CC BY
53Probl. Anal. Issues Anal. Vol. 11 (29), No 1, 2022, pp. 45-57 45
DOI: 10.15393/j3.art.2022.10370
UDC 517.988
B. S. Choudhury, N. Metiya, S. Kundu
A MULTIVALUED FIXED POINT RESULT WITH ASSOCIATED DATA DEPENDECE AND STABILITY
STUDY
Abstract. In this paper, our primary result is on the existence of non-empty fixed point sets for a new multivalued function defined here. An admissibility condition is also postulated and used in the main theorem. In two separate sections, we present data dependence and stability results for the non-empty fixed-point sets of these mappings. Some consequences of the main theorem are discussed. The analysis is in the most general setting of metric spaces. An illustrative example is discussed, which shows that our existence theorem effectively extends some results in this line of research.
Key words: fixed point, data dependence, stability, metric space, admissibility condition
2020 Mathematical Subject Classification: 47H10, 54H25, 54H10
1. Introduction and Mathematical Background. Multivalued contractions appeared first in fixed point theory in the work of Nadler [14]. This work was followed by a development of the branch of fixed point theory in the domain of set-valued analysis. The books [2], [12] describe this development to a considerable extent.
There have been several generalizations of the famous Banach's contraction mapping principle, which appeared in the famous work of Ba-nach [3] in 1922. Some of the generalizations are Boyd et al. [5] in 1969, Meir et al. [13] in 1969, Suzuki [18] in 2008, etc. This domain of mathematics is active in contemporary research as well. Also, many of these results have important applications in different problem of mathematics.
One of the recent generalizations is due to Feng and Liu [8] in 2006, which was followed by the introductions of several single and multivalued contractions, incorporating the basic idea of Feng and Liu. These
© Petrozavodsk State University, 2022
have come to be known as Feng-Liu-type contractions, which occupy an important place in metric fixed point theory.
Alongside, there has been another development in fixed point theory, which addresses the fact coming out of the observations: that in many established fixed point results the contraction condition between arbitrary pairs of points in the space is not utilized. There are scopes of restricting the domains of these contractions. Two approaches appeared for that, one is introduction of order relations on the spaces [9], [11], [15], and the other is the use of admissibility conditions, which are certain restrictions on the concerned mapping [1], [10], [17].
In this paper, we introduce a new multivalued contraction of Feng-Liu-type and a new admissibility condition. We utilize these concepts to prove a new result in fixed point theory. Further, we investigate the associated data dependence and stability problem of the fixed point sets. Some references on these topics are [4], [6], [7], [16].
We begin with the essentials of mathematics required for our purpose. Let (X, p) be a metric space. We denote the collection of nonempty subsets of X by V (X), the collection of nonempty bounded subsets of X by B(X), and the collection of nonempty closed and bounded subsets of X by CB(X).
Let z E X and A E B(X). The distance of A from z, denoted by V(z,A), is defined as V(z,A) = inf{p(z,y): y E A}. The distance between two subsets A,B E CB(X), denoted by %(A,B), is defined as %(A,B) = max {sup V(z,B), sup V(x,A)}. % is known as the Hausdorff
zea xeb
metric induced by the metric p on CB(X) [14]. Further, if (X, p) is complete then (CB(X), %) is also complete.
Definition 1. A point x E X is called a fixed point of a multivalued mapping T: X ^ V(X) if x E Tx. We denote the set of fixed points of T by FT.
Definition 2. A function f: X ^ R, where (X,p) is a metric space, is said to be lower semi-continuous if f (x) ^ liminf f (xn) for any convergent
sequence {xn} in X with limit x E X.
Definition 3. [17] A mapping f: X ^ X is said to be rj-admissible, where rq: X x X ^ [0, if rq(u,v) ^ 1, for u,v E X implies
V(fu,fv) ^ L
In a separate vein, the following definition was introduced in [1].
Definition 4. [1] Let X be a nonempty set and rj, 0: X ^ [0, A
function f: X ^ X is called a cyclic (rj — 0)-admissible mapping if
(i) rj(x) ^ 1, for some x E X, implies 0(fx) ^ 1,
(ii) 0(x) ^ 1, for some x E X, implies rj(fx) ^ 1.
In the following, we define cyclic ( rq — 0) admissibility for multivalued mappings.
