Some fixed point theorems via measure of noncompactness with applications to differential equations
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Some fixed point theorems via measure of noncompactness with applications to differential equations Shahram Banaei1 · Mohammad Mursaleen2,3,4 · Vahid Parvaneh5 Received: 2 January 2020 / Revised: 24 March 2020 / Accepted: 9 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this article, we prove some Darbo’s type fixed point theorems associated with measure of noncompactness via the concept of operators A( f ; .) and weakly J S-contractive condition in Banach space. Moreover, as an application of our results, we study the existence of solutions for a system of differential equations. Finally, we present an example to support the effectiveness of our results. Keywords Fixed point theorem · Measure of noncompactness · Condensing operator · Differential equation Mathematics Subject Classification Primary 47H08 · 47H10
1 Introduction Fixed point theory is one of the most effective and applicable tools used in analysis to study the existence of solutions for a system of differential equations. Kuratowski (1930) in 1930, opened up a new direction of research with the presentation of the notion of a measure of noncompactness (MNC). Darbo’s fixed point theorem (Darbo 1955) is an important application
Communicated by Carlos Conca.
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Mohammad Mursaleen [email protected] Shahram Banaei [email protected] Vahid Parvaneh [email protected]
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Department of Mathematics, Bonab Branch, Islamic Azad University, Bonab, Iran
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Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan
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Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
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Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
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Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran 0123456789().: V,-vol
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of this measure, which it generalizes Schauder’s fixed point theorem and Banach contraction principle. Thereafter, many articles have been published in the MNC and it’s application, (see Agarwal et al. 2004; Aghajania et al. 2014; Aghajani et al. 2011; Aghajani and Sabzali 2014; Banaei 2019, 2018; Banaei and Ghaemi 2019; Chandok et al. 2015 and Jleli et al. 2016; Hazarika et al. 2019; Mohiuddine et al. 2016; Mursaleen et al. 2017; Mursaleen and Mohiuddine 2012; Mursaleen and Rizvi 2016; Saadati et al. 2019; Srivastava et al. 2018). The aim of this work is to state and prove some Darbo type fixed point theorems associated with measure of noncompactness via the concept of operators A( f ; .) and weakly JS-contractive condition in Banach space. Moreover, we apply our obtained results to investigate the existence of solutions for a system of differential equations with the initial condition. Now, we present some facts and definitions which are used throughout this article. Throughout this article, let R, denotes the set of real numbers, R+ = [0, +∞). Let (E, ·) be a real Banach space. Moreover
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