Monotone iterative technique for Sobolev type fractional integro-differential equations with fractional nonlocal conditi
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Monotone iterative technique for Sobolev type fractional integro-differential equations with fractional nonlocal conditions Renu Chaudhary1 Received: 4 July 2019 / Accepted: 22 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract By applying monotone iterative technique coupled with the method of lower and upper solutions, sufficient conditions are obtained for the existence of mild solution of Sobolev type fractional integro-differential equations with fractional nonlocal conditions. The results are obtained under monotonicity and measure of noncompactness conditions of the nonlinear functions. At last, an example is given to show the applicability of main results. Keywords Fractional differential equations · Monotone iterative technique · Nonlocal conditions · Upper and lower solutions Mathematics Subject Classification 34A08 · 34G20 · 34K30
1 Introduction In recent few decades, fractional differential equations have been the subject of great deal of research due to its wide applicability in science and engineering such as physics [1], viscoelasticity [2], electrical engineering [3], signal processing [4], bioengineering [5], control systems [6] and so on. Fractional differential equations have been proved to be valuable tools in modeling of many real world phenomenon such as hereditary properties of various materials and processes, nonlinear oscillations of earthquake, dynamical processes in porous structures. For more details on fractional differential equations and applications we refer to the monographs [7–10] and references cited therein. The nonlocal Cauchy problem was first introduced by Byszewski [11–14]. The differential equations with nonlocal conditions describe many physical phenomena more practically and in turn to led better effects in applications than the classical ones, for instance, Deng [15] shows that the diffusion phenomenon of a small amount of gas in a transparent tube can be described using the nonlocal conditions better than using the local conditions. Therefore non local conditions can be applied in applied sciences, engineering and physical etc. with better effect. The differential
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Renu Chaudhary [email protected] School of Basic and Applied Sciences, GD Goenka University, Gurugram, India
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R. Chaudhary
equations with nonlocal conditions have been investigated by many researchers see [16–19] and references cited therein. The Sobolev type differential equations can be considered as an abstract formulation of partial differential equations which occurs in several applications such as in the flow of fluid through fissured rocks [20], thermodynamics [21], and shear in second order fluids [22]. Moreover, Sobolev type fractional integro-differential equations appear in the theory of control of dynamical systems, when the controlled system or/and the controller is characterized by a Sobolev type fractional integro-differential equation. Moreover, the mathematical modeling and simulations of systems and processes are based on the description of their
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