Nabla inequalities and permanence for a logistic integrodifferential equation on time scales

Hu, Meng; Wang, Lili

doi:10.1515/math-2017-0133

Cited by 1 publication

(2 citation statements)

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“…Note that inequality (48) and the delta inequality (15) obtained for the delta time scale calculus coincide. Therefore Theorem 15 is the diamond alpha unification of Theorem 1.…”

Section: Case R ≤ 0 < Q mentioning

confidence: 61%

“…[35][36][37][38][39][40] For some results about the nabla differential equations and nabla inequalities, see previous works. [41][42][43][44][45][46][47][48][49][50][51] Contrary to delta case, Bennett-Leindler type inequalities had not been considered until the paper 52 appeared. The nabla time scale unifications of the discrete Bennett-Leindler inequalities (9) and (10) and the continuous Bennett-Leindler inequalities ( 13) and ( 14) as well as the nabla versions of Theorems 1-4 for an arbitrary time scale can be seen in the next theorems.…”

Section: Introductionmentioning

confidence: 99%

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Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

Kayar

Kaymakçalan

Pelen

2021

Math Methods in App Sciences

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In this paper, two kinds of dynamic Bennett-Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett-Leindler type inequalities give diamond alpha Bennett-Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett-Leindler type inequalities. For the second type, choosing 𝛼 = 1 or 𝛼 = 0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them.Therefore, both kinds of our results expand some of the known delta and nabla Bennett-Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett-Leindler type inequalities.Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.

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“…Note that inequality (48) and the delta inequality (15) obtained for the delta time scale calculus coincide. Therefore Theorem 15 is the diamond alpha unification of Theorem 1.…”

Section: Case R ≤ 0 < Q mentioning

confidence: 61%