Nonlinear Analysis 71 (2009) 2335–2342 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Locally Lipschitz selections in Banach lattices Mariusz Michta, Jerzy Motyl ∗ Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland article info Article history: Received 13 November 2008 Accepted 12 January 2009 MSC: 49J53 60H20 93E03 Keywords: Order complete Banach lattice Upper separated multifunction Legendre–Fenchel transform Convex selection Differential inclusion Stochastic differential inclusion abstract Let X be a Banach space while (Y , ) is a Banach lattice. We consider the class of ‘‘upper separated’’ set-valued functions F : X → 2 Y and investigate the problem of the existence of convex and locally Lipschitz selections of F . We discuss some applications of the selection results obtained in the paper to the theory of deterministic and stochastic differential inclusions. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In general, for deterministic or stochastic differential inclusions, an appropriate kind of regularity of their multivalued structure is required. In particular, the properties such as the Lipschitz continuity, lower, upper semicontinuity or monotonicity of set-valued mappings have most often been considered (see e.g. [1,2] and references therein). One of the main reasons is that such regularities imposed on set-valued operators allow us to use results on the existence of exact or at least approximate selections having an appropriate kind of regularity, and therefore to reduce the multivalued problems to single-valued ones. Hence regular selections have attracted considerable interest as a useful tool for proving the existence of solutions of set-valued problems. In the paper, we deal with a new class of multifunctions with values in Banach lattices, which need not satisfy any of these properties mentioned above. For this class, we study necessary and sufficient conditions for the existence of convex and locally Lipschitz selections. In particular, the convexity of selections allows us to prove an infinite-dimensional version of the convex-type Sandwich Theorem. This result extends the one dimensional, necessary and sufficient conditions for the existence of a convex function which separates two given functions. On the other hand, having a local Lipschitz selection, we prove new existence results for deterministic and stochastic differential inclusions. We begin our considerations with auxiliary definitions and facts needed in the sequel. 2. Upper separated set-valued functions in Banach lattices Let X , Y be Banach spaces. Let K + denote a cone of positive elements in Y . We will use the notation x  y if y − x ∈ K + . We will always assume that (Y , ) is an order complete Banach lattice (i.e., every nonempty and majorized subset of ∗ Corresponding author. E-mail address: j.motyl@wmie.uz.zgora.pl (J. Motyl). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.067