) = 0,
(C2) ф is l.s.c.,
(C3) ф, ф are non-decreasing,
(C4) ф(Ф) > ф(Ф), for all t> 0,
(C5) limsup ф(Ф) < ф(с+), for all c> 0,
t——c+
(C6) limsupф(Ф) < liminfф(t).
t—0 t—e+
Main results
This section introduce a novel concept of a unified interpolative Kan-nan contraction condition and establish some fixed-point results for such contractions. Through an example, it will be demonstrated how the unified interpolative Kannan contraction condition extends the classical notions of contraction mappings defined in (Kannan, 1968; Karapinar, 2018; Nazam et al., 2023a).
Definition 8. Let (X,d, R) be an RMS. A self-mapping S defined on X is characterized as a unified interpolative Kannan contraction (UIKC, for brief) if there exist functions ф, ф : [0, +rc>) ^ [0, +rc>), and a function § : X x X ^ [0, +rc>), along with a parameter a e (0,1), such that
§(v, /Ф)ф(д^v, S/)) < ф (Q(d(v, Sv),d(/, S/))), for all v, / e XR, (6)
where Q : R2 ^ R be a function satisfying Q(v, /) < max {v, /, va/l-a).
Example 3. Let (X, d) be a metric space with X = [0, +rc>) and d is the usual metric, define the self-map S on X by,
v
5, if v < 1,
v2, if v > 1.
Then, it is important to note that S is not a Kannan contraction (Kannan, 1968). This is evident that when considering v = 1 and / = 2, as there
does not exist any X e [0, |) that satisfies (1). Additionally, for the same values of v = 2 and j = 2, there is no pair of X e [0,1) and a e (0,1) for which (2) holds. Consequently, S is notan interpolative Kannan contraction (Karapinar, 2018). Now, let us define the binary relation R on X as,
Observing the definition of R, it is evident that R is not an orthogonal relation. It is important to recall that a binary relation R is considered as an orthogonal relation if for any element v0 e X, either (for all j, (v0, j) e R) or (for all j, (j, v0) e R). As a consequence, the function S is not a (ф, ф)-orthogonal interpolative Kannan-type contraction (Nazam et al., 2023a). However, it will now be demonstrated that S is indeed a unified interpola-tive Kannan contraction. Consider § : X x X ^ [0, +rc>) defined by
Observing that §(v,j) > 1 for all v,j e X with (v,j) e R, and that (v, j) eR implies (Sv, Sj) e R, it follows that S is R-admissible. Suppose there exist functions ф, ф : [0, +rc>) ^ [0, +rc>) defined by ф(Ь) = 6,
The aim now is to show that S satisfies (6). Consider the function Q :
X x X ^ [0, +rc>) defined as Q(v,j) = . For every v,j e XR, the
following inequality holds,
Consequently, it is deduced that S is a unified interpolative Kannan contraction.
R = {(v,j) e X2 : max{v,j} < 1}.
4v,j№(d(Sv>SJ)) = 2ф ( 5 - j j
=50|v - j
< 1 (Q(d(v, Sv),d(j, Sj))) 6
= ф (Q(d(v, Sv),д(ц, Sj))).
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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Remark 2. From Example 1, Example 3, and Example 2.3 in Karapinar et al. (2018a), one arrives at the following conclusion
Now, let us proceed to establish this paper’s main results concerning the unified interpolative Kannan contraction maps.
Theorem 2. Consider the RMS (X, d, R) where R is a locally S-transitive binary relation. Suppose that S is a unified interpolative Kannan contraction and there exist functions ф, ф : [0, +rc>) ^ [0, +rc>) satisfying conditions C, (i = 1,2,3,4). Under the following conditions:
(D-ф) S is R-admissible,
(D2) there exists Y с X with S(X) c Y, such that (Y, d, R) is R-complete,
(D3) X(S, R) is non-empty,
(D4) either S is R\Y-continuous or R is d-self-closed, there exists at least one y e X such that y e F (S).
