The intention and novelty in the presented study were to develop the regularity analysis for a parabolic equation describing a type of Eyring-Powell fluid flow in two dimensions. We proved that, under certain general conditions involving the space of bounded mean oscillation (BMO) and the Lebesgue space L2, there exist bounded and regular velocity solutions under the L2 space scope. This conclusion was additionally supplemented by the condition of a finite square integrable initial data (also some of the obtained expressions involved the gradient and the laplacian of the initial velocity distribution). To make our results further general, the proposed analysis was extended to cover regularity results in Lp(p>2) spaces. As a remarkable conclusion, we highlight that the solutions to the two dimensional Eyring-Powell fluid flow did not exhibit blow up behaviour.
Citation: José Luis Díaz Palencia, Saeed Ur Rahman, Saman Hanif. Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium[J]. Electronic Research Archive, 2022, 30(11): 3949-3976. doi: 10.3934/era.2022201
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The intention and novelty in the presented study were to develop the regularity analysis for a parabolic equation describing a type of Eyring-Powell fluid flow in two dimensions. We proved that, under certain general conditions involving the space of bounded mean oscillation (BMO) and the Lebesgue space L2, there exist bounded and regular velocity solutions under the L2 space scope. This conclusion was additionally supplemented by the condition of a finite square integrable initial data (also some of the obtained expressions involved the gradient and the laplacian of the initial velocity distribution). To make our results further general, the proposed analysis was extended to cover regularity results in Lp(p>2) spaces. As a remarkable conclusion, we highlight that the solutions to the two dimensional Eyring-Powell fluid flow did not exhibit blow up behaviour.
As we all know, the vibration equation is one of the important research topics in mechanics, physics and other disciplines. In the recent decades, researchers have been paying more and more attentions to the fractional differential equations due to its wide applications on mechanics, physical science, biological sciences and engineering disciplines, etc., see [3,6,7,12,13,17,18,19,20,25] and the references therein. The development of fractional differential equations provides some new theoretical bases for the study of vibration problems. In [22], the vibration equations with fractional derivatives are used to describe the vibration behavior of viscoelastic polymers and good results are obtained. The theoretical study of vibration equations with fractional derivative has also been widely concerned. These studies include the mechanical properties, dynamic characteristics of the system and the correlation functions with various influences for the vibration equations with fractional derivatives, and so on see[15,16,21,22]. In addition, mutation often occurs in vibration system. These abrupt changes can be simulated by the impulsive vibration equation. And this simulation are effective in describing the behavior of real system. There have been a large number of references for the study of fractional impulsive differential equations, see [1,4,5,9,11,23,24,26].
In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives under the nonlinear boundary conditions
{x″(t)−λcDα0+x(t)=f(t,x(t),x′(t)),t∈J′,Δx(t)|t=tk=Ik(x(tk)),k=1,2,⋯,m,Δx′(t)|t=tk=Qk(x(tk)),k=1,2,⋯,m,g0(x(0),x(1))=0,g1(x′(0),x′(1))=0, | (1.1) |
where 0<α<1, λ>0 and cDα0+ is the Caputo derivative, 0=t0<t1<⋯<tm<tm+1=1. Set J=[0,1],J′=J∖{t1,t2,⋯,tm},J0=[0,t1],Jk=(tk,tk+1],k=1,2,⋯,m. Δx(tk)=x(t+k)−x(t−k),Δx′(tk)=x′(t+k)−x′(t−k). x(t−k),x(t+k) denote the left limit and the right limit of x(t) at t=tk. x′(t−k),x′(t+k) denote the left limit and the right limit of x′(t) at t=tk. Let x(tk)=x(t−k). f∈C(J×R2,R),Ik,Qk∈C(R,R),k=1,2,⋯, m. g0,g1∈C(R2,R) are given nonlinear functions. By using monotone iterative technique, some new results on multiplicity of boundary value problems are obtained, and the properties of the solutions are discussed. Finally, an example is given out to illustrate the applicability of our main results.
