In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following third-order generalized right-focal boundary value problem with p-Laplacian operator:The main tool is the monotone iterative technique.

Abstract.We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability are concluded for the system. Moreover, we show that the exponential stability is independent of the location of the joint. The range of the feedback gains that guarantee the system to be exponentially stable is identified.Mathematics Subject Classification. 93C20, 93C25, 35J10, 47E05

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The purpose of this work is to study the Riesz basis generation of the well-known SCOLE model. By using Guo's conclusion that the Riesz basis property holds for the general system if its associated characteristic equation is strongly regular, it is shown that the Riesz basis property can be established for a beam equation with an endmass . Furthermore, we get the conclusions that the system operator A generates a Co-semigroup eAt on state space and the spectrum-determined growth condition holds: s(A) = w(A).