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Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

100%
Li Y., Yang H.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 2
105-114
EN
The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2n+1)} = f(t,u,u',...,u^{(2n)})$, where $f: ℝ × ℝ^{2n+1} → ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f(t,x₀,x₁,...,x_{2n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].
2
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Existence of solutions for delay evolution equations with nonlocal conditions

100%
Zhang X., Li Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
616-627
EN
In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.
3
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Existence of positive radial solutions for the elliptic equations on an exterior domain

100%
Li Y., Zhang H.
Annales Polonici Mathematici
|
2016
|
tom 116
|
nr 1
67-78
EN
We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩$lim_{|x|→∞} u(x) = 0$, where $Ω = {x ∈ ℝ^{N}: |x| > r₀}$, N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0 < ∫_{r₀}^{∞} rK(r)dr < ∞$, f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point index theory in cones.
4
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Existence of mild solutions for fractional evolution equations with nonlocal initial conditions

81%
Chen P., Li Y., Li Q.
Annales Polonici Mathematici
|
2014
|
tom 110
|
nr 1
13-24
EN
This paper discusses the existence of mild solutions for a class of semilinear fractional evolution equations with nonlocal initial conditions in an arbitrary Banach space. We assume that the linear part generates an equicontinuous semigroup, and the nonlinear part satisfies noncompactness measure conditions and appropriate growth conditions. An example to illustrate the applications of the abstract result is also given.
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