| Summary: | We establish nonoscillation criterion for the even order half-linear differential equation <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>Φ</mi><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msup><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>Φ</mi><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>,</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> are real numbers, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced open="|" close="|"><mi>s</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>sgn</mi><mi>s</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub></semantics></math></inline-formula> is a regularly varying (at infinity) function of the index <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are replaced by the <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mi>α</mi></msup></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></msup></semantics></math></inline-formula>, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.
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