Thu Dau Mot University Journal of Science


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In this paper, we establish compactness and continuous dependence on parameters for solution-set of the second order differential inclusion including self-adjoint operator in the form \begin{align*} \left\{ \begin{gathered} \frac{\partial^2}{\partial t^2} u(t,x) +2\mathcal{A} \frac{\partial}{\partial t}u(t,x)+\mathcal{A}^{2} u(t,x) \in F(t,u(t),\mu),\,\, \hfill (t,x)\in [0,T)\times\Omega \\ u(0,x)=\frac{\partial }{\partial t}u(0,x)=0, \,\, \hfill x \in \Omega, \\ %u(T,x) = h(x), \,\, \hfill x\in\Omega, \end{gathered}\right. %\label{MainProblem} \end{align*} where $\mathcal A$ is a self-adjoint operator. We use the spectral theory on Hilbert spaces to obtain formulation for mild solutions. Using the mild solution formula together with a measure of non-compactness with values in an ordered space, we construct useful bounds for solution operators. Then, we establish necessarily upper semi-continuous and condensing settings, which mainly help to obtain the global existence of mild solutions and the compactness of the mild solution set. Finally, we provide a brief discussion on the continuous dependence of the solution-set on parameter $\mu$.

Publication Information

Publisher

Thu Dau Mot University, Viet Nam

Honorary Editor-in-Chief and Chairman of the Editorial Board

Assoc. Prof. Nguyen Van Hiep

Deputy Editor-in-Chief

PhD. Trần Hạnh Minh Phương
Thu Dau Mot University

Editorial Board

Prof. Tran Van Doan
Fujen University, Taiwan
Prof. Zafar Uddin Ahmed
Vietnam National University Ho Chi Minh City

Prof.Dr. Phillip G.Cerny
The University of Manchester, United Kingdom
Prof. Ngo Van Le
University of Social Sciences and Humanities (VNU-HCM)

Prof. Bui The Cuong
Southern Institute of Social Sciences​​​​​​​
Prof. Le Quang Tri
Can Tho University

Assoc. Prof. Nguyen Van Duc
Animal Husbandry Association of Vietnam
Assoc. Prof. Ted Yuchung Liu
National Pingtung University, Taiwan

PhD. Anita Doraisami
Economics Monash University, Australia
Prof. Dr. Andrew Seddon
Asia Pacific University of Technology & innovation (APU)

Assoc. Prof. Le Tuan Anh
Thu Dau Mot University
Prof. Abtar Darshan Singh
Asia Pacific University, Malaysia

Prof.Dr. Ron W.Edwards
The University of Melbourne, Australia
Assoc. Prof. Hoang Xuan Nien
Thu Dau Mot University

PhD. Nguyen Duc Nghia
Vietnam National University Ho Chi Minh City
PhD. Bao Dat
Monash University (Australia)

PhD. Raqib Chowdhury
Monash University (Australia)
PhD. Nguyen Hoang Tuan
Thu Dau Mot University

PhD. Nguyen Thi Lien Thuong
Thu Dau Mot University

Assistant

Nguyen Thi Man
Thu Dau Mot University