Citation: Xiao Tu, Qinran Zhang, Wei Zhang, Xiufen Zou. Single-cell data-driven mathematical model reveals possible molecular mechanisms of embryonic stem-cell differentiation[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5877-5896. doi: 10.3934/mbe.2019294
[1] | Ehsan Movahednia, Young Cho, Choonkil Park, Siriluk Paokanta . On approximate solution of lattice functional equations in Banach f-algebras. AIMS Mathematics, 2020, 5(6): 5458-5469. doi: 10.3934/math.2020350 |
[2] | Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen . Refined stability of the additive, quartic and sextic functional equations with counter-examples. AIMS Mathematics, 2023, 8(6): 14399-14425. doi: 10.3934/math.2023736 |
[3] | Supriya Kumar Paul, Lakshmi Narayan Mishra . Stability analysis through the Bielecki metric to nonlinear fractional integral equations of n-product operators. AIMS Mathematics, 2024, 9(4): 7770-7790. doi: 10.3934/math.2024377 |
[4] | Lingxiao Lu, Jianrong Wu . Hyers-Ulam-Rassias stability of cubic functional equations in fuzzy normed spaces. AIMS Mathematics, 2022, 7(5): 8574-8587. doi: 10.3934/math.2022478 |
[5] | Sizhao Li, Xinyu Han, Dapeng Lang, Songsong Dai . On the stability of two functional equations for (S,N)-implications. AIMS Mathematics, 2021, 6(2): 1822-1832. doi: 10.3934/math.2021110 |
[6] | Nazek Alessa, K. Tamilvanan, G. Balasubramanian, K. Loganathan . Stability results of the functional equation deriving from quadratic function in random normed spaces. AIMS Mathematics, 2021, 6(3): 2385-2397. doi: 10.3934/math.2021145 |
[7] | Kandhasamy Tamilvanan, Jung Rye Lee, Choonkil Park . Ulam stability of a functional equation deriving from quadratic and additive mappings in random normed spaces. AIMS Mathematics, 2021, 6(1): 908-924. doi: 10.3934/math.2021054 |
[8] | Murali Ramdoss, Divyakumari Pachaiyappan, Inho Hwang, Choonkil Park . Stability of an n-variable mixed type functional equation in probabilistic modular spaces. AIMS Mathematics, 2020, 5(6): 5903-5915. doi: 10.3934/math.2020378 |
[9] | Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for ψ-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244 |
[10] | J. Vanterler da C. Sousa, E. Capelas de Oliveira, F. G. Rodrigues . Ulam-Hyers stabilities of fractional functional differential equations. AIMS Mathematics, 2020, 5(2): 1346-1358. doi: 10.3934/math.2020092 |
In 1940, Ulam [25] proposed the following question concerning the stability of group homomorphisms: Under what condition does there is an additive mapping near an approximately additive mapping between a group and a metric group? In the next year, Hyers [7,8] answered the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately linear mappings was given by Rassias [22]. Since then, the stability problems of various functional equation have been extensively investigated by a number of authors (see [3,6,13,14,16,17,22,24,26,27,28]). By regarding a large influence of Ulam, Hyers and Rassias on the investigation of stability problems of functional equations the stability phenomenon that was introduced and proved by Rassias [22] in the year 1978 is called the Hyers-Ulam-Rassias stability.
Consider the functional equation
f(x+y)+f(x−y)=2f(x)+2f(y). | (1.1) |
The quadratic function f(x)=cx2 is a solution of this functional Equation (1.1), and so one usually is said the above functional equation to be quadratic [5,10,11,12]. The Hyers-Ulam stability problem of the quadratic functional equation was first proved by Skof [24] for functions between a normed space and a Banach space. Afterwards, the result was extended by Cholewa [2] and Czerwik [4].
Now, we consider the following functional equation:
f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y). | (1.2) |
It is easy to see that the function f(x)=cx4 satisfies the functional equation (1.2). Hence, it is natural that Eq (1.2) is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping (see [15,19]).
The theory of random normed spaces (briefly, RN-spaces) is important as a generalization of deterministic result of normed spaces and also in the study of random operator equations. The notion of an RN-space corresponds to the situations when we do not know exactly the norm of the point and we know only probabilities of possible values of this norm. Random theory is a setting in which uncertainty arising from problems in various fields of science, can be modelled. It is a practical tool for handling situations where classical theories fail to explain. Random theory has many application in several fields, for example, population dynamics, computer programming, nonlinear dynamical system, nonlinear operators, statistical convergence and so forth.
