Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions
Volume 14, Issue 3, pp 124--138
http://dx.doi.org/10.22436/jnsa.014.03.02
Publication Date: September 16, 2020
Submission Date: June 24, 2020
Revision Date: August 07, 2020
Accteptance Date: August 17, 2020
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Authors
Ahmad Y. A. Salamooni
- School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India.
D. D. Pawar
- School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606, India.
Abstract
This article contains a new discussion for Hilfer-Katugampola-type fractional derivative.
We establish an existence and uniqueness results of Hilfer-Katugampola-type fractional
derivative for implicit differential equations with the help of Schaefer's fixed point theorem and Banach contraction principle. Also,
we use the Gronwall's lemma for singular kernels to prove the Ulam-Hyers-Rassias stability results.
Further, the examples are given to illustrate our main results.
Share and Cite
ISRP Style
Ahmad Y. A. Salamooni, D. D. Pawar, Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions, Journal of Nonlinear Sciences and Applications, 14 (2021), no. 3, 124--138
AMA Style
Salamooni Ahmad Y. A., Pawar D. D., Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions. J. Nonlinear Sci. Appl. (2021); 14(3):124--138
Chicago/Turabian Style
Salamooni, Ahmad Y. A., Pawar, D. D.. "Existence and stability results for Hilfer-Katugampola-type fractional implicit differential equations with nonlocal conditions." Journal of Nonlinear Sciences and Applications, 14, no. 3 (2021): 124--138
Keywords
- Hilfer-Katugampola-type fractional derivative
- implicit differential equation
- Schaefer's fixed point theorem
- existence
- uniqueness
- Ulam stability
MSC
References
-
[1]
S. Abbas, M. Benchohra, J. E. Lagreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equ., 2017 (2017), 14 pages
-
[2]
S. Abbas, M. Benchohra, G. M. N’Guerekata, Topics in fractional differential equations, Developments in Mathematics, Springer, New York (2012)
-
[3]
R. Almeida, Variational problems involving a Caputo-type fractional derivative, J. Optim. Theory Appl., 174 (2017), 276--294
-
[4]
Sz. Andras, J. J. Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal. Theory Methods Appl., 82 (2013), 1--11
-
[5]
D. Baleanu, Z. B. Guvenc, J. A. Tenreiro Machado (Eds.), New trends in nanotechnology and fractional calculus applications, Springer, New York (2010)
-
[6]
M. Benchohra, J. E. Lazreg, Nonlinear fractional implicit differential equations, Commun. Appl. Anal., 17 (2013), 471--482
-
[7]
M. Benchohra, J. E. Lazreg, On stability for nonlinear fractional implicit differential equations, Matematiche (Catania), 70 (2015), 49--61
-
[8]
M. Benchohra, J. E. Lazreg, Existence and Ulam stability for nonlinear fractional implicit differential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math., 62 (2017), 27--38
-
[9]
S. P. Bhairat, Existence and stability of fractional differential equations involving generalized Katugampola derivative, Stud. Univ. Babes-Bolyai. Math., 65 (2020), 29--46
-
[10]
S. P. Bhairat, On stability of generalized Cauchy-type problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 27 (2020), 235--244
-
[11]
R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: modeling and control applications, World scientific publishing Co., (2010)
-
[12]
A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer-Verlag, Vienna (1997)
-
[13]
S. Das, Functional fractional calculus, Springer, Berlin (2011)
-
[14]
A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York (2003)
-
[15]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge (2000)
-
[16]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
-
[17]
R. W. Ibrahim, Generalized UlamHyers stability for fractional differential equations, Internat. J. Math., 23 (2012), 9 pages
-
[18]
R. W. Ibrahim, S. Harikrishnan, K. Kanagarajan, Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives, Stud. Univ. Babes-Bolyai Math., 63 (2018), 291--302
-
[19]
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135--1140
-
[20]
S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order II, Appl. Math. Lett., 19 (2006), 854--858
-
[21]
S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer, New York (2011)
-
[22]
U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860--865
-
[23]
U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1--15
-
[24]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)
-
[25]
F. Mainardi, Fractional calculus and waves in linear viscoelasticity, An introduction to mathematical models, Imperial College Press, London (2010)
-
[26]
P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102 (2015), 631--642
-
[27]
D. S. Oliveira, E. C. de Oliveira, Hilfer-Katugampola fractional derivative, Comput. Appl. Math., 37 (2018), 3672--3690
-
[28]
I. Podlubny, Fractional differential equations, Academic Press, San Diego (1999)
-
[29]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc, 72 (1978), 297--300
-
[30]
Th. M. Rassias, J. Brzdek, Functional equations in mathematical analysis, Springer, New York (2012)
-
[31]
I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103--107
-
[32]
J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado (Eds.), Advances in fractional calculus, Springer, Dordrecht (2007)
-
[33]
A. Y. A. Salamooni, D. D. Pawar, Unique positive solution for nonlinear Caputo-type fractional q-difference equations with nonlocal and Stieltjes integral boundary conditions, Fract. Differ. Calc., 9 (2019), 295--307
-
[34]
A. Y. A. Salamooni, D. D. Pawar, Existence and continuation of solutions of HilferKatugampola-type fractional differential equations, arXiv, 2020 (2020), 19 pages
-
[35]
S. M. Ulam, A collection of mathematical problems, Interscience Publishers, New York (1960)
-
[36]
S. M. Ulam, Problems in modern mathematics, John Wiley & Sons, New York (1964)
-
[37]
D. Vivek, K. Kanagarajan, E. M. Elsayed, Some existence and stability results for hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15 (2018), 21 pages
-
[38]
J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 10 pages
-
[39]
J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530--2538