In this paper, several metric characterizations of well-posedness for systems of generalized mixed quasivariational inclusion problems and for optimization problems with systems of generalized mixed quasivariational inclusion problems as constraints are given. The equivalence between the well-posedness of systems of generalized mixed quasivariational inclusion problems and the existence of solutions of systems of generalized mixed quasivariational inclusion problems is given under suitable conditions.

In this paper we present Lyapunov inequality for the following fractional boundary value problem are the left and right Riemann-Liouville fractional integrals of order 0 β < 1, respectively, and ω ∈ L 1 ([a, b], R). Using the obtained inequality, we provide lower bounds for the first eigenvalue of the fractional differential equations with homogeneous Dirichlet boundary problem.

The purpose of this paper is to find a solution of a general system of variational inequalities (for short, GSVI), which is also a unique solution of a hierarchical variational inequality (for short, HVI) for an infinite family of nonexpansive mappings in Banach spaces. We introduce general implicit and explicit iterative algorithms, which are based on the hybrid steepest-descent method and the Mann iteration method. Under some appropriate conditions, we prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the GSVI, which is also a unique solution of the HVI.

In this paper, we introduce and analyze a hybrid implicit steepest-descent algorithm for solving the triple hierarchical variational inequality problem with the hierarchical variational inequality constraint for finitely many nonexpansive mappings in a real Hilbert space. The proposed algorithm is based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method, and Halpern's iteration method. Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.