Minimal periods of solutions to higher-order functional differential equations

Abstract

Abstract. We show that a problem on minimal periods of solutions of Lipschitz functional differential equations is closely related to the unique solvability of the periodic problem for linear functional differential equations. Sharp bounds for minimal periods of non-constant solutions of higher order functional differential equations with different Lipschitz nonlinearities are obtained.

Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory

“…With 3 = max{ 3 , 3 }, the inequalities on the right hand of (28) hold. Thus we get (28), which implies that 2 | | ⩽ 3 | | for | | ⩽ 1 and that 1 < < < 2. The proof is complete.…”

“…where : R → R satisfies the Lipschitz condition and : R 1 → R 1 is a measurable function. The lower bounds for the periods of the periodic solutions to (5) and their special forms are estimated in [25][26][27][28][29][30][31]. From this perspective, Theorem 1 complements the information in the case of non-Lipschitzian differential equations.…”

“…From (25), it is easy to see that̃∈ 1 (R , R). Now we start to prove (28). Let be such a constant that | lñ( ) − ln | || ⩽ for ∈ 1 ≡ 1 .…”

“…The existence of the periodic solutions for delay differential equations has been extensively investigated by using various methods, including fixed point theorems [1][2][3][4][5], Hopf bifurcation theorems [6][7][8], variational methods [9][10][11][12][13][14], the methods of differential inequalities [15][16][17][18][19][20][21], and other effective approaches (e.g., see [22][23][24]). In [25][26][27][28][29][30][31], the minimal periods of the periodic solutions to Lipschitzian differential equations are estimated through the Lipschitz constants (see Remark 4).…”