
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12569-8
10.1016/j.heliyon.2024.e36538
e36538
Research Article
Boundary problems of sequential fractional differential equations having a monomial coefficient
Yan Debao yangdebao@hezeu.edu.cn

School of Mathematics and Statistics, Heze University, Heze City, Shandong Province, 274000, PR China
22 8 2024
15 9 2024
22 8 2024
10 17 e365384 6 2024
8 7 2024
19 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

MSC

34A08
34B10
34B15
34D20
Keywords

Sequential fractional differential equation
Monomial coefficient
Boundary value problem
Existence of solutions
Stability of Ulam-Hyers
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pmc1 Introduction

A substantial topic, fractional calculus(FC), is researched in academia and technology-oriented committees since the non-local nature of fractional derivatives plays a critical role in several implementations. FC is the generalization of the differentiation and integration to every non-integer order. Over the decades, FC has been an important tool for mathematicians and experts in various fields to improve effectively the mathematical models. At the moment, the differential operators with fractional order have been employed to widely characterize several processes' memory, hereditary features, and occurrences in various disciplines such as control theory, rheology, engineering, biophysics, signal and image processing, etc. [1], [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].

Though fractional order differential equations (FODEs) are implemented to account for several phenomena in science and engineering fields, as with conventional integer-order counterparts, the solutions to these equations cannot be precisely discovered in many cases. So, its qualitative theory and properties such as stability and asymptotics of solutions are substantial to explore. More references regarding those investigations can be found in [3], [14], [15], [16], [17], [18], [20], [21], [22], [23], [24], [30] and the references therein.

The sequential fractional derivatives were introduced [19]. Since fractional derivatives' non-local features, the boundary value (BV) and initial value (IV) problems of sequential fractional-order differential equations (SFODEs) have gained a vast amount of attention, and their stability theory requires further investigation. Thus, a stability called the Ulam-Hyers type creates quite interest. Some significant outcomes have been reached regarding these equations in [25], [26], [27], [28], [29], [31], [32], [33], [34], [37], [38], [39], [40], [42], [43], [44]. Some milestone outcomes are presented below.

SFODE's 3-point BV problem was studied in [31] and was defined byDαC(D+λ)x(t)=f(t,x(t)),0<t<1,1<α≤2,

x(0)=x′(0)=0,x(1)=βx(η),0<η<1,

where DαC represents Caputo's differentiation whose order is α, D denotes the usual differentiation, f:[0,1]×R→R is continuous, λ and β denotes R+ and R such thatβ≠λ+e−λ−1λη+e−λη−1.

Banach's contraction mapping principle (CMP) and Krasnoselskii's fixed-point theorem (FPT) give rise to the existence conditions.

SFODE's IV problems based on the Riemann-Liouville sequential fractional derivative were investigated by [37] and were delineated by(D0+2αy)(x)=f(x,y,D0+αy),x∈(0,T],

x1−αy(x)|x=0=y0,x1−α(D0+αy)(x)|x=0=y1.

Where D0+2α,D0+α denote Riemann-Liouville fractional derivatives whose orders are 2α and α, respectively. The existence of extreme solutions is researched by combining two methods called upper and lower solutions and their associated monotone iteration.

The BV problem of the Hilfer type fractional integral differential equations was researched in [38]:(Dα,βH+kHDα−1,β)x(t)=f(t,x(t),Iδx(t)),t∈[a,b],

x(a)=0,x(b)=∑i=1m−2ξiIϕix(θi).

Where Dα,βH shows the Hilfer fractional derivative operator whose orders are α(1<α<2) and β(0<β<1), k∈R, IΨ represents the Riemann-Liouville fractional integral whose order is Ψ>0,Ψ∈{ϕi,δ}. The BV problem's existence and uniqueness conditions were attained by employing Banach's FPT, nonlinear forms of the Leray-Schauder sort fractional derivative, and Krasnoselskii's FPT.

SFODE's existence and Ulam stability solutions of an IV problem were examined by [39] and were denoted by{(Da+αC+λ1CDa+α−1+λ2CDa+α−2)x(t)=f(t,x(t)),t∈J,α∈(2,3),x(k)(a)=bk,k=0,1,2,

where J=[a,T],T>a≥0,λ1,λ2 are nonzero constants, f:J×R⟶R presents a continuous mapping.

Most of the FODEs are presented with only constant coefficients. However, when those coefficients are considered variable, there is a limit to the amount of research results. In recent times, some remarkable work have been done on BV problems of FODEs with constant and variable coefficients (see [45], [46], [47], [48]).

This research aims to analyze SFODE's BV problems with variable coefficients. The generic form is represented by(Q) D0+θCx(t)+λCD0+θ−1[tx(t)]=f(t,x(t)),0<t<1

with boundary conditions(BCs) x(0)=x′(1)=0,x′(0)=a

where 2<θ≤3,t∈[0,1], λ>0,a∈R, f is a continuous mapping in [0,1].

