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Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays

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Abstract

In this work, necessary and sufficient conditions for oscillations of the solutions of a class of nonlinear neutral first-order differential equations with several delays of the form

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}[x(t)+r(t)x(t-\tau )]+ \sum _{i=1}^m \phi _i(t)H(x(t-\sigma _i))=0, \end{aligned}$$

are established under various ranges of r(t). Our main tools are Knaster–Tarski fixed point theorem and Banach’s fixed point theorem. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.

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Authors and Affiliations

  1. RUDN University, 6 Miklukho-Maklaya st, Moscow, Russia, 117198

    Sandra Pinelas

  2. Department of Mathematics, Sambalpur University, Sambalpur, 768019, India

    Shyam Sundar Santra

Authors
  1. Sandra Pinelas
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  2. Shyam Sundar Santra
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Correspondence to Sandra Pinelas.

Additional information

This work is supported by the Department of Science and Technology (DST), New Delhi, India, through the bank instruction order no. DST/INSPIRE Fellowship/2014/140, dated Sept. 15, 2014. The publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No 02.a03.21.0008).

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Pinelas, S., Santra, S.S. Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays. J. Fixed Point Theory Appl. 20, 27 (2018). https://doi.org/10.1007/s11784-018-0506-9

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  • DOI: https://doi.org/10.1007/s11784-018-0506-9

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