Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability

Huiwen Wang 1  and Fang Li 1
  • 1 School of Mathematics, Yunnan Normal University, 650092, Kunming, People's Republic of China
Huiwen Wang
  • School of Mathematics, Yunnan Normal University, Kunming, 650092, People's Republic of China
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and Fang Li
  • Corresponding author
  • School of Mathematics, Yunnan Normal University, Kunming, 650092, People's Republic of China
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Abstract

In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.

1 Introduction

The fractional differential equations can model some engineering and scientific disciplines in the fields of physics, chemistry, electrodynamics of complex medium, polymer rheology, etc. In particular, the forward and backward fractional derivatives provide an excellent tool for the description of some physical phenomena such as the fractional oscillator equations and the fractional Euler-Lagrange equations [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Recently, a linear boundary value problem (BVP) involving both the right Caputo and the left Riemann-Liouville fractional derivatives has been studied by some authors [10,11,12,13,14].

On the other hand, Ulam’s stability problem [15] has been attracted by many famous researchers (see [16,17] and references therein). Recently, studying the stability of Ulam-Hyers for fractional differential equations is gaining much importance and attention [18,19]. However, the Ulam-Hyers stabilities of differential equations involving with the forward and backward fractional derivatives have not yet been investigated.

In this article, we investigate the following BVP of the fractional differential equation with two different fractional derivatives:

D1βc(D0+αL+λ)u(t)=f(t,u(t)),tJ(0,1],
limt0+t1αu(t)=u0,(I0+1αu)(0)+(I0+qρu)(1)=0,
where α,β,α+β(0,1), λ,ρ,q>0, α+ρ>1. D1βc is the right Caputo fractional derivative of order β, D0+αL is the left Riemann-Liouville fractional derivative of order α, I0+1α is the Riemann-Liouville fractional integral and I0+qρ is the Katugampola fractional integral.

The rest of this article is organized as follows. In Section 2, we collect some concepts of fractional calculus. In Section 3, we prove some properties of classical and generalized Mittag-Leffler functions. In Section 4, we present the definition of solution to (1.1) and (1.2). In Section 5, we obtain the existence and uniqueness of solutions to problem (1.1) and (1.2). In Section 6, we present the Ulam-Hyers stability result for Eq. (1.1). An example is given in Section 7 to demonstrate the application of our result.

2 Preliminaries

In this section, we introduce some notations and definitions of fractional calculus. Throughout this article, we denote by C(J,) the Banach space of all continuous functions from J to , Lp(J,) the Banach space of all Lebesgue measurable functions l:J with the norm lLp=J|l(t)|pdt1p< and AC([a,b],) the space of all the absolutely continuous functions defined on [a,b].

Definition 2.1

[3,4] The left-sided and the right-sided fractional integrals of order γ for a function x(t)L1 are defined as

(Ia+γx)(t)=1Γ(γ)at(ts)γ1x(s)ds,t>a,γ>0,(Ibγx)(t)=1Γ(γ)tb(st)γ1x(s)ds,t<b,γ>0,
respectively. Here, Γ() is the gamma function.

Definition 2.2

[3,4] If x(t)AC([a,b],), then the left-sided and the right-sided Riemann-Liouville fractional derivatives Da+γLx(t) and DbγLx(t) of order γ exist almost everywhere on [a,b] and can be written as

(Da+γLx)(t)=1Γ(1γ)ddtat(ts)γx(s)ds=ddt(Ia+1γx)(t),t>a,0<γ<1,(DbγLx)(t)=1Γ(1γ)ddttb(st)γx(s)ds=ddt(Ib1γx)(t),t<b,0<γ<1,
respectively.

Definition 2.3

[3,4] If x(t)AC([a,b],), then the right-sided Caputo derivative Dbγcx(t) of order γ exists almost everywhere on [a,b] and can be written as

(Dbγcx)(t)=(DbγL[x(s)x(b)])(t),t<b,0<γ<1.

Definition 2.4

[5] For ρ,q>0, then the Katugampola fractional integral of y(t) can be defined as

(Ia+qρy)(t)=ρ1qΓ(q)at(tρτρ)q1τρ1y(τ)dτ,t>a,
if the integral exists.

