Stability and Convergence Analysis for Set-Valued Extended Generalized Nonlinear Mixed Variational Inequality Problems and Generalized Resolvent Dynamical Systems

In this paper, we study a set-valued extended generalized nonlinear mixed variational inequality problem and its generalized resolvent dynamical system. A three-step iterative algorithm is constructed for solving set-valued extended generalized nonlinear variational inequality problem. Convergence and stability analysis are also discussed. We have shown the globally exponential convergence of generalized resolvent dynamical system to a unique solution of set-valued extended generalized nonlinear mixed variational inequality problem. In support of our main result, we provide a numerical example with convergence graphs and computation tables. For illustration, a comparison of our three-step iterative algorithm with Ishikawa-type algorithm and Mann-type algorithm is shown.

Existence of solution and iterative approximation of a system of generalized variational-like inclusion problems in semi-inner product spaces

In this paper, we consider the system of generalized variational-like inclusion problems in semi-inner product spaces. We define a class of (H,?)-?-monotone operators and its associated class of generalized resolvent operators. Further, using generalized resolvent operator technique, we give the existence of solution of the generalized variational-like inclusion problems. Furthermore, we suggest an iterative algorithm and give the convergence analysis of the sequences generated by the iterative algorithm. The results presented in this paper extend and unify the related known results in the literature.

On fluttering modes for aircraft wing model in subsonic air flow

The paper deals with unstable aeroelastic modes for aircraft wing model in subsonic, incompressible, inviscid air flow. In recent author’s papers asymptotic, spectral and stability analysis of the model has been carried out. The model is governed by a system of two coupled integrodifferential equations and a two-parameter family of boundary conditions modelling action of self-straining actuators. The Laplace transform of the solution is given in terms of the ‘generalized resolvent operator’, which is a meromorphic operator-valued function of the spectral parameter λ, whose poles are called the aeroelastic modes. The residues at these poles are constructed from the corresponding mode shapes. The spectral characteristics of the model are asymptotically close to the ones of a simpler system, which is called the reduced model. For the reduced model, the following result is shown: for each value of subsonic speed, there exists a radius such that all aeroelastic modes located outside the circle of this radius centred at zero are stable. Unstable modes, whose number is always finite, can occur only inside this ‘circle of instability’. Explicit estimate of the ‘instability radius’ in terms of model parameters is given.