In this section, we present some basic definitions and lemmas, which will be used to prove our main results.
Definition 2.1. (See[8], P67) Let α,β>0. The function Eα,β is defined by
Eα,β(z)=∞∑j=0zjΓ(jα+β), |
whenever the series converge is called the two parameters Mittag-Leffler function with parameters α and β.
Lemma 2.1. (See[8], P68) Consider the two parameters Mittag-Leffler function Eα,β for some α,β>0. The power series defining Eα,β is convergent for all z∈C. In other words, Eα,β is an entire function.
Lemma 2.2. Let α,β>0,k=0,1,2,⋯,z∈R. Then
E(k)α,β(z)=∞∑j=0zjΓ(k+j+1)Γ(j+1)Γ(α(k+j)+β). |
Proof. By Definition 2.1 and Lemma 2.1, we get
E(k)α,β(z)=dkdzkEα,β(z)=∞∑j=0dkdzk(zjΓ(jα+β))=∞∑j=kΓ(j+1)zj−kΓ(j−k+1)Γ(αj+β)=∞∑j=0zjΓ(k+j+1)Γ(j+1)Γ(α(k+j)+β). |
Lemma 2.3. (See[14], P314) Let 0<α<β,n−1<α≤n,l−1<β≤l(n,l∈N,n≤l,λ∈R). Then
cDβ0+x(t)−λcDα0+x(t)=0(t>0) |
has its linearly independent solutions given by
xj(t)=tjEβ−α,j+1(λtβ−α)−λtβ−α+jEβ−α,β−α+j+1(λtβ−α)(j=0,1,⋯,n−1), | (2.1) |
xj(t)=tjEβ−α,j+1(λtβ−α)(j=n,⋯,l−1). | (2.2) |
Lemma 2.4. (See[14], P324) Let l−1<β≤l(l∈N),0<α<β be such that β−l+1≥α, λ∈R, and h(t) be a given real function defined on R+. The general solution to the nonhomogeneous linear differential equation
cDβ0+x(t)−λcDα0+x(t)=h(t)(t>0) |
is given by
x(t)=∫t0(t−s)β−1Eβ−α,β(λ(t−s)β−α)h(s)ds+l−1∑j=0cjxj(t), |
where xj(t) are given by (2.1) and (2.2), cj are arbitrary real constants (j=0,1,⋯,l−1).
Lemma 2.5. Let L[0,1] denote the space of Lebesgue integrable functions on [0, 1], h∈L[0,1] and 0<α<1, then the Cauchy problem of the second order vibration equation with fractional derivative
{x″(t)−λcDα0+x(t)=h(t),t∈J,x(ξ)=x0,x′(ξ)=x1,ξ∈J |
has a unique solution, which is given by
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫ξ0(ξ−s)E2−α,2(λ(ξ−s)2−α)h(s)ds+x0+tE2−α,2(λt2−α)−ξE2−α,2(λξ2−α)E2−α,1(λξ2−α)(x1−∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds), | (2.3) |
and x(t) is derivable while its derivative is given by
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+E2−α,1(λt2−α)E2−α,1(λξ2−α)(x1−∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds). | (2.4) |
Proof. In view of Lemma 2.4, for β=2,0<α<1, the general solution of the equation
x″(t)−λcDα0+x(t)=h(t) |
is given by
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds+c1tE2−α,2(λt2−α)+c0, |
where c0,c1∈R, and
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+c1E2−α,1(λt2−α). |
Therefore,
x(ξ)=∫ξ0(ξ−s)E2−α,2(λ(ξ−s)2−α)h(s)ds+c1ξE2−α,2(λξ2−α)+c0=x0,x′(ξ)=∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds+c1E2−α,1(λξ2−α)=x1. |
We get
c1=1E2−α,1(λξ2−α)(x1−∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds),c0=x0−∫ξ0(ξ−s)E2−α,2(λ(ξ−s)2−α)h(s)ds−c1ξE2−α,2(λξ2−α). |
So
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫ξ0(ξ−s)E2−α,2(λ(ξ−s)2−α)h(s)ds+x0+c1(tE2−α,2(λt2−α)−ξE2−α,2(λξ2−α))=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫ξ0(ξ−s)E2−α,2(λ(ξ−s)2−α)h(s)ds+x0+tE2−α,2(λt2−α)−ξE2−α,2(λξ2−α)E2−α,1(λξ2−α)(x1−∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds), |
and
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+E2−α,1(λt2−α)E2−α,1(λξ2−α)(x1−∫ξ0E2−α,1(λ(ξ−s)2−α)h(s)ds). |
The proof is completed.
