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In this paper, we investigate the convergence of approximate solutions for a class of first-order integro-differential equations with antiperiodic boundary value conditions. By introducing the definitions of the coupled lower and upper solutions which are different from the former ones and establishing some new comparison principles, the results of the existence and uniqueness of solutions of the problem are given. Finally, we obtain the uniform and rapid convergence of the iterative sequences of approximate solutions via the coupled lower and upper solutions and quasilinearization method. In addition, an example is given to illustrate the feasibility of the method. [ABSTRACT FROM AUTHOR]

Passive worms are prone to spreading through Peer-to-Peer networks, and they pose a great threat to the security of the network. In this paper, considering network heterogeneity and the number of hops a search can reach, we propose a novel mathematical model to study the dynamics of the propagation of passive worms. For the proposed model, the basic reproduction number R0 is derived by employing the existence of the positive equilibrium. And the stabilities of the worm-free equilibrium and positive equilibrium are analyzed. Moreover, we verify the rationality of the model established by comparing the stochastic simulation with the numerical simulation. Finally, we examine the effect of the number of hops on the spread of passive worms and discuss the various immunization strategies. We find that if R0>1, the propagation speed of passive worms is accelerated with the increase of hop count d; if R0<1, the number of infected peers decreases rapidly with the increase of the value of d and drops to zero eventually. Results show that the network topology and the number of hops can affect the spread of passive worms. [ABSTRACT FROM AUTHOR]