This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.
Citation: Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li. Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303
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This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.
We consider numerical invariants associated with polynomial identities of algebras over a field of characteristic zero. Given an algebra
limn→∞n√cn(A) | (1.1) |
exist and what are its possible values? In case of existence, the limit (1.1) is called the PI-exponent of
Nevertheless, the answer to Amitsur's question in the general case is negative: a counterexample was presented in [14]. Namely, for any real
The main goal of the present paper is to construct a series of unital algebras such that
Let
cn(A)=dimPnPn∩Id(A). |
If the sequence
exp_(A)=lim infn→∞n√cn(A),¯exp(A)=lim supn→∞n√cn(A), |
are well-defined. An existence of ordinary PI-exponent (1.1) is equivalent to the equality
In [14], an algebra
Clearly, polynomial identities of
f=∑fi1,…,ik,{i1,…,ik}⊆{1,…,n},0≤k≤n, | (2.1) |
where
Remark 2.1. A multilinear polynomial
The next statement easily follows from Remark 2.1.
Remark 2.2. Suppose that an algebra
Using results of [13], we obtain the following inequalities.
Lemma 2.1. ([13,Theorem 2]) Let
Lemma 2.2. ([13,Theorem 3]) Let
Given an integer
{a,b,zi1,…,ziT|i=1,2,…} |
and by the multiplication table
zija={zij+1ifj≤T−1,0ifj=T |
for all
ziTb=zi+11,i≥1. |
All other products of basis elements are equal to zero. Clearly, algebra
x1(x2x3)≡0 | (2.2) |
is an identity of
We will use the following properties of algebra
Lemma 2.3. ([14,Lema 2.1]) Let
Lemma 2.4. ([14,Lema 2.2]) Let
cn(BT)≥k!=(N−1T)!. |
Lemma 2.5. ([14,Lema 2.3]) Any multilinear identity
Let
QN=F[θ]0(QN+1), |
where
R=B(T1,N1)⊕B(T2,N2)⊕⋯, | (2.3) |
where
Let
Lemma 2.6. For any
(a) if
Pn∩Id(R)=Pn∩Id(B(Ti,Ni)⊕B(Ti+1,Ni+1))=Pn∩Id(BTi⊕BTi+1); |
(b) if
Pn∩Id(R)=Pn∩Id(B(Ti+1,Ni+1))=Pn∩(Id(BTi+1)). |
Proof. This follows immediately from the equality
The folowing remark is obvious.
Remark 2.3. Ler
Id(R♯)=Id(B(T1,N1)♯⊕B(T2,N2)♯⊕⋯). |
Theorem 3.1. For any real
Proof. Note that
cn(A)≤ncn−1(A) | (3.1) |
for any algebra
2m3<αm | (3.2) |
for all
cn(BT1)<αnfor alln≤N1−1andcN1(BT1)≥αN1. |
Consider an arbitrary
cn(R♯)≤n∑k=0(nk)ck(R)=Σ′1+Σ′2, |
where
Σ′1=N1∑k=0(nk)ck(R),Σ′2=n∑k=N1+1(nk)ck(R). |
By Lemma 2.6, we have
Σ1=N1∑k=0(nk)ck(BT1),Σ2=n∑k=0(nk)ck(BT2). |
Then for any
Σ2≤n∑k=0(nk)2k3≤2n3n∑k=0(nk)=2n32n, | (3.3) |
which follows from (3.2), provided that
Let us find an upper bound for
Σ1≤N1αN1N1∑k=0(nk) | (3.4) |
which follows from the choice of
From the Stirling formula
m!=√2πm(me)me112m+θm,0<θm<1, |
it follows that
(nk)≤√nk(n−k)⋅nnkk(n−k)n−k. | (3.5) |
Now we define the function
Φ(x)=1xx(1−x)1−x. |
It is not difficult to show that
(nk)≤√Φ(kn)⋅Φ(kn)n<2Φ(kn)n≤2Φ(N1n)n | (3.6) |
provided that
Σ1≤2N1αN1(N1+1)Φ(N1n)n,Σ2≤2n32n. |
Since
limn→∞Φ(N1n)n=1 |
and
2N1(N1+1)αN1Φ(N1n)n+2n32n<(2+12)n. | (3.7) |
Now we take
cn(R♯)<(2+12)n |
for
As soon as
cn(R)<αn+2n3 | (3.8) |
for all
αn≤cn(R)<αn+n(αn−1+2n3) | (3.9) |
for all
2Nj(Nj+1)αNjΦ(NjTj+1)Tj+1+2T3j+1⋅2Tj+1<(2+12j)Tj+1 | (3.10) |
for all
Let us denote by
cn(R♯α)<(2+12j)n | (3.11) |
if
exp_(R♯α)≤2. | (3.12) |
On the other hand, since
exp_(R♯α)≥1. | (3.13) |
Since the PI-exponent of non-nilpotent algebra cannot be strictly less than
exp_(Rα)=1,exp_(R♯α)=2. |
Finally, relations (3.8), (3.9) imply the equality
As a consequence of Theorem 3.1 we get an infinite family of unital algebras of exponential codimension growth without ordinary PI-exponent.
Corollary 1. Let
We would like to thank the referee for comments and suggestions.
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