R E S E A R C H
Open Access
A fixed point result and the stability
problem in Lie superalgebras
Madjid Eshaghi
1,2, Sadegh Abbaszadeh
1,3*, Manuel de la Sen
4and Zahra Farhad
1*Correspondence:
1Department of Mathematics,
Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, 35195-363, Iran
3Young Researchers and Elite Club,
Malayer Branch, Azad University, Malayer, Iran
Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the concept of normed Lie superalgebras and define the superhomomorphism and the superderivation in normed Lie superalgebras. We define a generalizedT-orbitally complete metric space and prove a new fixed point alternative concerning the stability problem of functional equations. Consequently, we deal with the stability of the superhomomorphism and the superderivation in Lie superalgebras using that new fixed point result. In addition, we find some conditions under which an approximate superhomomorphism or superderivation is an exact superhomomorphism or superderivation.
MSC: Primary 47H10; 39B82; secondary 16W25
Keywords: Lie superalgebra; superhomomorphism; superderivation;T-orbitally complete metric space
1 Introduction
Most physicists believe that all of the familiar particle interactions, weak and electromag-netic as well as strong interactions, are associated with a Lie algebra in similar ways. This suggests that it may be possible to unify all the particle interactions as different aspects of a single underlying interaction, based on a single simple Lie algebra. With an increas-ing amount of theory and applications concernincreas-ing Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications: Casimir operators [], derived, lower central and upper central sequences, the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [, ], and recently megaideals []. These characteristics are particularly crucial when considering possible connections among Lie algebras. Physically motivated relations between two Lie algebras, namely contractions and deformations, have been extensively studied; seee.g.[–].
The concept of Lie algebra can be expressed in several different ways. The most familiar are in terms of generators and relations and in terms of a bilinear bracket on a vector space Vsatisfying the Jacobi identity. In physical notation, letXabe a basis forV. The bracket [·,·] can be specified by structure constantsCcabvia the formula
[Xa,Xb] =Cabc Xc.
The structure constants are skew-symmetric in the lower indicesa,b.
The derivation of a new Lie superalgebra from given Lie algebras is particularly interest-ing in physics, since it allows us to find new physical theories from already known theories. The presentation of Lie superalgebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. It is very important, for instance, for the investigation of the particular Lie superalgebras aris-ing in different supersymmetric physical models. Palev [] showed that the position and momentum operators in quantum mechanics and fields in quantum field theory are the generators of finite-dimensional and infinite-dimensional orthosymplectic Lie superalge-bras, respectively.
In mathematical analysis, the stability problem is presented as follows: Assume that a mathematical object satisfies a certain property approximately according to some conven-tion. Is it then possible to find near this object an object satisfying the property accurately? The interested reader can find more information on such problems with the emphasis on functional equations in [–].
In the theory of functional equations, there are cases in which each approximate func-tion is actually a true funcfunc-tion. In such cases, we say that the funcfunc-tional equafunc-tion is hyper-stable. Indeed, a functional equation is hyperstable if every solution satisfying the equation approximately is an exact solution of it. For the history and various aspects of this theory we refer the reader to [–].
In this paper, we introduce the concept of normed Lie superalgebras. For a continuous self-mappingT, we define a generalizedT-orbitally complete metric space and prove a new fixed point result. We define a superhomomorphism and a superderivation and in-vestigate the stability of these linear mathematical objects in Lie superalgebras using the new fixed point alternative. Moreover, we find some conditions under which an approxi-mately object is an exact object.
The paper is organized as follows: The concept of normed Lie superalgebras is intro-duced in Section . In Section , generalizedT-orbitally complete metric spaces are de-fined and a new alternative of fixed point is presented. In Section , we begin to discuss the stability of the superhomomorphism in Lie superalgebras. Moreover, we find conditions under which an approximate superhomomorphism is an exact superhomomorphism. In Section , we deal with the stability of the superderivation in Lie superalgebras. Moreover, we find conditions under which an approximate superderivation is an exact superderiva-tion. In Section , an application of the new fixed point theorem to prove the stability of functional equations is stated.
2 Lie superalgebras
LetZ={¯,}¯ denote the group of two elements. A superalgebraG=G¯⊕G¯, sometimes
also called aZ-graded algebra, is a direct sum of two spacesG¯andG¯equipped with a
bilinear multiplication satisfyingGiGj⊆Gi+j, fori,j∈Z. Elementsx∈G¯are called even
or of degree|x|= while elementsx∈G¯are called odd or of degree|x|= . The interested reader can find more information about superalgebras in [–].
