Periodic solutions for a Liénard type p-Laplacian differential equation

By using topological degree theory and some analysis skill, we obtain sufficient conditions for the existence and uniqueness of periodic solutions for Liénard type p$p$-Laplacian differential equation.     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

Existence and uniqueness of periodic solutions for a p-Laplacian Duffing equation with a deviating argument

By means of Mawhin’s continuation theorem, we study a class of p-Laplacian Duffing type differential equations of the form

(_φp(x^′(t)))^′=Cx^′(t)+g(t,x(t),x(t−τ(t)))+e(t).${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }=C{x}^{\prime }\left(t\right)+g(t,x\left(t\right),x(t-\tau \left(t\right)))+e\left(t\right)\text{.}$

Periodic solutions for Liénard type p-Laplacian equation with two deviating arguments

In this paper, the Liénard type p$p$-Laplacian equation with two deviating arguments

(_φp(x^′(t)))^′+f(x(t))x^′(t)+_g1(t,x(t−_τ1(t)))+_g2(t,x(t−_τ2(t)))=e(t)${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+f\left(x\left(t\right)\right)x^{\prime }\left(t\right)+g_1(t,x(t-{\tau }_1\left(t\right)))+g_2(t,x(t-{\tau }_2\left(t\right)))=e\left(t\right)$

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coefficient of nonlinear term

As p-Laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Lu’s continuation theorem, which is an extension of Manásevich-Mawhin, we study the existence of periodic solutions for a Rayleigh type p-Laplacian equation

(φ_p(x^′(t)))^′+f(x^′(t))+g₁(x(t-τ₁(t,|x|_∞)))+β(t)g₂(x(t-τ₂(t,|x|_∞)))=e(t).     

Periodic solutions for a kind of Liénard equation

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.     

Periodic solutions for a kind of Liénard equation with a deviating argument

In this paper, the Liénard equation with a deviating argument

x^″(t)+_f1(t,x(t))x^′(t)+_f2(x(t))(x^′(t))²+g(t,x(t−τ(t)))=p(t)$x^{″}\left(t\right)+f_1(t,x\left(t\right))x^{\prime }\left(t\right)+f_2\left(x\left(t\right)\right){\left(x^{\prime }\left(t\right)\right)}^2+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$

Periodic solutions for a kind of Liénard equation with a deviating argument

By means of continuation theorem of coincidence degree theory, we study a kind of Liénard equation with a deviating argument as follows

     

New results on the existence of periodic solutions to a p-Laplacian differential equation with a deviating argument

By means of Mawhin's continuation theorem, a kind of p-Laplacian differential equation with a deviating argument as follows:

(φp(x^′(t))^)′=f(t,x(t),x(t−τ(t)),x^′(t))+e(t)     

Periodic solutions for generalized Liénard neutral equation with variable parameter

By means of Mawhin’s continuation theorem, existence criteria is established for the periodic solutions to a neutral equation with variable parameter. It is worth stating that the coefficient c$c$ is not a constant, which is different from the corresponding ones of past work.     

On existence of periodic solutions of a kind of Rayleigh equation with a deviating argument

Existence of periodic solutions for a kind of Rayleigh equation with a deviating argument

x^″(t)+f(x^′(t))+g(t,x(t−τ(t)))=p(t)$x^{″}\left(t\right)+f\left(x^{\prime }\left(t\right)\right)+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

On a periodic problem for higher-order differential equations with a deviating argument

For higher-order nonautonomous linear and nonlinear differential equations with deviating arguments, new sufficient conditions of existence and uniqueness of a periodic solution are found.     

The existence and uniqueness of periodic solutions for a kind of Duffing-type equation with two deviating arguments

By applying Mawhin’s continuation theorem and establishing new lemmas, some sufficient conditions for the existence and uniqueness of periodic solutions were obtained for a Duffing-type equation with two deviating arguments. Moreover, an example is given to illustrate the results.     

Stable phase-locked periodic solutions in a delay differential system

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory.