We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.

Using the fractional Caputo-Fabrizio derivative, we investigate a new version of the mathematical model of Rabies disease. Using fixed point results, we prove the existence of a unique solution. We calculate the equilibrium points and check the stability of solutions. We solve the equation by combining the Laplace transform and Adomian decomposition method. In numerical results, we investigate the effect of coefficients on the number of infected groups. We also examine the effect of derivation orders on the behavior of functions and make a comparison between the results of the integer-order derivative and the Caputo and Caputo-Fabrizio fractional-order derivatives.

By using the notion of endpoints for set-valued functions and some classical fixed point techniques, we investigate the existence of solutions for two fractional [Formula: see text]-differential inclusions under some integral boundary value conditions. By providing an example, we illustrate our main result about endpoint. Also, we give some related algorithms and numerical results.

Let S denote the class of functions f (z) = z + a 2 z 2 + ... analytic and univalent in the open unit disc D = {z ∈ C||z| < 1}. Consider the subclass and S * of S, which are the classes of convex and starlike functions, respectively. In 1952, W. Kaplan introduced a class of analytic functions f (z), called close-to-convex functions, for which there exists φ(z) ∈ C, depending on f (z) with Re( f (z) φ (z) ) > 0 in , and prove that every close-to-convex function is univalent. The normalized class of close-to-convex functions denoted by K. These classes are related by the proper inclusions C ⊂ S * ⊂ K ⊂ S. In this paper, we generalize the close-to-convex functions and denote K(λ) the class of such functions. Various properties of this class of functions is alos studied.

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