3 Properties of Mittag-Leffler functions

In this section, we prove some properties of the Mittag-Leffler functions.

Definition 3.1

[3,4] For μ,ν>0,z, the classical Mittag-Leffler functions Eμ(z) and the generalized Mittag-Leffler functions Eμ,ν(z) are defined by

Eμ(z)=k=0zkΓ(μk+1),Eμ,ν(z)=k=0zkΓ(μk+ν).
Clearly, Eμ,1(z)=Eμ(z).

Lemma 3.2

[4,9] Let α(0,1), θ>α be arbitrary. The functions Eα, Eα,α and Eα,θ are nonnegative and have the following properties:

  1. (i) For any tJ, Eα(λtα)1, Eα,α(λtα)1Γ(α), Eα,θ(λtα)1Γ(θ).
  2. (ii) For any t1,t2J,
|Eα(λt2α)Eα(λt1α)|=O(|t2t1|α),ast2t1,|Eα,α(λt2α)Eα,α(λt1α)|=O(|t2t1|α),ast2t1,|Eα,α+1(λt2α)Eα,α+1(λt1α)|=O(|t2t1|α),ast2t1.

Lemma 3.3

[4,9] For γ,μ,ν,λ>0,t>0, 0<α,β<1 , the usual derivatives of Eμ,ν and the fractional integrals and derivatives of Eμ,ν are expressed by

  1. (1) ddtEμ(λtμ)=λtμ1Eμ,μ(λtμ);
  2. (2) I0+γ(sν1Eμ,ν(λsμ))(t)=1Γ(γ)0t(ts)γ1sν1Eμ,ν(λsμ)ds=tγ+ν1Eμ,γ+ν(λtμ);
  3. (3) [Da+νL(sa)β1Eα,β(λ(sa)α)](t)=(ta)βν1Eα,βν(λ(ta)α).

Lemma 3.4

For ν,λ>0,t>0, 0<α,β<1 , the fractional integrals and derivatives of Eμ,ν are expressed by

  1. (1) [Da+αL(sa)α1Eα,α(λ(sa)α)](t)=λ(ta)α1Eα,α(λ(ta)α);
  2. (2) [D1βcD0+αLsαEα,α+1(λsα)](t)+λ[D1βcsαEα,α+1(λsα)](t)=0;
  3. (3) [D1βcD0+αLsα1Eα,α(λsα)](t)+λ[D1βcsα1Eα,α(λsα)](t)=0;
  4. (4) [I0+qρsν1Eα,ν(λsα)](t)=tqρ+ν1ρqΓ(q)01sν1ρ(1s)q1Eα,ν(λtαsαρ)ds, να.

Proof

  1. (1)By using Lemma 3.3, we find
    [Da+αL(sa)α1Eα,α(λ(sa)α)](t)=1Γ(1α)ddtat(ts)α(sa)α1Eα,α(λ(sa)α)ds=ddt[Eα(λ(ta)α)]=λ(ta)α1Eα,α(λ(ta)α).
    Similarly, we have
    [Da+αL(sa)αEα,α+1(λ(sa)α)](t)=Eα(λ(ta)α).
  2. (2)Using (3.1) and (D1βc1)(t)=0, we arrive at
    [D1βcD0+αLsαEα,α+1(λsα)](t)+λ[D1βcsαEα,α+1(λsα)](t)=[D1βc(Eα(λsα)+λsαEα,α+1(λsα))](t)=(D1βc1)(t)=0.
  3. (3)The proof of (3) is similar to that of (2).
  4. (4)Clearly, for να, the integral 0t(1τ)q1τν1ρEα,ν(λtαταρ)dτ exists, then we get
[I0+qρsν1Eα,ν(λsα)](t)=ρ1qΓ(q)0t(tρsρ)q1sρ+ν2Eα,ν(λsα)ds=ρ1qtρq+ν1ρΓ(q)01(1τ)q1τν1ρEα,ν(λtαταρ)dτ=tρq+ν1ρqΓ(q)01(1τ)q1τν1ρEα,ν(λtαταρ)dτ.

