A singular ABC-fractional differential equation with p-Laplacian operator
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 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 equationwhere is the ABC-fractional order differential operator and ϑ0 ∈ (0, 1), . 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:where and Λ(t, x(t)) ∈ C[0,1] are continuous functions. The is the ABC-fractional differential operator of order ϵ0 ∈ (0,1] and is the Caputo’s fractional order differential operators of order . The is the p-Laplacian-operator such that and . 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 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 bywhere is satisfied the property 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: Definition 1.3 [31], [32] The fractional Atangane–Baleanu integral of the function f ∈ H*(a, b), b > a, 0 < η* < 1 is given by Lemma 1.4 [10] The ABC fractional derivative and AB fractional integral of the function f satisfies the Newton–Leibniz formula
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, is given bywhere for Re(η*) > 0 we have Definition 1.6 The Caputo fractional derivative of a continuous function readsfor where [ϑ0] is the greatest integer less than ϑ0. Lemma 1.1 [4] For a fractional order the following is satisfiedwhere are constant.
To prove the existence of solution, we benefit from the following Guo–Krasnoselskii theorem. Theorem 1.2 Let a Banach space and be a cone. Suppose that are two bounded subsets of such that and let the operator be continuous such that if and if or if and if . Then has a fixed point in . Lemma 1.3 Let Ψp be a nonlinear operator. Then for and then If p > 2, and then
Section snippets
Green’s function and its properties
Theorem 2.1 For and Λ(t, x(t)) ∈ C[0, 1] such that , x(t) is a solution of (1.2) if and only ifwhere Proof If we apply the AB-fractional integral operator on (1.2) and make use of Lemma 1.4, then problem (1.2) becomes as belowThe conditions and
Existence results
Let be the Banach space endowed by the norm and P be the cone containing non-negative functions in the space defined by . Let .
With the help of Theorem 2.1, an alternate form of (1.2) isDefine byHence, the solution of (1.2), x(t), is nothing but the fixed point of i.e.,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 such that: ifthere is a u(t) satisfying
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 clearly . Presume a singular ABC-fractional DE with Ψp-operator:with . We consider:
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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