In 2008, Mihet and Radu [18] applied fixed point alternative method to prove the stability theorems of the Cauchy functional equation:
f(x+y)−f(x)−f(y)=0 |
in random normed spaces. In 2008, Najati and Moghimi [20] obtained a stability of the functional equation deriving from quadratic and additive function:
f(2x+y)+f(2x−y)+2f(x)−f(x+y)−f(x−y)−2f(2x)=0 | (1.3) |
by using the direct method. After that, Jin and Lee [9] proved the stability of the above mentioned functional equation in random normed spaces.
In 2011, Saadati et al. [21] proved the nonlinear stability of the quartic functional equation of the form
16f(x+4y)+f(4x−y)=306[9f(x+y3)+f(x+2y)]+136f(x−y)−1394f(x+y)+425f(y)−1530f(x) |
in the setting of random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the fixed point theory, the theory of intuitionistic spaces and the theory of functional equations were also presented. Azadi Kenary [1] investigated the Ulam stability of the following nonlinear function equation
f(f(x)−f(y))+f(x)+f(y)=f(x+y)+f(x−y), |
in random normed spaces.
In this note, we investigate the general solution for the quartic functional equation of the form
(3n+4)f(n∑i=1xi)+n∑j=1f(−nxj+n∑i=1,i≠jxi)=(n2+2n+1)n∑i=1,i≠j≠kf(xi+xj+xk)−12(3n3−2n2−13n−8)n∑i=1,i≠jf(xi+xj)+12(n3+2n2+n)n∑i=1,i≠jf(xi−xj)+12(3n4−5n3−7n2+13n+12)n∑i=1f(xi) | (1.4) |
(n∈N,n>4) and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces by using the direct approach and the fixed point approach.
In this part, we make some notations and basic definitions used in this article.
Definition 2.1. A function T:[0,1]×[0,1]→[0,1] is called a continuous triangular norm, if T satisfies the following condition
(a) T is commutative and associative;
(b) T is continuous;
(c) T(a,1)=a for all a∈[0,1];
(d) T(a,b)≤T(c,d) when a≤c and b≤d for all a,b,c,d∈[0,1].
Typical examples of continuous t−norms are Tp(a,b)=ab,Tm(a,b)=min(a,b) and TL(a,b)=max(a+b−1,0) (The Lukasiewicz t−norm). Recall [23] that if T is a t−norm and {xn} is a given sequence of numbers in [0,1], then Tni=1xn+i is defined recurrently by T′i=1xi=xi and Tni=1xi=T(Tn−1i=1xi,xn) for n≥2,T∞i=1xi is defined as T∞i=1xn+i. It is known that, for the Lukasiewicz t−norm, the following implication holds:
limn→∞(TL)∞i=1xn+i=1⇔∞∑n=1(1−xn)<∞. |
Definition 2.2. A random normed space (briefly, RN−space) is a triple (X,μ,T), where X is a vector space. T is a continuous t−norm and μ is a mapping from X into D+ satisfying the following conditions:
(RN1) μx(t)=ε0(t) for all t>0 if and only if x=0;
(RN2) μαx(t)=μx(t|α|) for all x∈X, and alpha∈ℜ with α≠0;
(RN3) μx+y(t+s)≥T(μx(t),μy(s)) for all x,y∈X and t,s≥0.
Definition 2.3. Let (X,μ,T) be an RN−space.
1). A sequence {xn} in X is said to be convergent to a point x∈X if, for any ε>0 and λ>0, there exists a positive integer N such that μxn−x(ε)>1−λ for all n>N.
2). A sequence {xn} in X is called a Cauchy sequence if, for any ε>0andλ>0, there exists a positive integer N such that μxn−xm(ε)>1−λ for all n≥m≥N.
3). An RN−space (X,μ,T) is said to be complete, if every Cauchy sequence in X is convergent to a point in X.
Throughout this paper, we use the following notation for a given mapping f:X→Y as
Df(x1,⋯,xn):=(3n+4)f(n∑i=1xi)+n∑j=1f(−nxj+n∑i=1,i≠jxi)−(n2+2n+1)n∑i=1,i≠j≠kf(xi+xj+xk)−12(3n3−2n2−13n−8)n∑i=1,i≠jf(xi+xj)+12(n3+2n2+n)n∑i=1,i≠jf(xi−xj)+12(3n4−5n3−7n2+13n+12)n∑i=1f(xi) |
for all x1,x2,⋯,xn∈X.
In this section we investigate the general solution of the n-variable quartic functional equation (1.4).