Let PC1(J)={x:J→R|x,x′∈C(J′,R),x(t+k),x(t−k),x′(t+k),x′(t−k)exist,andx(tk)=x(t−k),k=1,2,⋯,m} and endowed with the normal ‖x‖=max{supt∈[0,1]|x(t)|,supt∈[0,1]|x′(t)|}. Then PC1(J) is a Banach space.
For x∈PC1(J), by the Lagrange mean value theorem, there exists ξk∈[tk−ε,tk] such that
x(tk)−x(tk−ε)=x′(ξk)ε, |
and
x′−(tk)=limε→0+x(tk)−x(tk−ε)ε=limε→0+x′(ξk)εε=x′(t−k),k=1,2,⋯,m. |
Thus, for x∈PC1(J), we denote
x′(tk)=x′−(tk)=x′(t−k),k=1,2,⋯,m. | (2.5) |
Let P={x∈PC1(J)|x(t)≥0,x′(t)≥0,t∈J}. It is obvious that P⊂PC1(J) is a normal solid cone. We denote x≺_y∈PC1(J) if and only if x(t)≤y(t) and x′(t)≤y′(t) on t∈[0,1], i.e. y−x∈P. We denote x≺y if x≺_y∈PC1(J) and x≠y, and x≺≺y if y−x∈˚P.
Lemma 2.6. (See[10], P220, [2], P666) Let E be a Banach space, and P⊂E be a normal solid cone. Suppose that there exist α1,β1,α2,β2∈E with α1≺β1≺α2≺β2 and A:[α1,β2]→E is a completely continuous strongly increasing operator such that
α1≺_Aα1,Aβ1≺β1,α2≺Aα2,Aβ2≺_β2. |
Then the operator A has at least three distinct fixed points x1,x2,x3 on [α1,β2] such that
α1≺_x1≺≺β1,α2≺≺x2≺_β2,α2⊀_x3⊀_β1. |
In this section, we obtain the solution of the linear impulsive vibration equation and discuss the properties of its kernel function.
Lemma 3.1. For any pk,qk∈R (k=1,2,⋯,m), mi,ni∈R (i=1,2) and h∈L[0,1], the following boundary value problem of the second order impulsive vibration equation with fractional derivative
{x″(t)−λcDα0+x(t)=h(t),t∈J′,Δx(t)|t=tk=pk,k=1,2,⋯,m,Δx′(t)|t=tk=qk,k=1,2,⋯,m,m1x(0)+m2x(1)=γ0,n1x′(0)+n2x′(1)=γ1 | (3.1) |
has a unique solution, which is given by
x(t)=∫10G(t,s)h(s)ds+φ(t)+(tE2−α,2(λt2−α)n1+n2E2−α,1(λ)−m2E2−α,2(λ)(m1+m2)(n1+n2E2−α,1(λ)))γ1+γ0m1+m2,t∈J, | (3.2) |
where
G(t,s)={(t−s)E2−α,2(λ(t−s)2−α)−n2tE2−α,2(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)+m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ))−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2,0≤s≤t≤1,m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ))−n2tE2−α,2(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2,0≤t<s≤1, | (3.3) |
φ(t)=∑0<ti<tpi−m2m1+m2m∑i=1pi+∑0<ti<ttE2−α,2(λt2−α)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)qi+m∑i=1(m2n2E2−α,2(λ)E2−α,1(λ)(m1+m2)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)−m2(E2−α,2(λ)−tiE2−α,2(λt2−αi))(m1+m2)E2−α,1(λt2−αi)−n2E2−α,1(λ)tE2−α,2(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi))qi,t∈J. | (3.4) |
