Interval Oscillation Criteria for Second-Order Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Yuangong Sun

Abstract and Applied Analysis, 2011, vol. 2011, 1-14

Abstract:

By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form [ ð ‘ ( ð ‘¡ ) 𠜙 ð ›¼ ( ð ‘¥ Δ ( ð ‘¡ ) ) ] Δ + ð ‘ž ( ð ‘¡ ) 𠜙 ð ›¼ ∫ ( ð ‘¥ ( ð œ ( ð ‘¡ ) ) ) + ð ‘Ž 𠜎 ( ð ‘ ) ð ‘Ÿ ( ð ‘¡ , ð ‘ ) 𠜙 ð ›¾ ( ð ‘ ) ( ð ‘¥ ( ð ‘” ( ð ‘¡ , ð ‘ ) ) ) Δ 𠜉 ( ð ‘ ) = ð ‘’ ( ð ‘¡ ) , where ð ‘¡ ∈ [ ð ‘¡ 0 , ∞ ) ð •‹ = [ ð ‘¡ 0 â‹‚ ð •‹ , ∞ ) , ð •‹ is a time scale which is unbounded from above; 𠜙 ∗ ( ð ‘¢ ) = | ð ‘¢ | ∗ s g n ð ‘¢ ; ð ›¾ ∶ [ ð ‘Ž , ð ‘ ] ð •‹ 1 → â„ is a strictly increasing right-dense continuous function; ð ‘ , ð ‘ž , ð ‘’ ∶ [ ð ‘¡ 0 , ∞ ) ð •‹ → â„ , ð ‘Ÿ ∶ [ ð ‘¡ 0 , ∞ ) ð •‹ × [ ð ‘Ž , ð ‘ ] ð •‹ 1 → â„ , ð œ âˆ¶ [ ð ‘¡ 0 , ∞ ) ð •‹ → [ ð ‘¡ 0 , ∞ ) ð •‹ , and ð ‘” ∶ [ ð ‘¡ 0 , ∞ ) ð •‹ × [ ð ‘Ž , ð ‘ ] ð •‹ 1 → [ ð ‘¡ 0 , ∞ ) ð •‹ are right-dense continuous functions; 𠜉 ∶ [ ð ‘Ž , ð ‘ ] ð •‹ 1 → â„ is strictly increasing. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:719628

DOI: 10.1155/2011/719628

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