Theorem 3.1. Let X and Y be real vector spaces. If a mapping f:X→Y satisfies the functional equation (1.4) for all x1,⋯,xn∈X, then f:X→Y satisfies the functional equation (1.2) for all x,y∈X.
Proof. Assume that f satisfies the functional equation (1.4). Putting x1=x2=⋯=xn=0 in (1.4), we get
(3n+4)f(0)=(n2+2n+1){6n−20+4(n−4)(n−5)2+(n−4)(n−5)(n−6)6}f(0)−12(3n3−2n2−13n−8){4n−10+(n2−9n+20)2}f(0)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)−n2(3n4−5n3−7n2+13n+12)f(0) | (3.1) |
It follows from (3.1) that
f(0)=0. | (3.2) |
Replacing (x1,x2,⋯,xn) by (x,0,⋯,0⏟(n−1)−times) in (1.4), we have
(3n+4)f(x)+n4f(−x)+(n−1)f(x)=(n4−n3−3n2+n+22)f(x)−12(3n4−5n3−11n2+5n+8)f(x)+12(n4+n3+n2−n)f(x)+12(3n4−5n3−7n2+13n+12)f(x) | (3.3) |
for all x∈X. It follows from (3.3) that
f(−x)=f(x) |
for all x∈X. Setting x1=x2=⋯=xn=x in (1.4), we obtain
(3n+4)f(x)+nf(x)=(n2+2n+1){6n−20+4(27)(n−4)(n−5)2+(n−4)(n−5)(n−6)6}f(x)−12(3n3−2n2−13n−8){4n−10+(n2−9n+20)2}f(2x)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)+n2(3n4−5n3−7n2+13n+12)f(x) | (3.4) |
for all x∈X. It follows from (3.4) and (3.2) that
f(2x)=16f(x) |
for all x∈X. Setting x1=x2=⋯=xn=x in (1.4), we get
(3n+4)f(x)+nf(x)=(n2+2n+1){6n−20+4(n2−9n+20)2+(n2−9n+20)(n−6)6}f(3x)−12(3n3−2n2−13n−8){4n−10+16(n2−9n+20)2}f(x)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)+n2(3n4−5n3−7n2+13n+12)f(x) | (3.5) |
for all x∈X. It follows from (3.5) and (3.2) that
f(3x)=81f(x) |
for all x∈X. In general for any positive m, we get
f(mx)=m4f(x) |
for all x∈X. Replacing (x1,x2,⋯,xn) by (x,x,⋯,x⏟(n−1)−times,y) in (1.4), we get
(3n+4)(f((n−1)x+y)+(n−1)f(−2x+y))+f((n−1)x−ny)=(n2+2n+1)(3n−9+(n2−3n+2)2)f(2x+y)(n2+2n+1)(3n−11+(3n2−27n+602)+((n−4)(n−5)(n−6)6))f(3x)−12(3n3−2n2−13n−8){3n−9+(n2−9n+20)2}f(2x)−(n−1)2(3n3−2n2−13n−8)f(x+y)+12(n3+2n2+n)(n−1)f(x−y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.6) |
for all x,y∈X. It follows from (3.6) that
(3n+4)(f((n−1)x+y)+(n−1)f(−2x+y))+f((n−1)x−ny)=(n2+2n+1)(n2−3n+22)f(2x+y)+81(n2+2n+1)(n3−5n2+11n−66)f(x)−16(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)f(x+y)+12(n3+2n2+n)(n−1)f(x−y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.7) |
for all x,y∈X. Replacing y by −y in (3.7), we get
(3n+4)(f((n−1)x−y)+(n−1)f(2x+y))+f(()x+y)=(n2+2n+1)(n2−3n+22)f(2x−y)+81(n2+2n+1)(n3−5n2+11n−66)f(x)−16(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)f(x−y)+12(n3+2n2+n)(n−1)f(x+y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.8) |
for all x,y∈X. Adding (3.7) and (3.8), we get
(3n+4)((n−1)2{f(x+y)+f(x−y)}+2{f((n−1)x)−(n−1)2f(x)}+2{f(y)−(n−1)2f(y)})+(n−1)(f(−2x+y)+f(2x+y))+(f((n−1)x−ny)+f((n−1)x+ny))=(n2+2n+1)(n2−3n+22)(f(2x−y)+f(2x+y))+162(n2+2n+1)(n3−5n2+11n−66)f(x)−32(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)(f(x−y)+f(x+y))+12(n3+2n2+n)(n−1)(f(x+y)+f(x−y))+(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.9) |
for all x,y∈X. It follows from (3.9) that
(3n+4)((n−1)2{f(x+y)+f(x−y)}+2{f((n−1)x)−(n−1)2f(x)}+2{f(y)−(n−1)2f(y)})+(n−1)(f(2x−y)+f(2x+y))+n2(n−1)2(f(x+y)+f(x−y))+2{f((n−1)x)−n2(n−1)2f(x)+n4f(y)−n2(n−1)2f(y)}=(n2+2n+1)(n2−3n+22)(f(2x−y)+f(2x+y))+162(n2+2n+1)(n3−5n2+11n−66)f(x)−32(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)(f(x−y)+f(x+y))+12(n3+2n2+n)(n−1)(f(x+y)+f(x−y))+(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.10) |
for all x,y∈X. It follows from (3.10) that
−2f(2x+y)−2f(2x−y)=−8f(x+y)−8f(x−y)−48f(x)+12f(y) | (3.11) |
for all x,y∈X. From (3.11), we get
(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y) |
for all x,y∈X. Thus the mapping f:X→Y is quartic.