A Lie superalgebra is an algebraL=L¯⊕L¯ with bilinear multiplicationμ:L→L
satisfying the following two axioms: fora,b,c∈L¯∪L¯:
() Skew-supersymmetry:μ(a,b) = –(–)|a|·|b|μ(b,a).
Definition . LetL=L¯⊕L¯andL=L¯⊕L ¯
be Lie superalgebras with multiplications
μandν, respectively. Additive functionsHi:L→Lare called homomorphisms of degree iif
Hiμ(x,y)=νHi(x),Hi(y) ()
for allx,y∈L¯∪L¯. A superhomomorphismH:L→LbyH=H¯+H¯is the sum of a homomorphism of degree and a homomorphism of degree .
Definition . LetL=L¯ ⊕L¯ be a Lie superalgebra with multiplicationμ. Additive
functionsDi:L→Lare called derivation of degreeiif
Diμ(x,y)=μDi(x),y+ (–)i|x|μx,Di(y) ()
for allx,y∈L¯∪L¯. A superderivationD:L→LbyD=D+Dis the sum of a derivation
of degree and a derivation of degree .
Definition . LetLbe a Lie superalgebra with multiplicationμ. The commutant ofL is the subsetLcofLgiven by
Lc=a∈L:μ(x,a) =μ(a,x),x∈L.
3 A fixed point result
Let (X,d) be a metric space. The celebrated Banach contraction theorem (Banach [], ) ensures us of a unique fixed point if a mappingT :X→Xis a contraction,i.e., if there exists a positive numberq< such that
d(Tx,Ty)≤qd(x,y)
for allx,y∈X. The Banach contraction theorem formulated for a complete metric space is one of the most simple and, at the same time, the most important method for the ex-istence and uniqueness of solution of nonlinear problems arising in mathematics and its applications to engineering and natural sciences [–].
In , Baker [] used the Banach fixed point theorem to prove the Hyers-Ulam sta-bility. The method was generalized by Radu [] and since then has been used by many authors (see [–]). We prove a new theorem about this fundamental result as follows. LetXbe a nonempty set andd:X×X→[,∞] be a function satisfying the following conditions:
() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x)(symmetry);
() d(x,y)≤d(x,z) +d(z,y)(triangular inequality),
for allx,y,z∈X. Then (X,d) is called a generalized metric space.
Definition .(Ćirić [–]) Let T be a self-mapping of a generalized metric space (X,d). If for anyx∈X, every Cauchy sequence of the orbit
Ox(T) =x,Tx,Tx, . . .
Definition .(Ćirić [–]) LetT be a self-mapping of a metric space (X,d). We say T is orbitally continuous if and only iflimn→∞Tnx=u∈XimpliesTu=lim
n→∞Tnx.
Theorem .(Fixed point theorem) Suppose that(X,d)is a generalized T -orbitally com-plete metric space and T is an orbitally continuous self-mapping such that
dTx,Tx≤qd(x,Tx) ()
for any x∈X and fixed constant q with <q< .Then the following alternative holds: either
(a) for alln≥,d(Tnx,Tn+x) =∞,or
(b) there exists a positive integerNsuch thatd(Tnx,Tn+x) <∞for alln≥N.
In this case, the sequence (Tnx)n∈
N is convergent to a fixed point x∗ of T. Moreover, if
d(x,Tx) <∞,then
dx,x∗≤
–qd(x,Tx). ()
Proof For any x∈ X, consider the T-orbit {x,Tx,Tx, . . .}. Then either for all n≥,
d(Tnx,Tn+x) =∞, or there exists a positive integerNsuch that
dTnx,Tn+x<∞
for all n≥N. In order to show that (Tnx)n∈
N is convergent inX, we need to show that
(Tnx)
n∈Nis Cauchy inX. Using the inequality (), for alln≥Nwe have
dTnx,Tn+x≤qdTn–x,Tnx≤ · · · ≤qn–NdTNx,TN+x. Thus, for alln,m∈Nwithn>m≥N, we get
dTmx,Tnx≤dTmx,Tm+x+· · ·+dTn–x,Tnx
≤qm–NdTNx,TN+x+· · ·+qn–N–dTNx,TN+x =qm–N+· · ·+qn–N–dTNx,TN+x.
Asm,ntend to infinity, we conclude that (Tnx)n∈
Nis Cauchy. SinceXisT-orbitally
com-plete metric space, then (Tnx)n∈
N converges to somex∗∈X. We show thatx∗ is a fixed
point ofT. Indeed, by Definition . we have
Tx∗= lim n→∞T
n+(x) =x∗.
It remains to show that the inequality () is sharp. Indeed, for alln∈Nand fixed constant qwith <q< ,
dx,Tnx≤d(x,Tx) +· · ·+dTn–x,Tnx≤ +· · ·+qn–d(x,Tx).
As a result of Theorem ., we prove the Hyers-Ulam stability of the superhomomor-phism and the superderivation in Lie superalgebras, using the above mentioned fixed point alternative. For more details as regards the stability problems in Lie algebras, we refer the reader to [, ].
4 The superhomomorphism in Lie superalgebras
Letϕ:L→(,∞) andψ:L→[,∞) be functions for which there exists <q=q() <
such that
ϕx, y≤qϕ(x,y), ()
ψx,y≤qψ(x,y), ψx, y≤qψ(x,y) ()
for allx,y∈L¯∪L¯and∈ {–, }.
Theorem . LetL=L¯⊕L¯be a Lie superalgebra with multiplicationμand norm·L andL=L¯⊕L¯be a complete Lie superalgebra with multiplicationνand norm · L. Suppose that fi:L→L(i∈Z)are functions satisfying
fi(x+y) –fi(x) –fi(y)L≤ϕ(x,y), ()
fiμ(x,y)–νfi(x),fi(y)L≤ψ(x,y) ()
for all x,y∈L¯∪L¯,whereϕ:L→(,∞)andψ:L→[,∞)are functions
satisfy-ing the contractive conditions()and(),for <q=q() < .Then there exists a unique superhomomorphism H:L→Lwith H=H
¯
+H
¯
such that
H
i(x) –fi(x)L≤
q–
( –q)ϕ(x,x) ()
for all x∈L¯∪L¯,∈ {–, },and i∈Z.
Proof We defineE:={g:L→L:g() = }and a generalized metric onE:
d(g,h) = sup x∈L¯∪L¯
g(x) –h(x)L ϕ(x,x) .
Consider the linear mappingT:E→Edefined as
Tg(x) = –g
x
for allx∈L¯∪L¯and∈ {–, }. We can write, for anyf ∈E, Tf(x) –Tf(x)L
ϕ(x,x) =
–f(x) – –f(x) L ϕ(x,x)
=
–f(x) – –f(x) L ϕ(x,x)
≤qf(
x) – –f(x) L ϕ(x, x)
for allx∈L¯∪L¯and∈ {–, }. So,
dTf,Tf
≤qd(f,Tf).
This means thatT is an orbitally continuous self-mapping ofEsatisfying (). SinceLis complete, then (E,d) is a generalizedT-orbitally complete metric space.
Puttingy=xand dividing both sides of () by , we get
fi(x) –fi(x)
L≤ϕ(x,x)/ =q
ϕ(x,x)/
and then
Tfi(x) –fi(x)L≤qϕ(x,x)/ ()
for allx∈L¯∪L¯andi∈Z.
Replacingxandyin () by x, we get
fi(x) – fi x
L≤ϕ
x ,
x
≤qϕ(x,x)/
and then
fi(x) –T–fi(x)L≤qϕ(x,x)/ ()
for allx∈L¯∪L¯andi∈Z.
It follows from () and () that, for∈ {–, }andi∈Z,
d(fi,Tfi)≤q
–
ϕ(x,x)/ <∞.
Now, it follows from Theorem . that there exists a fixed pointH
i ofTin (E,d) such
that
Hi(x) = lim n→∞T
n
fi(x) =n→∞lim –n
finx
and
H
i(x) –fi(x)L≤
q–
( –q)ϕ(x,x)
for allx∈L¯∪L¯,∈ {–, }, andi∈Z.
Letxandybe any two points ofL¯∪L¯. It follows from () that
–n
fin(x+y)– –n
fin
x– –n
fin
yL
≤–nϕnx, ny≤qnϕ(x,y) andntending to infinity, we get
Hi(x+y) =Hi(x) +Hi(y)
Claim that for each∈ {–, }andi∈Z, the functionHisatisfies (). It follows from () that
Hiμ(x,y)–νHi(x),Hi(y)L
= lim n→∞
–n
fiμnx, ny–νfinx,finyL
≤ lim n→∞
–nψ
nx, ny
≤ lim n→∞q
nψ(x,y) =
for allx,y∈L¯∪L¯,∈ {–, }, andi∈Z. So, we can conclude that
Hiμ(x,y)=νHi(x),Hi(y)
for allx,y∈L¯∪L¯,∈ {–, }, andi∈Z. Now, it follows from Definition . thatH=
H¯ +H¯ is superhomomorphism and the proof is completed. Example . LetL=L¯⊕L¯be a Lie superalgebra with multiplicationμand norm · .
Consider the functionsfi:L→L(i∈Z) defined byfi(x) =x+c, where =c∈Lc, for all
x∈L¯∪L¯andi∈Z. Define the functionsϕ:L→(,∞) andψ:L→[,∞) by
ϕ(x,y) =ψ(x,y) =cex
for allx,y∈L¯∪L¯withx= . It is easy to see thatϕandψsatisfy the contractive
condi-tions () and () forq=q() = /. Forx,y∈L¯∪L¯withx= andi∈Zwe have
fi(x+y) –fi(x) –fi(y)=(x+y) +c– (x+c) – (y+c)
=c ≤ cex =ϕ(x,y)
and
fiμ(x,y)–μfi(x),fi(y)=μ(x,y)+c–μ(x+c,y+c)
=μ(x,y)+c–μ(x,y) –μ(x,c) –μ(c,y) –μ(c,c)
=c ≤ cex =ψ(x,y).
Thus, by Theorem ., there exists a unique superhomomorphismH:L→LwithH=
H¯ +H¯, whereH¯ andH¯are defined as
H(x) = lim n→∞
f(nx)
n =n→∞lim nx+c
n =x,
H(x) = lim n→∞
f(nx)
n =n→∞lim nx+c
n =x.
Moreover, () is sharp. Indeed,
H
i(x) –fi(x)=x+c–x=c ≤
(/)
( – /)ce
x =cex
In the next theorem, we prove the hyperstability of the superhomomorphism.
Theorem . LetL=L¯⊕L¯be a Lie superalgebra with multiplicationμand norm·L andL=L¯⊕L¯be a complete Lie superalgebra with multiplicationνand norm · L. Assume that fi:L→L(i∈Z)are functions such that fi(x) = fi(x)and
fi(x+y) –fi(x) –fi(y)
L≤ϕ(x,y), ()
fiμ(x,y)–νfi(x),fi(y)L≤ψ(x,y) ()
for all x,y∈L¯∪L¯,whereϕ:L→(,∞)andψ:L→[,∞)are functions satisfying
the contractive conditions()and(),for <q=q() < .Then f =f¯+f¯is
superhomo-morphism.
Proof If we replacexandyin () with nxand ny, respectively, and divide by nthe resulting inequality, then we have
fi(x+y) –fi(x) –fi(y)L≤–nϕ
nx, ny≤qnϕ(x,y) ()
for allx,y∈L¯∪L¯,i∈Z, andn∈N.
If we replacexandyin () with nxand ny, respectively, and divide by nthe resulting inequality, then we have
fi
μ(x,y)–νfi(x),fi(y)L≤–nψ
nx, ny≤qnψ(x,y) ()
for allx,y∈L¯∪L¯,i∈Zandn∈N. Fromn→ ∞in () and (), we conclude thatf¯
andf¯are homomorphisms and hencef =f¯+f¯is superhomomorphism.
5 The superderivation in Lie superalgebras
Theorem . LetL=L¯⊕L¯be a complete Lie superalgebra with multiplicationμand norm · L.Let fi:L→L(i∈Z)be functions satisfying
fi(x+y) –fi(x) –fi(y)L≤ϕ(x,y), ()
fi
μ(x,y)–μfi(x),y– (–)i|x|μx,fi(y)L≤ψ(x,y) ()
for all x,y∈L¯∪L¯,whereϕ:L→(,∞)andψ:L→[,∞)are functions
satisfy-ing the contractive conditions()and(),for <q=q() < .Then there exists a unique superderivation D:L→Lwith D=D
¯
+D
¯
such that
D
i(x) –fi(x)L≤ q–
( –q)ϕ(x,x)
for all x∈L¯∪L¯,∈ {–, },and i∈Z.
Proof Since the functionsfi(i∈Z) satisfy (), according to the corresponding part of
withD=D
¯
+D
¯
and defined by
D
i(x) =n→∞lim–nfi
nx
such that
Di(x) –fi(x)L≤ q –
( –q)ϕ(x,x)
for allx∈L¯∪L¯,∈ {–, }, andi∈Z. We only need to show thatD¯ andD
¯
satisfy ().
