Nonlinear Integral Riccati Systems and Comparison Theorems for Linear Differential Equations

1983 ◽  
Vol 14 (3) ◽  
pp. 463-473 ◽  
Author(s):  
G. J. Butler ◽  
L. H. Erbe
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Linear Differential

Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1185-1194
Author(s):  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Comparison Theory ◽  
Second Order Differential Equations ◽  
Nonlinear Version ◽  
Linear And Nonlinear ◽  
Linear Differential ◽  
Picone Identity

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)??(y'))' + Xn,i=1 qi(t)??i(y)=0(1) and the second is the half-linear differential equations (k(t)??(x'))'+ p(t)??(x) = 0 (2) where ?*(s) = |s|*-1s and ?1 >...> ?m > ? > ?m+1 > ... > ?n > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone?s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

scholarly journals Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 502
Author(s):  
Zuzana Pátíková
Keyword(s):  
Differential Equations ◽  
Comparison Theorem ◽  
Type Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Type Integral ◽  
Comparison Criteria ◽  
Linear Differential

In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation).

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