In this section, the Ulam-Hyers stability of the quartic functional equation (1.4) in RN−space is provided. Throughout this part, let X be a linear space and (Y,μ,T) be a complete RN−space.
Theorem 4.1. Let j=±1, f:X→Y be a mapping for which there exists a mapping η:Xn→D+ satisfying
limk→∞T∞i=0(η2(k+i)x1,2(k+i)x2,...,2(k+i)xn(24(k+i+1)jt))=1=limk→∞η2kjx1,2kjx2,..,2kjxn(24kjt) |
such that f(0)=0 and
μDf(x1,x2,...,xn)(t)≥η(x1,x2,...,xn)(t) | (4.1) |
for all x1,x2,...,xn∈X and all t>0. Then there exists a unique quartic mapping Q:X→Y satisfying the functional equation (1.4) and
μQ(x)−f(x)(t)≥T∞i=0(η2(i+1)jx,2(i+1)jx,,...,2(i+1)jx(14(3n5−5n4−11n3+5n2+8n)24(i+1)jt)) |
for all x∈X and all t>0. The mapping Q(x) is defined by
μC(x)(t)=limk→∞μf(2kjx)24kj(t) | (4.2) |
for all x∈X and all t>0.
Proof. Assume j=1. Setting (x1,x2,...,xn) by (x,x,...,x⏟n−times) in (4.1), we obtain
μ(3n5−5n4−11n3+5n2+8n)4f(2x)−4(3n5−5n4−11n3+5n2+8n)f(x)(t)≥η(x,x,...,x⏟n−times)(t) | (4.3) |
for all x∈X and all t>0. It follows from (4.2) and (RN2) that
μf(2x)16−f(x)(t)≥η(x,x,...,x⏟n−times)(24(3n5−5n4−11n3+5n2+8n)t) |
for all x∈X and all t>0. Replacing x by 2kx in (4.3), we have
μf(2k+1x)24(k+1)−f(2kx)24k(t)≥η(2kx,2kx,...,2kx⏟n−times)(24k(3n5−5n4−11n3+5n2+8n)16t)≥η(x,x,...,x⏟n−times)(24k(3n5−5n4−11n3+5n2+8n)16αkt) | (4.4) |
for all x∈X and all t>0. It follows from f(2nx)24n−f(x)=n−1∑k=0f(2k+1x)24(k+1)−f(2kx)24k and (4.4) that
μf(2nx)24n−f(x)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16)≥Tn−1k=0(ηx,x,0,...,0(t))=η(x,x,...,x⏟n−times)(t), |
μf(2nx)24n−f(x)(t)≥η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)24) | (4.5) |
for all x∈X and all t>0. Replacing x by 2mx in (4.5), we get
μf(2n+mx)24(n+m)−f(2mx)24m(t)≥η(x,x,...,x⏟n−times)(tn+m∑k=mαk24k(3n5−5n4−11n3+5n2+8n)16). | (4.6) |
Since η(x,x,...,x⏟n−times)(tn+m∑k=mαk24k(3n5−5n4−11n3+5n2+8n)16)→1 as m,n→∞, {f(2nx)24n} is a Cauchy sequence in (Y,μ,T). Since (Y,μ,T) is a complete RN-space, this sequence converges to some point C(x)∈Y. Fix x∈X and put m=0 in (4.6). Then we have
μf(2nx)24n−f(x)(t)≥η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16) |
and so, for every δ>0, we get
μC(x)−f(x)(t+δ)≥T(μQ(x)−f(2nx)24n(δ),μf(2nx)24n−f(x)(t))≥T(μQ(x)−f(2nx)24n(δ),η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16)). | (4.7) |
Taking the limit as n→∞ and using (4.7), we have
μC(x)−f(x)(t+δ)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)(24−α)t). | (4.8) |
Since δ is arbitrary, by taking δ→0 in (4.8), we have
