# Applied Analysis and Differential Equations: Iasi, Romania, by Ovidiu Carja, Ioan I. Vrabie

By Ovidiu Carja, Ioan I. Vrabie

This quantity includes refereed learn articles written by means of specialists within the box of utilized research, differential equations and comparable themes. recognized major mathematicians world wide and widespread younger scientists disguise a various variety of issues, together with the main interesting fresh advancements. A wide variety of themes of contemporary curiosity are handled: life, strong point, viability, asymptotic balance, viscosity suggestions, controllability and numerical research for ODE, PDE and stochastic equations. The scope of the publication is broad, starting from natural arithmetic to numerous utilized fields resembling classical mechanics, biomedicine, and inhabitants dynamics

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9. O. Cˆ arj˘ a and I. I. Vrabie, Differential Equations on Closed Sets, in Handbook of Differential Equations, Ordinary Differential Equations, 2, Edited by A. Ca˜ nada, P. Dr´ abek and A. , 2005), 147–238. 10. I. I. Vrabie, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. S ¸ tiint¸. Univ. Al. I. Cuza Ia¸si Sect¸. , 27 (1981), 117–125. ro The aim of this paper is to present a short survey of several new results concerning optimization of hyperbolic discrete inclusions.

On the other hand τ +h SA (h)ξ + hF (ξ, η) − uh ≤ SA (τ + h − s)F (ξ, η) − F (ξ, η) ds τ τ +h +M eah F (u(s), v(s)) − F (u(τ ), v(τ )) ds, τ where M ≥ 1 and a > 0 are the growth constants of the C0 -semigroup {SA (t) : X → X, t ≥ 0}. Since F , u and v are continuous we conclude that the first equality in (22) holds. Similarly, we get the second equality, and this completes the proof of the viability of K and consequently the viability of K. Let us remark that f and g have sublinear growth. In addition K satisfies: for each sequence ((tn , ξn , ηn )n ) in K with limn (tn , ξn ηn ) = (t, ξ, η) and t < TK , where TK = sup{t ∈ R; there exists (ξ, η)∈X×X, with (t, ξ, η)∈K}, it follows that (t, ξ, η) ∈ K.

Namah and observe that for any (x, t) ∈ IRN × IR we have α(−Mα − V (x)) + H(x, −Mα , 0) ≤ f (t) ≤ α(Mα − V (x)) + H(x, Mα , 0). Therefore we can construct the stationary viscosity solution Vα of α(Vα − V (x)) + H(x, Vα , DVα ) = f , x ∈ IRN , and the time periodic viscosity solution vα of α(vα − V (x)) + ∂t vα + H(x, vα , Dvα ) = f (t), (x, t) ∈ IRN × IR. 1 we obtain |vα (x, t)−V (x)| = |vα (x, t)−Vα (x)|≤ f − f L1 (0,T ) , ∀(x, t) ∈ IRN ×IR, (18) which implies that (vα )α is uniformly bounded vα L∞ (IRN ×IR) ≤ V L∞ (IRN ) + f− f L1 (0,T ) , ∀α > 0.