Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

The aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.     

Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y=Pφ 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 f∈L¹_loc 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 Pφ ergodic implies Pφ bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general .     

有序Banach空间常微分方程的正周期解

依据凝聚锥映射的一个krasnoselskii型不动点定理，在有序Banach空间中获得了二阶常微分方程．-u^n(t) Mu(t)＝f(t,u(t))正ω—周期解的存在性结果。     

Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces

In this paper, we prove that the space is complete. This not only gives an affirmative answer to a basic problem in this field, but also enables us to obtain an existence and uniqueness theorem of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. An example is given to illustrate our theorem. The work was supported partly by the National Natural Science Foundation of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.     

关于Banach空间隐式常微分方程的解的存在性

讨论了 Ｂａｎａｃｈ 空间中隐式常微分方程 Ｆ（ｔ,ｘ,ｘ′） ＝ ０, ｘ（ｔ０ ） ＝ ｘ０ , ｘ′（ｔ０） ＝ ｙ０ 的解的存在性,其中, Ｆ 的定义域可含无穷远点     

Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations

In this work we study the existence of almost periodic and asymptotically almost periodic solutions for partial neutral functional differential equations with unbounded delay.     

On the existence of periodic solutions for the third-order nonlinear ordinary differential equations

In this paper,the existence of periodic solution for the third-order nonlinear ordinarydifferential equation of the form -^x f(-^x) g(-^x) h(x)=p(t) is consldered,where f,g,h andp are the continuous functlons, and p (t T) =p (t). By using the Leray-Schauder degreemethod,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained. Some previous results are extended and improved.     

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Let E be a finite-dimensional Banach space, let C⁰(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C ⁰(R ,E) → C ⁰(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C ⁰(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation

On the existence of almost periodic solutions of some dissipative second order differential equations

Summary The search for almost periodic solutions of any dissipative equation of the form(1.1), in which p(t) is an almost periodic function, has come to be closely linked up with a number of standard ? convergence ? restrictions on f, g′, g″ and k (see, for example,[2] and[3]). The object of the present paper is to show that as far as the existence, alone, of an almost periodic solution of(1.1) is concerned these ? convergence ? restrictions on f, g′, g″ and k are quite unnecessary. The first result (Theorem 1) shows in fact that the conditions(1.2) alone are quite sufficient for the existence of an almost periodic solution of(1.1); and Theorem 2 extends this result (though under stronger conditions on f and g) to the case in which the forcing function depends on x and x as well. Partially supported by N.S.F. research project G-57 at The University of Michigan.