4 Solutions for BVP

In this section, we present the formulas of solutions to problem (1.1) and (1.2).

Lemma 4.1

[4] For θ>0 , a general solution of the fractional differential equation D1θcu(t)=0 is given by

u(t)=i=0n1Ci(1t)i,
where Ci,i=0,1,2,,n1(n=[θ]+1) , and [θ] denotes the integer part of the real number θ.

Lemma 4.2

For α,β(0,1), D1βc(D0+αL+λ)u(t)=h(t),tJ , then

u(t)=C0tαEα,α+1(λtα)+C1tα1Eα,α(λtα)+01K(t,τ)h(τ)dτ,tJ,
if the integral exists. Here,
K(t,τ)=1Γ(β)0τ(ts)α1(τs)β1Eα,α(λ(ts)α)ds,0<τ<t1,0t(ts)α1(τs)β1Eα,α(λ(ts)α)ds,0<t<τ1.

Formally, by Lemma 4.1, for C0, we obtain (D0+αL+λ)u(t)=C0+(I1βh)(t). Based on the arguments of [4], we derive

u(t)=C1tα1Eα,α(λtα)+0t(ts)α1Eα,α(λ(ts)α)(C0+(I1βh)(s))ds=C0tαEα,α+1(λtα)+C1tα1Eα,α(λtα)+1Γ(β)0t(ts)α1Eα,α(λ(ts)α)s1(τs)β1h(τ)dτds=C0tαEα,α+1(λtα)+C1tα1Eα,α(λtα)+1Γ(β)0th(τ)dτ0τ(ts)α1(τs)β1Eα,α(λ(ts)α)ds+t1h(τ)dτ0t(ts)α1(τs)β1Eα,α(λ(ts)α)ds=C0tαEα,α+1(λtα)+C1tα1Eα,α(λtα)+01K(t,τ)h(τ)dτ.
We define C1α([0,1],)={uC(J,):t1αu(t)C([0,1],)} with the norm u1α=maxt[0,1]t1α|u(t)| and we abbreviate C1α([0,1],) to C1α.

To prove our results, we make the following assumptions.

  1. (H1)Let f:J× be a function such that f(,u):J is measurable for all u and f(t,): is continuous for a.e. tJ, and there exists a function φL1p(J,+)(0<p<min{α,β}) such that
    |f(t,u(t))f(t,v(t))|φ(t)uv1α.
  2. (H2) suptJ|f(t,0)|<.

For convenience of the following presentation, we set

A(θ,t)=tqρ+θρqΓ(q)01sθρ(1s)q1Eα,θ+1(λtαsαρ)ds;Nθ=Bθρ+1,qρqΓ(q)1Γ(θ+1);M1=1ρqΓ(q+1)Γ(α+1)A(α,1)+1;M2=1β1pαp1p+1α1pβp1pφL1pΓ(α)Γ(β);M3=2Γ(α+1)Γ(β+1),
where B(,) is the Beta function.

For tJ,y>p, applying the Hölder inequality, there hold the following estimates:

at(ts)y1φ(s)dsat(ts)y11pds1pφL1p=1pyp1p(ta)ypφL1p,t>a,
tb(st)y1φ(s)dstb(st)y11pds1pφL1p=1pyp1p(bt)ypφL1p,t<b.

Lemma 4.3

0τ[(t1s)α1Eα,α(λ(t1s)α)(t2s)α1Eα,α(λ(t2s)α)](τs)β1dsO((t2t1)α),0<τ<t1<t21,
0t1(t1s)α1(t2s)α1(τs)β1ds(τt2)β1O((t2t1)α),0<t1<t2<τ1.

Proof

For 0<t1<t21, it follows from Lemma 3.2 and the mean value theorem that

|t1α1Eα,α(λt1α)t2α1Eα,α(λt2α)||t1α1t2α1|Eα,α(λt1α)+|Eα,α(λt1α)Eα,α(λt2α)|t2α1O((t2t1)α),
which yields
0τ[(t1s)α1Eα,α(λ(t1s)α)(t2s)α1Eα,α(λ(t2s)α)](τs)β1ds0τ(τs)β1dsO((t2t1)α)O((t2t1)α),for0<τ<t1<t21.

