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In this paper we prove that the problem x(t)=f(x(t))+h(t), x(0)=0, where f and h are non-negative, f is finite a.e. and , h are Lebesgue integrable, has an absolutely continuous solution.  相似文献   

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The paper is concerned with the delay differential equation u+λb(t)f(u(tτ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.  相似文献   

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Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that etf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .  相似文献   

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The classical criterion of asymptotic stability of the zero solution of equations x=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,bK, such that for some cK. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,TBH.  相似文献   

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The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y= of an almost periodic (ap) is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c0. This is here generalized in several directions: Instead of it holds also for φ defined only on a half-line , instead of ap functions abstract classes with suitable properties are admissible, can be weakened to φ in some “mean” class , then ; here contains all fL1loc with in for all h>0 (usually strictly); furthermore, instead of boundedness of y mean boundedness, y in some , or in , ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t+h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr-Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y=φ of degree m with ap φ is extended similarly for ; then provided, for example, y is in some with U=L or is totally ergodic and, for the half-line, Reλ?0 for all eigenvalues P(λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and ergodic implies bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general .  相似文献   

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Assume that and are uniformly continuous functions, where D1,D2X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x(x)+a and g(x)=x(x)+b with some xX and a,bR or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .  相似文献   

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Bi-Sobolev mappings have been defined as those homeomorphisms such that f and f−1 belong to . We deduce regularity properties of the distortion of f from the regularity of the differential matrix Df−1 and conversely.  相似文献   

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We consider the semilinear elliptic problem in Ω, u=0 on ∂Ω, where 0∈Ω is a smooth bounded domain in RN, N?4, , is the critical Sobolev exponent, f(x,⋅) has subcritical growth at infinity, K(x)>0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.  相似文献   

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