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Sturmian comparison theory for half‐linear and nonlinear differential equations via Picone identity
Author(s) -
Özbekler Abdullah
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4224
Subject(s) - mathematics , nonlinear system , type (biology) , differential equation , identity (music) , oscillation (cell signaling) , mathematical analysis , pure mathematics , combinatorics , ecology , physics , genetics , quantum mechanics , acoustics , biology
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 ) Φ β ( y ′ ) ) ′ + ∑ i = 1 nq i ( t ) Φα i( y ) = 0 and the second is the half‐linear differential equations( k ( t ) Φ β ( x ′ ) ) ′ + p ( t ) Φ β ( x ) = 0 where Φ α ( s ) = | s | α − 1 s and α 1 > ⋯ > α m > β > α m + 1 > ⋯ > α n > 0. Under the assumption that the solution of has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for . Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley & Sons, Ltd.