Yongxiang Li

Bio: Yongxiang Li is an academic researcher from Northwest Normal University. The author has contributed to research in topics: Banach space & Fixed-point theorem. The author has an hindex of 18, co-authored 86 publications receiving 917 citations.

Approximate Controllability of Non-autonomous Evolution System with Nonlocal Conditions

Abstract: In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution system of parabolic type with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. The discussions are based on Schauder’s fixed-point theorem as well as the theory of evolution family. An example is also given to illustrate the feasibility of our theoretical results.

Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators

Abstract: Abstract In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.

Monotone Iterative Technique for a Class of Semilinear Evolution Equations with Nonlocal Conditions

Abstract: This paper discusses the existence and uniqueness of mild solutions for a class of semilinear evolution equations with nonlocal conditions in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of the nonlinearity, a new monotone iterative method on the evolution equations with nonlocal conditions has been established. Particularly, an existence result without using noncompactness measure condition is obtained in ordered and weakly sequentially complete Banach spaces, which is very convenient for application. An example to illustrate our main results is also given.

Study on fractional non-autonomous evolution equations with delay

Abstract: In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ ( t , s ) , φ ( t , η ) and U ( t ) , which are generated by the operator − A ( t ) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k -set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator − A ( t ) generates an analytic semigroup for every t > 0 . The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

A blowup alternative result for fractional nonautonomous evolution equation of Volterra type

Abstract: In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space \begin{document}$E$\end{document} , where the operators in linear part (possibly unbounded) depend on time \begin{document}$t$\end{document} . Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [ 13 , J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

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