Caputo-Katugampola fractional Volterra functional differential equations with a vanishing lag function
Volume 13, Issue 5, pp 293--302
http://dx.doi.org/10.22436/jnsa.013.05.06
Publication Date: March 29, 2020
Submission Date: October 14, 2019
Revision Date: February 11, 2020
Accteptance Date: February 15, 2020
-
1389
Downloads
-
2982
Views
Authors
M. I. Youssef
- Department of Mathematics, College of Science, Jouf University, P. O. Box 2014, Sakaka, Saudi Arabia.
- Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt.
Abstract
In the present article, we study the solvability of a class of fractional functional integro-differential equations of the Caputo-Katugampola type. The existence of solutions is investigated under sufficient conditions as well as the assumptions which guarantee the uniqueness of the solution is explained. Also, we examine the continuous dependence of the solution on the initial condition, the lag function \(0 \leq \psi(t)\leq t\), and the considered nonlinear functional. We give an example to explain our results. The outcomes in this paper extend the results developed by El-Sayed et al. in [A. M. A. El-Sayed, R. G. Ahmed, J. Nonlinear Sci. Appl., \(\bf 13\) (2020), 1--8], recently.
Share and Cite
ISRP Style
M. I. Youssef, Caputo-Katugampola fractional Volterra functional differential equations with a vanishing lag function, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 5, 293--302
AMA Style
Youssef M. I., Caputo-Katugampola fractional Volterra functional differential equations with a vanishing lag function. J. Nonlinear Sci. Appl. (2020); 13(5):293--302
Chicago/Turabian Style
Youssef, M. I.. "Caputo-Katugampola fractional Volterra functional differential equations with a vanishing lag function." Journal of Nonlinear Sciences and Applications, 13, no. 5 (2020): 293--302
Keywords
- Volterra functional equation
- existence
- uniqueness
- fixed point principle
- delay function
MSC
References
-
[1]
R. Almeida, A. B. Malinowska, T. Odzijewicz, Fractional Differential Equations With Dependence on the Caputo–Katugampola Derivative, J. Comput. Nonlinear Dynam., 11 (2016), 11 pages
-
[2]
T. M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley & Sons, Hoboken (2014)
-
[3]
A. Bellen, M. Zennaro, Numerical methods for delay differential equations. Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York (2003)
-
[4]
A. Buica, Existence and continuous dependence of solutions of some functional-differential equations, Seminar on Fixed Point Theory (Babes-Bolyai Univ., Cluj-Napoca), 3 (1995), 1--14
-
[5]
V. Daftardar-Gejji, Fractional Calculus and Fractional Differential Equations, Birkhäuser, Singapore (2019)
-
[6]
J. Deng, J. Wang, Existence and approximation of solutions of fractional order iterative differential equations, Central Eur. J. Phys., 11 (2013), 1377--1386
-
[7]
E. Eder, The functional differential equation $x'(t)=x(x(t))$, J. Differential Equations, 54 (1984), 390--400
-
[8]
M. M. Elborai, M. A. Abdou, M. I. Youssef, On the existence, uniqueness, and stability behavior of a random solution to a non local perturbed stochastic fractional integro-differential equation, Life Sci. J., 10 (2013), 3368--3376
-
[9]
M. M. Elborai, M. A. Abdou, M. I. Youssef, On some nonlocal perturbed random fractional integro-differential equations, Life Sci. J., 10 (2013), 1601--1609
-
[10]
M. M. Elborai, M. I. Youssef, On stochastic solutions of nonlocal random functional integral equations, Arab J. Math. Sci., 25 (2019), 180--188
-
[11]
A. M. A. El-Sayed, R. G. Ahmed, Solvability of the functional integro-differential equation with self-reference and state-dependence, J. Nonlinear Sci. Appl., 13 (2020), 1--8
-
[12]
M. A. E. Herzallah, Notes on some fractional calculus operators and their properties, J. Fract. Calc. Appl., 5 (2014), 10 pages
-
[13]
H. Hochstadt, Integral Equations, John Wiley & Sons, New York (1988)
-
[14]
U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860--865
-
[15]
U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1--15
-
[16]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B. V., Amsterdam (2006)
-
[17]
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney (1978)
-
[18]
D. Poljak, C. M. Cvetković, C. V. Dorić, C. I. Zulim, C. Z. Đogaš, C. J. Haueisen, K. E. Drissi, Integral Equation Formulations and Related Numerical Solution Methods in Some Biomedical Applications of Electromagnetic Fields: Transcranial Magnetic Stimulation (TMS), Nerve Fiber Stimulation, Int. J. E-Health Medical Commun., 9 (2018), 65--84
-
[19]
A. Ricardo, Variational Problems Involving a Caputo-Type Fractional Derivative, J. Optim. Theory Appl., 174 (2017), 276--294
-
[20]
S. Stanek, Globel properties of solutions of the functional differenatial equation $x(t)x'(t)=kx(x(t)),~ 0 <|k|< 1$, Funct. Differ. Equ., 9 (2004), 527--550
-
[21]
P. Umamaheswari, K. Balachandran, N. Annapoorani, Existence of solutions of stochastic fractional integrodifferential equations, Discontin. Nonlinearity Complex., 7 (2018), 55--65
-
[22]
S. D. Zeng, D. Baleanu, Y. R. Bai, G. C. Wu, Fractional differential equations of Caputo Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549--554