On the Presence of Positive Solutions for Generalized Fractional Boundary Value Problems with Green’s Function

Ibrahim A. Al-Mania¹

Section:Research Paper, Product Type: Journal-Paper
Vol.11 , Issue.5 , pp.1-10, Oct-2024

Online published on Oct 31, 2024

Copyright © Ibrahim A. Al-Mania . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
The branch of mathematics that deals with the study of non-integer order derivatives and integrals is called fractional calculus. The interesting thing about this subject is that in contrast to the classical derivatives, the fractional derivatives are not a point quantity. Indeed, the fractional derivative of a function of order ? at some point is a local property only for ? being an integer. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. In this paper, the existence of positive solutions is established for boundary value problems defined within generalized Riemann–Louville and Caputo fractional operators. Our approach is based on utilizing the technique of fixed point theorems. For the sake of converting the proposed problems into integral equations, we construct Green’s function and study their properties for three different types of boundary value problems. An example is presented to demonstrate the validity of theoretical findings.

Key-Words / Index Term :
Existence of positive solutions; Fractional Calculus; Riemann-Liouville Differintegral; Generalized Fractional Boundary Value Problems

References :
[1] Bai, Chuanzhi. “Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator”. Advances in Difference Equations, 1: 4, 2018?.
[2] Yang, Chen. “Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter”. Hacettepe Journal of Mathematics and Statistics, 44.3: pp.659-667, 2015.?
[3] Sheng, K., Zhang, W., & Bai, Z. “Positive solutions to fractional boundary-value problems with p-Laplacian on time scales”. Boundary Value Problems, pp.1-15, 2018.?
[4] Nyamoradi, N., Baleanu, D., & Bashiri, T. “Positive solutions to fractional boundary value problems with nonlinear boundary conditions”. In: Abstract and Applied Analysis. Hindawi Publishing Corporation, p. 579740, 2013.?
[5] Sevinik Adigüzel, R., Aksoy, Ü., Karapinar, E., & Erhan, ?. M. “On the solution of a boundary value problem associated with a fractional differential equation”. Mathematical Methods in the Applied Sciences, 2020.?
[6] Wahash, H. A., Panchal, S. K., & Abdo, M. S. “Positive solutions for generalized Caputo fractional differential equations with integral boundary conditions”. Journal of Mathematical Modeling, 8.4, pp.393-414, 2020.?
[7] Daftardar-Gejji, V. (Ed.). “Fractional Calculus”. Alpha Science International Limited. ?2013.
[8] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. “Theory and applications of fractional differential equations”. Elsevier, 2006.?
[9] Diethelm, K., & Ford, N. J. “ Analysis of fractional differential equations”. Journal of Mathematical Analysis and Applications, 265.2: pp.229-248, 2002.?
[10] Jajarmi, A., & Baleanu, D. “A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems”. Frontiers in Physics, 8: 220, 2020.?
[11] Morita, T., & Sato, K. I. “Liouville and Riemann-Liouville fractional derivatives via contour integrals”. Fractional Calculus and Applied Analysis, 16.3: pp.630-653, 2013.?
[12] Zhou, Y. “Basic theory of fractional differential equations”. Singapore: World scientific, 2023.?
[13] Zhou, Y., Wang, J., Zhang, L. “Basic theory of fractional differential equations”. World scientific., pp.223-262, 2016.
[14] Coleman, R. “Calculus on normed vector spaces”. Springer Science & Business Media, 2012.?
[15] Jeelani, M. B., Saeed, A. M., Abdo, M. S., & Shah, K. “Positive solutions for fractional boundary value problems under a generalized fractional operator”. Mathematical Methods in the Applied Sciences, 44.11: pp.9524-9540, 2021.?
[16] Abdo, M. S., IBRAHIM, A. G., & Panchal, S. K. “Nonlinear implicit fractional differential equation involving-Caputo fractional derivative”. In: Proceedings of the Jangjeon Mathematical Society. pp.387-400, 2019.?
[17] Seemab, A., Ur Rehman, M., Alzabut, J., & Hamdi, A. “On the existence of positive solutions for generalized fractional boundary value problems”. Boundary Value Problems, pp.1-20, 2019.?
[18] Chen, H., & Stynes, M. Martin. “Blow-up of error estimates in time-fractional initial-boundary value problems”. IMA Journal of Numerical Analysis, 41.2: pp.974-997, 2021.?