Filomat 2023 Volume 37, Issue 2, Pages: 393-402
https://doi.org/10.2298/FIL2302393K
Full text (
226 KB)
Fractional differential equations with maxima on time scale via Picard operators
Karapınar Erdal (Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot City, Binh Duong Province, Vietnam + Department of Mathematics, Çankaya University, Etimesgut, Ankara, Turkey + Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan), erdalkarapinar@tdmu.edu.vn; erdalkarapinar@yahoo.c
Benkhettou Nadia (Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, Sidi Bel Abbes, Algeria), benkhettou_na@yahoo.fr
Lazreg Jamal Eddine (Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, Sidi Bel Abbes, Algeria), lazregjamal@yahoo.fr
Benchohra Mouffak (Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, Sidi Bel Abbes, Algeria), benchohra@yahoo.com
In this paper, we prove a result of existence and uniqueness of solutions for
the following class of problem of initial value for differential equations
with maxima and Caputo’s fractional order on the time scales: cΔωa u(ϑ) =
ζ(ϑ, u(ϑ), max ς∈[a,ϑ] u(ς)), ϑ ∈ J := [a, b]T, 0 < ω ≤ 1, u(a) = ϕ, We used
the techniques of the Picard and weakly Picard operators to obtain some data
dependency on the parameters results.
Keywords: Fractional differential equations, Existence, Time scale, Picard operator, Initial value problem, Maxima, Fixed point, Abstract comparison principle, Data dependance
Show references
S. Abbas, M. Benchohra, J. E. Lazreg and G. M. N’Guerekata, Coupled systems of Hilfer fractional differential equations with maxima, J. Nonlinear Evol. Equ. Appl. 2 (2018), 11-24.
S. Abbas, M. Benchohra, J.E. Lazreg and Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: analysis and stability, Chaos Solitons Fractals 102 (2017), 47-71.
S. Abbas, M. Benchohra, J.E. Lazreg, A Alsaedi and Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equ. (2017) 14p.
S. Abbas, M. Benchohra, J. R. Graef, and J. Henderson, Implicit Fractional Differential and Integral Equations Existence and Stability, De Gruyter, 2018.
S. Abbas, M. Benchohra and G M. N’Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.
S. Abbas, M. Benchohra and G M. N’Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2014.
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.6652
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions , RACSAM (2021) 115:155; https://doi.org/10.1007/s13398-021- 01095-3
R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math., V.20, N.2, 2021,313-333
R. P. Agarwal, U. Aksoy, E. Karapinar, and I. M. Erhan. F-contraction mappings on metric-like spaces in connection with integral equations on time scales. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114 (3) (2020), Paper No. 147, 12 pp.
B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon, Hadamard-type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham, 2017.
A. Ahmadkhanlu and M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc. 38 (2012), no. 1, 241-252.
B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in Qualitative theory of differential equations (Szeged, 1988), 37-56, Colloq. Math. Soc. János Bolyai, 53 North-Holland, Amsterdam, 1990.
D. D. Bainov, S. Hristova, Differential Equations with Maxima, Chapman and Hall/CRC, Boca Ratonv, 2011.
D. Baleanu, Z.B. Güvenc，, and J.A.T. Machado New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
N. R.O. Bastos, Fractional calculus on time scales, Ph.D. thesis, University of Aveiro, 2012.
N. Benkhettou, A.M.C. Brito da Cruz and D.F.M. Torres, A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, Signal Processing 107 (2015), 230-237.
N. Benkhettou, S. Hassani and D.F.M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud University (Science) 28 (1) (2016), 93-98.
M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2001.
M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.
A. Cernea, On a fractional differential inclusion with ”Maxima”, Frac. Calc. Appl. Anal. 19 (2016), no. 5, 1292-1305.
A. Chis-Novac, R. Precup and Ioan A. Rus, Data dependance of fixed points for non-self generalized contractions, Fixed Point Theory 10 (2009), no. 1, 73-87.
S. Georgiev, Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, 2018.
S. Hilger, Analysis on measure chains: a unified approach to continuous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18-56.
R. W. Ibrahim, Extremal solutions for certain type of fractional differential equations with maxima, Adv. Difference Equ. 2012, 7, 8pp.
K. Nisse and L. Nisse, An iterative method for solving a class of fractional functional differential equations with Maxima, Mathematics, 2017, 13pp.
D. Otrocol, Hybrid differential equations with maxima via Picard operators theory, Stud. Univ. Babes. Bolyai. Math. 61 (2016), 421-428.
D. Otrocol, Systems of functional differential equations with maxima of mixed type, Electr. J. Qual. Theory Differ. Equ. (2014), 5, 9pp.
D. Otrocol and I. A. Rus, Functional differential equations with Maxima of mixed type, Fixed point theory 9 (1) (2008), 207-200.
I. A. Rus, Fiber Picard operators theorem and applications, Stud. Univ. Babes. Bolyai. Math. 44 (3) (1999), 89-97.
I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 3 (1983), 1-130.
I. A. Rus, Weakly Picard mappings, Comment. Math. Univ. Caroline. 34 (4) (1993), 769-773.
I. A. Rus, Weakly Picard operators and applications, Semin. Fixed Point Theory, Cluj-Napoca 2 (2001), 41-57.
I. A. Rus, Functional differential equation of mixed point, via weakly Picard operator, Semin. Fixed Point Theory, Cluj University Press, Cluj-Napoca, 3 (2002), 335-345.
I. A. Rus, Picard operators and applications Sci. Math. Jpn. 58 (2003), 191-219.
I. A. Rus, Generalized contractions, Cluj University Press, 2001.
Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.