Elsevier

Chaos, Solitons & Fractals

Volume 129, December 2019, Pages 56-61
Chaos, Solitons & Fractals

A singular ABC-fractional differential equation with p-Laplacian operator

https://doi.org/10.1016/j.chaos.2019.08.017Get rights and content

Highlights

  • The ABC fractional derivative is applied for singular boundary value problems with p-Laplacian.

  • The Hyers–Ulam-stability is considered for such an ABC boundary value problem.

  • The existence and uniqueness is investigated for such an ABC boundary value problem.

  • An inclusive model was studied to postulate the applicability of the results.

  • We endorse the investigators for re-considerations of the problem for multiplicity and exponential stability.

Abstract

In this article, we have focused on the existence and uniqueness of solutions and Hyers–Ulam stability for ABC-fractional DEs with p-Laplacian operator involving spatial singularity. The existence and uniqueness of solutions are derived with the help of the well-known Guo-Krasnoselskii theorem. Our work is a continuation of the study carried out in the recently published article ” Chaos Solitons & Fractals. 2018;117:16-20.” To manifest the results, we include an example with specific parameters and assumptions.

Introduction

Among all areas of research in mathematics, one of the most rapidly growing area is the fractional calculus. One of the most important reasons of this growth are the results recorded by many researched when they benefited from the fractional operators for the sake of modelling real world problems [1], [2], [3], [4], [5]. Another interesting fact is that there are varieties of fractional operators with different kernels. This fact gives the researchers the opportunity to choose the operator which is the most appropriate for the model they investigate. Recently, ABC-fractional differential equations were studied in both the theoretical and applied aspects. In the theoretical aspect, existence and uniqueness of solutions to the ABC-fractional DEs are under development. To the best of our studies, no one considered the existence of positive solutions for the ABC-fractional DEs involving singularities.

Till 2015, all of the fractional operators had singularities in the kernel they embody their kernels. These singularities are believed to be troublesome especially when these operators are applied to model some physical phenomena. To overcome this deficiency, Caputo et al. [6] suggested a new fractional derivative with nonsingular kernel involving an exponential function. Just one year later, Atangana et al. [7] suggested a fractional differential derivative including the Mittag-Leffler functions which generalize the exponential function. Then researchers have started to discover the theoretical properties of these operators and their discrete analogue. For more details about both continuous and discrete case we suggest the reader inspect the work in [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].

The existence of solutions and stability analysis have been considered as hot research topics in fractional calculus and a lot of new ideas and notions have been introduced [19], [20], [21], [22]. In this paper, we are presenting an analytical study of fractional DEs including the existence of positive solutions and Hyer–Ulam stability. Here, we highlight some recent developments related to our work. Very recently, the authors in [12], [13] introduced and studied an extension by using generalized Mittag-Leffler kernel. In fact, the author in [13] analyzed the use of this extension in gaining a semigroup property for the correspondent integral operator and in releasing the vanishing condition of the right hand side in the ABC fractional dynamical system.

Existence and uniqueness of solutions to boundary value problems in the frame work of the singular Riemann–Liouville fractional derivative were analyzed in [23]. Using Schauder’s fixed point theorem and upper-lower solution techniques, existence-uniqueness analysis of fractional differential equations with nonlocal boundary values was executed in [24]. Existence and uniqueness of solutions to fractional differential equations with singular kernel and in the presence of p-Laplacian were considered in [25]. Meanwhile many researchers studied existence and uniqueness of positive solution to fractional differential equations with p-Laplacian operator. We refer the reader to [26] and the references therein. Some more interesting results for the existence and uniqueness of delay fractional differential equations were considered in [27], [28].

