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Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses
| Content Provider | Scilit |
|---|---|
| Author | Santra, Shyam Sundar |
| Copyright Year | 2020 |
| Description | In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system { ( r ( t ) ( z ′ ( t ) ) γ ) ′ + q ( t ) x α ( σ ( t ) ) = 0 , t ≥ t 0 , t ≠ λ k , Δ ( r ( λ k ) ( z ′ ( λ k ) ) γ ) + h ( λ k ) x α ( σ ( λ k ) ) = 0 , k ∈ \left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill & {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill & {k \in \mathbb{N}} \hfill \cr } } \right. are established, where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right) Under the assumption ∫ ∞ ( r ( η ) ) - 1 / α d η = ∞ \int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ>α and γ<α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem. |
| Related Links | https://content.sciendo.com/downloadpdf/journals/tmmp/76/1/article-p157.pdf |
| e-ISSN | 12103195 |
| DOI | 10.2478/tmmp-2020-0025 |
| Journal | Tatra Mountains Mathematical Publications |
| Issue Number | 1 |
| Volume Number | 76 |
| Language | English |
| Publisher | Walter de Gruyter GmbH |
| Publisher Date | 2020-11-04 |
| Access Restriction | Open |
| Subject Keyword | Journal: Tatra Mountains Mathematical Publications Tatra Mountains Mathematical Publications |
| Content Type | Text |
| Resource Type | Article |
