<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2016-0017_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mtext>d</mtext><mtext>dt</mtext></mfrac><mtext>x</mtext><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mtext>a</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo> </mo><mtext>h</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>x</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mfrac><mtext>d</mtext><mtext>dt</mtext></mfrac><mtext>Q</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo>,</mo><mo> </mo><mtext>x</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo>−</mo><mi>τ</mi><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mtext>G</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo>,</mo><mo> </mo><mtext>x</mtext><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><mtext>x</mtext><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo>−</mo><mi>τ</mi><mo> </mo><mo stretchy="false">(</mo><mtext>t</mtext><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow></math><tex-math>$${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$</tex-math></alternatives></disp-formula>
We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.</p></abstract>

In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation
ddtx(t)=−a (t) h (x (t))+ddtQ (t, x (t−τ (t)))+G (t, x(t), x (t−τ (t))).$${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$
We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

Applied Mathematics Lab., Department of Mathematics, Faculty of Sciences, Univ Annaba,

Department of Mathematics and Informatics, Univ Souk Ahras, Algeria Applied Mathematics Lab., Department of Mathematics, Faculty of Sciences, Univ Annaba,

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential...

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

Published Online: Jan 23, 2017

Page range: 255 - 270

Received: Dec 10, 2014

DOI: https://doi.org/10.1515/ausm-2016-0017

Keywords
Krasnoselskii-Burton’s theorem, large contraction, neutral differential equation, integral equation, periodic solution, positive solution

© 2016 Mouataz Billah Mesmouli et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

Published Online: Jan 23, 2017

Page range: 255 - 270

Received: Dec 10, 2014

DOI: https://doi.org/10.1515/ausm-2016-0017

KeywordsKrasnoselskii-Burton’s theorem, large contraction, neutral differential equation, integral equation, periodic solution, positive solution

© 2016 Mouataz Billah Mesmouli et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Krasnoselskii-Burton’s theorem, large contraction, neutral differential equation, integral equation, periodic solution, positive solution