The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping utt + ∆ 2 u − m(∇u 2)∆u + ut p ut + f (u) = h, in Ω × R + , p ≥ 0 on a bounded smooth domain Ω ⊂ R n with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient m(∇u 2) and the nonlinear term f (u), the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.

This paper is devoted to establishing the long-time behavior of extensible beam equation with the nonlocal weak damping on a bounded smooth domain of Rn with hinged (clamped) boundary condition. It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity f(u) is up to the critical case in natural energy space.

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In this paper, we consider extended Nash equilibriums of nonmonetized noncooperative games. By using a modified fixed point theorem of set-valued mappings on partially ordered sets, we prove an existence theorem of extended Nash equilibriums of the nonmonetized noncooperative game. Finally, an example is given to illustrate the advantages of our results.MSC: 06A06; 47H10; 58J20; 91A06; 91A10