Uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations

Uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations

This work is devoted to the study of the existence of uncountably many nonoscillatory bounded solutions tosecond-order nonlinear neutral dynamic equations by means of the Darbo fixed point theorem. We construct assumptionswithout sign conditions on the nonlinear part of the equation. Moreover, we prove the necessary condition for the existenceof an asymptotically zero solution to the problem under consideration.

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  • [1] Agarwal RP, Bohner M, Li WT. Nonoscillation and Oscillation Theory for Functional Differential Equations. New York, NY, USA: Marcel Dekker, 2003.
  • [2] Agarwal RP, Bohner M, R̆ehák P. Half-linear Dynamic Equations in Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday, Vol. 1, 2. Dordrecht, the Netherlands: Kluwer Academic Publishers, 2003, pp. 1-57.
  • [3] Agarwal RP, Grace SR, O’Regan D. Nonoscillatory solutions for discrete equations. Computers & Mathematics with Applications 2003; 45: 1297-1302.
  • [4] Appell J. Measures of noncompactness, condensing operators and fixed point: an application-oriented survey. Fixed Point Theory 2005; 6 (2): 157-229.
  • [5] Bohner M, Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston, MA, USA: Birkhäuser, 2001.
  • [6] Bohner M, Stević S. Asymptotic behavior of second-order dynamic equations. Applied Mathematics and Computation 2007; 188 (2): 1503-1512.
  • [7] Cheng SS. Existence of nonoscillatory solutions of a second-order linear neutral difference equation. Applied Mathematics Letters 1999; 12: 71-78.
  • [8] Cheng SS, Li HJ, Patula WT. Bounded and zero convergent solutions of second order difference equations. Journal of Mathematical Analysis and Applications 1989; 141: 463-483.
  • [9] Darbo G. Punti uniti in trasformazioni a codominio non compatto. Rendiconti del Seminario Matematico della Universit￿ di Padova 1955; 24: 84-92 (in Italian).
  • [10] Deng XH, Wang QR. Oscillation and nonoscillation for second-order nonlinear neutral functional dynamic equations on time scales. Electronic Journal of Differential Equations 2013; 234: 17.
  • [11] Deng XH, Wang QR. Nonoscillatory solutions to forced higher-order nonlinear neutral dynamic equations on time scales. Rocky Mountain Journal of Mathematics 2015; 45 (2): 475-507.
  • [12] Dunford N, Schwartz JT. Linear Operator Part 1: General Theory. New York, NY, USA: John Wiley & Sons, 1988.
  • [13] Galewski M, Jankowski R, Nockowska-Rosiak M, Schmeidel E. On the existence of bounded solutions for nonlinear second order neutral difference equations. Electronic Journal of Qualitative Theory of Differential Equations 2014; 72: 1-12.
  • [14] Gao J, Wang QR. Existence of nonoscillatory solutions to second-order nonlinear neutral dynamic equations on time scales. Rocky Mountain Journal of Mathematics 2013; 43: 1521-1535.
  • [15] Higgins R. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems (B) 2010; 13: 609-622.
  • [16] Jinfa C. Existence of a nonoscillatory solution of a second-order linear neutral difference equation. Applied Mathematics Letters 2007; 20: 892-899.
  • [17] Karpuz B. Asymptotic behavior of bounded solutions of a class of higher-order neutral dynamic equations. Applied Mathematics and Computation 2009; 215: 2174-2183.
  • [18] Karpuz B. Necessary and sufficient conditions on the asymptotic behavior of second-order neutral delay dynamic equations with positive and negative coefficients. Mathematical Methods in the Applied Sciences 2014; 37 (8): 1219-1231.
  • [19] Lalli BS, Zhang BG. On existence of positive solutions and bounded oscillations for neutral difference equations. Journal of Mathematical Analysis and Applications 1992; 166: 272-287.
  • [20] Liu Z, Kang SM, Ume JS. Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations. Applied Mathematics and Computation 2009; 213: 554-576.
  • [21] Liu Z, Xu Y, Kang SM. Global solvability for second order nonlinear neutral delay difference equation. Computers & Mathematics with Applications 2009; 57: 587-595.
  • [22] Liu Z, Zhao L, Kang SM, Ume JS. Existence of uncountably many bounded positive solutions for second order nonlinear neutral delay difference equations. Computers & Mathematics with Applications 2011; 61: 2535-2545.
  • [23] Migda M, Migda J. Bounded solutions of nonlinear discrete Volterra equations. Mathematica Slovaca 2016; 66 (5): 1169-1178.
  • [24] Migda J, Nockowska-Rosiak M. Asymptotic properties of solutions to difference equations of Sturm-Liouville type. Applied Mathematics and Computation 2019; 340: 126-137.
  • [25] Nockowska-Rosiak M, Hachuła P, Schmeidel E. Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with p-Laplacian. Discrete & Continuous Dynamical Systems B 2018; 23 (1): 369-375.
  • [26] Schmeidel E. An application of measures of noncompactness in investigation of boundedness of solutions of second order neutral difference equations. Advances in Difference Equations 2013; 91: 9.
  • [27] Sun T, Xi H, Peng X, Yu W. Nonoscillatory solutions for higher-order neutral dynamic equations on times scales. Abstract and Applied Analysis 2010; 2010: 428963.
  • [28] Ye H, Yin J, Jin C. Non-oscillatory solutions for nonlinear neutral delay differential equations. Applied Mathematics and Computation 2014; 235: 283-291.
  • [29] Zhenguo Z, Wenlei D, Haiyan L. Existence of nonoscillatory solutions of higher-order nonlinear neutral dynamic equations on times scales. Journal of Applied Mathematics and Computing 2008; 28 (1-2): 29-38.
  • [30] Zhu SL, Lian FY. Existence for nonoscillatory solutions of forced higher-order nonlinear neutral dynamic equations. Dynamics of Continuous, Discrete and Impulsive Systems 2013; 20: 179-196.
  • [31] Zhu ZQ. Existence for positive solutions of second-order neutral dynamic equations on time scales. Advances in Dynamical Systems and Applications 2007; 2 (2): 265-282.
  • [32] Zhu ZQ, Wang QR. Existence of nonoscillatory solutions to neutral dynamic equations on times scales. Journal of Mathematical Analysis and Applications 2007; 335: 751-762.