μQ(x)−f(x)(t)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)(24−α)t). | (4.9) |
Replacing (x1,x2,...,xn) by (2nx1,2nx2,...,2nxn) in (4.1), respectively, we obtain
μDf(2nx1,2nx2,...,2nxn)(t)≥η2nx1,2nx2,...,2nxn(24nt) |
for all x1,x2,...,xn∈X and for all t>0. Since
limk→∞T∞i=0(η2k+ix1,2k+ix2,...,2k+ixn(24(k+i+1)jt))=1, |
we conclude that Q fulfils (1.1). To prove the uniqueness of the quartic mapping Q, assume that there exists another quartic mapping D from X to Y, which satisfies (4.9). Fix x∈X. Clearly, Q(2nx)=24nQ(x) and D(2nx)=24nD(x) for all x∈X. It follows from (4.9) that
μQ(x)−D(x)(t)=limn→∞μQ(2nx)24nD(2nx)24n(t)μQ(2nx24n(t)≥min{μQ(2nx)24nf(2nx)24n(t2),μD(2nx)24nf(2nx)24n(t2)} |
≥η(2nx,2nx,...,2nx⏟n−times)(24n(3n5−5n4−11n3+5n2+8n)(24−α)t)≥η(x,x,...,x⏟n−times)(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn). |
Since limn→∞(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn)=∞, we get
limn→∞ηx,x,0,...,0(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn)=1.
Therefore, it follows that μQ(x)−D(x)(t)=1 for all t>0 and so Q(x)=D(x).
This completes the proof.
The following corollary is an immediate consequence of Theorem 4.1, concerning the stability of (1.4).
Corollary 4.2. Let Ω and ℧ be nonnegative real numbers. Let f:X→Y satisfy the inequality
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t)ηΩ(n∏i=1‖xi‖n℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t) |
for all x1,x2,x3,...,xn∈X and all t>0, then there exists a unique quartic mapping Q:X→Y such that
μf(x)−Q(x)(t)≥{ηΩ|60|(3n5−5n4−11n3+5n2+8n)(t)ηΩ‖x‖℧|4|(3n5−5n4−11n3+5n2+8n)|24−2℧|(t),℧≠4ηΩ‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4nη(n+1)Ω‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4n |
for all x∈X and all t>0.
In this section, we prove the Ulam-Hyers stability of the functional equation (1.4) in random normed spaces by the using fixed point method.
Theorem 5.1. Let f:X→Y be a mapping for which there exists a mapping η:Xn→D+ with the condition
limk→∞ηδkix1,δkix2,...,δkixn(δkit)=1 |
for all x1,x2,x3,...,xn∈X and all 0, where
δi={2i=0;12i=1; |
and satisfy the functional inequality
μDf(x1,x2,...,xn)(t)≥ηx1,x2,...,xn(t) |
for all x1,x2,x3,...,xn∈X and all 0. If there exists L=L(i) such that the function x→β(x,t)=η(x2,x2,...,x2⏟n−times)((3n5−5n4−11n3+5n2+8n)t) has the property that
β(x,t)≤L1δ4iβ(δix,t) | (5.1) |
for all x∈X and t>0, then there exists a unique quartic mapping Q:X→Y satisfying the functional equation (1.4) and
μQ(x)−f(x)(L1−i1−Lt)≥β(x,t) |
for all x∈X and t>0.
