Ultimate Boundedness of Solutions for Certain Third Order Nonlinear Differential Equations

We investigate in this paper, the ultimate boundedness of solutions for certain special class of third order nonlinear differential equations. Using suitable complete Lyapunov function, we obtain the criteria for the ultimate boundedness of solutions for this equation. Our result extends and improves on some well known results on boundedness of solutions of third order differential equations in the literature. Key words: Nonlinear differential equations; Third order; Ultimate boundedness of solutions; Lyapunov’s method

Ultimate boundedness and periodicity results for a certain system Of third-order nonlinear Vector delay differential equations

In the last years, there has been increasing interest in obtaining the sufficient conditions for stability, instability, boundedness, ultimately boundedness, convergence, etc. For instance, in applied sciences some practical problems concerning mechanics, engineering technique fields, economy, control theory, physical sciences and so on are associated with third, fourth and higher order nonlinear differential equations. The problem of the boundedness and stability of solutions of vector differential equations has been widely studied by many authors, who have provided many techniques especially for delay differential equations. In this work a class of third order nonlinear non-autonomous vector delay differential equations is considered by employing the direct technique of Lyapunov as basic tool, where a complete Lyapunov functional is constructed and used to obtain sufficient conditions that guarantee existence of solutions that are periodic, uniformly asymptotically stable, uniformly ultimately bounded and the behavior of solutions at infinity. In addition to being for a more general equation, the obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results

On Some Generalizations Bellman-Bihari Result for Integro-Functional Inequalities for Discontinuous Functions and Their Applications

We present some new nonlinear integral inequalities Bellman-Bihari type with delay for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniformly), stability by Lyapunov (uniformly), practical stability by Chetaev (uniformly) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations

Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of differential equations) is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.Comment: Communications in contemporary mathematics. Print ISSN: 0219-1997. Online ISSN: 1793-668

Stability and Liapunov Functionals for Fractional Differential Equations

Abstract. This project is devoted to developing Liapunov direct method for fractional differential equations and systems. The method (constructing a system related scalar function) enables investigators to analyze the qualitative behavior of solutions of a differential equation without actually solving it. We are able to convert some fractional differential equations (semi-linear) to integral equations with singular kernels and construct Liapunov functionals for the integral equations to deduce conditions for boundedness and stability of solutions. Extending such a method to fully nonlinear equations presents a significant challenge to investigators and will be a major area of research for many years to come

On the Global Existence and Boundedness of Solutions of Nonlinear Vector Differential Equations of Third Order

In this paper, we give some criteria to ensure the global existence and boundedness of solutions to a kind of third order nonlinear vector differential equations. By using the Lyapunov\u27s direct method, we obtain a new result on the topic and give an example for the illustrations. Our result includes, completes and improves some earlier results in the literature

Qualitative Properties Of Solutions Of Fully Nonlinear Equations And Overdetermined Problems

In section 2 of part I, We study the maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We obtain the radial symmetry and monotonicity properties for nonnegative viscosity solutions of fully nonlinear equations under some asymptotic decay rate at infinity. Our symmetry and monotonicity results also apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum principle for viscosity solutions to fully nonlinear elliptic equations is established. In section 3, We establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma, we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. Part II is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. These results answer two open questions W. Reichel to some extent

Інтегро-функціональні нерівності типу Беллмана–Біхарі для розривних функцій та їх застосування

Наведено новi нелiнiйнi iнтегральнi нерiвностi типу Беллмана–Бiхарi для розривних функцiй (iнтегро-сумарнi нерiвностi; iмпульснi iнтегральнi нерiвностi). Розглянуто застосування отриманих результатiв: умови обмеженостi (рiвномiрної), стiйкостi за Ляпуновим (рiвномiрної), практичної стiйкостi за Четаєвим (рiвномiрної) для розв’язкiв iмпульсних диференцiальних та iнтегро-диференцiальних систем звичайних диференцiальних рiвнянь.We present some new nonlinear integral inequalities of the Bellman–Bihari type for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniform), stability by Lyapunov (uniform), and practical stability by Chetaev (uniform) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations

Boundedness and Square Integrability in Neutral Differential Systems of Fourth Order

The aim of this paper is to study the asymptotic behavior of solutions to a class of fourth-order neutral differential equations. We discuss the stability, boundedness and square integrability of solutions for the considered system. The technique of proofs involves defining an appropriate Lyapunov functional. Our results obtained in this work improve and extend some existing well-known related results in the relevant literature which were obtained for nonlinear differential equations of fourth order with a constant delay. The obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results