Anti-periodic solutions of nonlinear first order impulsive functional differential equations

The existence of anti-periodic solutions of the following nonlinear impulsive functional differential equations $$ x'(t) + a(t)x(t) = f(t,x(t),x(\alpha _1 (t)), \ldots ,x(\alpha _n (t))),t \in \mathbb{R},\Delta x(t_k ) = I_k (x(t_k )),k \in \mathbb{Z} $$ is studied. Sufficient conditions for the existence of at least one anti-periodic solution of the mentioned equation are established. Several new existence results are obtained.     

Periodic solutions for systems of forced coupled pendulum-like equations

The existence of periodic solutions for systems of forced pendulum-like equations was studied in the papers by J. A. Marlin (Internat. J. Nonlinear Mech.3 (1968), 439–447) and J. Mawhin (Internat. J. Nonlinear Mech.5 (1970), 335–339). In both works some symmetry hypotheses on the forcing terms were considered. This paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms. Note that as a model example it is possible to consider the motion of N coupled pendulums (see the already mentioned paper by J. A. Marlin) or the oscillations of an N-coupled point Josephson junction with external time-dependent disturbances studied in the autonomous case by M. Levi, F. C. Hoppensteadt, and W. L. Miranker (Quart. Appl. Math.36 (1978), 167–198).     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

New results of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T$T$-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(t,x^′(t))+g(t,x(t))=e(t).$x^{″}+f(t,x^{\prime }\left(t\right))+g(t,x\left(t\right))=e\left(t\right)\text{.}$

Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method

In this paper, reproducing kernel theorem is employed to solve anti-periodic solutions for Rayleigh-type equations. A simple algorithm is given to obtain the approximate solutions of the equations. By comparing the approximate solution with the exact analytical solution, we find that the simple algorithm is of good accuracy and it can be also applied to some ordinary or partial differential equations with initial-boundary value conditions and nonlocal boundary value conditions.     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

Anti-periodic solutions for semilinear evolution equations

In this paper, we study the existence problem of anti-periodic solutions for the following first order evolution equation:

     

Anti-periodic solutions for second order differential equations

By using the bilinear form and the Leray-Schauder principle, we prove the new anti-periodic existence results for second order differential equations.     

Anti-periodic solutions to nonlinear evolution equations

We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.     

一类含强迫项的Rayleigh型方程的反周期解

利用双线性形式和重合度理论,研究了一类含有强迫项的Rayleigh曲型方程的反周期解,给出了反周期解存在和唯一的新的判据,所得结果改善了已有的工作.     

Anti-periodic solutions for semilinear evolution equations in Banach spaces

In this paper, we study the anti-periodic problem for semi-linear evolution equations in reflexive Banach spaces. Several existence results are obtained under suitable conditions.     

Anti-periodic solutions of some nonlinear evolution equations

Following a recent work of H. Okochi in the case of evolution equations generated by subdifferential operators in a real Hilbert space, we point out that many quasi-autonomous evolution equations of non monotone type associated to odd non linear operators have some anti-periodic solutions provided the forcing term is anti-periodic. This comes from the fact that the space of anti-periodic functions is transversal to the kernel of the linear part and stable under the action of odd non linear operators. The proofs of our results combine strong a priori estimates which depend very little on the non-linearities with an application of Schauder's fixed point theorem to some related dissipative equations.     

Anti-periodic mild solutions to semilinear fractional differential equations

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.     

Anti-periodic solutions for evolution equations associated with maximal monotone mappings

In this work, we study the anti-periodic problem for a nonlinear evolution inclusion where the nonlinear part is an odd maximal monotone mapping and the forcing term is an anti-periodic mapping. Several existence results are obtained under suitable conditions. An example is presented to illustrate the results.     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).