It follows from () that
Diμ(x,y)–μDi(x),y– (–)i|x|μx,Di(y)L
= lim n→∞
–n
fiμnx, ny–μfinx, ny– (–)i|x|μnx,finyL
≤ lim n→∞
–nψ
nx, ny
≤ lim n→∞q
nψ(x,y) =
for allx,y∈L¯∪L¯,∈ {–, }, andi∈Z. It is easy to see that
Diμ(x,y)=μDi(x),y+ (–)i|x|μx,Di(y)
for allx,y∈L¯∪L¯,∈ {–, }, andi∈Z. It follows from Definition . thatD=D¯+D
¯
is superderivation and the proof is completed.
In the next theorem, we present the hyperstability result concerning the superderivation in Lie superalgebras.
Theorem . LetL=L¯⊕L¯be a complete Lie superalgebra with multiplicationμand
norm · L.Let fi:L→L(i∈Z)be functions such that fi(x) = fi(x)and
fi(x+y) –fi(x) –fi(y)L≤ϕ(x,y), ()
fi
μ(x,y)–μfi(x),y
– (–)i|x|μx,fi(y)L≤ψ(x,y) ()
for all x,y∈L¯∪L¯,whereϕ:L→(,∞)andψ:L→[,∞)are functions satisfying
the contractive conditions()and(),for <q=q() < .Then f =f¯+f¯is superderivation.
Proof Replacingx andyin () with nx and ny, respectively, and dividing by nthe resulting inequality, we get
fi(x+y) –fi(x) –fi(y)L≤–nϕnx, ny, ≤qnϕ(x,y) ()
for allx,y∈L¯∪L¯,i∈Zandn∈N.
Replacingxandyin () with nxand ny, respectively, and dividing by nthe resulting inequality, we obtain
for allx,y∈L¯∪L¯,i∈Z, andn∈N. Fromn→ ∞in () and (), we conclude thatf¯
andf¯are derivations andf =f¯+f¯is a superderivation.
6 An application of Theorem 3.3
In , Rassias proved the following theorem concerning the stability of functional equa-tions.
Theorem .(Rassias []) Consider E,Eto be two Banach spaces,and let f :E→E
be a mapping such that f(tx)is continuous in t for each fixed x.Assume that there exist
θ≥and≤p< such that
f(x+y) –f(x) –f(y)E
xpE+yEp ≤θ for any x,y∈E. () Then there exists a unique linear mapping A:E→Esuch that
f(x) –A(x)E xpE ≤
θ
– p for any x∈E.
In this section, we show that Theorem ., concerning the stability of the Cauchy func-tional equation in Banach spaces, is a direct consequence of Theorem .. In general, al-most all of the theorems concerning the stability problem of different functional equations are consequences of Theorem ..
Proof Let us consider the setX:={g:E→E:g() = }and introduce a generalized
metric onX:
d(g,h) =sup x=
g(x) –h(x)E xp
E .
Consider the linear mappingT:X→Xdefined as
Tg(x) = –g(x)
for allx∈E. We can write, for anyf ∈X,
Tf(x) –Tf(x)E
xp E
=
–f(x) – –f(x)E
xp E
= p–f(x) –
–f(x)E
xp E
≤p–d(f,Tf)
for allx∈E. So,
dTf,Tf≤p–d(f,Tf) (≤p< )
By equation (), one can easily show that
f(x) –f(x)/
xp ≤θ<∞.
Now, it follows from Theorem . that there exists a unique fixed pointAofT in (X,d) such that
A(x) = lim n→∞T
nf(x) = lim n→∞
–nfnx
and
f(x) –A(x)E xp
E
≤
– p–d(f,Tf) =
– p–θ
for allx∈E. It is clear thatAis linear and the proof is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. The authors further confirm that the order of authors listed in the manuscript has been approved by all of them. All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan,
35195-363, Iran. 2Center of Excellence in Nonlinear Analysis and Applications, Semnan University, Semnan, Iran.3Young
Researchers and Elite Club, Malayer Branch, Azad University, Malayer, Iran.4Department of Electricity and Electronics,
University of the Basque Country, 466, Bilbao, 48080, Spain.
Acknowledgements
The authors are very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651. The authors are also grateful to the Basque Government through Grant IT 378-10.
