Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fra
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Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative Kishor D. Kucche1 · Ashwini D. Mali1 Received: 9 May 2019 / Revised: 29 September 2019 / Accepted: 3 November 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract We build the quasilinearization method with initial time difference for nonlinear fractional differential equations (FDEs) involving generalized Hilfer fractional derivative under various conditions on the nonlinear function involved in the right hand side of the equation. An essential comparison result concerning lower and upper solutions is obtained for this generalized FDEs without demanding the Hölder continuity assumption. Keywords Generalized Hilfer fractional derivative · Quasilinearization method · Initial time difference · Comparison theorems Mathematics Subject Classification 26A33 · 34A12 · 34A45 · 34C11 · 34A40
1 Introduction The existence, uniqueness and various qualitative properties of solutions to nonlinear fractional differential equations (FDEs) have been investigated very well and plenty of papers are available on it in the literature. We refer the reader to a few of them in Vatsala and Lakshmikantham (2008), Diethelem and Ford (2002), Daftardar-Gejji and Babakhani (2004), Daftardar-Gejji and Jafari (2007), Kucche and Trujillo (2017), Kucche et al. (2016), Zhou (2009), Agrawal et al. (2009), Agrawal et al. (2010) and the references therein. It will, in general, be seen that a greater part of work relating to existence and uniqueness has endeavored through the fixed-point method. Therefore it is imperative to develop the different sorts of method to examine the existence and uniqueness of nonlinear FDEs.
Communicated by Roberto Garrappa.
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Kishor D. Kucche [email protected] Ashwini D. Mali [email protected]
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Department of Mathematics, Shivaji University, Kolhapur, Maharashtra 416 004, India
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K. D. Kucche, A. D. Mali
Lakshmikantham and Vatsala (1998) have developed the method of quasilinearization to deal with the problems of existence and uniqueness of nonlinear differential equations. The pioneering work on the quasilinearization method for nonlinear ordinary differential equations can be found in Lakshmikantham (1994), Lakshmikantham and Köksal (1994), Lakshmikantham and Shahzad (1994). Noticing the significance of FDEs in modelling natural phenomena emerging in real world problem (Diethelm 2010; Kilbas et al. 2006; Podlubny 1999; Miller and Ross 1993; Lakshmikantham et al. 2009), the procedure of quasilinearization have been reached out to nonlinear FDEs (Devi and Suseela 2008; Denton et al. 2011; Vasundhara Devi et al. 2010; Wang and Hou 2013; Liu et al. 2014; Liu and Wang 2014). The technique of quasilinearization is a beneficial and constructive technique which gives an explicit analytic representation for the solution of any type of nonlinear differential equations. This method includes the construction o
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