SFODE's existence and uniqueness conditions regarding the (Q)-(BCs) are proved by implementing both the CMP and Krasnoselskii's FPT. Furthermore, the stability condition of the Ulam-Hyers was discussed in the article. The main novelty of this paper is that we consider a class of SFODEs with monomial coefficients, rather than general FODEs. Therefore, this work is a supplement and contribution to the theory of fractional calculus.

The rest of the article is structured as follows. Some necessary notations and preliminaries on FC are contained in section 2. The subsequent two sections are devoted to presenting the main outcomes, namely, the existence of solutions in section 3, as well as the solution's stability called Ulam-Hyers in section 4, for the considered BV problem of the (Q)-(BCs). An example is given in section 5, substantiating the theoretical results, followed by conclusions in section 6.

2 Preliminary

Several relevant notions, lemmas, and the fractional calculus's features are presented [1], [2].

Definition 2.1 Let set be [a1,a2](−∞<a1<a2<+∞)⊂R. The fractional integral called Riemann-Liouville with order ξ∈C(ℜ(ξ)>0) can be represented by(Ia1+ξf)(x):=1Γ(ξ)∫a1xf(t)(x−t)1−ξdt(x>a1;ℜ(ξ)>0)

and(Ia2−ξf)(x):=1Γ(ξ)∫xa2f(t)(t−x)1−ξdt(x<a2;ℜ(ξ)>0)

provided the above integrals exist, where Γ(⋅) represents the Gamma function.

Definition 2.2 If a=a(x)∈ACn[a1,a2], Caputo's derivatives (Da1+ξca)(x) and (Da2−ξca)(x) can be computed almost on the interval [a1,a2].

(a) When ξ∉N0, (Da1+ξca)(x) and (Da2−ξca)(x) are described as(Da1+ξca)(x)=1Γ(n−ξ)∫a1xa(n)(t)(x−t)ξ−n+1dt

and(Da2−ξca)(x)=(−1)nΓ(n−ξ)∫xa2a(n)(t)(t−x)ξ−n+1dt,

respectively, n=[ℜ(ξ)]+1, ξ∈C, ℜ(ξ)≥0.

(b) If ξ∈N0, then(Da1+nca)(x)=a(n)(x)

and(Da2−nca)(x)=(−1)(n)a(n)(x).

Definition 2.3 ([19]) For a sufficiently smooth function b=b(x), the sequential fractional derivative is defined asDιb(x)=Dι1Dι2⋯Dιkb(x),

where ι=(ι1,ι2,⋯,ιk) is a multi-index.

Lemma 2.1 The generic solution of the fractional-order equation (Dς+ξca)(x)=0,n−1<ξ<n , can be given by a(x)=∑j=0n−1a(j)(ς)j!(x−ς)j.

Especially, for ς=0 , the outcome can be denoted by a(x)=d0+d1x+d2x2+⋯+dn−1xn−1,

where dj=a(j)(0)j!(j=0,1,⋯n−1) stand for certain constants.

To write conveniently in the sequel, we introduce a function erfi(x), which is an imaginary error function defined byerfi(x)=2π∫0xet2dt,x>0.

Note that erfi(x) is nondecreasing function in [0,+∞).

Lemma 2.2 For x>0,λ>0 , the subsequent equality holds, ∫0xeλ2t2dt=π2λerfi(λ2t).

Lemma 2.3 Suppose that η(t)∈C[0,1] represents a certain mapping, then the unique solution of the equation (2.1) D0+θCx(t)+λCD0+θ−1[tx(t)]=η(t),0<t<1

subject to the boundary conditions (BCs) is defined by x(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2η(τ)dτds+(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2η(τ)dτds

(2.2) −eλ2λΓ(θ−1)∫01(1−τ)θ−2η(τ)dτ+c]+de−λ2t2erfi(λ2t)

where 2<θ≤3,t∈[0,1],λ>0,a∈R , and c=abλ,d=aπ2λ,b=λ2πerfi(λ2)−eλ2 .

Proof Implementing a fractional integral operator of order θ−1 on both sides of eq. (2.1), Lemma 2.1 gives rise to(2.3) x′+λ(tx)=1Γ(θ−1)∫0t(t−τ)θ−2η(τ)dτ+c0+c1t

Multiplying eq. (2.3)'s both sides by eλ2t2, Eq. (2.4) is obtained.(2.4) [eλ2t2x]′=eλ2t2Γ(θ−1)∫0t(t−τ)θ−2η(τ)dτ+c0eλ2t2+c1eλ2t2t

Integrating on both sides of equation (2.4) from 0 to t gives rise to(2.5) eλ2t2x(t)−x(0)=1Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2η(τ)dτds+c0∫0teλ2s2ds+c1∫0tseλ2s2ds

By calculating easily, one can get(2.6) ∫0tseλ2s2ds=1λ(eλ2t2−1)

Eq. (2.6) and Lemma 2.2 are used with the condition x(0)=0 in (BCs), Eq. (2.5) gives rise to(2.7) x(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2η(τ)dτds+c0π2λe−λ2t2erfi(λ2t)+c1λ(1−e−λ2t2)

Differentiating and rearranging Eq. (2.7) leads tox′(t)=−λte−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2η(τ)dτds+1Γ(θ−1)∫0t(t−τ)θ−2η(τ)dτ

(2.8) +c0[1−λ2πte−λ2t2erfi(λ2t)]+c1te−λ2t2

Utilizing x′(0)=a,x′(1)=0 in (BCs), we have from (2.8) c0=a, and−λe−λ2Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2η(τ)dτds+1Γ(θ−1)∫01(1−τ)θ−2η(τ)dτ

(2.9) +c0[1−λ2πe−λ2erfi(λ2)]+c1e−λ2=0

Let b=λ2πerfi(λ2)−eλ2, thenc1=λΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2η(τ)dτds−eλ2Γ(θ−1)∫01(1−τ)θ−2η(τ)dτ+ab.

