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Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations

Author

Listed:
  • Meysam Alvan

    (Department of mathematics, Central Tehran Branch Islamic Azad university, Tehran 13185/768, Iran)

  • Rahmat Darzi

    (Department of Mathematics, Neka Branch Islamic Azad University, Neka 48411-86114, Iran)

  • Amin Mahmoodi

    (Department of mathematics, Central Tehran Branch Islamic Azad univesity, Tehran 13185/768, Iran)

Abstract

In this article, we study the following fractional boundary value problem D 0 + α c u ( t ) + 2 r D 0 + α − 1 c u ( t ) + r 2 D 0 + α − 2 c u ( t ) = f ( t , u ( t ) ) , r > 0 , 0 t 1 , u ( 0 ) = u ( 1 ) , u ′ ( 0 ) = u ′ ( 1 ) , u ′ ( ξ ) + r u ( ξ ) = η , ξ ∈ ( 0 , 1 ) Where 2 ≤ α 3 , D 0 + α − i c ( i = 0 , 1 , 2 ) are the standard Caputo derivative and η is a positive real number. Some new existence results are obtained by means of the contraction mapping principle and Schauder fixed point theorem. Some illustrative examples are also presented.

Suggested Citation

  • Meysam Alvan & Rahmat Darzi & Amin Mahmoodi, 2016. "Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations," Mathematics, MDPI, vol. 4(1), pages 1-10, March.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:1:p:13-:d:65044
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    References listed on IDEAS

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    1. Azizollah Babakhani & Dumitru Baleanu & Ravi P. Agarwal, 2013. "The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, March.
    2. Bashir Ahmad & Juan J. Nieto, 2009. "Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2009, pages 1-9, July.
    3. Sergei Rogosin, 2015. "The Role of the Mittag-Leffler Function in Fractional Modeling," Mathematics, MDPI, vol. 3(2), pages 1-14, May.
    4. Bashir Ahmad & Juan J. Nieto, 2009. "Existence of Solutions for Nonlocal Boundary Value Problems of Higher‐Order Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2009(1).
    5. R. Darzi & B. Mohammadzadeh & A. Neamaty & D. Bǎleanu, 2013. "Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, September.
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    Cited by:

    1. Serik Aitzhanov & Kymbat Bekenayeva & Zamira Abdikalikova, 2023. "Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator," Mathematics, MDPI, vol. 11(18), pages 1-12, September.
    2. Luís P. Castro & Anabela S. Silva, 2023. "On the Existence and Stability of Solutions for a Class of Fractional Riemann–Liouville Initial Value Problems," Mathematics, MDPI, vol. 11(2), pages 1-22, January.

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