Recently, Jarad et al. [29] have recently studied the following equation{aABCDϑ0x(t)=Λ(t,x(t)),x(a)=x0,where aABCDϑ0 is the ABC-fractional order differential operator and ϑ0 ∈ (0, 1), aABCDϑ0x(t),Λ(t,x(t))C[a,b]. To the best of our knowledge, no one has investigated existence and uniqueness of solution to ABC-fractional DEs with Ψp-operator with singularity. In order to overcome the deficiency in the study, we follow [29] for the EPS and HU-stability of a nonlinear ABC-fractional DE given by:{ABC0Dϵ0[Ψp[0CDϑ0x(t)]]=Λ(t,x(t)),Ψp[0CDϑ0x(t)]|t=0=0,x(1)=x(n)(0),x(k)(0)=0=x(n)(1),where k=1,2,,n1, ϵ0,(0,1],ϑ0(n,n+1] and Λ(t, x(t)) ∈ C[0,1] are continuous functions. The 0ABCDϵ0 is the ABC-fractional differential operator of order ϵ0 ∈ (0,1] and 0CDϑ0 is the Caputo’s fractional order differential operators of order ϑ0(n,n+1]. The Ψp(r)=|r|p2r is the p-Laplacian-operator such that 1/p+1/q=1 and Ψp1=Ψq. By a positive solution x(t) of (1.2), we mean x(t) > 0 for t ∈ (0,1] satisfying (1.2). Our problem ABC-fractional DE with the operator Ψp (1.2), is more complicated and more general than (1.1).

To investigate the existence uniqueness of solution to (1.2) and its stability in the sense of Hyers and Ulam stability, we will transform the problem into an equivalent AB-fractional integral form by using the classical results and 0ABIϵ0,0Iϑ0 and a Green function. The Green’s function will be tested for its nature. For an application, we present and analyze an important example. The readers may reconsider (1.2) for the multiplicity results.

Definition 1.1 [7], [30]

The fractional Atangane–Baleanu derivative in the Caputo setting of the function f ∈ H*(a, b), where b > a, η* ∈ [0, 1] is defined byABCaDτη*f(τ)=B(η*)1μaτf(ϑ0)Eη*[η*(τs)μ1η*]ds,where B(η*) is satisfied the property B(0)=B(1)=1.

Definition 1.2 [7]

The fractional Atangana–Baleanu derivative in Riemann–Liouville settingss of the function f ∈ H*(a, b), where b > a, η* ∈ [0,1] is described as follows:ABRaDτη*f(τ)=B(η*)1η*ddτaτf(ϑ0)Eη*[η*(τs)η*1η*]dϑ.

Definition 1.3 [31], [32]

The fractional Atangane–Baleanu integral of the function f ∈ H*(a, b), b > a, 0 < η* < 1 is given byABaIτη*f(τ)=1η*B(η*)f(τ)+η*B(η*)Γ(η*)aτf(ϑ0)(τs)η*1dϑ.

Lemma 1.4

[10] The ABC fractional derivative and AB fractional integral of the function f satisfies the Newton–Leibniz formulaABaIτη*(ABCaDτη*f(τ))=f(τ)f(a).

The discrete version of (1.6) was announced in [10] as well, and the Caputo–Fabrizio analogous of both continuous and discrete cases were proved in [11].

Definition 1.5

The Riemann–Liouville fractional integral of a function ψ of order η* > 0, ψ:(0,+)R, is given by0Iη*ψ(t)=1Γ(η*)0t(ts)η*1ψ(s)ds,where for Re(η*) > 0 we haveΓ(η*)=0+essη*1ds.

Definition 1.6

The Caputo fractional derivative of a continuous function ψ(t):(0,+)R reads0Dϑ0ψ(t)=1Γ(kϑ0)0t(ts)kϑ01ψ(k)(s)ds,for k=[ϑ0]+1, where [ϑ0] is the greatest integer less than ϑ0.

Lemma 1.1

[4] For a fractional order η*(n1,n], ψCn1, the following is satisfied0Iη*0Dη*ψ(t)=ψ(t)+m0+m1t+m2t2++mn1tn1,where mi,i=0,1,,n1 are constant.

To prove the existence of solution, we benefit from the following Guo–Krasnoselskii theorem.

Theorem 1.2

Let Y a Banach space and P*Y be a cone. Suppose that B1*,B2* are two bounded subsets of Y such that 0B1*,B1*¯B2* and let the operator J*:P*(B2*¯B1*)P* be continuous such that

(A1) J*zz if zP*B1* and J*zz if zP*B2*, or

(A2) J*zz if zP*B1* and J*zz if zP*B2*.

Then J* has a fixed point in P*(B2*¯B1*).

Lemma 1.3

Let Ψp be a nonlinear operator. Then

  • (1)

    for 1<p2,η1*η2>0 and |η1*|,|η2*|ρ>0, then|Ψp(η1*)Ψp(η2*)|(p1)ρp2|η1*η2*|.