Plugging the expressions of c0 and c1 into eq. (2.7) leads tox(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2η(τ)dτds+aπ2λe−λ2t2erfi(λ2t)

(2.10) +1−e−λ2t2λ[λΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2η(τ)dτds−eλ2Γ(θ−1)∫01(1−τ)θ−2η(τ)dτ+ab]

Let c=abλ,d=aπ2λ, Lemma 2.3 is proven. □

The conclusions in our work are mainly based on the utilization of both the contraction mapping principle and Krasnoselskii's fixed point theorem.

Lemma 2.4 ( [35] ) (Krasnoselskii's FPT) Suppose that E is a closed, bounded, convex and non-empty subset of a Banach space B . Let Φ1 , Φ2 be two operators and meet the following conditions (1) Φ1x1+Φ2x2∈E , for x1,x2∈E ; (2) Φ1 is compact and continuous; (3) Φ2 represents a contraction mapping. Then at least one point x∈E exists such that x=Φ1x+Φ2x .

Some notions in Lγ space are shown and Hölder's inequality is introduced [36].

Suppose that an open (or measurable) set represented by D⊂Rn and a mapping with measurability property delineated on D is represented by m(x). |m(x)|γ is measurable on D for 1≤γ<∞, ∫D|m(x)|γdx exists. Then, a function space Lγ(D) is introduced as followsLγ(D)={m(x)|m(x) is measurable on D, ∫D|m(x)|γdx<∞}.

For m∈Lγ(D), the subsequent norm is delineated∥m∥γ=(∫D|m(x)|γdx)1/γ.

1<γ1, γ2<∞ are called conjugate exponentials of each other if 1γ1+1γ2=1.

Lemma 2.5 ( [36] ) (Hölder's inequality) Suppose that D⊂Rn denotes an open set, γ1,γ2 represent conjugate exponentials, m(x)∈Lγ1(D),n(x)∈Lγ2(D) , the m(x)n(x) is integrable on D, then the subsequent equality holds ∫D|m(x)n(x)|dx≤∥m∥γ1∥n∥γ2.

3 Outcomes of existence of solutions to (Q)-(BCs)

Suppose that C=C([0,1],R) represents the Banach space of all continuous mappings from [0,1]→R and equipped with a norm described as ∥x∥=maxt∈[0,1]|x(t)| for x=x(t)∈C.

In view of Lemma 2.3, the problem (Q)-(BCs) can be equivalently converted into subsequent representation,x=Ϝ(x),

where Ϝ:C→C is defined as(Ϝx)(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds+(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds

−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,x(τ))dτ+c]+d⋅e−λ2t2erfi(λ2t).

So, the BV problem of the (Q)-(BCs) has solutions iff the operator Ϝ attains constant points.

For the mapping f(t,x(t)), appearing in eq.(Q), the subsequent condition is assumed.

(C1) f(t,x(t)):[0,1]→R is a jointly continuous mapping, and a mapping χ(t)∈Lγ([0,1],R+)(γ>1) exists such that|f(t,x1(t))−f(t,x2(t))|≤χ(t)|x1−x2|,

for any x1,x2∈C and t∈[0,1].

For χ(t)∈Lγ([0,1],R+)(γ>1) and the parameters used in the research, the subsequent conditions are met:

(C2) 2<θ≤3,γ1>1,γ2>1,λ>0 such thatλΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2>0.

The solution's uniqueness to the BV problem of the (Q)-(BCs) is a concern.

Theorem 3.1 Suppose that the requirements (C1) and (C2) hold, and 1γ1+1γ2=1 , then the BV problem of the (Q)-(BCs) attains a unique solution if (3.1) [2λπerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1<1

Proof Suppose that N0=maxt∈[0,1]|f(t,0)| and Bρ={x∈C:∥x∥≤ρ}, ρ satisfiesρ≥λΓ(θ−1)1+γ1(θ−2)γ1{[2λπerfi(λ2)+eλ2]N0+Γ(θ)[|d|⋅erfi(λ2)+|c|]}Γ(θ){λΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2}.

In the first step, we shall show ϜBρ⊆Bρ. Since λ>0, so one can easily conclude 0<maxt∈[0,1]{e−λ2t2,1−e−λ2t2}≤1. For x∈Bρ, we have∥(Ϝx)(t)∥≤1Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2(|f(τ,x(τ))−f(τ,0)|+|f(τ,0)|)dτds+1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2(|f(τ,x(τ))−f(τ,0)|+|f(τ,0)|)dτds+eλ2λΓ(θ−1)∫01(1−τ)θ−2(|f(τ,x(τ))−f(τ,0)|+|f(τ,0)|)dτ+|d|⋅erfi(λ2)+|c|.