For 0<t1<t2<τ1, by the mean value theorem, one can see that

0t1[(t1s)α1(t2s)α1](τs)β1ds0t1[(t1s)α1(t2s)α1]ds(τt1)β11α[t2αt1α+(t2t1)α](τt1)β1(τt2)β1O((t2t1)α).

Let us define

k1(t,τ)=1Γ(β)0τ(ts)α1(τs)β1Eα,α(λ(ts)α)ds,0<τ<t1,k2(t,τ)=1Γ(β)0t(ts)α1(τs)β1Eα,α(λ(ts)α)ds,0<t<τ1,
clearly,
01K(t,τ)dτ=0tk1(t,τ)dτ+t1k2(t,τ)dτ,
one can obtain the following estimates.

Lemma 4.4

|k1(t,τ)|(tτ)α1Γ(β+1)Γ(α),0<τ<t1,
|k2(t,τ)|(τt)β1Γ(α+1)Γ(β),0<t<τ1,
01|K(t2,τ)K(t1,τ)|φ(τ)dτ=O((t2t1)αp)+O((t2t1)βp),
01|K(t2,τ)K(t1,τ)|dτ=O((t2t1)α)+O((t2t1)β),
01K(t,τ)φ(τ)dτM2,
01K(t,τ)dτM3.

Proof

Since

0τ(ts)α1(τs)β1ds0τ(tτ)α1(τs)β1ds=τβ(tτ)α1β,0<τ<t1,0t(ts)α1(τs)β1ds0t(ts)α1(τt)β1ds=tα(τt)β1α,0<t<τ1,
(4.5) and (4.6) hold. Moreover, (4.1) and (4.2) imply
t1t2k1(t2,s)φ(s)dst1t2(t2s)α1φ(s)Γ(β+1)Γ(α)ds1pαp1p(t2t1)αpΓ(β+1)Γ(α)φL1p,
t1t2k2(t1,s)φ(s)dst1t2(st1)β1φ(s)Γ(α+1)Γ(β)ds1pβp1p(t2t1)βpΓ(α+1)Γ(β)φL1p.
For 0<τ<t1<t21, by (4.3), we get
|k1(t2,τ)k1(t1,τ)|1Γ(β)0τ[(t2s)α1Eα,α(λ(t2s)α)(t1s)α1Eα,α(λ(t1s)α)](τs)β1ds=O((t2t1)α),
furthermore,
0t1|k1(t2,τ)k1(t1,τ)|φ(τ)dτφL1O((t2t1)α)O((t2t1)α),
0t1|k1(t2,τ)k1(t1,τ)|dτO((t2t1)α).

For 0<t1<t2<τ1, by (4.4) and Lemma 3.2, we find

|k2(t2,τ)k2(t1,τ)|1Γ(β)1Γ(α)0t1[(t1s)α1(t2s)α1](τs)β1ds+0t1(t2s)α1(τs)β1|Eα,α(λ(t1s)α)Eα,α(λ(t2s)α)|ds+1Γ(α)t1t2(t2s)α1(τs)β1ds(τt2)β1O((t2t1)α)+(τt2)β1Γ(α)Γ(β)t1t2(t2s)α1ds(τt2)β1O((t2t1)α),
then by (4.2), we arrive at
t21|k2(t2,τ)k2(t1,τ)|φ(τ)dτt21(τt2)β1φ(τ)dτO((t2t1)α)O((t2t1)α),
t21|k2(t2,τ)k2(t1,τ)|dτO((t2t1)α).

From (4.13), (4.15), (4.11) and (4.12), it follows that

01|K(t2,τ)K(t1,τ)|φ(τ)dτ0t1|k1(t2,s)k1(t1,s)|φ(s)ds+t1t2k1(t2,s)φ(s)ds+t21|k2(t2,s)k2(t1,s)|φ(s)ds+t1t2k2(t1,s)φ(s)ds=O((t2t1)αp)+O((t2t1)βp).
Similarly, one can conclude from (4.14), (4.16), (4.5) and (4.6) that
01|K(t2,τ)K(t1,τ)|dτO((t2t1)α)+O((t2t1)β).
By (4.5), (4.6), (4.1) and (4.2), we observe that
01K(t,τ)φ(τ)dτ=0tk1(t,τ)φ(τ)dτ+t1k2(t,τ)φ(τ)dτM2,01K(t,τ)dτM3,01K(t,τ)|f(τ,0)|dτ01K(t,τ)dτsuptJ|f(t,0)|M3suptJ|f(t,0)|.

