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Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator

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Abstract

A class of second-order singular differential equations with a linear difference operator is investigated in this paper. The novelty of the present article is that for the first time we show that weak and strong singularities enable the achievement of a new existence criterion of positive periodic solutions through an applications of a fixed point theorem of Krasnoselskii’s, i.e., our results of the existence of positive periodic solutions reveal a delicate relation between the value of external force e(t) and the velocity of nonlinear term \(f(t,x(t-\tau (t)))\) approaching towards infinity when \(x(t-\tau (t))\) tending to zero.

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Authors and Affiliations

  1. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454000, China

    Zhibo Cheng & Feifan Li

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    Zhibo Cheng

Authors
  1. Zhibo Cheng
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  2. Feifan Li
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Correspondence to Zhibo Cheng.

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Research is supported by NSFC Project (No. 11501170), China Postdoctoral Science Foundation funded project (No. 2016M590886), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).

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Cheng, Z., Li, F. Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator. J. Fixed Point Theory Appl. 21, 48 (2019). https://doi.org/10.1007/s11784-019-0687-x

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