By the condition (C1), the subsequent inequality is attained,∥(Ϝx)(t)∥≤|x(τ)|Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2χ(τ)dτds+N0Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2dτds+|x(τ)|Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2χ(τ)dτds+N0Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2dτds+eλ2|x(τ)|λΓ(θ−1)∫01(1−τ)θ−2χ(τ)dτ+eλ2N0λΓ(θ−1)∫01(1−τ)θ−2dτ+|d|⋅erfi(λ2)+|c|≤2ρΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2χ(τ)dτds+2N0Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2dτds+eλ2ρλΓ(θ−1)∫01(1−τ)θ−2χ(τ)dτ+eλ2N0λΓ(θ)+|d|⋅erfi(λ2)+|c|.

By Lemma 2.5 and Lemma 2.2, the subsequent inequality is attained,∥(Ϝx)(t)∥≤2ρΓ(θ−1)∫01eλ2s2[∫0s(s−τ)(θ−2)γ1dτ]1γ1[∫0sχγ2(τ)dτ]1γ2ds+2N0Γ(θ)∫01eλ2s2sθ−1ds+eλ2ρλΓ(θ−1)[∫01(1−τ)(θ−2)γ1dτ]1γ1[∫01χγ2(τ)dτ]1γ2ds+eλ2N0λΓ(θ)+|d|⋅erfi(λ2)+|c|≤2ρΓ(θ−1)∫01eλ2s2[∫01(1−τ)(θ−2)γ1dτ]1γ1[∫01χγ2(τ)dτ]1γ2ds+2N0Γ(θ)∫01eλ2s2ds+eλ2ρλΓ(θ−1)[∫01(1−τ)(θ−2)γ1dτ]1γ1[∫01χγ2(τ)dτ]1γ2ds+eλ2N0λΓ(θ)+|d|⋅erfi(λ2)+|c|≤2ρ∥χ∥γ2Γ(θ−1)1+γ1(θ−2)γ1∫01eλ2s2ds+2N0Γ(θ)∫01eλ2s2ds+eλ2ρ∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1+eλ2N0λΓ(θ)+|d|⋅erfi(λ2)+|c|≤2ρ∥χ∥γ2Γ(θ−1)1+γ1(θ−2)γ1π2λerfi(λ2)+2N0Γ(θ)π2λerfi(λ2)+eλ2ρ∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1+eλ2N0λΓ(θ)+|d|⋅erfi(λ2)+|c|=[2λπerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1ρ+[2λπerfi(λ2)+eλ2]λΓ(θ)N0+|d|⋅erfi(λ2)+|c|.

The selected criterion for ρ leads us to∥(Ϝx)(t)∥≤ρ.

So, ϜBρ⊆Bρ for any x∈Bρ.

To prove the contraction of the operator Ϝ on Bρ, for any x,y∈Bρ and t∈[0,1], by the condition (C1) the subsequent inequality is attained,∥(Ϝx)(t)−(Ϝy)(t)∥≤1Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2|f(τ,x(τ))−f(τ,y(τ))|dτds+1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2|f(τ,x(τ))−f(τ,y(τ))|dτds+eλ2λΓ(θ−1)∫01(1−τ)θ−2|f(τ,x(τ))−f(τ,y(τ))|dτ≤2Γ(θ−1)∫01eλ2s2∫01(1−τ)θ−2χ(τ)|x(τ)−y(τ)|dτds+eλ2λΓ(θ−1)∫01(1−τ)θ−2χ(τ)|x(τ)−y(τ)|dτ.

By using Lemma 2.5 and Lemma 2.2, we have∥(Ϝx)(t)−(Ϝy)(t)∥≤{2Γ(θ−1)∫01eλ2s2[∫01(1−τ)(θ−2)γ1dτ]1γ1[∫01χγ2(τ)dτ]1γ2ds+eλ2λΓ(θ−1)[∫01(1−τ)(θ−2)γ1dτ]1γ1[∫01χγ2(τ)dτ]1γ2ds}∥x−y∥=[2λπerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1∥x−y∥.

So Eq. (3.1) ensures Ϝ that contradicts. Therefore x=Ϝx contains a unique constant point by the CMP. The BV problem of the (Q)-(BCs) equivalently attains a unique solution x=x(t)∈Bρ. The proof is completed. □

The solution's existence to the BV problems of the (Q)-(BCs) is investigated.

Theorem 3.2 Assume that conditions (C1) and (C2) hold, and 1γ1+1γ2=1 , then at least one solution in [0,1] exists regarding the BV problem of the (Q)-(BCs) if (3.2) [λ2πerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1<1

Proof Take a constant ρ>0 and a ball Bρ as in the proof of Theorem 3.1. Two operators Ϝ1,Ϝ2 on Bρ are delineated by(Ϝ1x)(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds,(Ϝ2x)(t)=1−e−λ2t2Γ(θ−1)[∫01eλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds−eλ2λ∫01(1−τ)θ−2f(τ,x(τ))dτ]+c(1−e−λ2t2)+d⋅e−λ2t2erfi(λ2t).

