Some results for initial value problem of nonlinear fractional equation in Sobolev space

Boucenna, Djalal; Boulfoul, Ali; Chidouh, Amar; Makhlouf, Abdellatif Ben; Tellab, Brahim

doi:10.1007/s12190-021-01500-5

Cited by 13 publications

(14 citation statements)

References 10 publications

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“…Proof. First, let u ∈ C([0, 1], R) be a solution of the multi-term fractional boundary value problem (1), which it implies that 𝜿 = D…”

Section: Then the Function U Is A Solution Of (1) If And Only If 𝜿(T)...mentioning

confidence: 99%

“…Finally, we check the two boundary conditions of problem (1). From Equation (3) and by definition of the Riemann-Liouville integral, it follows that…”

Section: 𝜿(T) 𝜿(T))mentioning

confidence: 99%

“…), (1) and in the second, we will discuss some results of existence to the following semilinear fractional differential inclusions for multi-term boundary value problem…”

Section: Introductionmentioning

confidence: 99%

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Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

Hadjadj

Meftah

Bensayah

et al. 2022

Math Methods in App Sciences

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In the present paper, we study the existence and uniqueness of the solution for multi-term fractional boundary value problem under Riemann-Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi-term problem by using some notions and properties on set-valued maps together with classical fixed point theorems due to Leray-Schauder. The cases when the set-valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

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“…Proof. First, let u ∈ C([0, 1], R) be a solution of the multi-term fractional boundary value problem (1), which it implies that 𝜿 = D…”

Section: Then the Function U Is A Solution Of (1) If And Only If 𝜿(T)...mentioning

confidence: 99%

“…Finally, we check the two boundary conditions of problem (1). From Equation (3) and by definition of the Riemann-Liouville integral, it follows that…”

Section: 𝜿(T) 𝜿(T))mentioning

confidence: 99%

See 1 more Smart Citation

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

Hadjadj

Meftah

Bensayah

et al. 2022

Math Methods in App Sciences

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“…en, Ahmad and Ntouyas [5] and Alsaedi et al [6] investigated similar results regarding solutions of a sequential BVPs consisting of nonlocal integro-differential inclusions of the Caputo type. In [7], Boucenna et al defined a nonlinear P on the Sobolev space and utilized the special operators for proving theorems with the help of some tools in functional analysis. Similarly, Azzaoui et al [8] defined a Sobolev space again and derived the existence criterion for positive solutions on such a space.…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

Rezapour

Boulfoul

Tellab

et al. 2022

Journal of Function Spaces

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This work is reserved for the study of a special category of boundary value problems ( BVP s) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. This fractional model and its boundary value conditions (BVCs) involve different simple BVP s, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. The Banach principle gives a unique solution for this Liouville–Caputo BVP . Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view.

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“…Regarding applications of FDEs, one can observe some fields including aerodynamics, biochemistry, electro-chemistry, bioengineering, physics, viscoelasticity, mathematical biology, and so on (see [1,2]). In the last few years, a large number of studies regarding the existence theory for different FDEs have received much attentions from researchers and some examples include [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Since we can model some of applied phenomena in the framework of the fractional coupled systems, a large number of researchers have conducted many research studies on the existence of solution for such a type of systems (for instances, see [19][20][21][22][23]).…”

Section: Introductionmentioning

confidence: 99%

H-U-Type Stability and Numerical Solutions for a Nonlinear Model of the Coupled Systems of Navier BVPs via the Generalized Differential Transform Method

et al. 2021

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This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term fractional coupled system of Navier equations under the fractional boundary conditions. The existence theory is studied regarding solutions of the given coupled sequential Navier boundary problems via the Krasnoselskii’s fixed-point theorem on two nonlinear operators. Moreover, the Banach contraction principle is applied to investigate the uniqueness of solution. We then focus on the Hyers–Ulam-type stability of its solution. Furthermore, the approximate solutions of the proposed coupled fractional sequential Navier system are obtained via the generalized differential transform method. Lastly, the results of this research are supported by giving simulated examples.

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Some results for initial value problem of nonlinear fractional equation in Sobolev space

Cited by 13 publications

References 10 publications

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

H-U-Type Stability and Numerical Solutions for a Nonlinear Model of the Coupled Systems of Navier BVPs via the Generalized Differential Transform Method

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