Furthermore,
x′(t)=∫10G′t(t,s)h(s)ds+φ′(t)+E2−α,1(λt2−α)n1+n2E2−α,1(λ)γ1,t∈J, | (3.5) |
G′t(t,s)={E2−α,1(λ(t−s)2−α)−n2E2−α,1(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ),0≤s≤t≤1,−n2E2−α,1(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ),0≤t<s≤1, | (3.6) |
and
φ′(t)=∑0<ti<tE2−α,1(λt2−α)E2−α,1(λt2−αi)qi−m∑i=1n2E2−α,1(λ)E2−α,1(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)qi,t∈J. | (3.7) |
Proof. For t∈[0,t1], let ξ=0,x(0)=c0,x′(0)=c1, by Lemma 2.5, Cauchy problem
{x″(t)−λcDα0+x(t)=h(t),x(0)=c0,x′(0)=c1 |
has a unique solution
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds+c1tE2−α,2(λt2−α)+c0, |
and
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+c1E2−α,1(λt2−α),c0,c1∈R. |
So
x(t−1)=∫t10(t1−s)E2−α,2(λ(t1−s)2−α)h(s)ds+c1t1E2−α,2(λt2−α1)+c0,x′(t−1)=∫t10E2−α,1(λ(t1−s)2−α)h(s)ds+c1E2−α,1(λt2−α1). |
For t∈(t1,t2], let ξ=t1,x(t+1)=x(t−1)+p1,x′(t+1)=x′(t−1)+q1. By Lemma 2.5, we can obtain that
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫t10(t1−s)E2−α,2(λ(t1−s)2−α)h(s)ds+x(t+1)+tE2−α,2(λt2−α)−t1E2−α,2(λt2−α1)E2−α,1(λt2−α1)(x′(t+1)−∫t10E2−α,1(λ(t1−s)2−α)h(s)ds)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫t10(t1−s)E2−α,2(λ(t1−s)2−α)h(s)ds+x(t−1)+p1+tE2−α,2(λt2−α)−t1E2−α,2(λt2−α1)E2−α,1(λt2−α1)(x′(t−1)+q1−∫t10E2−α,1(λ(t1−s)2−α)h(s)ds)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds−∫t10(t1−s)E2−α,2(λ(t1−s)2−α)h(s)ds+p1+∫t10(t1−s)E2−α,2(λ(t1−s)2−α)h(s)ds+c1t1E2−α,2(λt2−α1)+c0+tE2−α,2(λt2−α)−t1E2−α,2(λt2−α1)E2−α,1(λt2−α1)(∫t10E2−α,1(λ(t1−s)2−α)h(s)ds+c1E2−α,1(λt2−α1)+q1−∫t10E2−α,1(λ(t1−s)2−α)h(s)ds)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds+tE2−α,2(λt2−α)c1+c0+p1+tE2−α,2(λt2−α)−t1E2−α,2(λt2−α1)E2−α,1(λt2−α1)q1, |
and
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+E2−α,1(λt2−α)E2−α,1(λt2−α1)(x′(t−1)+q1−∫t10E2−α,1(λ(t1−s)2−α)h(s)ds)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+E2−α,1(λt2−α)c1+E2−α,1(λt2−α)E2−α,1(λt2−α1)q1. |
For t∈(tk,tk+1], let ξ=tk,x(t+k)=x(t−k)+pk,x′(t+k)=x′(t−k)+qk,k=2,3,⋯,m. In the same way, we have
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds+tE2−α,2(λt2−α)c1+c0+∑0<ti<tpi+∑0<ti<ttE2−α,2(λt2−α)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)qi, |