For any x,y∈Bρ, to prove Theorem 3.2, the same procedure used in proof of Theorem 3.1 is employed again. Then,∥Ϝ1x+Ϝ2y∥≤ρ.

Based on Eq. (3.2), one can prove the contraction for Ϝ2 by taking the same process to proving the contraction for Ϝ in Theorem 3.1. The continuous property of Ϝ1 is derived directly from the continuous function f. Also, Ϝ1 is uniformly bounded on Bρ delineated by∥(Ϝ1x)(t)∥≤1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2|f(τ,x(τ))|dτds≤1Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2(χ(τ)|x(τ)|+N0)dτds≤ρ∥χ∥γ2Γ(θ−1)1+γ1(θ−2)γ1∫01eλ2s2ds+N0Γ(θ)∫01eλ2s2ds=[ρ∥χ∥γ2Γ(θ−1)1+γ1(θ−2)γ1+N0Γ(θ)]π2λerfi(λ2).

To show the compactness of Ϝ1, The subsequent equality is attained by differentiating Ϝ1,(Ϝ1x)′(t)=−λte−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds+1Γ(θ−1)∫0t(t−τ)θ−2f(τ,x(τ))dτ.

Then the following relation holds|(Ϝ1x)′(t)|≤λΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2|f(τ,x(τ))dτds+1Γ(θ−1)∫01(1−τ)θ−2|f(τ,x(τ))|dτ≤λΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2(|f(τ,x(τ))−f(τ,0)|+|f(τ,0)|)dτds+1Γ(θ−1)∫01(1−τ)θ−2(|f(τ,x(τ))−f(τ,0)|+|f(τ,0)|)dτ≤[ρ∥χ∥γ2Γ(θ−1)1+γ1(θ−2)γ1+N0Γ(θ)][1+λ2πerfi(λ2)]:=M.

So, for t1,t2∈[0,1],t1<t2 and x∈Bρ, the subsequent equality is derived,∥(Ϝ1x)(t2)−(Ϝ1x)(t1)∥=∥∫t1t2(Ϝ1x)′(t)dt∥≤∫t1t2|(Ϝ1x)′(t)dt|≤M(t2−t1).

Obviously, ∥(Ϝ1x)(t2)−(Ϝ1x)(t1)∥→0 independent of x as t2→t1. Therefore, Ϝ1 is relatively compact on Bρ. The theorem called Arzelá-Ascoli implies that Ϝ1 is compact on Bρ. Lemma 2.4 implies that Ϝ=Ϝ1+Ϝ2 attaining at least one constant point, this ensures the BV problem of the (Q)-(BCs) having at least one solution in [0,1]. □

Remark 3.1 If the Eq. (3.2) in Theorem (3.2) is replaced byπ2λerfi(λ2)Γ(θ−1)1+γ1(θ−2)γ1∥χ∥γ2<1,

and by proving Ϝ1 a contraction, Ϝ2 is continuous and compact. Thus, the conclusion of Theorem 3.2 still holds.

4 The analysis of Ulam-Hyers stability

The stability of Ulam-Hyers for BV problem of the (Q)-(BCs) is analyzed in this subsection. For the related information on this type of stability, a comprehensive reference to look at is in [41] and the references therein.

Firstly, we construct an inequality for some function y=y(t)∈C3([0,1],R) and any ε>0 as follows(4.1) |CD0+αy(t)+CD0+α−1[ty(t)]−f(t,y(t))|≤ε,t∈[0,1]

Remark 4.1 A mapping y=y(t)∈C3([0,1],R) presents a solution of Eq. (4.1) iff a function z=z(t)∈C([0,1],R) (which is dependent on y) exists such that(1) D0+αCy(t)+CD0+α−1[ty(t)]=f(t,y(t))+z(t),t∈[0,1];

(2) |z(t)|≤ε,t∈[0,1].

Therefore, we have a BV problem for Eq. (4.1) as follows:(Q)′ D0+αCy(t)+CD0+α−1[ty(t)]=f(t,y(t))+z(t)

with boundary conditions(BCs)′ y(0)=y′(1)=0,y′(0)=a

Definition 4.1 The BV problem of the (Q)-(BCs) is called Ulam-Hyers stability if one constant Mf>0 exists such that for any ε>0 and for every solution y=y(t)∈C3[0,1] of BV problem (Q)′−(BCs)′, a solution x=x(t)∈C3[0,1] of the (Q)-(BCs) exists with|y(t)−x(t)|≤Mfε.

Theorem 4.1 For the conditions in Theorem 3.1 , the BV problem of the (Q)-(BCs) is Ulam-Hyers stable.

