Nonoscillation of higher order half-linear differential equations

Weighted Second-Order Differential Inequality on Set of Compactly Supported Functions and Its Applications

“…First, we investigate inequality (3). Then, we apply the obtained results to study the oscillatory properties of the fourth-order differential equation…”

“…Criteria for the validity of inequality (6) under various boundary conditions on the function f are given in [1,2]. Following the ideas and research methods of [2], we find characterizations of inequality (3) in terms different from those in [2], which are convenient for studying the oscillatory properties of Equation ( 4) and the spectral properties of operator (5).…”

“…In the third method, due to the lack of characterizations of inequality (6) in general form, one of the coefficients of the equations has also been taken as a power function. In the papers [3][4][5][6], published in recent years, the oscillatory properties of the equations in form (7) have been established by the above three methods under the restriction that at least one of the coefficients is a power function. In the paper [7], the restrictions on the coefficients have been removed.…”

Nonoscillation of even order Euler type half-linear difference equations

“…we have so far only the following implication. Equation (1.6) is nonoscillatory if C n;p;˛> 0 (for general˛see [4,Theorem 3.2], for˛D 0 see the older result in [7, Theorem 9.4.5]).…”

“…In [4], we use the variational principle together with the Wirtinger inequality, which enables us to show positivity of the energy functional associated with equation (1.8).…”