Existence and continuous dependence of mild solutions for impulsive fractional integrodifferential equations in Banach s
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Existence and continuous dependence of mild solutions for impulsive fractional integrodifferential equations in Banach spaces Safwan Al-Shara1
· Ahmad Al-Omari1
Received: 28 March 2020 / Revised: 29 August 2020 / Accepted: 30 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In the present paper, we investigate the existence, uniqueness, and continuous dependence of mild solutions of impulsive neutral integrodifferential equations with nonlocal conditions in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results. An example is given to illustrate our results. Keywords Impulsive · Integrodifferential equation · Fixed point · Nonlocal condition · Hadamard fractional derivative · Mild solution Mathematics Subject Classification 34A08 · 34N05 · 34A12
1 Introduction In recent studies, the theory of fractional differential equations and inclusions has been into the focus of many of them. This is due to its extensive applications in numerous branches of applied sciences, such as physics, economics, and engineering sciences (see, for instance, (Ahmad et al. 2013; Anguraj and Karthikeyan 2009; Anguraj and Maheswari 2012; Deimling 1985; Pazy 1983; Yang et al. 2015, 2018; Yang and Machado 2019; Yang et al. 2017b) and references therein). Fractional differential equations and their applications make it possible to find appropriate models for describing real-world problems which cannot be described using classical integral-order differential equations. Some recent contributions to the subject can be found in Ahmad and Ntouyas (2015) and references therein. It has been noticed that most of the work on the topic is based on Riemann–Liouville and Caputo-type fractional
Communicated by José Tenreiro Machado.
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Safwan Al-Shara [email protected] Ahmad Al-Omari [email protected]
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Department of Mathematics, Faculty of Sciences, Al al-Bayt University, P.O. Box 130095, Al-Mafraq 25113, Jordan 0123456789().: V,-vol
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differential equations. However, there are other kinds of fractional derivatives that appear side by side with Riemann–Liouville and Caputo derivatives. The fractional derivative is due to Hadamard who introduced it in 1892 (Hadamard 1892), but which differs from the preceding ones, in the sense that the kernel of the integral (in the definition of the Hadamard derivative) contains a logarithmic function of arbitrary exponent. The details and properties of the Hadamard fractional derivative and integral can be found in Ahmad et al. (2013), Katatbeh and Al-Omari (2016), Kilbas (2001), Liu et al. (2020), Yang et al. (2020), Yang (2019) and Yang (2017); Yang et al. (2017a). Recently, Li et al. (2012) studied fractional integrodifferential equations of the type: t q D x(t) = Ax(t) + f (t, x(t) + k(t, s) h(t, s, x(s)) ds) 0
x(0) = g(x) + x0 , using measure of noncompactness and fixed point theorem of condensing maps. Also, Radhakrishnan and Bhalchandr
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