Differential equations of even-order with p-Laplacian like operators: qualitative properties of the solutions

Advances in Difference Equations

Qasem M. Al-Mdallal¹  Thabet Abdeljawad²  Omar Bazighifan³ 

[1] Department of Mathematical Sciences, United Arab Emirates University;Department of Mathematics and General Sciences, Prince Sultan University;Department of Mathematics, Faculty of Science, Hadhramout University;

关键词: Even-order;    Differential equation;    Oscillation;    p-Laplacian equation;   

DOI  :  10.1186/s13662-021-03254-7

来源: DOAJ

【 摘 要 】

Abstract In this paper, we study the oscillation of solutions for an even-order differential equation with middle term, driven by a p-Laplace differential operator of the form { ( r ( x ) Φ p [ z ( κ − 1 ) ( x ) ] ) ′ + a ( x ) Φ p [ f ( z ( κ − 1 ) ( x ) ) ] + ∑ i = 1 j q i ( x ) Φ p [ h ( z ( δ i ( x ) ) ) ] = 0 , j ≥ 1 , r ( x ) > 0 , r ′ ( x ) + a ( x ) ≥ 0 , x ≥ x 0 > 0 . $$ \textstyle\begin{cases} ( r ( x ) \Phi _{p}[z^{ ( \kappa -1 ) } ( x ) ] ) ^{\prime }+a ( x ) \Phi _{p}[f ( z^{ ( \kappa -1 ) } ( x ) ) ]+ \sum_{i=1}^{j}q_{i} ( x ) \Phi _{p}[h ( z ( \delta _{i} ( x ) ) ) ]=0, \\ \quad j\geq 1, r ( x ) >0, r^{\prime } ( x ) +a ( x ) \geq 0, x\geq x_{0}>0. \end{cases}$$ The oscillation criteria for these equations have been obtained. Furthermore, an example is given to illustrate the criteria.

【 授权许可】

Unknown