Proof. Let d be a general metric on Ω such that
d(p,q)=inf{k∈(0,∞)/μ(p(x)−q(x))(kt)≥β(x,t),x∈X,t>0}. |
It is easy to see that (Ω,d) is complete. Define T:Ω→Ω by Tp(x)=1δ4ip(δix) for all x∈X. Now assume that for p,q∈Ω, we have d(p,q)≤K. Then
μ(p(x)−q(x))(kt)≥β(x,t)⇒μ(p(x)−q(x))(Ktδ4i)≥β(x,t)⇒d(Tp(x),Tq(x))≤KL⇒d(Tp,Tq)≤Ld(p,q) | (5.2) |
for all p,q∈Ω. Therefore, T is a strictly contractive mapping on Ω with Lipschitz constant L. It follows from (5.2) that
μ(3n5−5n4−11n3+5n2+8n)4f(2x)−4(n3n5−5n4−11n3+5n2+8n)f(x)(t)≥η(x,x,...,x⏟n−times)(t) | (5.3) |
for all x∈X. It follows from (5.3) that
μf(2x)16−f(x)(t)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)16t) | (5.4) |
for all x∈X. By using (5.1) for the case i=0, it reduce to
μf(2x)16−f(x)(t)≥Lβ(x,t) |
for all x∈X. Hence we obtain
d(μTf,f)≤L=L1−i<∞ | (5.5) |
for all x∈X. Replacing x by x2 in (5.4), we get
μf(x)16−f(x2)(t)≥η(x2,x2,...,x2⏟n−times)((3n5−5n4−11n3+5n2+8n)16t) |
for all x∈X. By using (5.1) for the case i=1, it reduce to
μ16f(x2)−f(x)(t)≥β(x,t)⇒μTf(x)−f(x)(t)≥β(x,t) |
for all x∈X. Hence we get
d(μTf,f)≤L=L1−i<∞ | (5.6) |
for all x∈X. From (5.5) and (5.6), we can conclude
d(μTf,f)≤L=L1−i<∞ |
for all x∈X.
The remaining proof is similar to the proof of Theorem 4.1. So Q is a unique fixed point of T in the set such that
μ(f(x)−Q(x))(L1−i1−Lt)≥β(x,t) |
for all x∈X and t>0. This completes the proof of the theorem.
From Theorem 4.1, we obtain the following corollary concerning the stability for the functional equation (1.4).
Corollary 5.2. Suppose that a mapping f:X→Y satisfies the inequality
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t)ηΩ(n∏i=1‖xi‖℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t), |
for all x1,x2,...,xn∈X and all t>0, where Ω,℧ are constants with Ω>0. Then there exists a unique quartic mapping Q:X→Y such that
μf(x)−Q(x)(t)≥{ηΩ|60|(3n5−5n4−11n3+5n2+8n)(t)ηΩ‖x‖℧|4|(3n5−5n4−11n3+5n2+8n)|24−2℧|(t),℧≠4ηΩ‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4nη(n+1)Ω‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4n |
for all x∈X and all t>0.
Proof. Set
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t),ηΩ(n∏i=1‖xi‖℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t) |
for all x1,x2,...,xn∈X and all t>0. Then
η(δkix1,δkix2,...,δkixn)(δ4kit)={ηΩδ4ki(t),ηΩn∑i=1‖xi‖℧δ(4−n℧)ki(t),ηΩ(n∏i=1‖xi‖℧δ(4−n℧)ki)(t),ηΩ(n∏i=1‖xi‖℧δ(4−n℧)ki+n∑i=1‖xi‖n℧)(t), |
={→ 1 as k→∞→ 1 as k→∞→ 1 as k→∞→ 1 as k→∞. |
But we have β(x,t)=η(x2,x2,...,x2⏟n−times)(14(3n5−5n4−11n3+5n2+8n)t) has the property L1δ4iβ(δix,t) for all x∈X and t>0. Now
β(x,t)={η4Ω3n5−5n4−11n3+5n2+8n(t)η4Ω‖x‖℧24℧(3n5−5n4−11n3+5n2+8n)(t)η4Ω‖x‖n℧24℧(3n5−5n4−11n3+5n2+8n)(t)η4Ω‖x‖ns2n℧(3n5−5n4−11n3+5n2+8n)(t), |
L1δ4iβ(δix,t)={ηδ−4iβ(x)(t)ηδ℧−4iβ(x)(t)ηδ℧−4iβ(x)(t)ηδn℧−4iβ(x)(t). |
By (4.1), we prove the following eight cases:
L=2−4 ifi=0 and L=24 if i=1L=2℧−4 for ℧<4 ifi=0 and L=24−℧ for ℧>4 if i=1L=2n℧−4 for ℧<4n ifi=0 and L=24−n℧ for ℧>4n if i=1L=2n℧−4 for ℧<4n ifi=0 and L=24−n℧ for ℧>4n if i=1 |
Case 1: L=2−4 if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω60(3n5−5n4−11n3+5n2+8n))(t). |
Case 2: L=24 if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω−60(3n5−5n4−11n3+5n2+8n))(t). |
Case 3: L=2℧−4 for ℧<4 if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ωn‖x‖℧4(3n5−5n4−11n3+5n2+8n)(24−2℧))(t). |
Case 4: L=24−℧ for ℧>4 if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(nΩ‖x‖s4(3n5−5n4−11n3+5n2+8n)(2℧−24))(t). |
Case 5: L=2n℧−4 for ℧<4n if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω‖x‖n℧4(3n5−5n4−11n3+5n2+8n)(24−2n℧))(t). |
Case 6: L=24−n℧ for ℧>4n if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω‖x‖n℧−4(3n5−5n4−11n3+5n2+8n)(2n℧−24))(t). |
Case 7: L=2n℧−4 for ℧<4n if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η((n+1)‖x‖n℧4(3n5−5n4−11n3+5n2+8n)(24−2n℧))(t). |
Case 8 : L=24−n℧ for ℧>4n if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η((n+1)Ω‖x‖n℧−4(3n5−5n4−11n3+5n2+8n)(2n℧−24))(t). |
Hence the proof is complete.
