Nonoscillatory solutions of third-order nonlinear dynamic equations: Existence and nonexistence

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Öztürk Ö., Akin E., Pelen N. N.

Turkish Journal of Mathematics, cilt.49, sa.5, ss.585-602, 2025 (SCI-Expanded, Scopus) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 49 Sayı: 5
  • Basım Tarihi: 2025
  • Doi Numarası: 10.55730/1300-0098.3610
  • Dergi Adı: Turkish Journal of Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.585-602
  • Anahtar Kelimeler: 34A34, 34N05, 39A05, 39A06, fixed point theorem, Kneser solutions, oscillation, Third-order dynamic equation, time scales
  • Karadeniz Teknik Üniversitesi Adresli: Hayır

Özet

This paper investigates a third-order nonlinear dynamic equation on arbitrary time scales, a nonempty closed subset of the real numbers, unifying continuous and discrete analyses. We study the qualitative behavior of nonoscillatory solutions and their quasi-derivatives, focusing on their limiting behaviors. The existence of such solutions are established using improper integral criteria and Schauder’s and Knaster’s fixed point theorems. In addition, we establish the criteria for the nonexistence of nonoscillatory solutions. Furthermore, we prove the existence of Kneser-type solutions of the corresponding linear dynamic equation on isolated time scales, addressing an open problem in the literature. Several examples of theoretical results are illustrated on various time scales, including real numbers, integers, and the q-calculus time scale with q > 1.