Existence of solutions of nonlinear fractional pantograph equations

Balachandran, K.; Kiruthika, S.; Trujillo, Juan J.

doi:10.1016/s0252-9602(13)60032-6

Cited by 116 publications

(65 citation statements)

References 29 publications

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“…In [2], the authors studied the nonlinear pantograph equation, Eq. (2), under the following assumptions:…”

Section: Comparison With Other Resultsmentioning

confidence: 99%

“…Theorem 6.18 (Theorem 3.1 of [2]). Under assumptions (i)−(iv), (2) has a unique solution in C(J × X).…”

Section: Comparison With Other Resultsmentioning

confidence: 99%

“…This equation appears in different fields of pure and applied mathematics such as number theory, dynamical systems, probability, quantum mechanics, etc. Recently, in [2], the authors considered the fractional version of the pantograph equation, namely (2) D α 0 + u(t) = g t, u(t), u(λt) , t ∈ J = [0, T ], u(0) = u 0 , where α, λ ∈ (0, 1). The Banach contraction principle was the main tool used in this study.…”

Section: Introductionmentioning

confidence: 99%

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Existence of solutions for hybrid fractional pantograph equations

Darwish

Sadarangani

2015

APPL ANAL DISCRETE M

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In this paper, we study the existence of the hybrid fractional pantograph equationwhere α, µ, σ ∈ (0, 1) and D α 0 + denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.

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“…In [2], the authors studied the nonlinear pantograph equation, Eq. (2), under the following assumptions:…”

Section: Comparison With Other Resultsmentioning

confidence: 99%

“…Theorem 6.18 (Theorem 3.1 of [2]). Under assumptions (i)−(iv), (2) has a unique solution in C(J × X).…”

Section: Comparison With Other Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence of solutions for hybrid fractional pantograph equations

Darwish

Sadarangani

2015

APPL ANAL DISCRETE M

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“…Yu [12] used variational iteration method to solve the multipantograph delay equation, and sufficient conditions were given to assure the convergence of the method. Balachandran and Kiruthika [13] studied the existence of solutions of abstract fractional pantograph equations by using the fractional calculus and fixed point theorems. Peics [14] researched the asymptotic behaviour of solutions of difference equations with continuous argument and obtained asymptotic estimates.…”

Section: Introductionmentioning

confidence: 99%

The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

Bai-qi

2017

Advances in Mathematical Physics

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By using fractional calculus and the summation by parts formula in this paper, the asymptotic behaviours of solutions of nonlinear neutral fractional delay pantograph equations with continuous arguments are investigated. The asymptotic estimates of solutions for the equation are obtained, which may imply asymptotic stability of solutions. In the end, a particular case is provided to illustrate the main result and the speed of the convergence of the obtained solutions.

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“…The pantograph equation has the form Recently, in [11], the authors considered the fractional version of the pantograph equation, namely D α…”

Section: Introductionmentioning

confidence: 99%

About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation

Karimov¹,

López²,

Sadarangani³

2016

Fractional Differential Calculus

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Abstract. The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equationwhere α ∈ (0,1) , ϕ and ρ are functions from [0,1] into itself and D α 0+ denotes the RiemannLiouville fractional derivative. The main tool of our study is a generalization of Darbo's fixed point theorem associated to measures of non-compactness. Also, we present an example illustrating our results.

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Existence of solutions of nonlinear fractional pantograph equations

Cited by 116 publications

References 29 publications

Existence of solutions for hybrid fractional pantograph equations

Existence of solutions for hybrid fractional pantograph equations

The Asymptotic Behaviours of a Class of Neutral Delay Fractional-Order Pantograph Differential Equations

About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation

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