Proof. Under the assumption (D3), suppose that v0 e X(S, R). Define the sequence {vn} of Picard iterates with the initial point v0, i.e. vn = Snv0 for all n e N0. As (v0, Sv0) e R and S is R-admissible, using (n) it follows that
(Snv0, Sn+lv0) e R. Consequently, (Vn,Vn+i) e R for all n e N0, and this yields that the sequence {vn} is R-preserving and from (r2) there holds $(vn,vn+l) > 1. Let dn = d(vn,vn+l); applying contractive condition (1) yields that
Ф (dn) < ^(иа-1,иа)ф (d(SVn-l, SVn))
< ф (Q(d(Vn-i, SVn-i),d(Vn, SVn)))
= ф(П (dn-i,dn))
< ф (max {dn-i,dn, d^_i ■ dn_a})
< ф (max {dn_i, dn, d%_i ■ dn_a}) • (7)
By the monotonicity of the function ф one obtains
dn < max {dn-i,dn, dfc-1 • dl~a} . (8)
Now suppose there exists n e N for which dn-i < dn, then from (8) it yields that dn < dn, a contradiction. Therefore dn < dn-1, now it can be concluded that {vn} is a non-increasing sequence and thus a non-negative constant C exists such that, lim dn = C+. Suppose if possible C > 0,
n—
then from (7), it can be deduced that
Ф(С+) < lim inf ф(дп) < lim sup ф(дп-1) < ф(С+),
but, from (C4) there exists ф(и) > ф(и) for all v > 0, therefore C must be 0, i.e. lim dn = 0. The next objective is to establish that the sequence
n—
{vn} is Cauchy. For the sake of contradiction, suppose it is not; then there exists a positive real number e > 0 along with sub-sequences {vnk} and {vmk} of {vn}, with nk > mk > k, such that
d(vmk ,Vnk) > e, for all k e N. (9)
Selecting nk as the smallest integer exceeding mk such that (9) holds, it is deduced that
d(vmk,Vnk-i) < e. (10)
Using triangular inequality and (9), (10) one obtains that
e < d(vmk ,vnk ) < d(vmk ,vnk-1) + d(vnk-1, vnk )
< e + d(vnk-1,vnk ).
on taking the limit k ^ and utilizing the fact that lim dn = 0, one
n—
gets
lim d(vmk ,vnk )= e + . (11)
k—+<x>
By using triangular inequality, one obtains that
|d (vmk + 1, vnk+1) _ d (vmk , vnk )l < dvmk + dvnk ■
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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Letting limit к ^ in the above inequality and employing (11) yields the following:
lim d(vmk+i,Vnk+\) = lim d(vmk,Vuk)= e + • (12)
k—+<X> k— + <X>
Since {vn} c S(X) and {vn} is R-preserving, the local S-transitivity of R leads to the implication that (vmk ,vnk) e R. Thus, it can be deduced
Ф(д("тк+1 ,vnk+i)) < $(vmk ,vnk )ф(d(Svmk , Svnk ))
< ф (П (d(vmk , S vmk ),d(vnk , S vnk )))
= ф(П (dmk ,dnk ))
<ф max dmk,dnk,dm • dn-“}) •
Taking the limit as к ^ in the aforementioned inequality leads to the conclusion that e < 0, a contradiction. Hence, {vn} is the R- preserving Cauchy sequence in Y. The R-completeness of the metric space (Y, d, R) now guarantees the existence of a point y e Y such that, lim vn =
n—^+^0
First, one assumed that S is R-continuous; one can deduce that lim vn+1 = lim Svn = S7. Applying the uniqueness of the limit, one
n—^+^0 n—^+^0
consequently establishes that S7 = 7, indicating that 7 e F(S). Alternatively, let R\Y is d-self-closed. The fact that {vn} is R-preserving and {vn} ^ y can be utilized again. This implies the existence of a subsequence {vnk} of {vn} with [vnk, y] e R, for all к e N0. If (vnk ,y) e R, then since S is a unified interpolative Kannan contraction, there exists
Ф(д(Svnk , SY)) < V(vnk ,Y)ф(д(Svnk , SY))
< ф(П(д(vnk, Svnk),d(Y,SY)))
= №(dnk,d(Y,SY)))
< ф max dnk,d(Y, SY),dak • dSY)1_^) , (13)
on taking the limit к ^ +rc>, in (13), one obtains
ф(дф/, Sy)) < ф(д(y, Sy))• (14)
It is important to note that in equation (14), if d(y, Sy) = 0, it is contradictory to (C4). Similarly, if (Y,vnk) e R, then by utilizing the symmetry of d, we once again encounter a contradiction of (C4). Therefore, d(y, Sy) = 0, implying y e F(S).
