Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions

Chergui, Djamila; Oussaeif, Taki-Eddine; Merad, Ahcene

doi:10.3934/math.2019.1.112

Cited by 12 publications

(3 citation statements)

References 7 publications

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“…At the present day, there are numerous results on the existence and uniqueness of solutions for fractional differential equations. For greater details, the readers are cited the previous research [22,23,29,36] and the references therein. However, due to the fact that in lots of conditions, which include nonlinear analysis and optimization, locating the exact solution of differential equations is almost tough or impossible, we don't forget approximate solutions.…”

Section: Introductionmentioning

confidence: 99%

A Coupled System of Nonlinear Langevin Fractional q- Difference Equations Associated with Two Different Fractional Orders in Banach Space

BOUTIARA

2024

KgJMat

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In this research article, we study the coupled system of nonlinear Langevin fractional q-diﬀerence equations associated with two diﬀerent fractional orders in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with Mönch ﬁxed point theorem, the implementation Banach contraction principle ﬁxed point theorem, and the employment of Urs’s stability approach. Two examples illustrating the eﬀectiveness of the theoretical results are presented.

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Section: Introductionmentioning

confidence: 99%

A Coupled System of Nonlinear Langevin Fractional q- Difference Equations Associated with Two Different Fractional Orders in Banach Space

BOUTIARA

2024

KgJMat

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“…Integro-differential equations are a combination of derivatives and integrals which are appealing to both researchers and scientists for their applications in many areas [6][7][8][9]. Numerous mathematical formulations of physical phenomena include integro-differential equations, which may arise in modelling biological fluid dynamics [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Aicha

Merad

2021

AJMS

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PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

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“…In 2018, Al-Omari et al 37 discussed q analogs of Laplace-type integrals on diverse types of q special functions involving Fox's H q functions, which some of the assumed functions were the q Bessel functions of the first kind, the q Bessel functions of the second kind, the q Bessel functions of the third kind, and the q Struve functions as well. In 2019, Chergui et al 38 reviewed the existence and uniqueness of solution for the nonlinear fractional boundary value problem D q x(t) = f(t, x(t), D r x(t)) with nonseparated type integral boundary conditions…”

Section: Introductionmentioning

confidence: 99%

Solutions of sum‐type singular fractional q integro‐differential equation with m‐point boundary value problem using quantum calculus

Ahmadian

Rezapour

Salahshour

et al. 2020

Math Methods in App Sciences

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Nowadays, many researchers have considerable attention to fractional calculus as a useful tool for modeling of different phenomena in the world. In this work, we investigate the sum‐type singular nonlinear fractional q integro‐differential equations with m‐point boundary value problem. The existence of positive solutions is obtained by the properties of the Green function, standard Caputo q derivative, Riemann–Liouville fractional q integral, and a fixed point theorem on a real Banach space scriptX, which has a partial order by using a cone P⊂scriptX. The proofs are based on solving the operators equation. By providing seven algorithms, four tables, and three figures, we give two numerical examples to illustrate our main result.

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Existence and uniqueness of solutions for nonlinear fractional differential equations depending on lower-order derivative with non-separated type integral boundary conditions

Cited by 12 publications

References 7 publications

A Coupled System of Nonlinear Langevin Fractional q- Difference Equations Associated with Two Different Fractional Orders in Banach Space

A Coupled System of Nonlinear Langevin Fractional q- Difference Equations Associated with Two Different Fractional Orders in Banach Space

Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Solutions of sum‐type singular fractional q integro‐differential equation with m‐point boundary value problem using quantum calculus

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