Analysis of Volterra integrodifferential equations with nonlocal and boundary conditions via Picard operator
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Analysis of Volterra integrodifferential equations with nonlocal and boundary conditions via Picard operator Pallavi U. Shikhare1 · Kishor D. Kucche1
· J. Vanterler da C. Sousa2
Received: 6 November 2019 / Revised: 10 June 2020 / Accepted: 20 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This article investigates the existence and uniqueness of solutions to the second-order Volterra integrodifferential equations with nonlocal and boundary conditions through its equivalent integral equations and fixed point of Banach. Furthermore, utilizing the Picard operator theory, we obtain the dependence of solutions on the initial nonlocal data and on functions involved on the right-hand side of the equations. Keywords Nonlocal conditions · Integrodifferential equations · Fixed point theorem · Picard operator · Data dependency Mathematics Subject Classification 45J05 · 34G20 · 47H10 · 34B15 · 65L10
1 Introduction Over the years, the study of solutions of differential equations has been the subject of research, and continues for a number of reasons, from theoretical comfort, which is about existence, uniqueness, controllability, among others, and in the practical sense, involving the stability and continuous dependence on data (Byszewski 1991, 1999; Bednarz and Byszewski 2015; Byszewski 2017; Balachandran and Park 2003; Bednarz and Byszewski 2018). Therefore, there are several attractive points to investigate the solutions of the various types of differential equations. On the other hand, we can also highlight the integrodifferential equations that have gained prominence in the scientific community (Balachandran and Park 2003; Lin and Liu
Communicated by Hui Liang.
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Kishor D. Kucche [email protected] Pallavi U. Shikhare [email protected] J. Vanterler da C. Sousa [email protected]
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Department of Mathematics, Shivaji University, Kolhapur, Maharashtra 416 004, India
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Department of Applied Mathematics, Imecc-Unicamp, Campinas, SP 13083-859, Brazil 0123456789().: V,-vol
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P. U. Shikhare et al.
1996; Muresan 2007, 2004; Kucche and Shikhare 2018, a, b, 2019; Kucche and Dhakne 2013). The nonlocal condition is a generalization of the classical initial condition. Studies with the nonlocal conditions are driven by theoretical premium, yet additionally, there are several events that happened in engineering, physics and life sciences that can be described by means of differential equations subject to nonlocal conditions (Bates 2006; Delgado et al. 2019). Therefore, differential equations with nonlocal condition have turned into an active zone of research. In 1991, Byszewski (1991) introduced the nonlocal Cauchy problem in abstract spaces. In the following years, Byszewski (1999), carried out the work pertaining to the existence and uniqueness of classical solutions of nonlocal Cauchy type problem in a Banach space. In the literature, many researchers have been commented on nonlocal conditions and investigated the various class of different
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