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On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations

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Abstract

In this manuscript, we study the existence, uniqueness and various kinds of Ulam stability including Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability of the solutions to a nonlinear coupled systems of implicit fractional differential equations involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach contraction principle and Leray–Schauder of cone type. For stability, we utilize classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

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Acknowledgements

We are very thankful to the anonymous referees for their careful reading and suggestions which improved the quality of this paper.

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    Authors and Affiliations

    1. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

      Zeeshan Ali & Akbar Zada

    2. Department of Mathematics, University of Malakand, Lower Dir, Khyber Pakhtunkhwa, Pakistan

      Kamal Shah

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    1. Zeeshan Ali
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    2. Akbar Zada
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    3. Kamal Shah
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    Correspondence to Kamal Shah.

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    Communicated by Norhashidah Hj. Mohd. Ali.

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    Ali, Z., Zada, A. & Shah, K. On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations. Bull. Malays. Math. Sci. Soc. 42, 2681–2699 (2019). https://doi.org/10.1007/s40840-018-0625-x

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    • DOI: https://doi.org/10.1007/s40840-018-0625-x

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