In this note we investigated the general solution for the quartic functional equation (1.4) and also investigated the Hyers-Ulam stability of the quartic functional equation (1.4) in random normed space using the direct approach and the fixed point approach. This work can be applied to study the stability in various spaces such as intuitionistic random normed spaces, quasi-Banach spaces and fuzzy normed spaces. Moreover, the results can be applied to investigate quartic homomorphisms and quratic derivations in Banach algebras, random normed algebras, fuzzy Banach algebras and C∗-ternary algebras.
This work was supported by Incheon National University Research Grant 2020-2021. We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.
The authors declare that they have no competing interests.
[1] | J. A. Thomson, J. Itskovitz-Eldor, S. S. Shapiro, et al., Blastocysts Embryonic Stem Cell Lines Derived from Human, Science, 282 (1998), 1145–1147. |
[2] | C. E. Murry, M. A. Laflamme, X. Yang, et al., Human embryonic-stem-cell-derived cardiomyocytes regenerate non-human primate heart, Nature, 510 (2014), 273–277. |
[3] | P. P. Tam and D. A. Loebel, Gene function in mouse embryogenesis: get set for gastrulation, Nat. Rev. Genet., 8 (2007), 368–381. |
[4] | A. Adamo, I. Paramonov, M. J. Barrero, et al., LSD1 regulates the balance between self-renewal and differentiation in human embryonic stem cells, Nat. Cell Biol., 13 (2011), 652–659. |
[5] | L. Chu, J. Zhang, J. A. Thomson, et al., Single-cell RNA-seq reveals novel regulators of human embryonic stem cell differentiation to definitive endoderm, Genome Biol., 17 (2016), 173. |
[6] | S. Larabee, H. Coia, G. Gallicano, et al., miRNA-17 Members that Target Bmpr2 Influence Signaling Mechanisms Important for Embryonic Stem Cell Differentiation In Vitro and Gastrulation in Embryos, Stem Cells Dev., 24 (2015), 354–371. |
[7] | R. A. Young, L. A. Boyer, T. I. Lee, et al., Core Transcriptional Regulatory Circuitry in Human Embryonic Stem Cells, Cell, 122 (2005), 947–956. |
[8] | L. W. Jeffrey, T. A. Beyer, J. L. Wrana, et al., Switch enhancers interpret TGF- and Hippo signaling to control cell fate in human embryonic stem cells, Cell Rep., 5 (2013), 1611–1624. |
[9] | J. Rossant, J. S. Draper, A. Nagy, et al., Establishment of endoderm progenitors by SOX transcription factor expression in human embryonic stem cells, Cell Stem Cell, 3 (2008), 182–195. |
[10] | N. Ivanova, Z. Wang, S. Razis, et al., Distinct lineage specification roles for NANOG, OCT4, and SOX2 in human embryonic stem cells, Cell Stem Cell, 10 (2012), 440–454. |
[11] | A. F. Schier, A. Regev, D. Gennert, et al., Spatial reconstruction of single-cell gene expression data, Nat. Biotechnol., 33 (2015), 495–502. |
[12] | J. L. Rinn, C. Trapnell, T. S. Mikkelsen, et al., The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells, Nat. Biotechnol., 32 (2014), 381–386. |
[13] | J. Tan and X. Zou, Complex dynamical analysis of a coupled network from innate immune responses, Int. J. Bifurcat. Chaos, 23 (2013), 1350180. |
[14] | S. Jin, D. Wang and X. Zou, Trajectory control in nonlinear networked systems and its applications to complex biological systems, SIAM J. Appl. Math, 78 (2018), 629–649. |
[15] | S. Jin, F. Wu and X. Zou, Domain control of nonlinear networked systems and applications to complex disease networks, Discrete Cont. Dyn. B, 22 (2017), 2169–2206. |
[16] | X. Shu, D. Pei, S. Wei, et al., A sequential EMT-MET mechanism drives the differentiation of human embryonic stem cells towards hepatocytes, Nat. Commun., 8 (2017), 15166. |
