Title: p-integrable solution of boundary fractional differential and integro-differential equations with Riemann derivatives of order (n − 1 < δ ≤ n)
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00050; Volume 4 / Issue 2 / Year 2022, Pages 1-10
Document Type: Research Paper
aDepartment of Mathematics, College of Science, University of Duhok, Duhok 42001, IRAQ
bDepartment of Mathematics, College of Science, University of Duhok, Duhok 42001, IRAQ
Received: 25 July 2021, Accepted: 9 March 2022, Published: 12 April 2022.
Corresponding Author: Shayma Adil Murad (Email address: firstname.lastname@example.org)
Full Text: PDF
This paper considers the existence of Lp-solutions of certain fractional differential and integro-differential equations involving the Riemann derivatives of order (n − 1 < δ ≤ n), with boundary conditions. The results are established by means of the Hölder’s inequality in a Banach space. Some special cases and examples are given to explain the main results.
Keywords: Fracional differential equations, integro-differential equations, Riemann fractional derivatives, existence and uniqueness, Hölder’s inequality, Lp spaceReferences:
- M. I. Abbas, Existence and uniqueness of solution for a boundary value problem of fractional order involving two Caputo’s fractional derivatives, Adv. Differ. Equ. 2015 (252), 1–19, 2015.
- J. Gh. Abdulahad and Sh. A. Murad, Local existence theorem of fractional differential equations in Lp space, Raf. J. of Comp. & Math’s. 9 (2), 71–78, 2012.
- D. N. Abdulqader and Sh. A. Murad, Existence and uniqueness results for certain fractional boundary value problems, Journal of Duhok University 22 (2), 76–88, 2019.
- A. Aghajani, Y. Jalilian and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal. 15 (1), 44–69, 2012.
- R. P. Agrwal, A. Asma, V. Lupulescu and D. O’Regan, Lp-solutions for a class of fractional integral equations, J. Integral Equations Appl. 29 (2), 251–270, 2017.
- H. Ahmed, H. Boulares, A. Ardjouni and A. Chaoui, On the study of fractional differential equations in a weighted sobolevs pace, Bull. Int. Math. Virtual Inst. 9, 333–343, 2019.
- S. Arshad, V. Lupulescu and D. O’Regan, Lp-solutions for fractional integral equations, Fract. Calc. Appl. Anal. 17 (1), 259–276, 2014.
- T. A. Barton and I. K. Purnaras, Lp-solutions of singular integro-differential equations, J. Math. Anal. Appl. 386, 830–841, 2012 .
- T. A. Barton and B. Zhang, Lp-solutions of fractional differential equations, Nonlinear Stud. 19 (2), 161–177, 2012.
- L. Ibnelazyz, K. Guida, S. Melliani and K. Hilal, On a nonlocal multi point and integral boundary value problem of nonlinear fractional integrodifferential equations, J. Funct. Spaces. 2020, 1–8, 2020.
- A. Karoui and A. Jawahdou. Existence and approximate Lp and continuous solutions of nonlinear integral equations of the Hammerstein and Volterra types, Appl. Math. Comput. 216 (7), 2077–2091, 2010.
- A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V, Amsterdam, 2006.
- Sh. A. Murad and Z. A. Ameen, Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives, AIMS Math. 7 (4), 6404–6419, 2022.
- Sh. A. Murad and S. Hadid, Existence and uniqueness theorem for fractional differential equation with integral boundary condition, J. Frac. Calc. Appl. 3 (6), 1–9, 2012.
- Sh. A. Murad and A. S. Rafeeq, Existence of solutions of integro-fractional differential equation when α ∈ (2, 3] through fixed point theorem, J. Math. Comput. Sci. 11 (5), 6392–6402, 2021.
- Sh. A. Murad, H. J. Zekri and S. Hadid, Existence and uniqueness theorem of fractional mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions, Int. J. Differ. Equ. 2011, 1–15, 2011.
- K. B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, London, 1974.
- I. Podlubny, Fractional differential equation, Mathematics in Science and Egineering, Academic Press, San Diego, 1999.