Definition 5. Let X be a nonempty set and rj, 0: X ^ [0, f: X ^ V(X). We say that f is a cyclic (rj — 0)-admissible mapping if
(i) rj(x) ^ 1, for some x E X, implies 0(u) ^ 1, for u E fx,
(ii) 0(x) ^ 1, for some x E X, implies rj(u) ^ 1, for u E fx.
Example. Define T: X ^ V(X), where X = [0, 1] is the usual metric space, as Tx = [0, |]. Define rj, 0: X ^ [0, as
fex,
i0,
,^,if 0 ^ x ^ 1, 1 \ I cosh x, if 0 ^ x ^ 2, r](x) = < 2 and p(x) = < 2
0, otherwise 0, otherwise.
Let x e X and rj(x) ^ 1. Therefore, x e [0, 1 ] and Tx = [0, f] c [0, 1 ]. Then 0(u) ^ 1, for all u E Tx. Similarly, if y E X and 0(y) ^ 1, it can be shown that rj(v) ^ 1, for all v E Ty. Therefore, T is a cyclic (rj — 0)-admissible mapping.
In the following, we define a generalized Feng-Liu-type contraction for multivalued mapping.
Definition 6. Let (X, p) be a metric space, T: X ^ CB(X) be a multivalued mapping and rj, 0: X ^ [0, Let b, c E (0,1) with c < b. Then T is said to be a generalized Feng-Liu-type contraction if for x E X with A(x) ^ 1 there is a y E Tx, such that
bp(x, y) ^ fx and f y ^ c M(x, y), where
M {X,V) = max ^ ^, ^ },
A(x) = rj(x) or 0 (x) and f: X ^ R is given by fx = V(x,Tx),x EX.
Definition 7. Let (X, p), (Y, p\) be two metric spaces and % be the Hausdorff metric on CB(Y). A multivalued mapping T: X ^ CB(Y) is
said to be continuous at z E X if for any sequence {zn} in X, lim %(Tz, Tzn) = 0, whenever p(z, zn) ^ 0 as n ^
2. Multivalued fixed point results.
Theorem 1. Let T: X ^ CB(X) be a multivalued mapping and rq, ft: X ^ [0, where (X, p) is a complete metric space. Suppose that:
(i) there exists x0 E X, such that rq(x0) ^ 1 or ft(x0) ^ 1; (ii) T is cyclic (rq — ft)-admissible; (iii) T is continuous, or f, where f (x) = V(x, Tx) for all x E X, is lower semi-continuous; (iv) there exist b, c E (0, 1) with c < b, such that T is a generalized Feng-Liu-type contraction. Then Ft is non-empty.
Proof. Since Tx E CB(X), for x E X and b E (0, 1), we have the following:
for any x E X there exists y E Tx, such that p(x,y) ^ -V(x,Tx), that is,
b
b p(x, y) ^ V(x, Tx) = f (x). (1)
By the assumption (i), there exists x0 E X, such that rq(x0) ^ 1 (the proof is similar if ft(x0) ^ 1). Then A(x0) ^ 1 and, hence, by (1) and the assumption (iv), we choose x1 E Tx0 such that
b p(x0,x1) ^ fx0 and fx\ ^ cM(x0,x1).
As rq(x0) ^ 1 and x1 E Tx0, by the assumption (ii), we have ft(x1) ^ 1 and, hence, A(xi) = ft(x1) ^ 1. Then, by (1) and the assumption (iv), we choose x2 E Tx1 such that
b p(x1,x2) ^ fx1 and fx2 ^ cM(x1,x2).
As ft(x1) ^ 1 and x2 E Tx1, by the assumption (ii), we have rq(x2) ^ 1 and, hence, A(x2) = rq(x2) ^ 1. Then, by (1) and the assumption (iv), we choose x3 E Tx2 such that
bp(x2,x3) ^ fx2 and fx3 ^ cM(x2,x3).
As rq(x2) ^ 1 and x3 E Tx2, by the assumption (ii), we have ft(x3) ^ 1 and, hence, A(x3) = ft(x3) ^ 1. Then, by (1) and the assumption (iv), we choose x4 E Tx3 such that
bp(x3,x4) ^ fx3 and fx4 ^ cM(x3,x4).