Proof By Lemma 2.3, the solution of (Q)′−(BCs)′ can be transformed into an equivalently subsequent representation,y(t)=e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2[f(τ,y(τ))+z(τ)]dτds+(1−e−λ2t2){1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2[f(τ,y(τ))+z(τ)]dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2[f(τ,y(τ))+z(τ)]dτ+c}+d⋅e−λ2t2erfi(λ2t).

So by Remark 4.1, the subsequent expression is attained,|y(t)−e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,y(τ))dτ+c]−d⋅e−λ2t2erfi(λ2t)|≤e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2|z(τ)|dτds+1−e−λ2t2Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2|z(τ)|dτds+eλ2(1−e−λ2t2)λΓ(θ−1)∫01(1−τ)θ−2|z(τ)|dτ≤2εΓ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2dτds+eλ2ελΓ(θ−1)∫01(1−τ)θ−2dτ=2εΓ(θ)∫01eλ2s2sθ−1ds+eλ2ελΓ(θ)≤2εΓ(θ)∫01eλ2s2ds+eλ2ελΓ(θ)=2λπerfi(λ2)+eλ2λΓ(θ)ε.

Now, x=x(t),y=y(t)∈C3([0,1],R) implies the following equation,|y(t)−x(t)|=|y(t)−e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds−(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,x(τ))dτ+c]−d⋅e−λ2t2erfi(λ2t)|

=|y(t)−e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,y(τ))dτ+c]−d⋅e−λ2t2erfi(λ2t)+e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds+(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,y(τ))dτ]−e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds−(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,x(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,x(τ))dτ]|≤|y(t)−e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−(1−e−λ2t2)[1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2f(τ,y(τ))dτds−eλ2λΓ(θ−1)∫01(1−τ)θ−2f(τ,y(τ))dτ+c]−d⋅e−λ2t2erfi(λ2t)|+e−λ2t2Γ(θ−1)∫0teλ2s2∫0s(s−τ)θ−2|f(τ,y(τ))−f(τ,x(τ))|dτds+1−e−λ2t2Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2|f(τ,y(τ))−f(τ,x(τ))|dτds+(1−e−λ2t2)eλ2λΓ(θ−1)∫01(1−τ)θ−2|f(τ,y(τ))−f(τ,x(τ))|ds≤2λπerfi(λ2)+eλ2λΓ(θ)ε+1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2χ(τ)|y(τ)−x(τ)|dτds+1Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2χ(τ)|y(τ)−x(τ)|dτds+eλ2λΓ(θ−1)∫01(1−τ)θ−2χ(τ)|y(τ)−x(τ)|ds.

The calculation of the norm on both sides of the above equation leads to∥y−x∥≤2λπerfi(λ2)+eλ2λΓ(θ)ε+2∥y−x∥Γ(θ−1)∫01eλ2s2∫0s(s−τ)θ−2χ(τ)dτds+eλ2∥y−x∥λΓ(θ−1)∫01(1−τ)θ−2χ(τ)ds≤2λπerfi(λ2)+eλ2λΓ(θ)ε+[2λπerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1∥y−x∥.

Transferring and arranging the expression above leads to∥y−x∥≤Γ(θ−1)1+γ1(θ−2)γ1[2λπerfi(λ2)+eλ2]Γ(θ){λΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2}ε.

Taking Mf=Γ(θ−1)1+γ1(θ−2)γ1[2λπerfi(λ2)+eλ2]Γ(θ){λΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2}, the subsequent inequality is attained,|y(t)−x(t)|≤∥y−x∥≤Mfε.

Thus, the BV problem of (Q)-(BCs) is Ulam- Hyers stable. The proof is finished. □

5 An illustration

An illustration is provided.

Example 5.1 The fractional BV problem is assumed.(5.1) {D0+2.89Cx(t)+0.5CD0+1.89[tx(t)]=1(1+t)18arctan[sinx(t)],0<t<1,x(0)=x′(1)=0,x′(0)=1.

where f(t,x(t))=1(1+t)18arctan[sinx(t)], θ=2.89, λ=0.5, a=1. fx′(t,x(t))=1(1+t)18cosx(t)1+sin2x(t). So for any x,y∈C3([0,1],R), the Lagrange's mean value theorem ensures that there exists a function σ=σ(t) whose value is between x(t) and y(t), such that|f(t,x)−f(t,y)|=1(1+t)18|arctan[sinx(t)]−arctan[siny(t)]|=1(1+t)18|cosσ(t)|1+sin2σ(t)|x−y|≤1(1+t)18|x−y|.

Then χ(t)=1(1+t)18∈Lγ([0,1],R+) for any γ>1.

Then e is assigned to 2.71828, and γ1=10,γ2=109,π=3.1415926,Γ(θ)=Γ(2.89)=1.8114,Γ(θ−1)=Γ(1.89)=0.9584, we have2λπ=1.7725,eλ2=1.2840,∥χ∥γ2=0.07065,1+γ1(θ−2)γ1=1.25766,erfi(λ2)=0.61496.

λΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2=0.451925>0.