[17] | M. A. Nieto, J. P. Thiery, R. Y. Huang, et al., Epithelial-mesenchymal transitions in development and disease, Cell, 139 (2009), 871–890. |
[18] | J. P. Thiery and J. P. Sleeman, Complex networks orchestrate epithelial-mesenchymal transitions, Nat. Rev. Mol. Cell Biol., 7 (2006), 131–142. |
[19] | H. Peinado, F. Portillo and A. Cano, Transcriptional regulation of cadherins during development and carcinogenesis, Int. J. Dev. Biol., 48 (2004), 365–375. |
[20] | A. Voulgari and A. Pintzas, Epithelial mesenchymal transition in cancer metastasis: mechanisms, markers and strategies to overcome drug resistance in the clinic, Biochim. Biophys. Acta, 1796 (2009), 75–90. |
[21] | J. Zhao, X. Wang, M. Hung, et al., Krüppel-Like Factor 8 Induces Epithelial to esenchymal Transition and Epithelial Cell Invasion, Cancer Res., 67 (2007), 7184–7193. |
[22] | Y. Ma, X. Zheng, K. Chen, et al., ZEB1 promotes the progression and metastasis of cervical squamous cell carcinoma via the promotion of epithelial-mesenchymal transition, Int. J. Clin. Exp. Pathol., 8 (2015), 11258–11267. |
[23] | D. Chen, Y. Chu, S. Li, et al., Knock-down of ZEB1 inhibits the proliferation, invasion and migration of gastric cancer cells, Chin. J. Cell. Mol. Immunol., 33 (2017), 1073–1078. |
[24] | B. L. Li, J. M. Cai, F. Gao, et al., Inhibition of TBK1 attenuates radiation-induced epithelial-mesenchymal transition of A549 human lung cancer cells via activation of GSK-3β and repression of ZEB1, Lab. Invest., 94 (2014), 362–370. |
[25] | J. Comijn, G. Berx, Roy F. van, et al., The two-handed E box binding zinc finger protein SIP1 down-regulates E-cadherin and induces invasion, Mol. Cell, 7 (2001), 1267–1278. |
[26] | M. Moes, E. Friederich, A. Sol, et al., A novel network integrating a mi-RNA 203/SNAI1 feedback loop which regulates epithelial to mesenchymal transition, PloS One, 7 (2012), e35440. |
[27] | U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, 1st edition, Taylor & Francis Inc, Boca Raton, FL, 2006. |
[28] | D. Chu, N. R. Zabet and B. Mitavskiy, Models of transcription factor binding: Sensitivity of activation functions to model assumptions, J. Theor. Biol., 257 (2009), 419–429. |
[29] | J. Kennedy and R. Eberhart, Particle swarm optimization, Proceedings of ICNN'95, International Conference on Neural Networks, Perth, WA, Australia, 4 (1995), 1942–1948. |
[30] | D. Gillespie, Exact Stochastic Simulation of Coupled Chemical Reactions, J. Phys. Chem., 81 (1977), 2340–2361. |
[31] | X.Xiang, Y.Chen, X.Zou, etal., UnderstandinginhibitionofviralproteinsontypeIIFNsignaling pathways with modeling and optimization, J. Theor. Biol., 265 (2010), 691–703. |
[32] | S. Shin, O. Rath, K. Cho, et al., Positive- and negative-feedback regulations coordinate the dynamic behavior of the Ras-Raf-MEK-ERK signal transduction pathway, J. Cell Sci., 122 (2009), 425–435. |
[33] | T. Tian and K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch, Proc. Natl. Acad. Sci., 103 (2006), 8372–8377. |
[34] | M. R. Birtwistle, J. Rauch, B. N. Kholodenko, et al., Emergence of bimodal cell population responses from the interplay between analog single-cell signaling and protein expression noise, BMC Syst. Biol., 16 (2012), 109. |
[35] | L. K. Nguyen, M. R. Birtwistle, B. N. Kholodenko, et al., Stochastic models for regulatory networks of the genetic toggle switch, Proc. Natl. Acad. Sci., 103 (2006), 8372–8377. |
1. | Lingxiao Lu, Jianrong Wu, Hyers-Ulam-Rassias stability of cubic functional equations in fuzzy normed spaces, 2022, 7, 2473-6988, 8574, 10.3934/math.2022478 |