DSpace at Texas State UniversityThe University Scholarship digital repository system captures, stores, indexes, preserves, and distributes digital research material.https://digital.library.txstate.edu:4432022-08-18T13:43:31Z2022-08-18T13:43:31ZHigher order criterion for the nonexistence of formal first integral for nonlinear systemsXu, ZhiguoLi, WenleiShi, Shaoyunhttps://digital.library.txstate.edu/handle/10877/160752022-08-17T21:13:39Z2017-11-05T00:00:00ZHigher order criterion for the nonexistence of formal first integral for nonlinear systems
Xu, Zhiguo; Li, Wenlei; Shi, Shaoyun
The main purpose of this article is to find a criterion for the nonexistence of formal first integrals for nonlinear systems under general resonance. An algorithm illustrates an application to a class of generalized Lokta-Volterra systems. Our result generalize the classical Poincare's nonintegrability theorem and the existing results in the literature.
2017-11-05T00:00:00ZCharacterization of solutions to equations involving the p(x)-Laplace operatorStircu, IuliaUta, Vasile Florinhttps://digital.library.txstate.edu/handle/10877/160742022-08-17T21:00:44Z2017-11-05T00:00:00ZCharacterization of solutions to equations involving the p(x)-Laplace operator
Stircu, Iulia; Uta, Vasile Florin
In this article we study two problems, a nonlinear eigenvalue problem involving the p(x)-Laplacian and a subcritical boundary value problem for the same operator. We work on the variable exponent Sobolev spaces and use one of the variants of the Mountain-Pass Lemma.
2017-11-05T00:00:00ZApproximate controllability of fractional control systems with time delay using the sequence methodLi, XiuwenLiu, ZhenhaiTisdell, Christopherhttps://digital.library.txstate.edu/handle/10877/160732022-08-17T20:45:34Z2017-11-05T00:00:00ZApproximate controllability of fractional control systems with time delay using the sequence method
Li, Xiuwen; Liu, Zhenhai; Tisdell, Christopher
The aim of this article is to establish sufficient conditions for the approximate controllability of fractional control systems with time delay in Hilbert spaces. By the technique of sequential approach, we prove that the fractional control systems with time delay are approximately controllable. Finally, an example is provided to illustrate our main results.
2017-11-05T00:00:00ZClassical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditionsJawad, Salihhttps://digital.library.txstate.edu/handle/10877/160722022-08-17T18:45:24Z2017-10-31T00:00:00ZClassical-regular solvability of initial boundary value problems of nonlinear wave equations with time-dependent differential operator and Dirichlet boundary conditions
Jawad, Salih
<p>This article concerns the nonlinear wave equation</p>
<pre>u<sub>tt</sub> - <sup>n</sup>∑<sub>i,j=1</sub> ∂/∂x<sub>i</sub> {α<sub>ij</sub>(t, x) ∂u/∂x<sub>j</sub>} + c(t, x)u + λu<br>
+ <b>F′</b>(|u|<sup>2</sup>)u + ζu = 0, t ∈ [0, ∞), x ∈ Ω̅<br>
u(0, x) = ϕ, u<sub>t</sub>(0, x) = ψ, u|<sub>∂Ω</sub> = 0.</pre>
<p>Essentially this article ascertains and proves the important mapping property</p>
<pre><b>M</b> : D(<b>A</b><sup>(k″<sub>0</sub>+1/2</sup>(0)) → D(<b>A</b><sup>k″<sub>0</sub>/2</sup>(0)), D(<b>A</b>(0)) = H<sup>1</sup><sub>0</sub>(Ω) ∩ H<sup>2</sup>(Ω),</pre>
<p>as well as the associated Lipschitz condition</p>
<pre>∥<b>A</b><sup>k″<sub>0</sub>/2</sup>(0)(<b>M</b>u - <b>M</b>v)∥<br>
≤ k(∥<b>A</b><sup>(k″<sub>0</sub>+1)/2</sup>(0)u∥ + ∥<b>A</b><sup>k″<sub>0</sub>+1)/2</sup>(0)v∥) ∥<b>A</b><sup>k″<sub>0</sub>+1)/2</sup>(0) (u - v)∥,</pre>
<p>where</p>
<pre><b>A</b>(t) ≔ - <sup>n</sup>∑<sub>i,j=1</sub> ∂/∂x<sub>i</sub> {α<sub>ij</sub>(t, x) ∂/∂x<sub>j</sub>} + c(t, x) + λ,<br>
<b>M</b>u ≔ <b>F′</b> (|u|<sup>2</sup>)u + ζu,<br>
k″ ∈ ℕ, k″ > n/2 + 1, k″<sub>0</sub> ≔ min{k″},</pre>
<p>and k(⋅) ∈ C<sup>0</sup><sub>loc</sub> (ℝ⁺, ℝ⁺⁺) is monotonically increasing. Here are ℝ⁺ = [0, ∞), ℝ⁺⁺ = (0, ∞). This mapping property is true for the dimensions n ≤ 5. But we investigate only the case n = 5 because the problem is already solved for n ≤ 4, however, without the mapping property.</p>
<p>With the proof of the mapping property and the associated Lipschitz condition, the problem becomes considerably comparable with a paper from von Wahl, who investigated the same problem as Cauchy problem and solved it for the dimensions n ≤ 6, i.e. without boundary condition. In the case of the Cauchy problem there are no difficulties with regard to the mapping property.</p>
2017-10-31T00:00:00Z