Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities

Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form (r(t)@F"@a(x^@D(t)))^@D+p"0(t)@F"@a(x(@t"0(t)))+@?i=1np"i(t)@F"@b"""i(x(@t"i(t)))=e(t),t@?[t"0,~)"T where T is a time scale, t"0@?T a fixed number; [t"0,~)"T is a time scale interval; @F"*(u)=|u|^*^-^1u; the functions r,p"i,e:[t"0,~)"T->R are right-dense continuous with r>0 nondecreasing; @t"k:T->T are nondecreasing right-dense continuous with @t"k(t)@?t, lim"t"->"~@t"k(t)=~; and the exponents satisfy @b"1>...>@b"m>@a>@b"m"+"1>...@b"n>0. All results are new even for T=R and T=Z. Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives. Two examples are provided to illustrate one of the theorems.

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