Abstract
In this paper, we study the existence of periodic solutions to the following prescribed mean curvature Lienard equation with a singularity and a deviating argument: $$\biggl(\frac{u'(t)}{\sqrt{1+(u'(t))^{2}}}\biggr)'+f\bigl(u(t)\bigr)u'(t)+g \bigl( u(t-\sigma)\bigr)=e(t), $$ where g has a strong singularity at $x=0$ and satisfies a small force condition at $x=\infty$ . By applying Mawhin’s continuation theorem, we prove that the given equation has at least one positive T-periodic solution. We will also give an example to illustrate the application of our main results.
Highlights
In recent years, there have been many papers about the existence of periodic solutions for the second order differential equations with a singularity, especially for the Liénard equations
In [ ], Feng discussed the periodic solution for the prescribed mean curvature Liénard equation of the form u (t)
By applying Mawhin’s continuation theorem, we prove that Eq ( . ) has at least one positive T-periodic solution
Summary
There have been many papers about the existence of periodic solutions for the second order differential equations with a singularity, especially for the Liénard equations. The existence of periodic solutions of the Liénard equations with a deviating argument has been studied widely (see [ – ]). In [ ], Feng discussed the periodic solution for the prescribed mean curvature Liénard equation of the form u (t). +u (t) sufficient conditions on the existence of periodic solutions by using Mawhin’s continuation theorem. Liang and Lu [ ] studied the homoclinic solution for the prescribed mean curvature Duffing-type equation of the form u (t). We consider the following prescribed mean curvature Liénard equation with a singularity and a deviating argument:. The interest is that the conditions imposed on f , g and the approaches to estimate a priori bounds of periodic solutions are different from It is obvious that X and Y are Banach spaces
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