and
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds+E2−α,1(λt2−α)c1+∑0<ti<tE2−α,1(λt2−α)E2−α,1(λt2−αi)qi. |
Hence,
x(1)=∫10(1−s)E2−α,2(λ(1−s)2−α)h(s)ds+E2−α,2(λ)c1+c0+m∑i=1pi+m∑i=1E2−α,2(λ)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)qi,x′(1)=∫10E2−α,1(λ(1−s)2−α)h(s)ds+E2−α,1(λ)c1+m∑i=1E2−α,1(λ)E2−α,1(λt2−αi)qi. |
By the boundary conditions m1x(0)+m2x(1)=γ0,n1x′(0)+n2x′(1)=γ1, we can get that
{−(m1+m2)c0−m2E2−α,2(λ)c1=m2∫10(1−s)E2−α,2(λ(1−s)2−α)h(s)ds+m2m∑i=1pi+m2m∑i=1E2−α,2(λ)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)qi−γ0,−(n1+n2E2−α,1(λ))c1=n2∫10E2−α,1(λ(1−s)2−α)h(s)ds+m∑i=1n2E2−α,1(λ)E2−α,1(λt2−αi)qi−γ1. |
So
c0=∫10(−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2+m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ)))h(s)ds−m2m1+m2m∑i=1pi+m∑i=1(m2n2E2−α,2(λ)E2−α,1(λ)(m1+m2)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)−m2(E2−α,2(λ)−tiE2−α,2(λt2−αi))(m1+m2)E2−α,1(λt2−αi))qi−m2E2−α,2(λ)γ1(m1+m2)(n1+n2E2−α,1(λ))+γ0m1+m2, |
c1=−n2n1+n2E2−α,1(λ)(∫10E2−α,1(λ(1−s)2−α)h(s)ds+m∑i=1E2−α,1(λ)E2−α,1(λt2−αi)qi)+γ1n1+n2E2−α,1(λ). |
Therefore,
x(t)=∫t0(t−s)E2−α,2(λ(t−s)2−α)h(s)ds+∫10(m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ))−n2tE2−α,2(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2)h(s)ds+∑0<ti<tpi−m2m1+m2m∑i=1pi+∑0<ti<ttE2−α,2(λt2−α)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)qi+m∑i=1(m2n2E2−α,2(λ)E2−α,1(λ)(m1+m2)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)−m2(E2−α,2(λ)−tiE2−α,2(λt2−αi))(m1+m2)E2−α,1(λt2−αi)−n2E2−α,1(λ)tE2−α,2(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi))qi+γ0m1+m2+(tE2−α,2(λt2−α)n1+n2E2−α,1(λ)−m2E2−α,2(λ)(m1+m2)(n1+n2E2−α,1(λ)))γ1=∫10G(t,s)h(s)ds+φ(t)+(tE2−α,2(λt2−α)n1+n2E2−α,1(λ)−m2E2−α,2(λ)(m1+m2)(n1+n2E2−α,1(λ)))γ1+γ0m1+m2,t∈[0,1]. |
And
x′(t)=∫t0E2−α,1(λ(t−s)2−α)h(s)ds−∫10n2E2−α,1(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)h(s)ds+∑0<ti<tE2−α,1(λt2−α)E2−α,1(λt2−αi)qi−m∑i=1n2E2−α,1(λ)E2−α,1(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)qi+E2−α,1(λt2−α)n1+n2E2−α,1(λ)γ1=∫10G′t(t,s)h(s)ds+φ′(t)+E2−α,1(λt2−α)n1+n2E2−α,1(λ)γ1,t∈[0,1]. |
Therefore, boundary value problem (3.1) has a unique solution x=x(t) which is given by (3.2), and G(t,s),φ(t) are given by (3.3) and (3.4), respectively. Furthermore, x′(t) is also established.
