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In this paper, we study the problem of periodic solutions for fourth-order nonlinear differential equations. Under proper conditions, we employ a novel proof to establish some criteria to ensure the existence and uniqueness of $T$-periodic solutions. Moreover, we give two examples to illustrate the effectiveness of our main results.

Sufficient criteria are established for the existence of periodic solutions to a type of Duffing equation with state-dependent delay, which improve and generalize some related results in the literature. The approach is based on Mawhin’s continuation theorem. The significance of the present paper is that our results are relevant to delay by Lemma 2.1, which is different from the corresponding results of past work.

In this paper, we consider a general high-order delay differential equation in the following form: x^(^n^)(t)=F(t,x(t),x(t-@t(t)),x^'(t),...,x^(^n^-^1^)(t)), where @t is a continuous periodic function defined on R with period T; F is a continuous function defined on R^n^+^2 and periodic to t, F(t,c,c,0,...,0)@?0 for all c@?R. By applying Mawhin's continuation theorem and a new inequality established in this paper, we obtain sufficient conditions for the existence of periodic solutions for the equation. Our results are new and the methods are different from previous works. Moreover, an example to illustrate the results is given.

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) is studied. Some new results on the existence of periodic solutions are obtained. The main results (Theorems 3.2 and 3.3) are all related to the deviating argument τ(t). Meanwhile, the degrees with respect to the variables x0,x1 of f(t,x0,x1,x2) are allowed to be grater than p−1, which is different from the corresponding conditions of known literature.

In this paper, we consider a kind of neutral functional differential equation as follows: x(t)−cx(t−τ)′=gt,xt−μ(t)+e(t) in the critical case |c|=1. By employing the topological degree theory and some analysis techniques, we obtain some new results on the existence of periodic solutions.

By means of Mawhin's continuation theorem, we study m-point boundary value problem at resonance in the following form: x(k)(t)=f(t,x(t),x′(t),…,x(k−1)(t))+e(t),t∈(0,1),x′(0)=0,x″(0)=0,…,x(k−1)(0)=0,x(1)=∑i=1m−2aix(ξi), where m⩾3, k⩾2 are two integers, ai∈R, ξi∈(0,1) (i=1,2,…,m−2) are constants satisfying ∑i=1m−1ai=1 and 0

By means of Mawhin’s continuation theorem, we study a kind of third-order p-Laplacian functional differential equation with distributed delay in the form: ϕp(x (t)) = g t, 0 −τ x(t + s) dα(s) + e(t), some criteria to guarantee the existence of periodic solutions are obtained., {"references":["F.X. Zhang and Y. Li, Periodic solutions for a kind of Duffing type p-\nLaplacian equation, Nonl. Anal.: RWA 9 (2008) 985-989.","K. Wang and Y.L. Zhu, Periodic solutions for a higher order p-\nLaplacian neutral functional differential equation with a deviating argument,\nNonl.Anal. 71 (2009) 3906-3913.","M.D. Pino, M. Elgueta and R. Manasevich, A homotopic deformation\nalong p of a Leray-Schaulder degree result and existence for\n(|u|p−2u) + f(t, u) = 0, u(0) = u(T) = 0, p > 1, J.Diff.Equs.\n80 (1989) 1-13.","R.E. Gaines and J.L. Mawhin, Coincidence Degree and Nonlinear Differential\nEquations, Springer-Verlag, Berlin, 1977.","R. Manasevich and J.L. Mawhin, Periodic solutions for nonlinear systems\nwith p-Laplacian-like operators, J.Diff.Equs. 145 (1998) 367-393.","S. Jin and S.P. Lu, Periodic solutions for a fourth-order p-Laplacian\ndifferential equation with a deviating argument, Nonl.Anal. 69 (2008)\n1710-1718.","S.P. Lu and Z.J. Gui, On the existence of periodic solutions to a p-\nLaplacian Rayleigh differential equation with a delay, J.Math.Anal.Appl.\n325 (2007) 685-702.","W.S. Cheng and J.L. Ren, Periodic solution for p-Laplacian Li'enard\nequation with a deviating argument, Nonl.Anal. 59 (2004) 107-120.","W.S. Cheng and J.L. Ren, On the existence of periodic solutions for\np-Laplacian generalized Li'enard equation, Nonl.Anal. 60 (2005) 65-75.\n[10] Y.L. Zhu and S.P. Lu, Periodic solutions for p-Laplacian neutral functional\ndifferential equation with deviating arguments, J.Math.Anal.Appl.\n325 (2007) 377-385.\n[11] Y. Tang and Y.Q. Li, New results of periodic solutions for a kind of\nDuffing type p-Laplacian equation, J.Math.Anal.Appl. 340 (2008) 1380-\n1384.\n[12] Z.B. Cheng and J.L. Ren, Periodic solutions for a fourth order Rayleigh\ntype p-Laplacian delay equation, Nonl.Anal. 70 (2009) 516-523."]}

Perodic solutions of discrete Rayleigh equations with variable delays can be related to steady state solutions of discrete time neural networks. By means of Mawhin’s continuation theorem, and sharp a priori estimates, existence of periodic solutions is established.

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence of T-periodic solutions for fourth-order nonlinear differential equations. Two illustrative examples are provided to demonstrate that the results in this paper hold under weaker conditions than the existing ones, and are more effective.

By means of Mawhin's continuation theorem, we study some second order differential equations with a deviating argument: x ″ ( t ) = f ( t , x ( t ) , x ( t − τ ( t ) ) , x ′ ( t ) ) + e ( t ) . Some new results on the existence of periodic solutions are obtained. The interest is that we allow the degree with respect to the variables x 0 , x 1 , x 2 of f ( t , x 0 , x 1 , x 2 ) to be greater than 1; and also the result (Theorem 3.2) is related to the deviating argument τ ( t ) . Meanwhile, we give an example to demonstrate our result.