Abstract
By using fractional calculus and fixed point theorems, existence of mild solution for nonlinear Hilfer fractional integro-partial differential system is studied. In addition, sufficient conditions for controllability of Hilfer fractional integro-partial differential system is established.
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References
B. Bonilla, M. Rivero, L. Rodriguez-Germa, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Appl. Math. Comput. 187, 79–88 (2007).
V. Lakshmikantham, “Nonlinear analysis, theory of fractional functional differential equations,” Nonlin. Anal. 69, 3337–3343 (2008).
V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlin. Anal. 69, 2677–2682 (2008).
I. Podlubny, Fractional Differential Equations (Academic, San Diego, 1999).
Fang Wang, Ping Wang, and Zhengan Yao, “Approximate controllability of fractional partial differential equation,” Adv. Differ. Equat. 367 (2015).
K. M. Furati, M. D. Kassim, and N. e-Tatar, “Existence and uniqueness for a problem involving Hilfer factional derivative,” Comput. Math. Appl. 64, 1612–1626 (2012).
H. Gu and J. Trujillo, “Existence of mild solution for evolution equation with Hilfer fractional derivative,” Appl. Math. Comput. 257, 344–354 (2015).
K. Balachandran and R. Sakthivel, “Controllability of integrodifferential systems in Banach spaces,” Appl. Math. Comput. 118, 63–71 (2001).
H. M. Ahmed, “On some fractional stochastic integral equations,” Int. J. Math. Anal. 2, 299–306 (2008).
H. M. Ahmed, “Semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions,” J. Theor. Probab. 26 (4) (2013).
V. Dhanapalan, M. Thamilselvan, and M. Chandrasekaran, “Nonlocal fractional semilinear integrodifferential equations in separable Banach spaces,” Am. J. Appl. Math. 2, 60–63 (2014).
K. Balachandran and R. Sakthivel, “Controllability of delay integrodifferential systems in Banach spaces,” Libertas Math. 13, 119–127 (1998).
H. O. Fattorini, “Boundary control systems,” SIAM J. Control 6, 349–384 (1968).
V. Barbu, “Boundary control problems with convex cost criterion,” SIAM J. Control Optimiz. 18, 227–248 (1980).
K. Balachandran and E. R. Anandhi, “Boundary controllability of delay integrodifferential systems in Banach spaces,” J. Korean Soc. Ind. Appl. Math. 4, 67–75 (2000).
K. Balachandran and E. R. Anandhi, “Boundary controllability of integro-differential systems in Banach spaces,” Proc. Ind. Acad. Sci., Math. Sci. 111, 127–135 (2001).
R. Sakthivel and Y. Ren, “Approximate controllability of fractional differential equations with state-dependent delay,” Results Math. 63 949–963 (2013).
H. M. Ahmed, “Boundary controllability of nonlinear fractional integrodifferential systems,” Adv. Differ. Equat. 2010, 279493 (2010).
H. M. Ahmed, “Controllability for Sobolev type fractional integro-differential systems in a Banach space,” Adv. Differ. Equat. 167 (2012).
H. M. Ahmed, “Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion,” IMA J. Math. Control Inform. 32, 781–794 (2015).
H. M. Ahmed, “Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space,” Adv. Differ. Equat. 2014, 113 (2014).
Yong Zhuo, Basic Theory of Fractional Differential Equation (World Scientific, China, 2014).
R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
R. Hilfer, “Experimental evidence for fractional time evolution in glass materials,” Chem. Phys. 284, 399–408 (2002).
H. M. Ahmed, M. M. El-Borai, H. M. El-Owaidy, and A. S. Ghanem, “Impulsive Hilfer fractional differential equations,” Adv. Differ. Equat. 2018, 226 (2018).
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Ahmed, H.M., El-Owaidy, H.M. & AL-Nahhas, M.A. Nonlinear Hilfer Fractional Integro-Partial Differential System. Lobachevskii J Math 40, 115–126 (2019). https://doi.org/10.1134/S1995080219020021
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DOI: https://doi.org/10.1134/S1995080219020021