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lements of Nonsmooth Analysis
Published in Mircea Sofonea, Stanisław Migórski, Variational-Hemivariational Inequalities with Applications, 2017
Mircea Sofonea, Stanisław Migórski
Bochner-Lebesgue spaces. The Bochner-Lebesgue spaces are defined on the basis of the Bochner integral which represents a natural generalization of the Lebesgue integral of a scalar-valued function. We recall below only the very basic definitions and properties that we need in the following.
Solvability and optimal control of fractional differential hemivariational inequalities
Published in Optimization, 2021
Guangming Xue, Funing Lin, Bin Qin
Throughout this paper, let X be a real separable and reflexive Banach spaces. denotes the norm of X. denotes the dual of X. denotes the duality pairing between and X. The weak convergence is denoted by . The open ball of centre with radius r>0 is denoted by , and the open unit ball in X centred at 0 is denoted by B. denotes the Banach space consist of all continuous functions from T into X with the norm . denotes the Banach space consist of all Bochner integrable vector-valued functions from T to X. For a Banach space X, the symbol stands for X equipped with the weak topology.
Global exponential periodicity and stability of memristor-based complex-valued delayed neural networks
Published in International Journal of Systems Science, 2018
First, we will prove system (12) exists at least one ω-periodic solution. We denote be the solution of system (12) through (0, Φ), where Φ = ((φR)T, (φI)T)T. Let denote the Banach space of all functions which are Bochner integrable. Let , , and for all , then by the literatures (Dugundji & Grranas, 1986; Papini & Taddei, 2005; Xue & Yu, 2007), we know is not only a linear operator but also bijective. As result of that, L−1 is completely continuous.