Theorem 3. Consider the RMS (X, d, R) where R is a locally S-transitive binary relation. Suppose that S is a unified interpolative Kannan contraction and there exist functions ф, ф : [0, +rc>) ^ [0, +rc>) satisfying conditions C, (i = 3,4,5,6) and Dj, (j = l, 2,3,4) holds. Then, there exists at least one Y e X such that y e F(S).
Proof.Following the steps of the previous theorem, one can obtain an R-preserving and non-increasing sequence {vn} such that there exists some C > 0 and vn converges to C + as n ^ +rc>. Suppose C > 0, then (7) implies that
Ф(С+) < limsupф(дп)
п—+ж
< limsupф (max{дп-\,дп,дПь-1 ■ д1~а})
п—+ж
< limsupф(к),
k—C+
a contradiction of (C5), thus C = 0 i.e. lim дп = 0. Now, to establish that
п—^+^o
the sequence {ип} is Cauchy, one makes a counter assumption. Suppose it is not Cauchy, then following the steps outlined in the previous theorem, there exists a positive real number e > 0, along with sub-sequences {гпк} and {vmk} of {гп}, where nk > mk > k, satisfying condition (12). Since {гп} c S(X) and {гп} is R-preserving, the local S-transitivity of R leads to the implication that (vmk ,гпк) e R. Thus, it can be deduced
ф(д(^шк+1 ,Гпк+1)) < V^mk ,"пк )ф(д(Svmk , Svпк ))
< ф max {dmk ,дпк ,дшк ■ дп-“}) >
on taking the limit k ^ in the above equation, it implies that
lim mf ф(а) < lim inf ф(д(Гтк+1, Гпк+г )) a—c+ к—+ж
< lim sup ф (max {дтк , дпк , дтк ■ дп-“})
к—+ж
< limsupф(а).
a—0
This results in a contradiction of (C6), thus establishing that the {гп} is an R-preserving Cauchy sequence is in Y. Given that (Y,d, R) is an R-complete metric space, there exists y e Y such that lim гп = y■ If the
п—-+ж
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
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self-mapping S is R-continuous, the desired conclusion can be derived, as demonstrated in the previous theorem.
Alternatively, let R\Y be d-self-closed then utilizing the fact that {vn} is R-preserving and {vn} ^ 7. This implies the existence of a sub-sequence
{vnk} of {vn} with [vnk e R, for all k e N0. One claims that d(y, Sy) = 0.