For convenience, we give out the following hypothesis:
(H1) The constants mi,ni∈R(i=1,2) satisfy m2(m1+m2)<0 and n2(n1+n2E2−α,1(λ))<0.
Lemma 3.2. Suppose that (H1) holds. Then functions G and φ defined by (3.3) and (3.4) satisfy the following properties:
(1) G(t,s) is continuous for t,s∈[0,1].
(2) G(t,s)>0 for t,s∈[0,1] and maxt∈[0,1]G(t,s)=G(1,s),mint∈[0,1]G(t,s)=G(0,s).
(3) G′t(t,s)>0 for t,s∈[0,1] and maxt∈[0,1]G′t(t,s)=G′t(1,s),mint∈[0,1]G′t(t,s)=G′t(0,s).
(4) If pk≥0,qk≥0,k=1,2,⋯,m, then φ(t)≥0,φ′(t)≥0, for t∈Jk.
Proof. (1) By the definition of G(t,s),G∈C([0,1]×[0,1]) is obvious.
(2) By (H1), for 0≤s≤t≤1,
G′t(t,s)=∂G(t,s)∂t=E2−α,1(λ(t−s)2−α)−n2E2−α,1(λt2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)>0. |
Then G(s,s)≤G(t,s)≤G(1,s), for s∈[0,1] and t∈[s,1]. And
G(s,s)=m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ))−n2sE2−α,2(λs2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2>0. |
Hence, G(t,s)>0 for 0≤s≤t≤1 and G(t,s)≤G(1,s) for t∈[s,1].
For 0≤t<s≤1,
∂G(t,s)∂t=−n2E2−α,1(λt2−α)E2−α,1(λ(1−s)2−α)n1−n2E2−α,1(λ)>0, |
we can get that G(0,s)≤G(t,s)<G(s,s), for s∈[0,1] and t∈[0,s). And
G(0,s)=m2n2E2−α,2(λ)E2−α,1(λ(1−s)2−α)(m1+m2)(n1+n2E2−α,1(λ))−m2(1−s)E2−α,2(λ(1−s)2−α)m1+m2>0. |
Hence, G(t,s)>0 for 0≤t<s≤1 and G(t,s)<G(s,s) for t∈[0,s).
Therefore, G(t,s)>0 for any t,s∈[0,1]. And G(t,s) is monotone increasing with respect to t∈[0,1], so maxt∈[0,1]G(t,s)=G(1,s),mint∈[0,1]G(t,s)=G(0,s).
(3) Since
(E2−α,1(λt2−α))′=(∞∑k=0(λt2−α)kΓ((2−α)k+1))′=∞∑k=1(2−α)kλkt(2−α)k−1Γ((2−α)k+1)≥0,t∈[0,1]. |
By (H1), for 0≤s≤t≤1,
∂2G(t,s)∂t2=(E2−α,1(λ(t−s)2−α))′−n2E2−α,1(λ(1−s)2−α)(E2−α,1(λt2−α))′n1+n2E2−α,1(λ)≥0. |
Then G′t(s,s)≤G′t(t,s)≤G′t(1,s), for any s∈[0,1],t∈[s,1]. Because
G′t(s,s)=1−n2E2−α,1(λs2−α)E2−α,1(λ(1−s)2−α)n1+n2E2−α,1(λ)>0, |
we can get that G′t(t,s)>0 for 0≤s≤t≤1.
For 0≤t<s≤1,
∂G2(t,s)∂t2=−n2E2−α,1(λ(1−s)2−α)(E2−α,1(λt2−α))′n1−n2E2−α,1(λ)≥0, |
we can get that G′t(0,s)≤G′t(t,s)≤G′t(s,s) for any s∈[0,1] and t∈[0,s). Since
G′t(0,s)=−n2E2−α,1(λ(1−s)2−α)n1−n2E2−α,1(λ)>0, |
we have G′t(t,s)>0 for 0≤t<s≤1.
