Perturbations of the Half-Linear Euler Differential Equation

De la Vallée Poussin type inequality and eigenvalue problem for generalized half-linear differential equation

“…, we refer to [18] concerning the adjective "critical" in the Euler equation. Let w be a solution of the Riccati equation associated with (24) w + c(t) + (p − 1) |w| q = 0 and put…”

“…in [13,14,18,23], is the special form of the Riccati type differential equation associated with (1), see the later given equation (10) and also equation (28) from Section 4. Modified Riccati equation turned out to be a very useful tool in the oscillation theory of (4) since it essentially substitutes the missing transformation theory for (4), see [9,25] and also [11,Sec.…”

Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients

“…(2) is oscillatory for all > 0 and nonoscillatory for all < 0 : The constant 0 is called an oscillation constant (more precisely, oscillation constant of c with respect to r) of this equation. Considerable effort has been made over the years to extend oscillation constant theory of half linear differential equation (1), see [5][6][7][8][9][10] and reference therein. According to our knowledge, the first attempt to this problem was made by Kneser in [11], where the oscillation constant for Cauchy-Euler differential equation…”

“…for p and q are conjugate numbers, i.e., (4), whose oscillatory properties were studied in detail [4,7] and references given therein.…”

“…(6) If the functions r; c; and d are positive constants, then Eq. (7) is reduced to the half-linear Rimann-Weber equation (5), whose oscillatory properties are studied in detail [4,7] and references given therein.…”