As ¡3(x3) ^ 1 and x4 E Tx3, by the assumption (ii), we have rq(x4) ^ 1 and, hence, A(x4) = rq(x4) ^ 1. In this way, we construct a sequence {xn}, such that
xn+\ E Txn with ^(x2n) ^ 1, P(%2n+i) ^ 1, for all n ^ 0. (2)
By (2), either rq(xn) ^ 1 or ¡3(xn) ^ 1 for all n ^ 0 and, hence, A(xn) ^ 1, for all n ^ 0. Applying the assumption (iv), we have
b p(xn,xn+i) ^ fxn and fxn+\ ^ cM(xn,xn+i), for all n ^ 0, (3)
where fxn = V(xn,Txn) and
M (xn,xn+i) =
_ ( V(xn,Txn)+ V(xn+i,Txn+i) V(xn,Txn+i) \ = ma^p(xn, xn+l),-^-,-^-J ^
^ f / \ P(xn,xn+i)+ P(xn+i,xn+2) P(xn,xn+2)>\ ^ max|p(xn,xn+i), -2-, -2-J =
p(xn, xn+i) + p(xn+i,xn+2)
= max | p(xn,xn+i), ^ ;
p(xn, xn+i) + p(xn+i,xn+2) 1 <
2 J <
< max{p(xn, xn+i), p(xn+i, xn+2)}. (4)
Suppose that 0 < p(xn,xn+1) < p(xn+i,xn+2). Then, from (3) and (4), we have
bp(xn+i,xn+2) < fxn+i < cM(xn,xn+i) <
< c max{p(xn,xn+i), p(xn+i,xn+2)} < cp(xn+i,xn+2).
As c < b, it is a contradiction. Thus, we have p(xn+i,xn+2) < p(xn,xn+i), for all n ^ 0. Therefore, using (3) and (4), we come to
bp(xn,xn+i ) < V(xn,Txn), V(xn+i,Txn+i) < cp(xn,xn+i), for all n ^ 0. (5)
From (2) and (5), we have
c
bp(Xn+i,Xn+2) < c p{xn,Xn+i) , that ^ p{xn+i,Xn+2) <7 P(xn ,xn+i)
b
c
V(xn+i,Txn+i) < cp(xn,xn+i), that is, V(xn+i,Txn+i) < T V(xn,Txn).
b
By repeated application of the above inequalities, we get
r C -| ra+l
P(xra+l,Xra+2) p(Xo,Xi), for all n E N,
LoJ
rcnra+l
Dixn+i,Txn+i) ^ - V(xo,Tx0), for all n E N. LoJ
c
As c < b, we have a = - < 1. Hence, an ^ 0 as n ^ For m, n E N
b
with m > n, we have
p(xn, xm) ^ p(xn,xn+i)+ p(xn+i,xn+2) + ... + p(xm-i,xm) ^ ^ anp(xo,x1) + an+1p(xo,x1) + ... + am-1 p(x0,x1) ^
+TO an
^ } akp(xo,xi) = --- p(x0,x1) ^ 0 as n ^
— a)
k=n K '
Hence, [xn] is a Cauchy sequence. As (X,p) is complete, there exists a point x E X, such that
xn ^ x as n ^ (8)
We consider the two following cases.
Case (i): Suppose that T is continuous.
Using (8) and the continuity of T, we get H(Txn,Tx) ^ 0 as n ^ which implies that V(xn+l,Tx) ^ 0 as n ^ that is, V(x,Tx) = 0. Since Tx E CB(X), Tx = Tx, where Tx denotes the closure of Tx. Now, V(x, Tx) = 0 implies that x E Tx = Tx, that is, x E FT. Therefore, Ft is non-empty.
Case (ii): Suppose that f, where f(x) = V(x,Tx) for all x E X, is lower semi-continuous.
As f < 1, we have from (7) that lim fxn = lim V(xn,Txn) = 0.
0 ra^+TO ra^+TO
Then 0 ^ V(x,Tx) = fx ^ liminffxn = 0, that is, V(x,Tx) = 0.
ra^+TO
Arguing as above, we get x E Ft, that is, Ft is non-empty. □
3. Data dependence of fixed point sets. Here we prove a data dependence result for the mappings described in the previous section. The problem of data dependence is to estimate the distance between the fixed point set of an operator with the corresponding fixed point set of
another mapping where the functional value of the mapping at every point is different from that of the given operator by a magnitude less than a given number. It is generally considered for multivalued operators. In that case, the Hausdorff distance is often used for estimating distance between two functional values, wherever it is applicable.