Therefore, G′t(t,s)>0 for any t,s∈[0,1] and maxt∈[0,1]G′t(t,s)=G′t(1,s),mint∈[0,1]G′t(t,s)=G′t(0,s).
(4) If pk,qk≥0, k=1,2,⋯,m, by (H1), (3.4) and (3.7), we can easily get that
φ(t)≥0,φ′(t)≥0,t∈Jk. |
The proof is completed.
In this section, we will establish the existence results of the solutions for the boundary value problem (1.1).
For any u∈PC1(J), we consider the following boundary value problem
{x″(t)−λcDα0+x(t)=f(t,u(t),u′(t)),t∈J′,Δx(t)|t=tk=Ik(u(tk)),k=1,2,⋯,m,Δx′(t)|t=tk=Qk(u(tk)),k=1,2,⋯,m,m1x(0)+m2x(1)=g0(u(0),u(1))+m1u(0)+m2u(1):=γu,0,n1x′(0)+n2x′(1)=g1(u′(0),u′(1))+n1u′(0)+n2u′(1):=γu,1. | (4.1) |
By Lemma 3.1, we can get that boundary value (4.1) is equivalent to the following integral equation
x(t)=∫10G(t,s)f(s,u(s),u′(s))ds+φu(t)+(tE2−α,2(λt2−α)n1+n2E2−α,1(λ)−m2E2−α,2(λ)(m1+m2)(n1+n2E2−α,1(λ)))γu,1+γu,0m1+m2,t∈J, |
and
x′(t)=∫10G′t(t,s)f(s,u(s),u′(s))ds+φ′u(t)+E2−α,1(λt2−α)n1+n2E2−α,1(λ)γu,1,t∈J, |
where
φu(t)=∑0<ti<tIi(u(ti))−m2m1+m2m∑i=1Ii(u(ti))+∑0<ti<ttE2−α,2(λt2−α)−tiE2−α,2(λt2−αi)E2−α,1(λt2−αi)Qi(u(ti))+m∑i=1(m2n2E2−α,2(λ)E2−α,1(λ)(m1+m2)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)−m2(E2−α,2(λ)−tiE2−α,2(λt2−αi))(m1+m2)E2−α,1(λt2−αi)−n2E2−α,1(λ)tE2−α,2(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi))Qi(u(ti)),t∈J | (4.2) |
and
φ′u(t)=∑0<ti<tE2−α,1(λt2−α)E2−α,1(λt2−αi)Qi(u(ti))−m∑i=1n2E2−α,1(λ)E2−α,1(λt2−α)(n1+n2E2−α,1(λ))E2−α,1(λt2−αi)Qi(u(ti)),t∈J. | (4.3) |
We define an operator by
By Lemma 3.1 and (3.5),
We can easily get that the following Lemma 4.1 holds.
Lemma 4.1. The function is the solution of boundary value problem (1.1) if and only if is a fixed point of the operator in .
Lemma 4.2. If (H1) holds, then is completely continuous.
Proof. Step 1: is a continuous operator.
Suppose that and there exists such that . Then there exists a constant such that
Since , and , then
By (H1), Lemma 3.2 and Lebesgue dominated convergence theorem, for any , we have
and
So as , which means is continuous.
Step 2: is relatively compact.
Let be a bounded set. By the continuity of the functions , and , there exists a constant , for any and ,
Then
By Lemma 3.2, for any ,
and
Therefore, the operator is uniformly bounded.
Because is continuous on , then it is uniformly continuous on . Thus, for any , there exists a constant such that for any , , whenever , we can get that
Denote
By (3.6),
Similarly, for the , there exists a constant such that
whenever and .
By the uniformly continuity of functions and on , we can show that for the , there exists a constant such that
and
whenever and . Hence, by (4.2) and (4.3), we have
and
By the uniformly continuity of on , for the , there exists a constant , for , and , we have
Then
We take . Therefore, for any , there exists a constant such that for whenever and any , we can get that
and
Thus, the operator is equicontinuous on every interval .