Let us assume that d(7, S7) > 0, if (vnk ,7) e R, then since S is a unified interpolative Kannan contraction, there exists
Ф(д(^ак+1, SY)) < #(vnk ,7)^(d(Svnk , SY))
< ф (tt(d(vnk,Svnk),d(7, SY)))
= ф(П(аПк ,d(Y,SY)))
< ф (max {dnk,d(Y,SY),dak ' ^, SY)1-a})
<ф max dnk ,d(Y,Sy)> dnk • d(Y,SY)1-a ,
by using (C3) and taking the limit as k ^ +rc>, one deduces d(y, Sy) < d(y, Sy), which leads to a contradiction. Furthermore, if (Y,vnk) e R, then by utilizing the symmetry of d, one encounters again a contradiction. Hence, d(Y, Sy) = 0, implying y e F(S)
Theorem 4. Consider the RMS (X, d, R), where R is a locally S-transitive and S-closed. Suppose the conditions Dj, (j = 1,2,3,4) hold and there exist the functions ф, ф : [0, +rc>) ^ [0, +rc>) satisfying the conditions C,
(i = 1,2,3,4) or (i = 3,4,5,6), such that,
ф(д^и, S л)) < ф (Q(d(v, Sv),d(л, S /л))), forall v, /л e XR (15)
Then there exists at least one y e X such that y e F(S).
By considering the specific values of the functions ф, ф, Q, and v, in Theorem 4, one can derive the following relational theoretic versions of Kannan fixed-point results and Interpolative Kannan fixed-point results respectively.
Corollary 1. Let (X, d, R) be an R-complete RMS, where R is a locally S-transitive and S-closed. Suppose that the conditions Dj, (j = 1,2,3) hold and there exists a parameter 0 < X < 2, such that
d(Sv, Sл) < X [d(v, Sv) + д(л, Sл)}, for all v, л e XR.
Then there exists at least one y e X such that y e F(S).
Corollary 2. Let (X, d, R) be an R-complete RMS, where R is a locally S-transitive and S-closed. Suppose that the conditions Dj, (j = 1,2,3) hold and there exists a pair of constants a,X e [0,1) with a = 0, satisfying
d(Sv, Sу) < \ [d(v, Sv)a ■ д(у, Sу)1-а] , for all ve XR.
Then there exists at least one y e X such that y e F(S).
An application
In this section, the authors have applied their research findings to derive a result concerning the existence of solutions for a nonlinear matrix equation. In this context, let the set denoted as M(n) encompasses all square matrices with dimensions of n x n, while H(n), P(n), and K(n), respectively represent the sets of Hermitian matrices, positive definite positive, and semi-definite matrices. When there is a matrix C from H(n), one uses the notation ||C||tr to refer to its trace norm, which is the sum of all its singular values. If there are matrices P and Q from H(n), the notation P f Q signifies that the matrix P - Q is an element of the set K(n), while P у Q indicates that P - Q belongs to the set P(n). The upcoming discussion relies on the significance of the following lemmas.
Lemma 1. (Ran & Reurings, 2002) If X e H(n) satisfies X y In, then
l|XII < 1.
Lemma 2. (Ran & Reurings, 2002) For n x n matrices X f O and Y f O, the following inequalities hold:
0 < tr(XY) < ||XIltr(Y).
Examine now the following nonlinear matrix equation,
u V
X = A + C*Tk (X )Cj (16)
i=1 k=1
In the above equation, A is defined as a Hermitian and positive definite matrix. Additionally, the notation Cj refers to the conjugate transpose of a square matrix Cj of size n x n. Furthermore, Yk represents continuous functions that preserve order, mapping from H(n) to P(n). It is noteworthy that Y(O) = O, where O represents a zero matrix.
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
Theorem 5. Consider the nonlinear matrix equation expressed in (16) and assume the following:
(Hi) there exists A e P(n) with YU=1 Yvk=1 C*Yk(A)Cj у 0;
(H2) for every X,Y e P(n), X y y implies
и v и v
EE C*Tk(X)Cj y EE CjYk (Y )Cj;
j=i k=i j=i k=i
(H3) Y^=1 Cj C* y N In, for some positive number N, and for all X,Y e P (n) with X y Y, the following inequality holds
maxk(tr(Yk(Y) - Yk(X))) <
1
2Nv
xmax
tr X -A- Cjrk (X)Cj ,tr Y -A- Cj,rk(Y )Cj
j=1 fc=l j=1 k=1
tr X -A- C-Tk (X)Cj
j=1 k=1
A- PYYY)C
j = 1 k = 1
1
2
Then, there exists at least one solution of the nonlinear matrix equation (16). Moreover, the iteration
Xr = a + C* Yk (Xr-i)Cj, (17)
j=i k=i
where Xo e P (n) satisfies Xo y A+YU=1 Yvk=1 C*Yk (Xo )Cj, Convergence towards the solution of the matrix equation, in the context of trace norm
II • Ik.
