Interval-valued functional integro-differential equations
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RESEARCH
Open Access
Interval-valued functional integro-differential equations Ngo Van Hoa1 , Nguyen Dinh Phu2 , Tran Thanh Tung3* and Le Thanh Quang2 * Correspondence: [email protected]; [email protected] 3 Faculty of Natural Science and Technology, Tay Nguyen University, 567 Le Duan Road, Buon Ma Thuot City, Daklak Province, Vietnam Full list of author information is available at the end of the article
Abstract This paper is devoted to studying the local and global existence and uniqueness results for interval-valued functional integro-differential equations (IFIDEs). In the paper, for the local existence and uniqueness, the method of successive approximations is used and for the global existence and uniqueness, the contraction principle is a good tool in investigating. Some examples are given to illustrate the results. MSC: 34G20; 34A12; 34K30 Keywords: interval-valued differential equations; generalized Hukuhara derivative; functional integro-differential equations
1 Introduction Functional differential equations (or, as they are called, delay differential equations) play an important role in an increasing number of system models in biology, engineering, physics and other sciences. There exists an extensive amount of literature dealing with functional differential equations and their applications; the reader is referred to the monographs [–] and the references therein. The set-valued differential and integral equations are an important part of the theory of set-valued analysis. They have an important value in theory and application in control theory; and they were studied in by De Blasi and Iervolino []. Recently, setvalued differential equations have been studied by many authors due to their application in many areas. For many results in the theory of set-valued differential and integral equations, the readers can be referred to the following books and papers [–] and the references therein. The interval-valued analysis and interval-valued differential equations (IDEs) are the particular cases of the set-valued analysis and set differential equations, respectively. In many cases, when modeling real-world phenomena, information about the behavior of a dynamic system is uncertain, and we have to consider these uncertainties to gain more models. The interval-valued differential and integro-differential equations can be used to model dynamic systems subject to uncertainties. Recently, many works have been done by several authors in the theory of interval-valued differential equations (see, e.g., [–]). These equations can be studied with a framework of the Hukuhara derivative []. However, it causes that the solutions have increasing length of their values. Stefanini and Bede [] proposed to consider the so-called strongly generalized derivative of interval-valued functions. The interval-valued differential equations with this deriva©2014 Hoa et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses
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