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We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results. MSC: Primary 34A08; secondary 34A12Keywords: Caputo fractional derivative; Hybrid fractional differential equation and inclusion; Thermostat modelingh(t,z(t)) ] + g(t, z(t)) = 0 (t ∈ [0, 1], p ∈ (1, 2]), z(0) = z(1) = 0.

Highlights COVID-19 is transmitted from asymptomatic individuals to susceptible individuals. COVID-19 is transmitted from symptomatic individuals to susceptible individuals. Since R0=1.6 is greater than 1, the COVID-19 will spread exponentially. If COVID-19 is not controlled, it is estimated that about 20 million people will become infected in the next three years.

In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundaryvalue problem D α u(t) = f (t, u(t)) with a RiemannLiouville fractional derivative via the different boundary-value problems u(0) = u(T), and the threepoint boundary condition u(0) = β 1 u(η) and u(T) = β 2 u(η), where T > 0, t ∈ I = [0, T], 0 < α < 1, 0 < η < T, 0 < β 1 < β 2 < 1.

By using the fractional Caputo-Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

By mixing the idea of 2-arrays, continued fractions, and Caputo-Fabrizio fractional derivative, we introduce a new operator entitled the infinite coefficient-symmetric Caputo-Fabrizio fractional derivative. We investigate the approximate solutions for two infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential problems. Finally, we analyze two examples to confirm our main results. MSC: 30B70; 34A08

By using the fractional Caputo-Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order fractional integro-differential equations. We illustrate our results by providing two examples.

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