Here we investigate the data dependence of fixed point sets of the setvalued contractions mentioned in Section 2.
Theorem 2. Let (X, p) be a complete metric space, rq, ft: X ^ [0, Let Tl, T2: X ^ CB(X) be two multivalued mappings, such that H(Tlx, T2x) ^ K for all x E X, where K > 0 is a fixed a real number. Suppose that T2 satisfies the assumptions (ii), (iii) and (iv) of Theorem 1, Ft1 = 0 and rq(x) ^ 1 or ft(x) ^ 1 for any x E Ft1 . Then Ft2 = 0 and K
sup V(x, FT2) -.
xEFTl 0 — C
Proof. By the assumption of the theorem, Ft1 = 0. Let y0 E Ft1 , that is, y0 E Tly0. Then 'q(y0) ^ 1 or ft(y0) ^ 1. Therefore, we know that there exists y0 E X, such that 'q(y0) ^ 1 or ft(y0) ^ 1, and T2 satisfies the assumptions (ii), (iii) and (iv) of Theorem 1. By Theorem 1, F?2 is nonempty, that is, Ft2 = 0. Assuming that rq(y0) ^ 1 (the proof is similar if ft(yo) ^ 1) and arguing similarly as in the proof of Theorem 1, we construct a sequence {yra}, such that
Vra+i E T2yn; rq(y2n) > 1, ft(y2ra+i) ^ 1, for all n ^ 0, (9)
bp(yra,yra+l) ^ T>(yra,T2Vra), 'D(Vra+l, T2Vra+l) ^ Cp(y„,, Vra+l) , for all U ^ 0
(10)
and
rC1 ra+l
p(Vra+l,Vra+2) ^7 P(V0,Vl), for all U ^ 0,
rq«+l > (11)
V(yn+l,T2yn+l) ^ V(yo,T2Vo), for all n ^ 0.
Like in the proof of Theorem 1, we prove {yra} to be a Cauchy sequence in X and that there exists u E X, such that
yra ^ u as n ^ (12)
and u is a fixed point of T2, that is, u E T2u. Following (10), we have
1 1 K
P(Vo,Vl) ^ ^V(Vo,T2Vo) ^ ^H(TlVo,T2yo) ^ y. (13)
Using (11), we have
n n ^
p(yo,u) C ^p(Уi, yi+i)+p{yn+\,u) C ^ ^ p(yo, yi)+p{yn+\,u).
i=0 i=0
Taking limit as n ^ in the above inequality and using (12) and (13) we have
n 1 b K K o ,vm c --— p( v0, vl) C —-r — C
p(yo,u) C ^ [b p{yo, yi) C (1 _ £) P(Уo, yi) C i=0 '
(1 - IV (b - c) b b-c
K
Thus, given y0 E FTl, we have u E FT2, for which p(y0,u) C ,
b — c
K
holds. Hence, V(y0,FT„) C -• As y0 E FTl is arbitrary, we get
b — c
K
supxeFTl
V(x,ft2) C □
4. Stability of fixed point sets. In this section, we have a stability result, in which we answer the question whether for a convergent sequence of functions of the category considered in Section 2, the corresponding fixed point sets also converge to the fixed point set of the limiting function.
Theorem 3. Let (X, p) be a complete metric space, rj, ft: X ^ [0, Suppose there exists x0 E X, such that r](x0) ^ 1 or ft(x0) ^ 1. Let {Tn: X ^ CB(X): n E N} be a sequence of continuous multivalued mappings, which uniformly converges to T: X ^ CB(X). Suppose that each Tn (n E N) and T satisfy the assumptions (ii) and (iv) of Theorem 1. Then FTn = 0, for each n E N and FT = 0. Furthermore, suppose that ft(x) ^ 1 or r](x) ^ 1 for any x belonging to FTn (n E N) and FT. Then limn—H(FTn, FT) = 0, that is, the fixed point sets of the sequence of mappings {Tn} are stable.
