Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations
Abstract
:1. Introduction
- (i)
- Hilfer and Caputo fractional nonlocal integro-differential boundary value problem if
- (ii)
- -Hilfer and Caputo-type fractional nonlocal integro-differential boundary value problem if
- (iii)
- -Hilfer and Caputo-type nonlocal integro-differential boundary value problem if
2. Preliminaries
- (i)
- ;
- (ii)
3. Main Results
- There exists such thatand .
- There exists a continuous function which is nondecreasing and two continuous functions such thatfor all and
- There exists a positive constant K such that
4. Some Special Cases
- (a)
- If , where M is a positive constant, then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J.
- (b)
- If , , , where , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J if
- (c)
- If , , , where , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution if
5. Illustrative Examples
- (i)
- If the function is defined by
- (ii)
- Let the function be defined as
- (iii)
- (iv)
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Gaul, L.; Klein, P.; Kempfle, S. Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5, 81–88. [Google Scholar] [CrossRef]
- Glockle, W.G.; Nonnenmacher, T.F. A fractional calculus approach of self-similar protein dynamics. Biophys. J. 1995, 68, 46–53. [Google Scholar] [CrossRef] [PubMed]
- Hilfer, R. Experimental evidence for fractional time evolution in glass forming materials. J. Chem. Phys. 2002, 284, 399–408. [Google Scholar] [CrossRef]
- Mainardi, F. Fractional calculus: Some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer: Vienna, Austria, 1997; pp. 291–348. [Google Scholar]
- Metzler, F.; Schick, W.; Kilian, H.G.; Nonnenmacher, T.F. Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 1995, 103, 7180–7186. [Google Scholar] [CrossRef]
- Diethelm, K. The Analysis of Fractional Differential Equations; Lecture Notes in Mathematics; Springer: New York, NY, USA, 2010. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of the Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Differential Equations; John Wiley: NewYork, NY, USA, 1993. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
- Zhou, Y. Basic Theory of Fractional Differential Equations; World Scientific: Singapore, 2014. [Google Scholar]
- Ahmad, B.; Ntouyas, S.K. Nonlocal Nonlinear Fractional-Order Boundary Value Problems; World Scientific: Singapore, 2021. [Google Scholar]
- Hilfer, R. (Ed.) Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
- Kamocki, R. A new representation formula for the Hilfer fractional derivative and its application. J. Comput. Appl. Math. 2016, 308, 39–45. [Google Scholar] [CrossRef]
- Vanterler da Sousa, J.; de Oliveira, E.C. On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 2018, 60, 72–91. [Google Scholar] [CrossRef]
- Vanterler da Sousa, J.; Kucche, K.D.; de Oliveira, E.C. On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Methods Appl. Sci. 2019, 42, 3021–3032. [Google Scholar] [CrossRef]
- Vanterler da Sousa, J.; de Oliveira, E.C. On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ–Hilfer operator. J. Fixed Point Theory Appl. 2018, 20, 96. [Google Scholar] [CrossRef]
- Ahmad, B.; Ntouyas, S.K.; Alsaedi, A.; Alotaibi, F.M. Existence results for a ψ-Hilfer type nonlocal fractional boundary value problem via topological degree theory. Dyn. Syst. Appl. 2021, 30, 1091–1103. [Google Scholar] [CrossRef]
- Sitho, S.; Ntouyas, S.K.; Samadi, A.; Tariboon, J. Boundary value problems for ψ-Hilfer type sequential fractional differential equations and inclusions with integral multi-point boundary conditions. Mathematics 2021, 9, 1001. [Google Scholar] [CrossRef]
- Kiataramkul, C.; Ntouyas, S.K.; Tariboon, J. Existence results for ψ-Hilfer fractional integro-differential hybrid boundary value problems for differential equations and inclusions. Adv. Math. Phys. 2021, 2021, 9044313. [Google Scholar] [CrossRef]
- Kiataramkul, C.; Ntouyas, S.K.; Tariboon, J. An existence result for ψ-Hilfer fractional integro-differential hybrid three-point boundary value problems. Fractal. Fract. 2021, 5, 136. [Google Scholar] [CrossRef]
- Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef]
- Mallah, I.; Ahmed, I.; Akgul, A.; Jarad, F.; Alha, S. On ψ-Hilfer generalized proportional fractional operators. AIMS Math. 2021, 7, 82–103. [Google Scholar] [CrossRef]
- Granas, A.; Dugundji, J. Fixed Point Theory; Springer: New York, NY, USA, 2003. [Google Scholar]
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Sitho, S.; Ntouyas, S.K.; Sudprasert, C.; Tariboon, J. Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics 2023, 11, 867. https://doi.org/10.3390/math11040867
Sitho S, Ntouyas SK, Sudprasert C, Tariboon J. Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics. 2023; 11(4):867. https://doi.org/10.3390/math11040867
Chicago/Turabian StyleSitho, Surang, Sotiris K. Ntouyas, Chayapat Sudprasert, and Jessada Tariboon. 2023. "Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations" Mathematics 11, no. 4: 867. https://doi.org/10.3390/math11040867
APA StyleSitho, S., Ntouyas, S. K., Sudprasert, C., & Tariboon, J. (2023). Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics, 11(4), 867. https://doi.org/10.3390/math11040867