Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative

Shammakh, Wafa; Alzumi, Hadeel Z.

doi:10.1186/s13662-019-2038-z

Cited by 16 publications

(9 citation statements)

References 20 publications

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“…Remark 5. According to Lemma 2 the initial value condition in Equation ( 15) could be replaced by equality (16) and the impulsive conditions could be replaced by…”

Section: Define the Setmentioning

confidence: 99%

“…Later, Jarad et al [10] introduced a new generalized proportional derivative which is well-behaved and has several advantages over the classical derivatives such as meaning that it generalizes formerly known derivatives in the literature. For recent contributions relevant to fractional differential equations via generalized proportional derivatives, see [11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

Hristova

Abbas

2021

Symmetry

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The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.

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“…Remark 5. According to Lemma 2 the initial value condition in Equation ( 15) could be replaced by equality (16) and the impulsive conditions could be replaced by…”

Section: Define the Setmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

Hristova

Abbas

2021

Symmetry

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“…These generalizations are helpful to reach a better form of the real models that appeared in various fields such as physics, chemistry, aerodynamics, and electrorheology employing fractional differential equations (see [8][9][10][11][12][13][14][15][16][17]). The existence and uniqueness results of fractional boundary value problems involving fractional derivatives and derivative of a generalized type were established by many authors (for example, see [18][19][20][21]).…”

Section: Introductionmentioning

confidence: 99%

On More General Fractional Differential Equations Involving Mixed Generalized Derivatives with Nonlocal Multipoint and Generalized Fractional Integral Boundary Conditions

Shammakh

Alzumi

AlQahtani

2020

Journal of Function Spaces

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This paper deals with the existence of solutions for a new boundary value problem involving mixed generalized fractional derivatives of Riemann-Liouville and Caputo supplemented with nonlocal multipoint boundary conditions. The existence results for inclusion case are also discussed. The nonlinear term belongs to a general abstract space, and our results rely on modern theorems of fixed point. Ulam stability is also presented. We provide some examples that well-illustrate our main results.

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“…It was a new idea and had useful applications in imaging via strip-detectors [16] and acoustics [17]. For examples of boundary value problems for nonlinear differential equations, one can see [18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

On multi-term proportional fractional differential equations and inclusions

2020

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The aim of this paper is to study new nonlocal boundary value problems of fractional differential equations and inclusions supplemented with slit-strips integral boundary conditions. Based on the functional analysis tools, the existence results for a nonlinear boundary value problem involving a proportional fractional derivative are presented. In addition to that, the extension of the problem at hand to its inclusion case is discussed. The obtained results are very interesting and are well illustrated with examples.

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Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative

Cited by 16 publications

References 20 publications

Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

On More General Fractional Differential Equations Involving Mixed Generalized Derivatives with Nonlocal Multipoint and Generalized Fractional Integral Boundary Conditions

On multi-term proportional fractional differential equations and inclusions

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