According to the Arzela-Ascoli theorem, is relatively compact.
Therefore, is completely continuous.
In the following, we give out some hypotheses.
(H2) if , , . Furthermore, if and .
And for ,
(H3.1) If , then for , ,
and
(H3.2) If , then for , ,
and
(H3.3) If , then for , ,
and
(H3.4) If , then for , ,
and
Remark. It is easy to see that if one of (H3.1)–(H3.4) holds, then (H1) holds.
Lemma 4.3. For , if one of the following conditions holds, then and for .
(1) and satisfies
(2) and satisfies
(3) and satisfies
(4) and satisfies
Proof. (1) Denote Then .
By Lemma 3.1, the following boundary value problem
has a unique solution
and
It's easy to see that and for .
Similarly, (2)–(4) are easy to be proved.
Lemma 4.4. Suppose (H2) and one of (H3.1)-(H3.4) holds, then is a strongly increasing operator.
Proof. Here, we will only prove the conclusion when (H3.1) holds, and other situations are similar.
For any , and which implies that and for . By (H2), for any ,
Since , there exists an interval such that for . It follows from (H2)
(4.4) |
By (H3.1), we can get that
By (4.2), (4.3), (H3.1) and (H2), we can get that for ,
and
By Lemma 3.2, (H2) and (4.4), for any and , as , we have
and
Thus,
which implies that is a strongly increasing operator.
The proof is completed.
Theorem 4.5. If (H2) and (H3.1) hold, there exist , with such that
(4.5) |
and
(4.6) |
where , and are not the solutions of boundary value problem (1.1). Then boundary value problem (1.1) has at least three distinct solutions and satisfies
Proof. By Lemma 4.2 and Lemma 4.4, we can get that is a completely continuous strongly increasing operator.
By the definition of operator , we can show that
Let . By (4.5),
Similarly, we can get that
By (1) in Lemma 4.3, we have
Therefore, .
It is similar that we can obtain . Because is not the solution of boundary value problem (1.1), then . Thus, .
By using the same method, we can easily get .
By using Lemma 2.6, we can get that the operator has at least three distinct fixed points , and satisfies
Therefore, by Lemma 4.1, we can obtain that boundary value problem (1.1) has at least three distinct solutions , and
The proof is completed.
In the similar way, the following three theorems can be established.
Theorem 4.6. If (H2) and (H3.2) hold, there exist , with such that
and
where , and are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions and satisfies
Theorem 4.7. If (H2) and (H3.3) hold, there exist , with such that
and
where , and are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions and satisfies
Theorem 4.8. If (H2) and (H3.4) hold, there exist , with such that
and
where , and are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions and satisfies
In this section, we discuss the applicability of our main results.
Consider the following the boundary value problem
(5.1) |
where .
Obviously, are nonlinear functions. And satisfy (H2).
Let . Since
then satisfy (H3.1).
For , we take
We can easily get that
and
So
and
We can easily get that satisfy (4.5) and satisfy (4.6) with .
The conditions of Theorem 4.5 are all satisfied. So by Theorem 4.5, the boundary value problem (5.1) has at least three distinct solutions and moreover,
In this work, we investigate the existence of solutions for a class of second order impulsive vibration equation with fractional derivatives. Some sufficient conditions for existence of the multiplicity solutions are established by applying monotone iterative technique. Finally, a concrete example is given to illustrate the wide range of potential applications of our main results.
Further extensions of this paper are to study the motion state of the vibrator in the system described by boundary value problem (1.1) and the existence of solutions to the boundary value problems with other boundary conditions. Moreover, fractional differential equation models such as the rheological model of the fractional derivative and singular systems model of fractional differential equations have real world applications. So, we also can consider using the boundary value problem of impulsive differential equations to simulate the abrupt changes in the systems described by these models.
The authors would like to thank college mathematics characteristic pilot team of University of Shanghai for science and technology for its support to this project.
The authors declare that they have no conflicts of interest.
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