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In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann-Liouville fractional derivative. Applying the Banach contraction principle and the Leray-Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

We study the existence, uniqueness, and various kinds of Ulam-Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii's fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ($$\textsc {S}$$ S ), exposed ($$\textsc {E}$$ E ), asymptomatic infected ($$\textsc {I}_1$$ I 1 ), symptomatic infected ($$\textsc {I}_2$$ I 2 ), and recovered ($$\textsc {R}$$ R ) classes named $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model, using the Caputo fractional derivative. Here, $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model describes the effect of asymptomatic and symptomatic transmissions on coronavirus disease outbreak. The existence and uniqueness of the solution are studied with the help of Schaefer- and Banach-type fixed point theorems. Sensitivity analysis of the model in terms of the variance of each parameter is examined, and the basic reproduction number $$(R_{0})$$ ( R 0 ) to discuss the local stability at two equilibrium points is proposed. Using the Routh–Hurwitz criterion of stability, it is found that the disease-free equilibrium will be stable for $$R_{0} < 1$$ R 0 < 1 whereas the endemic equilibrium becomes stable for $$R_{0} > 1$$ R 0 > 1 and unstable otherwise. Moreover, the numerical simulations for various values of fractional-order are carried out with the help of the fractional Euler method. The numerical results show that asymptomatic transmission has a lower impact on the disease outbreak rather than symptomatic transmission. Finally, the simulated graph of total infected population by proposed model here is compared with the real data of second-wave infected population of COVID-19 outbreak in India.

We present some results on the existence, uniqueness and Hyers-Ulam stability to the solution of an implicit coupled system of impulsive fractional differential equations having Hadamard type fractional derivative. Using a fixed point theorem of Kransnoselskii's type, the existence and uniqueness results are obtained. Along these lines, different kinds of Hyers-Ulam stability are discussed. An example is given to illustrate the main theorems.