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In the present paper, we consider the existence of solutions with a prescribed l 2 -norm for the following discrete Schrödinger equations, { − Δ 2 u k − 1 − f (u k) = λ u k k ∈ Z , ∑ k ∈ Z | u k | 2 = α 2 , where Δ 2 u k − 1 = u k + 1 + u k − 1 − 2 u k , f ∈ C (R) , α is a fixed constant, and λ ∈ R arises as a Lagrange multiplier. To get the solutions, we investigate the corresponding minimizing problem with the l 2 -norm constraint: E α = inf { 1 2 ∑ | Δ u k − 1 | 2 − ∑ F (u k) : ∑ | u k | 2 = α 2 }. An elaborative analysis on a minimizing sequence with respect to E α is obtained. We prove that there is a constant α 0 ≥ 0 such that there exists a global minimizer if α > α 0 , and there exists no global minimizer if α < α 0 . It seems that it is the first time to consider the solution with a prescribed l 2 -norm of the discrete Schrödinger equations. [ABSTRACT FROM AUTHOR]

In this paper, we address the existence and multiplicity of 2-periodic solutions to differential equations with a distributed delay of the form x′(t)=f[∫t-1tg(x(s))ds],x∈RN.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} x^{\prime }(t)=f\Big [\int _{t-1}^t g\big (x(s)\big ) d s\Big ],\quad x \in \textbf{R}^N. \end{aligned}$$\end{document}Combining Kaplan–Yorke’s method with pseudoindex theory, we estimate the number of periodic solutions when the equations are both resonant and nonresonant. More specifically, we define two indices using asymptotic linear coefficient matrices at the origin and at infinity. Then the lower bound on the number of periodic solutions to the equations is estimated by the indices. Finally, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the following discrete Schrödinger equations − (a + b ∑ k ∈ Z | Δ u k − 1 | 2) Δ 2 u k − 1 + V k u k = f k (u k) k ∈ Z , where a, b are two positive constants and V = { V k } is a positive potential. Δ u k − 1 = u k − u k − 1 and Δ 2 = Δ (Δ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any T > 0 , there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere. [ABSTRACT FROM AUTHOR]

This study aimed to evaluate the T2 relaxation time of the brain in severely scalded rats using a magnetic resonance (MR) T2 mapping sequence, and to investigate the correlation between T2 relaxation time and plasma glucose level. Twenty-eight Wistar rats were randomly divided into the scalded group ( n=21) and control group ( n=7). Magnetic resonance scans were performed with T1WI, T2WI, and T2-mapping sequences in the scalded group; the scans were performed 1 day prior to scalding and 1, 3, 5, and 7 days post-scalding; in addition, identical MR scans were performed in the control group at the same time points. T2-maps were generated and T2 relaxation times were acquired from the following brain regions: the hippocampus, thalamus, caudate-putamen, and cerebrum. Pathological changes of the hippocampus were observed. The plasma glucose level of each rat was measured before each MR scan, and a correlation analysis was performed between T2 relaxation time and plasma glucose level. We found that conventional T1WI and T2WI did not reveal any abnormal signals or morphological changes in the hippocampus, thalamus, caudate-putamen, or cerebrum post-scalding. Both the T2 relaxation times of the selected brain regions and plasma glucose levels increased 1, 3, and 5 days post-scalding, and returned to normal levels 7 days post-scalding. The most marked increase of T2 relaxation time was found in the hippocampus; similar changes were also revealed in the thalamus, caudate-putamen, and cerebrum. No correlation was found between T2 relaxation time and plasma glucose level in scalded rats. Pathological observation of the hippocampus showed edema 1, 3, and 5 days post-scalding, with recovery to normal findings at 7 days post-scalding. Thus, we concluded that T2 mapping is a sensitive method for detecting and monitoring scald injury in the rat brain. As the hippocampus is the main region for modulating a stress reaction, it showed significantly increased water content along with an increased plasma glucose level post-scalding. [ABSTRACT FROM AUTHOR]

By applying the method based on the usage of the equivariant gradient degree introduced by Gęba () and the cohomological equivariant Conley index introduced by Izydorek (), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant $${\omega(\nabla \Phi)}$$, expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of $${\Phi}$$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation (here $${f : V \rightarrow \mathbb{R}}$$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (). The equivariant invariant $${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$$ in the case n ≠ 0 (for maximal twisted orbit types ( H)) guarantees the existence of at least | n| different G-orbits of periodic solutions with symmetries at least ( H). This result generalizes the result by Guo and Yu () obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of $${\omega(\,\nabla \Phi)}$$ evaluated. [ABSTRACT FROM AUTHOR]