Positive Eigenvalues for the Adjoint of an Increasing Operator

Nguyen Hoang Tuan; Vo Viet Tri; Anh Tuan Nguyen

doi:10.61383/ejam.202532109

Authors

Nguyen Hoang Tuan Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, Dien Hong Ward, Ho Chi Minh City, Vietnam; Vietnam National University Ho Chi Minh City, Linh Xuan Ward, Ho Chi Minh City, Vietnam https://orcid.org/0000-0003-4354-2937
Vo Viet Tri Division of Applied Mathematics, Thu Dau Mot University, Ho Chi Minh City, Vietnam https://orcid.org/0000-0003-4830-4954
Nguyen Anh Tuan Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam; Faculty of Applied Technology, Van Lang School of Technology, Van Lang University, Ho Chi Minh City, Vietnam Corresponding Author https://orcid.org/0000-0002-8757-9742

DOI:

https://doi.org/10.61383/ejam.202532109

Keywords:

eigenvalue, eigenvector, eigen-pair, fixed point index, dual operator

Abstract

This paper develops a general framework for investigating the eigenvalue problem of the extended adjoint mapping \(A^\ast:E_b^\ast\to E_b^\ast\) associated with a possibly nonlinear operator \(A:E\to E\). By employing the theory of relative topological degree for condensing mappings with respect to a measure of noncompactness, we establish sufficient conditions ensuring the existence of positive eigenvalues of \(A^\ast\). The extended dual space \(E_b^\ast\) is strictly larger than the classical dual \(E^\ast\), and the two coincide only when \(E\) is finite-dimensional. In this special case, our results recover the classical eigenvalue theory for linear compact operators. The framework introduced here extends the topological degree approach to a broader nonlinear setting and provides a foundation for further applications to convex processes and control problems.

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