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Impulsive Differential Equations and Inclusions
... Introduction. Fractional calculus extends the traditional derivative and integral concepts to non-integer orders [3,4,15]. Numerous works, including books and articles, explore linear and nonlinear problems related to fractional differential equations and various types of fractional derivatives, see [1,5,6,9,10,14] for more details. ...
... The extensive study of fractional differential equations involving abrupt and instantaneous impulses covers various aspects of solution existence and qualitative properties [15,19,32]. In pharmacotherapy, the dynamics of certain processes, such as the gradual absorption of drugs into the bloodstream for hemodynamic equilibrium, cannot be accurately described by instantaneous impulses. ...
... We provide the initial explanation for the model in the hope that neuroscience specialists will accept and adapt our suggestions further. The symmetry will open up new possibilities for productive application of the methods introduced and developed within the last few years for various types of impulsive systems [20][21][22] and networks [23][24][25]. Another interesting opportunity involves combining methods for discontinuous dynamics with those for synchronization [26,27]. ...
The purpose of this paper is to study the dynamics of Hopfield neural networks with impulsive effects, focusing on Poisson stable rates, synaptic connections, and unpredictable external inputs. Through the symmetry of impulsive and differential compartments of the model, we follow and extend the principal dynamical ideas of the founder. Specifically, the research delves into the phenomena of unpredictability and Poisson stability, which have been examined in previous studies relating to models of continuous and discontinuous neural networks with constant components. We extend the analysis to discontinuous models characterized by variable impulsive actions and structural ingredients. The method of included intervals based on the B-topology is employed to investigate the networks. It is a novel approach that addresses the unique challenges posed by the sophisticated recurrence.
... Hence, it is highly important to study impulsive generalizations of the HCV models. Such impulsive generalizations are represented by systems of impulsive differential equations [24][25][26][27][28]. Impulsive control systems [29,30] may offer a convenient tool for designing efficient therapy for HCV disease. ...
In this paper, we offer a new modelling approach for hepatitis C virus (HCV) infection models with cytotoxic T lymphocytes (CTL) immune responses. The standard neural network models with reaction-diffusion terms are generalized to the discontinuous case considering impulsive controllers. Also, the conformable approach is proposed as a more adjustable modelling apparatus which overcomes existing difficulties in the application of the fractional-order modelling technique. In addition, the existence of integral manifolds for the proposed model is investigated and efficient criteria are established. Since stability and control of a neural network model are essential for its proper application, stability results for the constructed integral manifold are presented. These results extend and generalize the stability results for steady states of proposed HCV models with CTL immune responses and contribute to the development of new innovative modelling approaches in biology. The qualitative analysis is performed via the conformable Lyapunov function method. The established results are demonstrated through an example.
The obscure mealybug, Pseudococcus viburni, is a serious agricultural pest worldwide. The biological control in commercial fields of P. viburni relies on predators and parasitoids, in particular the generalist coccidophagous ladybird Cryptolaemus montrouzieri and the specific parasitoid Acerophagus flavidulus. However, these two natural enemies can establish an intraguild predation interaction, reducing the efficiency of biological control. Cryptolaemus montrouzieri may negatively impact the population dynamics of A. flavidulus if it feeds indifferently on healthy and parasitized mealybugs. With the aim of improving the biological control of P. viburni, in this work, we studied the feeding behavior of C. montrouzieri in the absence or presence of A. flavidulus larvae of different age within mealybugs, in laboratory conditions. Subsequently, with the data obtained, we mathematically modeled the dynamics of P. viburni to study the impact on P. viburni control of different field implementation schedules for the release of ladybird and parasitoid populations. The ladybird fed on parasitized P. viburni but reduced its consumption when they were infested by parasitoids aged of 4 days or more. Modeling results suggest that these feeding preferences of predators may have a positive impact on pest control, that releasing predators and parasitoids together is in general more effective than releasing them independently, and that releasing highly effective predators alone could be the best choice. Modeling results also provide information on different release schedules.
In this paper, we study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear impulsive Sturm–Liouville boundary value problem with a parameter. By using a variational method, we prove that the problem has at least two positive solutions for the parameter λ∈(0,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \in (0,\Lambda )$\end{document}, one positive solution for λ=Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda =\Lambda $\end{document}, and no positive solution for λ>Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >\Lambda $\end{document}, where Λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Lambda >0$\end{document} is a constant.
In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.
In this paper, we shall establish sufcient conditions for the existence, approximate controllability, and Ulam-Hyers-Rassias stability of solutions for impulsive integrodiferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fxed point with measures of noncompactness. An example is presented to illustrate the efciency of the result obtained.
A double fixed point theorem is applied to obtain the existence of at least two positive solutions for a right focal two point boundary value problem for a second order ordinary differential equation with impulse effect.
In this paper, a nonlinear alternative of the Leray Schauder type for multivalued maps combined with upper and lower solutions are used to investigate the existence of solutions for first order impulsive differential inclusions with nonlinear boundary conditions and variable moments.
The aim of the paper is to discuss the existence of maximal and minimal solutions of a nonlinear nonlocal hyperbolic differential problem. A monotone iterative technique, considered in [G. S. Ladde, V. Lakshmihantham and A. S. Vatsala, Monotone iterative techniques for nonlinear differential equations (1985; Zbl 0658.35003)], is used to prove this result.
We investigate the existence of mild solutions to first-order semilinear delay integrodifferential inclusions of Sobolev type, in Banach spaces with nonlocal conditions. We rely of a fixed-point theorem for condensing map due to Martelli.
Preface. 1: 1.0. Introduction. 1.1. Measure Chains and Time Scales. 1.2. Differentiation. 1.3 Mean Value Theorem and Consequences. 1.4. Integral and Antiderivative. 1.5. Notes. 2: 2.0. Introduction. 2.1. Local Existence and Uniqueness. 2.2. Dynamic Inequalities. 2.3. Existence of Extremal Solutions. 2.4. Comparison Results. 2.5. Linear Variation of Parameters. 2.6. Continuous Dependence. 2.7. Nonlinear Variation of Parameters. 2.8. Global Existence and Stability. 2.9. Notes. 3: 3.0. Introduction. 3.1. Comparison Theorems. 3.2. Stability Criteria. 3.3. A Technique in Stability Theory. 3.4. Stability of Conditionally Invariant Sets. 3.5. Stability in Terms of Two Measures. 3.6. Vector Lyapunov Functions and Practical Stability. 3.7. Notes. 4: 4.0. Introduction. 4.1. Monotone Iterative Technique. 4.2. Method of Quasilinearization. 4.3. Monotone Flows and Stationary Points. 4.4. Invariant Manifolds. 4.5. Practical Stability of Large-Scale Uncertain Dynamic Systems. 4.6. Boundary Value Problems. 4.7. Sturmian Theory. 4.8. Convexity of Solutions Relative to the Initial Data. 4.9. Invariance Principle. 4.10. Notes. References. Subject Index.
The monotone method for finding maximal (minimal) solutions of an ODE involves finding a monotone sequence of upper (lower) solutions of the given ODE as the solutions of a linear differential equation which converges to the maximal (minimal) solution. We adopt the exact same method to find the maximal (minimal) solution of an impulsive differential equation as the limit of a sequence of its upper (lower) solutions which themselves are solutions of a linear impulsive differential equation. Conditions are given under which the sequences are monotone and convergent.