So the conditions (C1) and (C2) are satisfied. Then[2λπerfi(λ2)+eλ2]∥χ∥γ2λΓ(θ−1)1+γ1(θ−2)γ1=0.2707<1,

andMf=Γ(θ−1)1+γ1(θ−2)γ1[2λπerfi(λ2)+eλ2]Γ(θ){λΓ(θ−1)1+γ1(θ−2)γ1−[2λπerfi(λ2)+eλ2]∥χ∥γ2}=3.4956.

are attained. It follows from Theorem 3.1 and Theorem 4.1 that a unique solution called the Ulam-Hyers stable is derived from Eq. (5.1).

6 Conclusions

In this paper, we developed quantitative conclusions and stability result for the boundary problems considered. Specifically, for the boundary value problem (Q)-(BCs) proposed in this paper, we proved the uniqueness and existence results via the Banach's contraction mapping principle and Krasnoselskii's fixed -point theorem, and we provided additionally the criteria for Ulam-Hyers stability. Finally, we presented an example to demonstrate the effectiveness and validity of the obtained results. Moving forward, the interested researchers can further investigate various problems of multi-term sequential fractional differential equations with general variable coefficients.

Funding

This work has not received any funding.

CRediT authorship contribution statement

Debao Yan: Writing – review & editing, Writing – original draft, Methodology, Formal analysis.

Declaration of Competing Interest

The author declares he has no competing interests for his manuscript titled “Boundary Problems of Sequential Fractional Differential Equations Having a Monomial Coefficient”.

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Acknowledgements

The author is very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper.
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References

