Abstract-The central idea of this paper is the modeling andimplementation of a real-time dc servomotor angular speed control system with an unknown bounded uncertainty using a sliding mode observer (SMO) control strategy. We prefer to use a SMO in our approach due to its great potential in fault detection and isolation (FDI) of the actuators and sensor faults subjected frequently to several failures due to an abnormal change in their operating conditions or parameters. We use for this purpose the most suitable real time implementation tool MATLAB/SIMULINK software package. It provides special features for real time implementation by its extensions RealTime Workshop (RTW) and the Real-Time Windows Target (RTWT). The novelty of our paper is to prove in an extensive simulation MATLAB/SIMULINK frame the real time implementation potential of a most recently sliding mode observer (SMO) control strategy applied to a particular case study, namely for a dc servomotor angular speed control system. The proposed real-time Sliding Mode Observer (SMO) consists of an embedded nonlinear Sliding Mode Observer (SMO) with the dc servomotor actuator in an integrated control system structure to estimate its angular speed and armature current and to implement the sliding mode control law.

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Using some of the extended fixed point results for Geraghty contractions in b-metric spaces given by Faraji, Savić and Radenović and their idea to apply these results to nonlinear integral equations, in this paper we present some existence and uniqueness conditions for the solution of a nonlinear Fredholm–Volterra integral equation with a modified argument.

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In this paper we will use the Picard operators technique, in order to establish the existence and uniqueness, data dependence and Gronwall-type results for the solutions of a Fredholm-Volterra functional-integral equation. The paper ends with a result of the Ulam-Hyers stability of this integral equation.

The following delay integral equationhas been proposed by Cooke and Kaplan to describe the spread of certain infectious diseases with periodic contact rate that varies seasonally. This mathematical model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to present an approximating algorithm for the continuous positive solution of this integral equation from the theory of epidemics. This algorithm is obtained by applying the successive approximations method and the rectangle formula, used for the calculation of the approximate value of integrals which appear in the right-handside of the terms of the sequence of successive approximations. In order to establish this approximating algorithm, we will suppose that this integral equation has a unique solution. The main result contains also the error of approximation of the solution obtained by applying this approximating algorithm.

Using the fixed point theorem given by [Rus, I. A., A Fiber generalized contraction theorem and applications, Mathematica, 41(64) (1999), No. 1, 85–90] and an idea of [Sotomayor, J., Smooth dependence of solution of differential equation on initial data: a simple proof, Bol. Soc. Brasil., 4 (1973), No. 1, 55–59] we establish some conditions of differentiability of the solution for the following system of integral equations: ... and such we obtain two theorems of differentiability. Finally, two examples are given.