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The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved. [ABSTRACT FROM AUTHOR]

In this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: { D T α t (1 (h (t)) p − 2 ϕ p (h (t) 0 C D t α u (t))) + a (t) ϕ p (u (t)) = λ f (t , u (t)) , a.e. t ∈ [ 0 , T ] , α 1 ϕ p (u (0)) − α 2 t D T α − 1 (ϕ p (0 C D t α u (0))) = 0 , β 1 ϕ p (u (T)) + β 2 t D T α − 1 (ϕ p (0 C D t α u (T))) = 0 , where D t α 0 C , D T α t are the left Caputo and right Riemann–Liouville fractional derivatives of order α ∈ (1 2 , 1 ] , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained. [ABSTRACT FROM AUTHOR]

The paper is concerned with the solvability for several nonlinear boundary value problems of fractional p‐Laplacian differential equation involving the right‐handed Riemann‐Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, sufficient conditions for existence and uniqueness of extremal solutions are obtained and they extend existing results. At last, two examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solvability for fractional Sturm-Liouville boundary value problems with $p(t)$ -Laplacian operator at resonance using Mawhin's continuation theorem. Sufficient conditions for the existence of solutions have been acquired, and they would extend the existing results. Furthermore, an example is provided to illustrate the main result. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic boundary value problems for the coupled systems of fractional implicit differential equations. Basing on the coincidence degree theory, we establish the existence and uniqueness theorems. Further, we provide several examples to show our main results. [ABSTRACT FROM AUTHOR]