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Asymptotic representation of solutions for second-order impulsive differential equations
Abstract
We obtain sufficient conditions which guarantee the existence of a solution of a class of second order nonlinear impulsive differential equations with fixed moments of impulses possessing a prescribed asymptotic behavior at infinity in terms of principal and nonprincipal solutions. An example is given to illustrate the relevance of the results.
... can be found in [15][16][17]. Because the time scales calculus combines the continuous and discrete cases, the impulsive dynamic equations have also attracted the attention of many researchers; see [18][19][20][21][22][23] and the references cited therein. ...
... If conditions (6)- (8) and (16) hold, then each operator F k defined by (12)-(15) has the following properties: ...
The asymptotic integration problem has a rich historical background and has been extensively studied in the context of ordinary differential equations, delay differential equations, dynamic equations, and impulsive differential equations. However, the problem has not been explored for impulsive dynamic equations due to the lack of essential tools such as principal and nonprincipal solutions, as well as certain compactness results. In this work, by making use of the principal and nonprincipal solutions of the associated linear dynamic equation, recently obtained in [Acta Appl. Math. 188, 2 (2023)], we investigate the asymptotic integration problem for a specific class of nonlinear impulsive dynamic equations. Under certain conditions, we prove that the given impulsive dynamic equation possesses solutions with a prescribed asymptotic behavior at infinity. These solutions can be expressed in terms of principal and nonprincipal solutions as in differential equations. In addition, the necessary compactness results are also established. Our findings are particularly valuable for better understanding the long-time behavior of solutions, modeling real-world problems, and analyzing the solutions of boundary value problems on semi-infinite intervals.
... These kinds of equations have close relation with several natural phenomena such as electrodynamics, epidemiology, biology, etc. For some various properties of solutions for several iterative functional differential equations, we refer the interested reader to [4][5][6], [3,[7][8][9][10][11], [12], [13][14][15][16], [17]. ...
In this paper, we use the method of lower and upper solutions to study the maximal and minimal nondecreasing bounded solutions of iterative functional differential equations x ′ (t) = g(t, x [1] (t), x [2] (t),. .. , x [n] (t)).
Principal and nonprincipal solutions of differential equations play a critical role in studying the qualitative behavior of solutions in numerous related differential equations. The existence of such solutions and their applications are already documented in the literature for differential equations, difference equations, dynamic equations, and impulsive differential equations. In this paper, we make a contribution to this field by examining impulsive dynamic equations and proving the existence of such solutions for second-order impulsive dynamic equations. As an illustration, we prove the famous Leighton and Wong oscillation theorems for impulsive dynamic equations. Furthermore, we provide supporting examples to demonstrate the relevance and effectiveness of the results.
Thesis (M.S.)--University of Southern California, 1991. Includes bibliographical references (leaf 48).
Let u and v denote respectively the principal and nonprincipal solutions of the second-order linear equation (p(t)x′)′+q(t)x=0(p(t)x′)′+q(t)x=0 defined on some half-line of the form [t∗,∞)[t∗,∞).It is shown that for any given real numbers a and b the nonlinear equation(p(t)x′)′+q(t)x=f(t,x)(p(t)x′)′+q(t)x=f(t,x)has a solution x(t)x(t) asymptotic to av(t)+bu(t)av(t)+bu(t) as t→∞t→∞ under some reasonable conditions on the function f. Examples are given to illustrate the results.
For a class of second order nonlinear differential equations, sufficient conditions are presented to ensure that some, respectively all solutions are asymptotic to lines.
We study asymptotic properties of solutions for certain classes of sec-ond order nonlinear differential equations. The main purpose is to investigate when all continuable solutions or just a part of them with initial data satisfying an additional con-dition behave at infinity like nontrivial linear functions. Making use of Bihari's inequal-ity and its derivatives due to Dannan, we obtain results which extend and complement those known in the literature. Examples illustrating the relevance of the theorems are discussed.
We introduce the concept of principal and nonprincipal solutions for second order differential equations having fixed moments of impulse actions is obtained. The arguments are based on Polya and Trench factorizations as in non-impulsive differential equations, so we first establish these factorizations. Making use of the existence of nonprincipal solutions we also establish new oscillation criteria for nonhomogeneous impulsive differential equations. Examples are provided with numerical simulations to illustrate the relevance of the results.
We show that the Arzelà–Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.
We initiate a study of the asymptotic integration problem for second-order nonlinear impulsive differential equations. It is shown that there exist solutions asymptotic to solutions of an associated linear homogeneous impulsive differential equation as in the case for equations without impulse effects. We introduce a new constructive method that can easily be applied to similar problems. An illustrative example is also given.
We obtain sufficient conditions for existence and uniqueness of solutions for a class of second order nonlinear singular impulsive boundary value problems with fixed moments of impulses.
We shall study the asymptotic behavior for t → ∞ of solutions of the following nonlinear differential equation: (1) u” + f(t, u) = 0. We suppose that f(t, u) satisfies the following conditions: H-1: f(t, u) is continuous in D : t ≥ 0, - ∞ < u < ∞. H-2: The derivative of f_u exists on D and satisfies f_u(t, u) > 0 on D. H-3: |f(t, u(t))| ≤ f_u(t, 0) | u(t) | on D.
This work is concerned with the behavior of solutions of a class of second-order nonlinear differential equations locally near infinity. Using methods of the fixed point theory, the existence of solutions with different asymptotic representations at infinity is established. A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymptotic behavior.