Proof.Let T : P(n) ^ P(n) be a mapping defined by
uv
T(X) = A + EE C*Yk(X)Cj, for all X e P(n).
j=i k=i
Consider R = {(X,Y) e P(n) x P(n) : X y y}. Consequently, the fixed point of T serves as a solution to the nonlinear matrix equation (16). It is pertinent to mention that R is T-closed and T is well-defined as well as R-continuous. From condition (H1) there is Y^Yk^ C*Yk (X )Cj у 0 for some X e P(n), thus (X, T(X)) e R and consequently P(n)(T, R) is nonempty.
Define д : P(n) x P(n) ^ R+ by
d(X, Y) = ||X - Y\\tr, for all X,Y e P(n). Then (P(n), д, R) is R-complete RMS. Then
\T(Y) - T(X)||tr = tr(T(Y) - T(X))
tr I EE C*(Yk(Y) - Yk(X))Cj
j=i k=i
u v
tr(CjCj(Yk(Y) - Yk(X)))
j=1 k=1
tr ( [£ Cj C*)Y,(Yk (Y) - Yk (X))
j=1 j ti
<
Cj Cj
j=1
x v x max\\(Yk(Y) - Yk(X))||tr
< ^ x max ||X - TX\tr, ||Y - TY\tr,
IIX - TXHI ■ \\Y - TY||tr
= 1(Q(||X - TXHtr, || Y - TY||tr))
(18)
Now, when considering ф(и) = v, ф(и) = 2, then equation (18) becomes
ф(д(TX, TY)) < ф (Q (d(X, T(X)),d(Y, T(Y)))).
Consequently, upon fulfilling all the hypotheses stated in Theorem 2, it can be deduced that there exists an element X* e P(n) for which T(X*) = X* holds good. As a result, the matrix equation (16) is guaranteed to possess a solution within the set P(n). □
Example 4. Consider the nonlinear matrix equation (16) for u = v = 2, and n = 3, with Yi(X) = X4, Y2(X) = X1, i.e.,
X
A + C*X 4 Ci + C* X 5 Ci + C* X 4 C2 + C* X 5 C2
(19)
where
995
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
e
A
0.177855454222667
0.001123532012243
0.144562121365390
0.001123654123643
0.177856213654500
0.133214526352116
0.144563214565439'
0.133214521452362
0.266521364125960
Ci
0.222353216521933
0.277652136521619
0.144563125462493
0.104402312563210
0.122365475632174
0.111232145236838
0.077854213651530
0.066321541236599
0.244512365214147
C2
0.255541232145296
0.074456321236541
0.155462136521421
0.177563214532317
0.222351452365355
0.133652123652627
0.277854621452056
0.100321256321427
0.199663251400003
By taking N = |, the conditions specified in Theorem 5 can be validated numerically by evaluating various specific values for the matrices involved. For example, they can be tested (and verified to be true) for
X
0.285221251452362
0.192072365214523
0.232862136541254
0.123815632145236
0.219152365214523
0.172062136521452
0.016912136521452
0.026932365214569
0.096802123652145
Y
0.385224563214521
0.192076541236541
0.232861236541256
0.123811236521452
0.319150000000000
0.172061236521452
0.016912365214896
0.026931236541526
0.196823652145230
To ascertain the convergence of {Xn} defined in (17), one commences with three distinct initial values.
