Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

Upper and Lower Solutions Method for Impulsive Differential Inclusions With Nonlinear Boundary Conditions

“…For general information the reader can see [13]. [2]) A family of functions S is said to be quasiequicontinuous on J, if for every ε > 0 there exists δ > 0 such that if x ∈ S, k = 0, 1, .…”

Nonlinear Differential Inclusions of Semimonotone and Condensing Type in Hilbert Spaces

“…Then, {H n } ∞ n=1 is an increasing sequence of finite dimensional subspaces of H with the property ∪ ∞ n=1 H n = H. If P n is the orthogonal projection of H onto H n , then each P n induces a corresponding projection of L 2 l0 (I; H, M ) onto subspace L 2 l0 (I; H n , M ), we denote it again by P n : (P n u)(t) = P n u(t) ∈ H n , ∀t ∈ I, u ∈ L 2 l0 (I; H, M ). We have P * n = P n → I as n → ∞ where P * n is the adjoint operator of P and I is the identity map on H. For each n ≥ 1, we define the set-valued map [21] yields that P n F is a Caratheodory map and since…”

“…Second, geometrical conditions such as compactness, connectedness and convexity of the values of F . Existence of solution for differential inclusions have been studied by many authors in the past half century with different application-directed motivations, including the issues of control and optimization, dynamical systems and even biological sciences [11,12,15,16,21]. In the earlier works, set-valued maps have been usually considered with convex values [6,7].…”

“…Also, in [2,3] differential inclusion with convex and nonconvex values on infinite dimensional Banach spaces have systematically been studied. In the case that set-valued map F is nonexpansive or Lipschitz in Hausdorff metric topology, several inequalities such as the Gronwall inequality give us a lot of useful information about properties of the set of trajectories of solutions [2,7,21,24]. However, nonexpansive and Lipschitz conditions are very strong and are not of practical importance.…”

Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

“…Several singularly perturbed control problems where studied in the papers by Gaitsgory [28], Grammel [40], Gaitsgory & Leizarowitz [31], Gaitsgory & Grammel [30], Gaitsgory & Nguyen [32], Gaitsgory & Nguyen [33], Gaitsgory [29], Grammel [43], Wang et al [124,125]. Kamenski [54,51,52] applied the averaging method to singularly perturbed systems of semilinear differential inclusions (see also the book by Kamenski et al [53]). …”