For the sake of convenience, we adopt the following notation:

(Fu)(t)=01K(t,τ)f(τ,u(τ))dτ=0tk1(t,τ)f(τ,u(τ))dτ+t1k2(t,τ)f(τ,u(τ))dτ.
Since
|f(t,u(t))||f(t,u(t))f(t,0)|+|f(t,0)|φ(t)u1α+|f(t,0)|,
then by (4.9) and (4.10),
|(Fu)(t))|01K(t,τ)φ(τ)dτu1α+01K(t,τ)|f(τ,0)|dτM2u1α+M3suptJ|f(t,0)|.

Lemma 4.5

Assume that (H1) and (H2) hold. For uC1α, tJ, (Fu)(t) satisfies the following relations:

  1. (1) (Fu)(t)AC(J,);
  2. (2) [D0+αL(Fu)](t)=λ(Fu)(t)+(I1βf)(t);
  3. (3) [D1βcD0+αL(Fu)](t)+λ[D1βc(Fu)](t)=f(t,u(t));
  4. (4) [I0+qρ(Fu)](t)=tqρρqΓ(q)01(1s)q1(Fu)(ts1ρ)ds.

Proof

For every finite collection {(aj,bj)}1jn on J with j=1n(bjaj)0, noting that (4.7) and (4.8), we derive

j=1n|(Fu)(bj)(Fu)(aj)|=j=1n01K(bj,τ)f(τ,u(τ))dτ01K(aj,τ)f(τ,u(τ))dτj=1n01|K(bj,τ)K(aj,τ)|φ(τ)dτu1α+j=1n01|K(bj,τ)K(aj,τ)|dτsuptJ|f(t,0)|0.
Hence, (Fu)(t) is absolutely continuous on J. Furthermore, for almost all tJ, [LD0+α(Fu)(s)](t) exists and from the fact (Fu)(s)=0s(sτ)α1Eα,α(λ(sτ)α)(I1βf)(τ,u(τ))dτ and Lemma 3.3, we get
[D0+αL(Fu)(s)](t)=1Γ(1α)ddt0t0s(ts)α(sτ)α1Eα,α(λ(sτ)α)(I1βf)(τ)dτds=1Γ(1α)ddt0t(I1βf)(τ)τt(ts)α(sτ)α1Eα,α(λ(sτ)α)dsdτ=ddt0t(I1βf)(τ)Eα(λ(tτ)α)dτ=λ(Fu)(t)+(I1βf)(t).
Thus,
[D1βcD0+αL(Fu)(s)](t)+λ[D1βc(Fu)(s)](t)=[D1βc(λ(Fu)(s)+(I1βf)(s))](t)+λ[D1βc(Fu)(s)](t)=f(t,u(t)).

It follows from (4.17) that 01(1s)q1(Fu)(ts1ρ)ds exists and

[I0+qρ(Fu)](t)=ρ1qΓ(q)0t(tρsρ)q1sρ1(Fu)(s)ds=tqρρqΓ(q)01(1s)q1(Fu)(ts1ρ)ds.

Lemma 4.6

Assume that (H1) and (H2) hold. A function u is a solution of the following fractional integral equation:

u(t)=(Pu)(t)+(Qu)(t)
if and only if u is a solution of problem (1.1 ) and ( 1.2), where
(Pu)(t)=u0Γ(α)(1+A(α1,1))A(α,1)tαEα,α+1(λtα)+u0Γ(α)tα1Eα,α(λtα),(Qu)(t)=1ρqΓ(q)01(1τ)q1(Fu)(τ1ρ)dτA(α,1)tαEα,α+1(λtα)+(Fu)(t).