Proof. Since each Tn (n E N) is continuous and Tn ^ T uniformly as n ^ T is continuous. Then each Tn (n E N) and T satisfy the
assumptions (ii), (iii) and (iv) of Theorem 1. By Theorem 1, we have FTn = 0, for each n E N and FT = 0. Let Kn = supxeX H(Tnx,Tx), where n E N. Since the sequence {Tn} is uniformly convergent to T, we have
lim Kn = lim sup H(Tnx,Tx) = 0. (14)
n—n—
By Theorem 2, we have sup V(x,FT) ^ ^ and sup V(x,FTn) ^ ^,
xeFt„ XeFT
for all n E N. Combining these two inequalities, we get
K
K(FTn ,FT) ^ , for all n E N. b — c
Taking limit as n ^ in the above inequality and using (14), we have
K
lim n{FTn ,FT) ^ lim -—— = 0,
o — C
that is, lim %(Ft„ , Ft) = 0. So, the fixed point sets of the sequence of mappings {Tn: n E N} are stable. □
5. Consequences and illustration. We have the following consequent results of our main theorem.
Theorem 4. Let (X, p) be a complete metric space and T: X ^ CB(X) be a multivalued mapping, such that T is continuous or f, where f (x) = V(x,Tx) for all x E X is lower semi-continuous. Suppose that there exists c E (0,1), such that for all x,y E X
U(Tx,Ty) ^ cM(x,y).
Then Ft is nonempty.
Proof. Let us define rj, ft: X ^ [0, as rq(x) = ft(x) = 1, for all x E X. Then the assumptions (i) and (ii) of Theorems 1 are trivially satisfied. Let x E X and b E (0,1) with c < b. As b E (0,1), there exists y E Tx, such that bp(x,y) ^ V(x,Tx) = fx. Hence, by contractive inequality of the theorem, we have fy = V(y,Ty) ^ %(Tx,Ty) ^ cM(x,y). Therefore, for x E X with A(x) ^ 1, there is a y E Tx, such that bp(x,y) ^ fx and fy ^ cM(x,y). Hence, by an application of Theorem 1, we conclude that FT
is nonempty. □
Theorem 5. [14] Let (X,p) be a complete metric space and T: X ^ CB(X) be a multivalued mapping. Suppose that there exists c E (0,1), such that for all x,y E X,
K(Tx,Ty) ^ cp(x,y).
Then Ft is nonempty.
Proof. The inequality of the theorem implies that mapping T is continuous. Then the proof of the theorem (Nadler's result) follows from that of Theorem 4. □
Example. Take the metric space (X, p), where X = {0,1, 2,... ,n,...} and
I 0, if x = y, i x + y, if x = y.
p(x, y)
Let T: X ^ CB(X) be defined as T(x)
{4,5}, if x = 0, 4, 5, {2, 3}, if x =1, 2, 3, {7}, if x = 6, 8, {8}, if x = 7, {x,x +1,x + 2}, if x ^ 9.
Define 7], ft: X ^ [0, +^>) as eX, if x = 0,1,..., 5,
r](x)
X, 0,
0, otherwise
and ft(x)
!
2, if x = 0,1,..., 5, 0, otherwise.
Take = 0.7 and = 0.5. Then all the conditions of Theorem 1 are satisfied and here FT = X — {0,1, 6, 7, 8} is the fixed point set of T.
Remark 1. Let us take x = 0 and y = 1 in the example above. Then
H(Tx, Ty) = H(T0,T 1) = 7 > 1 = p(0,1) = p(x, y),
which shows that there exists no c E (0,1), such that H(Tx,Ty) c c p(x, y). Therefore, the inequality of Theorem 5 is not satisfied and, hence, Theorem 5 is not applicable to this example. So, Theorem 1 is a proper generalization of Theorem 5, that is, Theorem 5 in [14].
Remark 2. Let us take x = 6 in the example above. Then y = 7 is the only member in Tx and b p(x, y) = 0.7x 13 = 9.1 < f(x) = D(x, Tx) = 13. But f(y) = D(y ,Ty) = 15 > 13 = p(x, y). There exists no c E (0,1), such that f(y) = D(y ,Ty) C cp(x, y). Therefore, Theorem 3.1 in [8] is not applicable to this example.
Remark 3. The result of Feng and Liu (Theorem 3.1 of [8]) is derived for mappings that are from X to C (X), where C (X) is the set of all closed subsets of X. Our result (Theorem 1) generalizes the result mentioned
above for the case where the functional values are restricted to CB(X). The above Remark 2 shows that in this case the generalization is exact.
Acknowledgments. The authors gratefully acknowledge the suggestions made by the learned referee.