Vo
Uo
'~20 0 °' 0 ill 0 0 0 15
0.500354112000372
0.022141236541532
0.054621236525374
0.454632123005061
0.151234561235184
0.045213625456758
0.398954120000949
0.104256348563137
0.103456212563418
Wo
0.100963214521244
0.255632122000784
0.111232152412246
0.066321213621732
0.210032145632300
0.080032356212332
0.005445632123530
0.288632512325983
0.177521363201611
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
VOJNOTEHNICKI GLASNIK/ MILITARY TECHNICAL COURIER, 2024, Vol. 72, Issue 3
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W & W15
0.683105295075270
0.342270819337423
0.548632937937303
0.342270947154744
0.468237110723960
0.450115485038123
0.548634067672523
0.450115496164186
0.678165537275706
with error 5.3354 x 10_12.
Figure 1 is a graphical illustration of the convergence phenomenon.
References
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Alam, A. & Imdad, M. 2017. Relation-theoretic metrical coincidence theorems. Filomat, 31(14), pp.4421-4439. Available at: https://doi.org/10.2298/FIL1714421A.
Alam, A. & Imdad, M. 2018. Nonlinear contractions in metric spaces under locally T-transitive binary relations. Fixed Point Theory, 19(1), pp.13-24. Available at: https://doi.org/10.24193/fpt-ro.2018.1.02.
Banach, S. 1922. Sur les operations dans les ensembles abstraits et leur applications aux equations integrales. Fundamenta Mathematicae, 3, pp.133-181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181.
Debnath, P., Konwar, N. & Radenovic, S. 2021. Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences. Springer Verlag, Singapore. Available at: https://doi.org/10.1007/978-981-16-4896-0.
Debnath, P., Mitrovic, Z.D. & Radenovic, S. 2020. Interpolative Hardy-Rogers and Reich-Rus-Cirictype contractions in Ь-metric spaces and rectangular 6-metric spaces. Matematicki vesnik, 72(4), pp.368-374 [online]. Available at: https://www.vesnik.math.rs/landing.php?p=mv204.cap&name=mv20409 [Accessed: 28 May 2024].
Hammad, H.A., Aydi, H. & Kattan, D.A. 2023. Hybrid interpolative mappings forsolving fractional Navier-Stokes and functional differential equations. Boundary Value Problems, 2023, art.number:116. Available at: https://doi.org/10.1186/s13661-023-01807-1.
Jain, S. & Radenovic, S. 2023. Interpolative fuzzy Z-contraction with its application to Fredholm non-linear integral equation. Gulf Journal of Mathematics, 14(1), pp.84-98. Available at: https://doi.org/10.56947/gjom.v14i1.1009.
Jain, S., Stojiljkovic, V.N. & Radenovic, S.N. 2022. Interpolative generalised Meir-Keeler contraction. Vojnotehnicki glasnik/Military Technical Courier, 70(4), pp.818-835. Available at: https://doi.org/10.5937/vojtehg70-39820.
Kannan, R. 1968. Some results on fixed points. Bulletin of the Calcutta Mathematical Society, 60, pp.71-76.
Karapinar, E. 2018. Revisiting the Kannan type Contractions via Interpolation. Advances in the Theory of Nonlinear Analysis and its Application, 2(2), pp.85-87. Available at: https://doi.org/10.31197/atnaa.431135.
Karapinar, E. 2021. Interpolative Kannan-Meir-Keeler type contraction. Advances in the Theory of Nonlinear Analysis and its Application, 5(4), pp.611-614. Available at: https://doi.org/10.31197/atnaa.989389.
Karapinar, E., Agarwal, R. & Aydi, H. 2018a. Interpolative Reich-Rus-Ciric Type Contractions on Partial Metric Spaces. Mathematics, 6(11), art.number:256. Available at: https://doi.org/10.3390/math6110256.
Karapinar, E., Alqahtani, O. & Aydi, H. 2018b. On Interpolative Hardy-Rogers Type Contractions. Symmetry, 11(1), art.number:8. Available at: https://doi.org/10.3390/sym11010008.