  • (2)

    If p > 2, and |η1*|,|η2*|ρ*, then|Ψp(η1*)Ψp(η2*)|(p1)ρ*p2|η1*η2*|.

Section snippets

Green’s function and its properties

Theorem 2.1

For k=1,2,,n1, ϵ0,(0,1],ϑ0(n,n+1] and Λ(t, x(t)) ∈ C[0, 1] such that Λ(0,x(0))=0 , x(t) is a solution of (1.2) if and only ifx(t)=01Hϑ0(t,s)Ψq(0ABIϵ0[Λ(t,x(t))])dt,whereHϑ0(t,s)={(1s)ϑ01Γ(ϑ0)+tnn!(1s)ϑ0n1Γ(ϑ0n)(ts)ϑ01Γ(ϑ0),st,(1s)ϑ01Γ(ϑ0)+tnn!(1s)ϑ0n1Γ(ϑ0n),st.

Proof

If we apply the AB-fractional integral operator 0ABIϵ0 on (1.2) and make use of Lemma 1.4, then problem (1.2) becomes as belowΨp[0CDϑ0x(t)]=0ABIϵ0[Λ(t,x(t))]+d0.The conditions Ψp[C0Dϑ0x(t)]|t=0=0 and Λ(0,x(0))=0,

Existence results

Let Y be the Banach space C[0,1] endowed by the norm x=maxt[0,1]{|x(t)|:xY} and P be the cone containing non-negative functions in the space Y defined by P={xY:x(t)tϑ0x,t[0,1]}. Let W(r)={xP:x<r},W(r)={xP:x=r}.

With the help of Theorem 2.1, an alternate form of (1.2) isx(t)=01Hϑ0(t,s)Ψq(0ABIϵ0[Λ(ξ,x(ξ))])ds.Define J*:P{0}Y byJ*x(t)=01Hϑ0(t,s)Ψq(0ABIϵ0[Λ(ξ,x(ξ))])ds.Hence, the solution of (1.2), x(t), is nothing but the fixed point of J, i.e.,x(t)=J*x(t).Assume the following

Stability analysis

We have reserved this section for the Hyers–Ulam stability of our suggested nonlinear singular ABC-fractional DE with Ψp-operator (1.2). For this purpose, we follow our recently published articles [19], [26] and some more related notions in the literature.

Definition 4.1

We say that the fractional integral equations (3.1) is Hyers–Ulam stable if for every λ > 0, there is a constant D*>0, such that: if|x(t)01Hϑ0(t,s)Ψq(0ABIϵ0Λ(ζ,x(ζ)))ds|λ,there is a u(t) satisfyingu(t)=01Hϑ0(t,s)Ψq(0ABIϵ0Λ(ζ,u(ζ)))ds,

Illustrative example

In this section, an application of the results which have proved in the Sections 3 and 4, is provided.

Example 5.1

For t[0,1],p=3,q=1.5,n=4,4.5=ϑ0(4,5],b=1,ϵ0=0.5,a=0.1, y1*(t,x(t))=x+1x335, clearly ΛC([0,1]×(0,+),[0,+). Presume a singular ABC-fractional DE with Ψp-operator:{ABC0Dϵ0[Ψp[0Dϑ0x(t)]]+t[x+1x335]=0,Ψp[0CDϑ0x(t)]|t=0=0,x(1)=x(n)(0),x(k)(0)=0=x(n)(1),with t[x(t)+1x(t)335]|t=0=0, k=1,2,3. We consider:Ψmax(t,η)=max{tx+t5x335:t53ηxη}η+151t17η335,Ψmin(t,η)=min{tx+t5x335:t53ηxη}t56η0.5+15η335,

Conclusion

In this paper we have considered a more general class of nonliear ABC-fractional DE with nonlinear operator Ψp and singularity (1.2) for the EU of solutions and Hyers–Ulam-stability. For these determinations, we applied the standard results and got a substitute integral scheme of the singular ABC-fractional DE (1.2). The equation is based with a positively diminishing Green’s function. Then using Guo-Krasnoselskiis fixed point process, the EU of solutions were evaluated and then

Author’s contributions

All the authors claim equal contributions in this article.

Funding source

Not applicable.

Declaration of Competing Interest

There is no competing interest among the authors regarding the publication of the article.

Acknowledgment

All the authors are grateful to the reviewer and the editorial board whose guidance improved the quality of the paper. The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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