Dynamics and stability of Hilfer-Hadamard type fractional pantograph equations with boundary conditions

Vivek, D.; Kanagarajan, K.; Harikrishnan, S.

doi:10.5899/2018/jnaa-00387

Cited by 4 publications

(1 citation statement)

References 22 publications

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“…Pantograph equations play a pivotal role in pure and applied mathematics and physics. Motivated by their significance, a ton of scientists generalized these equations into different types and presented the solvability aspect of such problems both numerically and theoretically; for additional subtleties, see [40][41][42][43][44][45][46] and the references therein. Besides, some authors applied various kinds of fractional derivatives and studied the existence and stability of Ulam-Hyers, which can be found in [47][48][49][50][51].…”

Section: Introductionmentioning

confidence: 99%

On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

et al. 2021

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In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.

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Section: Introductionmentioning

confidence: 99%

On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

et al. 2021

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Stability results for fractional integral pantograph differential equations involving two Caputo operators

Moumen

Shafqat

Hammouch³

et al. 2022

MATH

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<abstract><p>In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.</p></abstract>

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Existence and uniqueness results for mixed derivative involving fractional operators

Elaiw

Hafeez

Jeelani

et al. 2023

MATH

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<abstract><p>In this article, we discuss the existence and uniqueness results for mix derivative involving fractional operators of order $ \beta\in (1, 2) $ and $ \gamma\in (0, 1) $. We prove some important results by using integro-differential equation of pantograph type. We establish the existence and uniqueness of the solutions using fixed point theorem. Furthermore, one application is likewise given to represent our fundamental results.</p></abstract>

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Dynamics and stability of Hilfer-Hadamard type fractional pantograph equations with boundary conditions

Cited by 4 publications

References 22 publications

On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

Stability results for fractional integral pantograph differential equations involving two Caputo operators

Existence and uniqueness results for mixed derivative involving fractional operators

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