Investigating at Least One Solution to a Systems of Multi-Points Boundary Value Problems

Abstract

AABSTRACTCurrent article is concerned with the investigating a coupled systems of boundary value problems(BVPs) of fractional order differential equations (FDEs). By means of classical fixed point theorems, we prove the existence and uniqueness of positive solution. We include a numerical problem to verify the investigated results. IntroductionIn recent years, the theory on existence and uniqueness of solutions to (FDEs) has got considerable attentions. In this regard, plenty of research papers and books are available in literature dealing various aspects of (FDEs) (Ahmad and Sivasaundaran, 2010;Rehman and Khan, 2010;Zhang, 2010). Existence and uniqueness of solutions for multi-point initial /(BVPs) have been studied by many researchers (Yang and Ge, 2009;El-Sayed and Bin-Taher, 2013;Cui et al., 2012;Salem, 2009;Han and Wang, 2011;El-Shahed and Nieto, 2010). On the other hand, coupled systems of (BVPs) for non-linear (FDEs) are also studied by many researchers and large numbers of papers can be found dealing with existence and uniqueness of solutions (Gafiychuk et al., 2009;Haq et al., 2016;Shah and Khan, 2015a;Shah et al., 2015b;Shoab et al., 2016). The area devoted to the study of coupled systems of both classical and arbitrary order differential equations is an active area of research. Because the concerned systems modeled variety of scientific problems of dynamical systems, psychological and biological phenomenon and chemical process (Benchohra et al., 2008;Perov, 1964;Han and Wang, 2011;. Authors some appropriate conditions for existence of solution to the given coupled systems of four point (BVPs) is the standard Caputo derivative . Some researcher has obtained enough conditions related to the existence theory of positive solutions to the given coupled systems of nonlinear three-point (BVPs)are continuous (Ahmad and Nieto, 2009).Motivated by the above studies, we develop some new existence and uniqueness results for the following coupled systems of nonlinear (BVPs) of (FDEs) . We apply Banach fixed point theorem and Leray-Schauder fixed point theorem to obtain proper conditions for existence and uniqueness results . We also provide a numerical problem to demonstrate the establish results. Basic materialWe recall some fundamental results and definitions (Benchohra eta al., 2008;Ahmad and Sivasaundaram., 2010;Zhang, 2010;.Definition 2.2. The Caputo fractional order derivative of a function 2) There existis equivalent to the following coupled system of FredHolm integralProof. By means of Definition 2.2 and Theorem 2.3, the solution ofNow applying the boundary conditionsSimilarly, we can get second equation of (2.2) by solving second equation of system of BVP (1.1) asTherefore, in view of Theorem 2.5, the considered coupled system (1.1) is equivalent to the following coupled system of Fred Holm integral equations Remark 2.6. We will usethroughout this paper. Method and discussion Define. Then, the product spacesTo proceed further the following assumptions hold:Define the operatorAlso we can easily show th...