A coincidence point theorem and its applications to fractional differential equations
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Journal of Fixed Point Theory and Applications
A coincidence point theorem and its applications to fractional differential equations Maher Berzig and Marwa Bouali Abstract. We establish a coincidence point theorem in complete metric spaces. As a consequence, we show the existence of solutions to a system of initial value problem of fractional differential equations involving Riemann–Liouville fractional derivatives. Next, we derive several coincidence point theorems for new classes of sublinear and superlinear operators, in the context of ordered Banach spaces. Finally, we apply these results to discuss the existence of positive solutions to a class of initial value problem of fractional differential equations. Mathematics Subject Classification. 54H25, 34A08, 47N20, 45M20. Keywords. Coincidence point theorems, fractional differential equations, integral equations, positive solutions.
Introduction In [1], Birkhoff has shown that theorems of Perron–Frobenius and Jentzch may be obtained via Hilbert’s projective metric using the well-known Banach’s contraction principle. The same approach has since been developed by many authors, namely by Stetsenko and Imomnazarov [2], Bushell [3], Potter [4], Krause [5] and Nussbaum [6], to cite only a few. Recently, more results are obtained and applied to study various nonlinear equations, see, for instance, [7–19]. The innovation of this approach lies in the reformulation of some operator inequalities in ordered Banach spaces into contractions of certain fixed point theorems in complete metric spaces. For instance, Krasnoselskii [7] showed that certain concave monotone operators satisfy the generalized contraction principle. In [9,11], Chen proved that certain classes of sublinear operators satisfy the Krasnoselskii, Edelstein and Caristi fixed point theorems. The operators related to a variant of Meir–Keeler fixed point theorem have been investigated in [10,16]. 0123456789().: V,-vol
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M. Berzig, M. Bouali
The purpose of this paper is to extend this approach for coincidence problems, and to apply it to study the existence of solutions to some fractional differential equations involving Riemann–Liouville fractional derivatives. First, we establish a new coincidence point theorem for mappings satisfying certain nonlinear contractions. We investigate then the existence of solutions to an initial value problem of a system of fractional differential equations. Next, in ordered Banach spaces, we show that certain sublinear and superlinear operators have coincidence points with super-homogeneous and weak-homogeneous ones. Then we discuss the existence of positive solutions to some classes of initial value problems of fractional differential equations, where the operators satisfy a property of inverse monotonicity. Furthermore, we provide examples supporting the results. This paper is divided into two sections. In Sect. 1, we establish the main coincidence point theorem, and we investigate the existence of solutions to a system of initial value problems of fractional differentia
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