Proof

(Sufficiency) Let u be the solution of (1.1) and (1.2), Lemmas 3.3, 3.4, 4.2 and 4.5 imply

u(t)=atαEα,α+1(λtα)+btα1Eα,α(λtα)+(Fu)(t),tJ,(I0+1αu)(t)=atEα,2(λtα)+bEα(λtα)+[I0+1α(Fu)](t),[I0+qρu](t)=aA(α,t)+bA(α1,t)+tqρρqΓ(q)01(1s)q1(Fu)(ts1ρ)ds,
where a,b are constants. Using the boundary value condition (1.2), we derive that b=u0Γ(α) and
aA(α,1)+u0Γ(α)(1+A(α1,1))+1ρqΓ(q)01(1τ)q1(Fu)(τ1ρ)dτ=0,
which means
a=u0Γ(α)(1+A(α1,1))+1ρqΓ(q)01(1τ)q1(Fu)(τ1ρ)dτA(α,1).
Now we can see that (4.18) holds.

(Necessity) Let u satisfy (4.18). According to Lemma 3.4(2), (3) and Lemma 4.5(3), [D1βcD0+αLu](t) exists and D1βc(D0+αL+λ)u(t)=f(t,u(t)) for tJ. Clearly, the boundary value condition (1.2) holds and hence the necessity is proved.□

5 Existence and uniqueness result

In this section, we deal with the existence and uniqueness of solutions to problem (1.1) and (1.2).

Theorem 5.1

Assume that (H1) and (H2) are satisfied, then problem (1.1 ) and ( 1.2) has a unique solution uC1α if M1M2<1.

Proof

We consider an operator :C1αC1α defined by

(u)(t)=(Pu)(t)+(Qu)(t).
Clearly, is well defined and the fixed point of is the solution of problem (1.1) and (1.2).

Let Br={uC1α:u1αr} be a bounded set in C1α, where

r|u0|α1+Nα1A(α,1)+α+M1M3suptJ|f(t,0)|1M1M2.

For uBr, it follows from Lemmas 3.2(i) and (4.17) that

Pu1α|u0|α1+Nα1A(α,1)+α,Qu1αM1[M2r+M3suptJ|f(t,0)|]
and hence
(u)1αPu1α+Qu1α|u0|α1+Nα1A(α,1)+α+M1[M2r+M3suptJ|f(t,0)|]r.
Then for tJ, uBr, uBr.

For any u,vBr, tJ, by (4.9),

|(Fu)(t)(Fv)(t)|01K(t,τ)|f(τ,u(τ))f(τ,v(τ))|dτ01K(t,τ)φ(τ)dτuv1αM2uv1α,
and thus
|(u)(t)(v)(t)|01(1τ)q1(Fu)(τ1ρ)(Fv)(τ1ρ)dτA(α,1)Γ(α+1)ρqΓ(q)+|(Fu)(t)(Fv)(t)|M1M2uv1α.
Furthermore, uv1α<uv1α. The proof now can be finished by using the Banach contraction mapping principle.□

6 Ulam-Hyers stability

Let ϵ˜ be a positive real number. We consider Eq. (1.1) with inequality

|D1βc(D0+αL+λ)v(t)f(t,v(t))|ϵ˜,tJ.

Definition 6.1

Eq. (1.1) is Ulam-Hyers stable if there exists a real number c>0 such that for each solution v(t) of inequality (6.1) there exists a solution u of Eq. (1.1) with

|v(t)u(t)|cϵ˜,tJ.

Remark 6.2

A function vC1α is a solution of inequality (6.1) if and only if there exists a function gC1α such that (i) |g(t)|ϵ˜; (ii) D1βc(D0+αL+λ)v(t)=f(t,v(t))+g(t).

Let

v˜(t)=u0Γ(α)(1+A(α1,1))A(α,1)tαEα,α+1(λtα)+u0Γ(α)tα1Eα,α(λtα)1ρqΓ(q)01(1τ)q1(Fv)(τ1ρ)dτA(α,1)tαEα,α+1(λtα)+(Fv)(t),
where
(Fv)(t)=01K(t,τ)f(τ,v(τ))dτ.

