In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.
Abstract. Sufficient conditions for the uniqueness, global existence and for the convergence to zero when t -> oo of solutions of an integral equation related to an epidemic model are proved. The existence result is proved by applying the Banach fixed point theorem and for the proof of the convergence result a new type of integral inequality is used.
IntroductionG. Gripenberg studied in the paper [3] the qualitative behaviour of solutions of the equationwhich arises in the study of the spread of an infectious disease that does not induce the permanent immunity. The existence of a nonnegative, continuous and bounded solution of the equation (1)
In this paper we reformulate the axioms of the well-known Solow macroeconomic growth model by means of the mathematical calculus on time scales. We derive a system of differential equations on a time scale T which is a generalization of the classical Solow fundamental differential equation for the continuous case as well as its discrete version. We also prove sufficient conditions for the exponential stability of equilibrium points of this system having positive coordinates. Applications of these results to the case of the Cobb-Douglas production function are given.
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