Karapinar, E., Fulga, A. & Yesilkaya, S.S. 2021. New Results on Perov-Interpolative Contractions of Suzuki Type Mappings. Journal of Function Spaces, 2021(1), art.number:9587604. Available at: https://doi.org/10.1155/2021/9587604.
Nazam, M., Aydi, H. & Hussain, A. 2023a. Existence theorems for (Ф, Ф)-orthogonal interpolative contractions and an application to fractional differential equations. Optimization, 72(7), pp.1899-1929. Available at: https://doi.org/10.1080/02331934.2022.2043858.
Nazam, M., Javed, K. & Arshad, M. 2023b. The (Ф, Ф)-orthogonal interpolative contractions and an application to fractional differential equations. Filomat, 37(4), pp.1167-1185. Available at: https://doi.org/10.2298/FIL2304167N.
Ran, A.C.M. & Reurings, M.C.B. 2002. On the nonlinear matrix equation X + A*F(X)A = Q : solutions and perturbation theory. Linear Algebra and its Applications, 346(1-3), pp.15-26. Available at: https://doi.org/10.1016/S0024-3795(01)00508-0.
Samet, B., Vetro, C. & Vetro, P. 2012. Fixed point theorems for «-^-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), pp.2154-2165. Available at: https://doi.org/10.1016/j.na.2011.10.014.
Teoremas de existencia para una contraction unificada Kannan interpolativa con una aplicacion en ecuaciones matriciales no lineales
Koti N. V. V. Vara Prasad3, Vinay Mishraa, autor de corresponden-cia, Stojan Radenovicb
a Guru Ghasidas Vishwavidyalaya, Departamento de Matematicas, Bilaspur, RepOblica de la India
b Universidad de Belgrado, Facultad de Ingenieria Mecanica,
Belgrado, RepOblica de Serbia
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
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CAMPO: matematicas
TIPO DE ARTiCULO: art^culo cientifico original Resumen:
Introduccion/objetivo: Este artfculo establecio un nuevo marco matematico al descubrir las relaciones entre las contracciones Kannan y las contracciones Kannan interpolativas. El concepto de contracciones Kannan interpolativas unificadas se introdujo en el marco de un espacio metrico relacional. Ademas, el estu-dio tuvo como objetivo ampliar el concepto de admisibilidad alfa incorporando ideas metricas relacionales especfficas.
Metodos: Se realize una exploracion detallada de las propieda-des y caracterfsticas de las contracciones Kannan y las contracciones Kannan interpolativas. La investigacion introdujo el concepto de contracciones Kannan interpolativas unificadas y for-mulo nuevos resultados de punto fijo para estas asignaciones.
Resultados: El estudio establecio con exito resultados de punto fijo para las contracciones unificadas Kannan interpolativas den-tro del marco de los espacios metricos relacionales. Ademas, se proporciono una aplicacion de estos resultados para resolver un problema relacionado con ecuaciones matriciales no lineales, enfatizando aun mas su importancia.
Conclusion: Los hallazgos de este estudio han permitido avan-zar significativamente en la comprension de las contracciones Kannan y las contracciones Kannan interpolativas, ofreciendo un marco unificado para su analisis. La introduccion de con-tracciones unificadas Kannan interpolativas y la expansion de la admisibilidad alfa tienen amplias implicaciones para el campo de las matematicas.
Palabras claves: contraccion interpolativa unificada de Kannan, R-admisible, espacio metrico relacional.