Received: 23 April 2015 Accepted: 16 November 2015 References
1. Abellanas, L, Alonso, L: A general setting for Casimir invariants. J. Math. Phys.16, 1580-1584 (1975) 2. Jacobson, N: Lie Algebras. Dover, New York (1979)
3. Rand, D, Winternitz, P, Zassenhaus, H: On the identification of Lie algebra given by its structure constants I. Direct decompositions, Levi decompositions and nilradicals. Linear Algebra Appl.109, 197-246 (1988)
4. Popovych, R, Boyko, V, Nesterenko, M, Lutfullin, M: Realizations of real low-dimensional Lie algebras. J. Phys. A36, 7337-7360 (2003)
5. Montigny, M, Patera, J: Discrete and continuous graded contractions of Lie algebras and superalgebras. J. Algebra24, 525-547 (1991)
6. Lada, T, Stasheff, J: Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys.32, 1087-1103 (1993) 7. de Azcárraga, JA, Bueno, JCP: Higher-order simple Lie algebras. Commun. Math. Phys.184, 669-681 (1997) 8. Palev, TD: Lie superalgebras, infinite-dimensional algebras and quantum statistics. Rep. Math. Phys.31, 241-262
(1992)
9. Ulam, SM: A Collection of Mathematical Problems. Interscience, New York (1960)
10. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA27, 222-224 (1941) 11. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.72, 297-300 (1978) 12. Baker, JA: The stability of certain functional equations. Proc. Am. Math. Soc.112, 729-732 (1991)
13. Baker, JA, Lawrence, J, Zorzitto, F: The stability of the equationf(x+y) =f(x)f(y). Proc. Am. Math. Soc.74, 242-246 (1979)
14. Rassias, TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl.251, 264-284 (2000) 15. Šemrl, P: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral
Equ. Oper. Theory18, 118-122 (1994)
16. Gordji, ME: Nearly involutions on Banach algebras: a fixed point approach. J. Math. Phys.14, 117-124 (2013) 17. Montgomery, S: Constructing simple Lie superalgebras from associative graded algebras. J. Algebra195, 558-579
18. Fošner, M: On the extended centroid of prime associative superalgebras with applications to superderivations. Commun. Algebra32, 689-705 (2004)
19. Chen, TS: Supercentralizing superderivations on prime superalgebras. Commun. Algebra33, 4457-4466 (2005) 20. Zusmanovich, P: Onδ-derivations of Lie algebras and superalgebras. J. Algebra324, 3470-3486 (2010)
21. Banach, S: Sur les operations dans les ensembles abstraites et leurs applications. Fundam. Math.3, 133-181 (1922) 22. Agarwal, RP, Balaj, M, O’Regan, D: A common fixed point theorem with applications. J. Optim. Theory Appl.163,
482-490 (2014)
23. Cho, YJ, Saadati, R, Wang, S: Common fixed point theorems on generalized distance in ordered cone metric spaces. Comput. Math. Appl.61, 1254-1260 (2011)
24. Pathak, HK, Agarwal, RP, Cho, YJ: Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions. J. Comput. Appl. Math.283, 201-217 (2015)
25. Golkarmanesh, F, Al-Mazrooei, AE, Parvaneh, V, Latif, A: Fixed point results for generalized mappings. Fixed Point Theory Appl.2014, 217 (2014)
26. Radu, V: The fixed point alternative and the stability of functional equations. Fixed Point Theory4, 91-96 (2003) 27. Gordji, ME, Najati, A: ApproximatelyJ∗-homomorphisms: a fixed point approach. J. Geom. Phys.60, 809-814 (2010) 28. Chauhan, S, Pant, BD: Fixed point theorems for compatible and subsequentially continuous mappings in Menger
spaces. J. Nonlinear Sci. Appl.7, 78-89 (2014)
29. Mihe¸t, D: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. J. Nonlinear Sci. Appl.6, 35-40 (2013)
30. ´Ciri´c, LB: Fixed and periodic points of almost contractive operators. Math. Balk.3, 33-44 (1973) 31. ´Ciri´c, LB: Fixed points for generalized multi-valued contractions. Mat. Vesn.9, 265-272 (1972) 32. ´Ciri´c, LB: On contraction type mappings. Math. Balk.1, 52-57 (1971)
33. Eshaghi, M, Abbaszadeh, S: Approximate generalized derivations close to derivations in LieC∗-algebras. J. Appl. Anal. 21, 37-43 (2015)
