We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y）...We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.展开更多
In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic s...In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic solution was obtained.展开更多
Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N...Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.展开更多
This paper is considered the existence of positive solutions for a class of generalized quasilinear Schrödinger equations with nonlocal term in R<sup>N</sup> which have appeared from plasma physic...This paper is considered the existence of positive solutions for a class of generalized quasilinear Schrödinger equations with nonlocal term in R<sup>N</sup> which have appeared from plasma physics, as well as high-power ultrashort laser in matter. We use a charge of variables and obtain the existence of solutions via minimization argument.展开更多
Sufficient conditions are given to assert that two differentiable mappings between Banach spaces have common values. The proof is essentially based upon continuation methods.
Mixing and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Recently the notion of coherence has come to a more rigorous footing, in particular,...Mixing and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Recently the notion of coherence has come to a more rigorous footing, in particular, within the studies of finite-time nonautonomous dynamical systems. Here we recall “shape coherent sets” which is proven to correspond to slowly evolving curvature, for which tangency of finite time stable foliations (related to a “forward time” perspective) and finite time unstable foliations (related to a “backwards time” perspective) serve a central role. We compare and contrast this perspective to both the variational method of geodesics [17], as well as the coherent pairs perspective [12] from transfer operators.展开更多
文摘We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.
文摘In this paper, the well known implicit function theorem was applied to study existence and uniqueness of periodic solution of Duffing-type equation. Un-der appropriate conditions around the origin, a unique periodic solution was obtained.
基金the National Natural Science Foundation of China(No.10571174,10631030)Chinese Academy oF Sciences grant KJCX3-SYW-S03.
文摘Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.
文摘This paper is considered the existence of positive solutions for a class of generalized quasilinear Schrödinger equations with nonlocal term in R<sup>N</sup> which have appeared from plasma physics, as well as high-power ultrashort laser in matter. We use a charge of variables and obtain the existence of solutions via minimization argument.
文摘Sufficient conditions are given to assert that two differentiable mappings between Banach spaces have common values. The proof is essentially based upon continuation methods.
文摘Mixing and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Recently the notion of coherence has come to a more rigorous footing, in particular, within the studies of finite-time nonautonomous dynamical systems. Here we recall “shape coherent sets” which is proven to correspond to slowly evolving curvature, for which tangency of finite time stable foliations (related to a “forward time” perspective) and finite time unstable foliations (related to a “backwards time” perspective) serve a central role. We compare and contrast this perspective to both the variational method of geodesics [17], as well as the coherent pairs perspective [12] from transfer operators.
