In this paper, some best proximity point theorems for generalized weakly contractive mappings which satisfy certain conditions by using three control functions in partially ordered Menger PM-spaces are obtained, and sufficient conditions to guarantee the uniqueness of the best proximity points are also given. Moreover, some corollaries are derived as consequences of the main results.
We first introduce the concept and the existence criterion of a random uniform exponential attractor for a non-autonomous random dynamical system. Then we prove the existence of a random uniform exponential attractor for FitzHugh-Nagumo system with additive noise and quasi-periodic external forces defined in Rn.
In this paper, the notion of a generalized central α-Armendariz ring is introduced, and this paper also obtains some basic properties of such rings. At the same time, the relations between generalized central α-Armendariz rings and other rings are studied.
We obtain a sharp second order subcritical Adams inequality in Lorentz Sobolev space W 2L2,q(R4). Moreover, the lower and upper bounds asymptotically for the subcritical Adams functional is obtained. Our approach is based on the rearrangement free argument developed by Lam and Lu[A new approach to sharp MoserTrudinger and Adams type inequalities:a rearrangement-free argument, J. Diff. Equ., 2013, 255(3):298-325].
There are two open problems on the global existence results of non-isentropic gas dynamics. One is whether the weak solutions exist globally with small initial data containing vacuum, the other is whether the global existence results hold with arbitrary large initial data. By introducing a scaling framework, we give the equivalence of the two problems above. For vanishing viscosity solutions, the positive answer to the first question naturally implies the positive answer to the second one. And this scaling framework can be applied to most systems of conservation laws with physical background.
We consider a new form of the localized Baskakov operators, and obtain some convergence properties of the new operators. We also obtain a new estimate for the kernel of the Baskakov operators by making use of one of the central limit theorems in probability theory.
We study the following fractional Schrödinger-Poisson system
where s ∈ (4/3, 1), t ∈ (0, 1), the continuous function f is superlinear at zero and subcritical at infinity and the exponent q ≥ 2s*=(3-2s)/6. We obtain a positive solution of the above problem for small λ > 0 via the variational method. Our main contribution is that we can deal with the supercritical case.
Some geometric conditions in terms of the characteristic of convexity, the normal structure coefficient, the James type constant and the García-Falset coefficient were considered in the paper, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some well known results and give affirmative answers to some open questions.
We consider the following Hausdorff operator Hφf(x)=∫Rnφ(u1,..., un) · f(u1x1,..., unxn)du1 · · · dun, where φ can be considered as a distribution on Rn. When n ≥ 2 and φ is a Schwartz function, we show that Hφ is bounded on Hp(Rn) for some p ∈ (0, 1) if and only if φ ≡ 0. Furthermore, when n ≥ 2, if φ is just a continuous function and Hφ can be defined suitable, then we can also prove that Hφ is bounded on Hp(Rn) for some p ∈ ((n+1)/n, 1) if and only if φ equals to a constant. These facts mean that Hφ is very complicated on Hp(Rn) (n ≥ 2). Moreover, we establish a result of the boundedness of Hφ on Lp(Rn), p > 1. The key idea used here is to reformulate Hφ as a convolution operator.
We describe a close relation between the p-fusion frames and the p-frames on Banach spaces. Using the analysis operators and the synthesis operators, we provide the equivalent descriptions of the p-fusion Bessel sequences, p-fusion frames and q-fusion Riesz bases.
Let R be a commutative ring. Then the small finitistic projective dimension of R is defined as fPD(R)=sup{pdRM|M ∈ FPR}. In this paper, it is shown that if R is a connected strong Prüfer ring, then fPD(R) ≤ 1. It is also shown that if R is a strong Prüfer ring, and if M is a Q-torsion module with M ∈ FPR, then pdRM ≤ 1.
In this paper, we study the following fractional Kirchhoff-type equation with critical growth ε2sM(ε2s-3 ∫∫R3×R3(|u(x)?u(y)|2)/(|x-y|3+2s)dxdy)(-Δ)su + V (x)u=λW(x)·f(u) + K(x)|u|2s*-2u, x ∈ R3, where M is a continuous and positive Kirchhoff function, λ > 0 is a parameter, (-Δ)s is the fractional Laplace operator with 3/4 < s < 1, V (x), W(x) and K(x) are all positive potentials. Under some assumptions on potentials, we obtain the existence of a positive ground state solution for ε > 0 small and λ large. Moreover, we show that these ground state solutions concentrate at a special set characterized by potentials. Finally, we study the decay estimate of ground state solutions.