Теоремы существования унифицированного интерполяционного сокращения Каннана с применением нелинейных матричных уравнений
Коти Н. В. В. Вара Прасад3, Винай Мишра3, корреспондент, Стоян Раденович6
a Биласпурский унуиверситет Гуру Гасидас Вишвавидьялайя, математический факультет, г. Биласпур, Республика Индия
6 Белградский университет, машиностроительный факультет, г Белград, Республика Сербия
РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций ВИД СТАТЬИ: оригинальная научная статья
Резюме:
Введение/цель: В данной статье были обозначены новые математические рамки, освещающие взаимосвязи между сокращениями Каннана и интерполяционными сокращениями Каннана. Концепция унифицированных интерполяционных сокращений Каннана была введена в рамках реляционного метрического пространства. Помимо того, целью исследования было развитие концепции альфа-допустимости за счет включения конкретных идей в отношении относительных показателей.
Методы: Было проведено подробное исследование
свойств и характеристик сокращений Каннана и интерполяции сокращений Каннана. В ходе исследования была представлена концепция унифицированной интерполяции сокращений Каннана и сформулированы новые результаты с фиксированными точками.
Результаты: Исследование дало успешные результаты с фиксированной точкой для унифицированных интерполяционных сокращений Каннана в рамках реляционных метрических пространств. Помимо того, было представлено применение этих результатов для решения задачи, касающейся нелинейных матричных уравнений, тем самым подчеркивая их значимость.
Выводы: Результаты данного исследования значительно улучшили понимание сокращений Каннана и интерполяционных сокращений Каннана, представив единую основу для их анализа. Введение унифицированных интерполяционных сокращений Каннана и расширение допустимости альфа-допустимости широко применяются в области математики.
Ключевые слова: унифицированное интерполяционное
сжатие Каннана, R-допустимое, реляционное метрическое пространство.
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
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Теореме nocTOja^a за ]единствену интерполативну Кананову контракци]у са применама код нелинеарних матричних]едначина
Коти Н. В. В. Вара Прасад3, Вина] Мишраа, ауторза преписку, CmojaH Раденови^5
а Гуру Гасидас Вишвавид]ала]а, Оде^е^е математике,
Биласпур, Република Инди]а б Универзитету Београду, Машински факултет,
Београд, Република Срби]а
ОБЛАСТ: математика
КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад Сажетак:
Увод/цил>: Овим радом успоставъен je нови математички оквир откривашем односа измену Кананове контракци-je и иегове интерполативне контракци]е. Концепт об-]едишене интерполативне Кананове контракци]е уведен ]е у оквиру релационог метричког простора. Поред тога, студи]а ]е имала за циъ да прошири концепт алфа-прихватъивости угра^ивашем специфичних релационих метричких иде]а.
Методе: Детаъно истраживаше свортава и карактери-стика Кананове контракци]е и шегове интерполативне контракци]е били су и рани]е разматрани. Овим истражи-вашем уведен ]е концепт унифициране интерполаци]е Кананове контракци]е чиме су формулисани нови резултати фиксне тачке за ших.
Резултати: Студи]а ]е успешно потврдила резултате фиксне тачке за унифициране интерполативне Кананове контракци]е у оквиру релационих метричких простора. По-ред тога, примена ових резултата за решаваше проблема ко]и се тиче нелинеарних матричних]едначина додатно наглашава шихов знача].
Закъучак: Налази ове студи]е знача]но су унапредили не-довоъно разумеваше Кананових контракци]а и шегових ин-терполативних контракци]а, нудеЬи ]единствен оквир за шихову анализу. Увод у унифициране интерполативне Кананове контракци]е и проширеше алфа-прихватъивости има широку примену у области математике.
Къучне речи: унифицирана интерполативна Кананова кон-тракци]а, R-допустив, релациони метрички простор.
Paper received on: 30.04.2024.
Manuscript corrections submitted on: 20.09.2024.
Paper accepted for publishing on: 21.09.2024.
© 2024 The Authors. Published by Vojnotehnicki glasnik/Military Technical Courier (http://vtg.mod.gov.rs, http://BTr.MO.ynp.cp6}. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).
Vara Prasad, K.N.V.V. et al., Existence theorems for a unified interpol. Kannan contraction with an applic. on nonlin. matrix equ., pp.980-1003