References
[1] Alizadeh S., Moradlou F., Salimi P. Some fixed point results for (a — ft) -(■0 — - contractive mappings. Filomat, 2014, vol. 28, no. 3, pp. 635-647. DOI: 10.2298/FIL1403635A
[2] Amar A. B., O'Regan D. Topological fixed point theory for singlevalued and multivalued mappings and applications. Springer International Publishing, Switzerland, 2016.
DOI: https://doi.org/10.1007/978-3-319-31948-3
[3] Banach S. Sur les operations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund Math., 1922, vol. 3, pp. 133-181.
[4] Bose R. K., Mukherjee R. N. Stability of fixed point sets and common fixed points of families of mappings. Indian J. Pure Appl. Math., 1980, vol. 11, no. 9, pp. 1130-1138.
[5] Boyd D. W., Wong J. S. W. On nonlinear contractions. Proc. Amer. Math. Soc., 1969, vol. 20, pp. 458-464.
DOI: https://doi.org/10.1090/S0002-9939-1969-0239559-9
[6] Chifu C., Petrusel G. Coupled fixed point results for (<p,G)-contractions of type (b) in b-metric spaces endowed with a graph. J. Nonlinear Sci. Appl., 2017, vol. 10, pp. 671-683. DOI: 10.22436/jnsa.010.02.29
[7] Choudhury B. S., Metiya N., Kundu S. Existence, data-dependence and stability of coupled fixed point sets of some multivalued operators. Chaos, Solitons and Fractals, 2020, vol. 133.
DOI: 10.1016/j.chaos.2020.109678.
[8] Feng Y., Liu S. Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl., 2006, vol. 317, pp. 103-112.
DOI: https://doi.org/10.1016Zj.jmaa.2005.12.004
[9] Harjani J., Sadarangani K. Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal., 2009, vol. 71, pp. 3403-3410. DOI: https://doi.org/10.1016/j.na.2009.01.240
[10] Hussain N., Karapinar E., Salimi P., Akbar F. a-admissible mappings and related fixed point theorems. J. Inequal. Appl., 2013, Article number: 114(2013). DOI: https://doi.org/10.1186/1029-242X-2013-114
[11] Kadelburg Z., Radenovic S. Fixed point and tripled fixed point theorems under Pata-type conditions in ordered metric spaces. International Journal of Analysis and Applications, 2014, vol. 6, no. 1, pp. 113-122.
[12] Kirk W., Shahzad N. Fixed point theory in distance spaces. Springer International Publishing, Switzerland, 2014.
DOI: https://doi.org/10.1007/978-3-319-10927-5
[13] Mier A., Keeler E. A theorem on contraction mappings. J. Math. Anal. Appl., 1969, vol. 28, pp. 326-329.
DOI: https://doi.org/10.1016/0022-247X(69)90031-6
[14] Nadler S. B. Jr. Multivalued contraction mapping. Pac. J. Math., 1969, vol. 30, no. 2, pp. 475-488.
[15] Ran A. C. M., Reurings M. C. B. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc., 2004, vol. 132, no. 5, pp. 1435-1443.
DOI: https://doi.org/10.1090/S0002-9939-03-07220-4
[16] Rus I. A., Petrusel A., Sintamarian A. Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Anal., 2003, vol. 52, pp. 1947-1959. DOI: 10.1016/S0362-546X(02)00288-2
[17] Samet B., Vetro C., Vetro P. Fixed point theorems for a — contractive type mappings. Nonlinear Anal., 2012, vol. 75, pp. 2154-2165.
DOI: https://doi.org/10.1016Zj.na.2011.10.014
[18] Suzuki T. A generalized Banach contraction principle that characterizes metric completeness. Proc. Amer. Math. Soc., 2008, vol. 136, no. 5, pp. 1861-1869.
DOI: https://doi.org/10.1090/S0002-9939-07-09055-7
Received June 1, 2021. In revised form, October 18, 2021. Accepted October 22, 2021. Published online November 16, 2021.
Binayak S. Choudhury
Indian Institute of Engineering Science and Technology,
Shibpur, Howrah - 711103, West Bengal, India
E-mail: [email protected], [email protected]
Nikhilesh Metiya Sovarani Memorial College,
Jagatballavpur, Howrah-711408, West Bengal, India E-mail: [email protected], [email protected]
Sunirmal Kundu
Government General Degree College,
Salboni, Paschim Mednipur - 721516, West Bengal, India.
E-mail: [email protected]
