Half-linear differential equations of fourth order: oscillation criteria of solutions

Abstract

In this paper, we are concerned with the oscillation of solutions to a class of fourth-order delay differential equations with p-Laplacian like operators $( r ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{\prime \prime \prime } ( t ) ) ^{\prime }+q ( t ) \vert x ( \tau ( t ) ) \vert ^{p_{2}-2}x ( \tau ( t ) ) =0$ ( r ( t ) | x ‴ ( t ) | p 1 − 2 x ‴ ( t ) ) ′ + q ( t ) | x ( τ ( t ) ) | p 2 − 2 x ( τ ( t ) ) = 0 and $( r ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{\prime \prime \prime } ( t ) ) ^{\prime }+\sigma ( t ) \vert x^{\prime \prime \prime } ( t ) \vert ^{p_{1}-2}x^{ \prime \prime \prime } ( t ) +q ( t ) \vert x ( \tau ( t ) ) \vert ^{p_{2}-2}x ( \tau ( t ) ) =0$ ( r ( t ) | x ‴ ( t ) | p 1 − 2 x ‴ ( t ) ) ′ + σ ( t ) | x ‴ ( t ) | p 1 − 2 x ‴ ( t ) + q ( t ) | x ( τ ( t ) ) | p 2 − 2 x ( τ ( t ) ) = 0 . New oscillation criteria are presented by the comparison technique and employing the Riccati transformation. Moreover, our results are an extension and complement to previous results in the literature. Two examples are shown to illustrate the conclusions.

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