In this article we study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence scattering solutions. The two main ingredients proof are space-time resonance method boundedness wave operators for linear Schr\"{o}dinger potential.

Fractional differential equations and fractional integral equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines. Gronwall-Bellman inequalities are important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of Fractional differential equations and fracti...

In [6], the second author proved general metatheorems for the extraction of effective uniform bounds from ineffective existence proofs in functional analysis, more precisely from proofs in classical analysis A(:= weakly extensional Peano arithmetic WE-PA in all finite types + quantitifer-free choice + the axiom schema of dependent choice DC) extended with (variants of) abstract bounded metric s...

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the equation to system first order, suitable complete Lyapunov constructed and employed obtain sufficient conditions that guarantee existence unique periodic solution, asymptotic uniform stability zero solutions, boundedness ultimate solutions. The obtained results are new include many promin...

in this paper, a complete description concerning linear operators of banach spaces with range in lipschitz algebras $lip_al(x)$ is provided. necessary and sufficient conditions are established to ensure boundedness and (weak) compactness of these operators. finally, a lower bound for the essential norm of such operators is obtained.

Abstract A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest each and the effect of time delay for interaction between are considered, which makes more applicable real situations. Specifically, we address global dynamics behavior, including existence positive solutions, ultimate boundedness, extinction with probability one, persistence mean stability. ...