1 Kilbas A.A. Srivastava H.M. Trujillo J.J. Theory of Fractional Differential Equations North-Holland Mathematics Studies vol. 204 2006 Elsevier B.V. Amsterdam
2 Podlubny I. Fractional Differential Equations 1999 Academic Press San Diego
3 Idczak D. Kamocki R. On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn Fract. Calc. Appl. Anal. 14 4 2011 538 553 10.2478/s13540-011-0033-5
4 Povstenko Z. Fractional Thermoelasticity 2015 Springer New York
5 Aslefallah M. Shivanian E. A nonlinear partial integro-differential equation arising in population dynamic via radial basis functions and theta-method J. Math. Comput. Sci. 13 2014 14 25
6 Das S. Functional Fractional Calculus for System Identification and Controls 2008 Springer New York
7 Djordjevic V. Jaric J. Fabry B. Fractional derivatives embody essential features of cell rheological behavior Ann. Biomed. Eng. 31 2003 692 699 10.1114/1.1574026 12797619
8 Meral F. Royston T. Magin R. Fractional calculus in viscoelasticity: an experimental study Commun. Nonlinear Sci. Numer. Simul. 15 2010 939 945 10.1016/j.cnsns.2009.05.004
9 Oldham K. Fractional differential equations in electrochemistry Adv. Eng. Softw. 41 2010 9 12 10.1016/j.advengsoft.2008.12.012
10 Balachandran K. Matar M. Trujillo J.J. Note on controllability of linear fractional dynamical systems J. Control Decis. 3 2016 267 279 10.1080/23307706.2016.1217754
11 Shunan L. Bingyang C. Beyond phonon hydrodynamics: nonlocal phonon heat transport from spatial fractional-order Boltzmann transport equation AIP Adv. 10 2020 10.1063/5.0021058
12 Marawan A.-R. Applying fractional quantum mechanics to systems with electrical screening effects Chaos Solitons Fractals 150 2021 10.1016/j.chaos.2021.111209
13 Dubey S. Chakraverty S. Hybrid techniques for approximate analytical solution of space- and time-fractional telegraph equations Pramana J. Phys. 97 11 2023 10.1007/s12043-022-02482-0
14 Lin W. Global existence theory and chaos control of fractional differential equations J. Math. Anal. Appl. 332 2007 709 726 10.1016/j.jmaa.2006.10.040
15 Uǧur S. Müfit S. Some analysis on a fractional differential equation with a right-hand side which has a discontinuity at zero Hacet. J. Math. Stat. 49 2020 1718 1725 10.15672/hujms.512563
16 Yan D. Solutions for a category of singular nonlinear fractional differential equations subject to integral boundary conditions Bound. Value Probl. 3 2022 10.1186/s13661-022-01585-2
17 Yan D. Existence results of fractional differential equations with nonlocal double integral boundary conditions Math. Biosci. Eng. 20 3 2023 4437 4454 10.3934/mbe.2023206 36896507
18 Padhi S. Graef J.R. Pati S. Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions Fract. Calc. Appl. Anal. 21 2018 716 745 10.1515/fca-2018-0038
19 Miller K.S. Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley and Sons New York
20 Agarwal R. Hristova S. O'Regan D. Mittag-Leffler stability for impulsive Caputo fractional differential equation Differ. Equ. Dyn. Syst. 29 2021 689 705 10.1007/s12591-017-0384-4
21 Nanware J.A. Dhaigude D.B. Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions J. Nonlinear Sci. Appl. 7 2014 246 254
22 Abuasbeh K. Shafqat R. Niazi A.U.K. Local and global existence and uniqueness of solution for class of fuzzy fractional functional evolution equation J. Funct. Spaces Appl. 2022 2022 1 14 10.1155/2022/7512754
23 Cong N. Tuan H.T. Trinh H. On asymptotic properties of solutions to fractional differential equations J. Math. Anal. Appl. 484 2 2020 10.1016/j.jmaa.2019.123759
24 Brandibu O. Kaslik E. Stability analysis of multi-term fractional-differential equations with three fractional derivatives J. Math. Anal. Appl. 495 2 2021 10.1016/j.jmaa.2020.124751
25 Ahmada B. Luca R. Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions Appl. Math. Comput. 339 2018 516 534 10.1016/j.amc.2018.07.025
26 Ahmad B. Alsaedi A. Aljoudi S. Ntouyas S.K. A six-point nonlocal boundary value problem of nonlinear coupled sequential fractional integro-differential equations and coupled integral boundary conditions J. Appl. Math. Comput. 56 1–2 2018 367 389 10.1007/s12190-016-1078-8
27 Ahmad B. Luca R. Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions Chaos Solitons Fractals 104 2017 378 388 10.1016/j.chaos.2017.08.035
28 Ahmad B. Ntouyas S.K. Existence results for Caputo type sequential fractional differential inclusions with nonlocal integral boundary conditions J. Appl. Math. Comput. 50 1–2 2016 157 174 10.1007/s12190-014-0864-4
29 Alsaedi A. Ntouyas S.K. Agarwal R.P. Ahmad B. On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions Adv. Differ. Equ. 33 2015 1 12 10.1186/s13662-015-0379-9
30 Tuan H.T. Thai H.D. Garrappa R. An analysis of solutions to fractional neutral differential equations with delay Commun. Nonlinear Sci. Numer. Simul. 100 2021 10.1016/j.cnsns.2021.105854
31 Ahmad B. Nieto J.J. Sequential fractional differential equations with three-point boundary conditions Comput. Math. Appl. 64 10 2012 3046 3052 10.1016/j.camwa.2012.02.036
32 Saber H. Imsatfia M. Boulares H. Moumen A. Alraqad T. On the existence and Ulam stability of BVP within kernel fractional time Fractal Fract. 7 12 2023 852 10.3390/fractalfract7120852
33 Wang J. Li X. Ulam-Hyers stability of fractional Langevin equations Appl. Math. Comput. 258 9 2015 72 83 10.1016/j.amc.2015.01.111
34 Wang J. Lv L. Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative Electron. J. Qual. Theory Differ. Equ. 63 2011 1 10 https://real.mtak.hu/22564/1/p1003.pdf
35 Krasnoselskii M.A. Two remarks on the method of successive approximations Usp. Mat. Nauk 10 1955 123 127
36 Wang Y. Xu J. Sobolev Space 2003 Southeast University Press (in Chinese)
37 Wei Z. Li Q. Che J. Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative J. Math. Anal. Appl. 367 2010 260 272 10.1016/j.jmaa.2010.01.023
38 Phuangthong N. Ntouyas S.K. Tariboon J. Nonlaopon K. Nonlocal sequential boundary value problems for Hilfer type fractional integro-differential equations and inclusions Mathematics 9 2021 615 10.3390/math9060615
39 Ahmad B. Matar M.M. El-Salmy O.M. Existence of solutions and Ulam stability for Caputo type sequential fractional differential equations of order α∈(2,3) Int. J. Anal. Appl. 15 1 2017 86 101
40 Shah K. Hussain W. Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam stability by means of topological degree theory Numer. Funct. Anal. Optim. 40 12 2019 1355 1372 10.1080/01630563.2019.1604545
41 Rus I.A. Ulam stability of ordinary differential equations Stud. Univ. Babeş–Bolyai, Math. 54 2009 125 133
42 Benkerrouche A. Souid M.S. Etemad S. Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam-Hyers-Rassias stability Fractal Fract. 5 2021 108 10.3390/fractalfract5030108
43 Rezapour S. Tellab B. Deressa C.T. H-U-type stability and numerical solutions for a nonlinear model of the coupled systems of Navier BVPs via the generalized differential transform method Fractal Fract. 5 2021 166 10.3390/fractalfract5040166
44 Etemad S. Ntouyas S.K. Stamova I. On solutions of two post-quantum fractional generalized sequential Navier problems: an application on the elastic beam Fractal Fract. 8 2024 236 10.3390/fractalfract8040236
45 Shah K. Rahmat A.K. Study of solution to a toppled system of fractional differential equations with integral boundary conditions Int. J. Appl. Comput. Math. 3 2017 2369 2388 10.1007/s40819-016-0243-y
46 Shah K. Ahmad I. Nieto J.J. Qualitative investigation of nonlinear fractional coupled pantograph impulsive differential equations Qual. Theory Dyn. Syst. 21 2022 131 10.1007/s12346-022-00665-z
47 Ertürk V.S. Ali A. Shah K. Existence and stability results for nonlocal boundary value problems of fractional order Bound. Value Probl. 2022 2022 25 10.1186/s13661-022-01606-0
48 Asma Ali A. Shah K. Ulam-Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions Adv. Differ. Equ. 2019 2019 7 10.1186/s13662-018-1943-x