Remark 6.3

Let vC1α be a solution of inequality (6.1) with limt0+t1αv(t)=u0,(I0+1αv)(0)+(I0+qρv)(1)=0. Then, v is a solution of the inequality |v(t)v˜(t)|M1M3ϵ˜.

Indeed, by Remark 6.2, we have

D1βc(D0+αL+λ)v(t)=f(t,v(t))+g(t),
limt0+t1αv(t)=u0,(I0+1αv)(0)+(I0+qρv)(1)=0.
Applying the same arguments as in the proof of Lemma 4.6, we obtain
v(t)=u0Γ(α)(1+A(α1,1))A(α,1)tαEα,α+1(λtα)+u0Γ(α)tα1Eα,α(λtα)1ρqΓ(q)01(1τ)q1(Gv)(τ1ρ)dτA(α,1)tαEα,α+1(λtα)+(Gv)(t),
where
(Gv)(t)=01K(t,τ)[f(τ,v(τ))+g(τ)]dτ.
Therefore, by (4.10), we conclude that |v(t)v˜(t)|M1M3ϵ˜.

Theorem 6.4

Assume that (H1) and (H2) are satisfied, then Eq. ( 1.1) is Ulam-Hyers stable if M1M2<1.

Proof

Let vC1α be a solution of inequality (6.1) with limt0+t1αv(t)=u0,(I0+1αv)(0)+(I0+qρv)(1)=0. u denotes the unique solution of the following problem:

D1βc(D0+αL+λ)u(t)=f(t,u(t)),tJ,limt0+t1αu(t)=u0,(I0+1αu)(0)+(I0+qρu)(1)=0.
It is easy to check that
|v(t)u(t)||v(t)v˜(t)|+|v˜(t)u(t)|M1M3ϵ˜+01(1τ)q1|(Fv)(τ1ρ)(Fu)(τ1ρ)|dτA(α,1)Γ(α+1)ρqΓ(q)+|(Fv)(t)(Fu)(t)|M1M3ϵ˜+M1M2vu1α,
then
vu1αM1M3ϵ˜1M1M2,
that is, Eq. (1.1) is Ulam-Hyers stable.□

7 Example

As an example, we consider here the following BVP of the fractional differential equation with two different fractional derivatives:

D115c(D0+35L+3)u(t)=130t10sin(t25u(t)+t34),tJ(0,1],
limt0+t25u(t)=u0,(I0+25u)(0)+(I0+32u)(1)=0.
Corresponding to (1.1) and (1.2), we recognize that α=35, β=15, λ=3, ρ=2, q=3 and
f(t,u(t))=130t10sin(t25u(t)+t34).
The space C1α={uC(J,):t25u(t)C([0,1],)} with the norm u25=maxt[0,1]t25|u(t)|.

It is not difficult to obtain

|f(t,u(t))f(t,v(t))|φ(t)uv25,
where φ(t)=130t10L1p[0,1](p=320) and φL1p=332030. Moreover,
|f(t,0)|=130t10|sint34|130,
and thus suptJ|f(t,0)|130.

By direct computation, we find

A35,1=18Γ(3)01s310(1s)2E35,85(3s310)ds0.005,M1=1ρqΓ(q+1)Γ(α+1)A(α,1)+1=18Γ(4)Γ85A35,1+15.6;M2=1β1pαp1p+1α1pβp1pφL1pΓ(α)Γ(β)=51791720+53(17)1720332030Γ35Γ150.16,
consequently, M1M20.9<1, by Theorems 5.1 and 6.4, problem (7.1) and (7.2) has a unique solution and Eq. (7.1) is Ulam-Hyers stable.

References

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    Kenneth S. Miller and Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

  • [2]

    Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

  • [3]

    Igor Podlubny, Fractional Differential Equations, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

  • [4]

    Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.

  • [5]

    Udita N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1–15.

  • [6]

    Fang Li, Jin Liang, and Hong-Kun Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), no. 2, 510–525, .

    • Crossref
    • Export Citation
  • [7]

    Jin Liang, Yunyi Mu, and Ti-Jun Xiao, Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, Banach J. Math. Anal. 13 (2019), no. 4, 745–768, .

