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By a solution of (1), (2) we understand a function u ∈ X such that u ′ ∈ X and (1), (2) hold true.
By a Filippov solution of (4), we understand a function (x cdot colon J rightarrowmathbb{R}) with absolutely continuous derivative, satisfying problem (5), almost everywhere on J.
By a solution of problem (2.1), we understand a function u ∈ E with φ p ( u ′ ) absolutely continuous which satisfies problem (2.1).
By a (bivariate) mean we understand a function m defined on ((0,infty)thatsatisfies)) thet satisfollowingfollowing double inequality: forall a,b>0, quad min a,b leq m a,b leqmax a,b).
By a solution to (1), (2) we understand a function ( u i ) i = 1 n ∈ A C ( [ a, b ] ; R n ) satisfying (1) almost everywhere in [ a, b ] and (2).
By a solution to (1.2), (1.3) we understand a function u : [ 0, ω ] → R which is positive, absolutely continuous together with its first derivative, satisfies (1.2) almost everywhere on [ 0, ω ], and verifies (1.3).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com