    • Crossref
    • Export Citation
  • [8]

    Jin Liang, James H. Liu, and Ti-Jun Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 3, 475–485, .

    • Crossref
    • Export Citation
  • [9]

    Yunsong Miao and Fang Li, Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients, Adv. Difference Equ. 2017 (2017), 190, .

    • Crossref
    • Export Citation
  • [10]

    Om P. Agrawal, Fractional variational calculus and transversality condition, J. Phys. A: Math. Gen. 39 (2006), no. 33, 10375–10384, .

    • Crossref
    • Export Citation
  • [11]

    Teodor M. Atanackovic and Bogoljub Stankovic, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal. 10 (2007), no. 2, 139–150.

  • [12]

    Tomasz Blaszczyk and Mariusz Ciesielski, Fractional oscillator equation – transformation into integral equation and numerical solution, Appl. Math. Comput. 257 (2015), 428–435, .

    • Crossref
    • Export Citation
  • [13]

    Assia Guezane-Lakoud, Rabah Khaldi, and Adem Kiliçman, Solvability of a boundary value problem at resonance, SpringerPlus 5 (2016), 1504, .

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Rabah Khaldi and Assia Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. 2017 (2017), 30, .

    • Crossref
    • Export Citation
  • [15]

    Stanisław M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.

  • [16]

    Donald H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), no. 4, 222–224, .

    • Crossref
    • Export Citation
  • [17]

    Donald H. Hyers, George Isac, and Themistocles M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.

  • [18]

    Wei Wei, Xuezhu Li, and Xia Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), no. 10, 3468–3476, .

    • Crossref
    • Export Citation
  • [19]

    JinRong Wang and Xuezhu Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput. 258 (2015), 72–83, .

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Kenneth S. Miller and Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

  • [2]

    Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

  • [3]

    Igor Podlubny, Fractional Differential Equations, vol. 198, Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

  • [4]

    Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.

  • [5]

    Udita N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), no. 4, 1–15.

  • [6]

    Fang Li, Jin Liang, and Hong-Kun Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), no. 2, 510–525, .

    • Crossref
    • Export Citation
  • [7]

    Jin Liang, Yunyi Mu, and Ti-Jun Xiao, Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, Banach J. Math. Anal. 13 (2019), no. 4, 745–768, .

    • Crossref
    • Export Citation
  • [8]

    Jin Liang, James H. Liu, and Ti-Jun Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S 10 (2017), no. 3, 475–485, .

    • Crossref
    • Export Citation
  • [9]

    Yunsong Miao and Fang Li, Boundary value problems of the nonlinear multiple base points impulsive fractional differential equations with constant coefficients, Adv. Difference Equ. 2017 (2017), 190, .

    • Crossref
    • Export Citation
  • [10]

    Om P. Agrawal, Fractional variational calculus and transversality condition, J. Phys. A: Math. Gen. 39 (2006), no. 33, 10375–10384, .

    • Crossref
    • Export Citation
  • [11]

    Teodor M. Atanackovic and Bogoljub Stankovic, On a differential equation with left and right fractional derivatives, Fract. Calc. Appl. Anal. 10 (2007), no. 2, 139–150.

  • [12]

    Tomasz Blaszczyk and Mariusz Ciesielski, Fractional oscillator equation – transformation into integral equation and numerical solution, Appl. Math. Comput. 257 (2015), 428–435, .

    • Crossref
    • Export Citation
  • [13]

    Assia Guezane-Lakoud, Rabah Khaldi, and Adem Kiliçman, Solvability of a boundary value problem at resonance, SpringerPlus 5 (2016), 1504, .

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Rabah Khaldi and Assia Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. 2017 (2017), 30, .

    • Crossref
    • Export Citation
  • [15]

    Stanisław M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1960.

  • [16]

    Donald H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), no. 4, 222–224, .

    • Crossref
    • Export Citation
  • [17]

    Donald H. Hyers, George Isac, and Themistocles M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.

  • [18]

    Wei Wei, Xuezhu Li, and Xia Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), no. 10, 3468–3476, .

    • Crossref
    • Export Citation
  • [19]

    JinRong Wang and Xuezhu Li, Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput. 258 (2015), 72–83, .

    • Crossref
    • Export Citation
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