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Abstract

We prove that the Quasi Differential of Bayoumi of maps between locally bounded F-spaces may not be Fréchet-Differential and vice versa. So a new concept has been discovered with rich applications (see [1–6]). Our F-spaces here are not necessarily locally convex

[1] Albiac F., The role of local convexity in Lipschitz maps, J. Convex. Anal., 2011, 18(4), 983–997 [2] Aoki T., Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 1942, 18, 588–594 http://dx.doi.org/10.3792/pia/1195573733 [3] Bayoumi A., Mean value theorem for complex locally bounded spaces, Comm. Appl. Nonlinear Anal., 1997, 4(4), 91–103 [4] Bayoumi A., Foundations of Complex Analysis in Non Locally Convex Spaces, North-Holland Math. Stud., 193, Elsevier, Amsterdam, 2003 [5] Bayoumi A., Bayoumi quasi-differential is different from Fréchet-differential

of this theorem can easily be obtained by showing the weak lower semi-continuity of J α same as in [ 10 ]. Theorem 3 Let the assumptions of Theorem 2 remain valid. The minimizing element v ∗ of V ad is characterized by π v ∗ , v − v ∗ ≥ L v − v ∗ $$\begin{array}{} \displaystyle \pi \left(v^*,v-v^*\right)\, \ge L\left(v-v^*\right) \end{array}$$ (11) for ∀ v ∈ V ad [ 10 ] . Inequalities of the type given by (11) are termed variational inequalities. 3 Frechet differential of the cost functional and numerical example Let us introduce the Lagrangian L ( u , v

optimal control, and the convergence of the sequence of discrete optimal control problems to the continuous optimal control problem was proved. Our goal in this work is to prove the Frechet differentiability and derive the formula for the Frechet differential under the minimal regularity assumptions on the data. This presents a significant technical challenge. Since in the new variational formulation the free boundary is treated as a control parameter, variation of the cost functional reflects the high sensitivity of the solution to the PDE problem with respect to the

, 28 two-dimensional Rossby waves equation, 27 Lemma Lions compactness lemma, 606 Operator boundedly Lipschitz continuous, 71 completely continuous, 71 Fréchet derivative, 597 Gâteaux differential, 596 locally uniform Fréchet differential, 72 monotonic mapping, 70 Nemytskii operator, 586 semicontinuous, 71 symmetric, 72 the Fréchet differential, 71 weakly lower semicontinuous, 71 Sequence -weakly converging, 583 weakly converging, 583 Sobolev spaces, 581 Solution classical solution, 485, 592 strong generalized solution, 102, 127, 197, 203, 244, 273, 322, 330, 338

>:=-FHF +ZxE + Z2GHG. ergibt sich für ε 2 : ε2 = — sup {— ζ , £ , — z 2 E t x + inf S[<frS]}. ζ > ο S > Ο Die notwendigen und hinreichenden Bedingungen, damit das Inf imum von S [ 4 > 5 ] an der Stelle Ξ 0 angenommen wird, lauten: 5 S ( 5 0 , ς ) > 0 und 6 S ( 5 o , 5 o ) = 0 für alle ξ > 0. Hierbei wird mit 8 S ( £ 0 , ξ) das Frechet-Differential des Funkt iona is S an der Stelle Ξ 0 mit dem Inkrement ξ sym- bolisiert. Als notwendige und hinreichende Bedingung erhalten wir dann für Φ die Restriktion Φ > 0! Dieses Ergebnis er- scheint einleuchtend, wenn man

) obviously implies that A£ (·,/?, V) is a convex function being even uniformly convex in case of V p.d. PROPOSITION 2. Let (/?, V) G Ω with V n.n.d. be given. Then for any £>, ~D G ftsxn and any μ G [0; 1] the inequality , ,V) > 2 Linear estimation in regression analysis 109 holds, where Όμ := μΌ 4- (l - μ)Ό , \mm( ,V) denotes the smallest eigenvalue of the n.n.d. matrix V + X 'X', and \\-\\ is the Euclidean norm. Proof. Since Ri(D, ,V) can be written as RL(D, /?, V) = Tr(DVD') + 1V((£X - L) '(DX - L)') we obtain the second (Frechet-) differential of Ri(D, ,V] with respect

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Principle: An Alternative Formulation 107 §14. The Mixing Value: Statement of Results 113 §15. The Mixing Value: Proof of Theorem E 116 §16. An Alternative Proof for Example 9.4 123 Chapter III . The Asymptotic Approach 126 §17. Introduction and Statement of Results 126 §18. Proof of Theorem F 128 §19. More on Example 9.4 and Related Set Functions 135 Chapter IV. Values and Derivatives 141 §20. Introduction IUl §21. Statement of Results IUU §22. Extensions: The Axiomatic Approach 1U5 §23. Proof of Theorem II 153 §24. Frechet Differentials 156 §25. Extensions

− vr r2 ) vIr + + ( ∆vϕ − vϕ r2 ) vIϕ + ∆vz · vIz + ∆T · T I } rdr, ~V I ∈ M. Following [7], problem (2.1)–(2.2) can be reduced to the nonlinear ope- rator equation ~V = λK~V . (2.7) The linearized problem (2.3)–(2.4) and its conjugate problem will respec- tively satisfy the operator equations ~U = λA~U, (2.8) ~Ψ = λA∗~Ψ. (2.9) Applying the results of [7, 8], we easily ascertain that the operators K, A, and A∗ are completely continuous in the space H1. The operator A is the Frechet differential of the operator K at the point ~V = 0, and A∗ is the conjugate operator

) + n∑ j 6=i ∥∥ :xj∥∥pj/qi Hkj (Ω) + ∥∥ :xi∥∥pi−1Hki (Ω)  . (3.3) Then the functional F : W 3 x→ ∫ T 0 g ( t, x, : x ) dt ∈ R 144 M. GALEWSKI is continuously Fréchet differentiable. Moreover the Fréchet differential W 3 h→ F ′ (x)h ∈ R is given by the formula F ′ (x)h = n∑ i=1 ∫ T 0 〈 ∇xig ( t, x (t) , : x (t) ) , hi (t) 〉 Hki (Ω) + 〈 ∇ :xig ( t, x (t) , : x (t) ) , : hi (t) 〉 Hki (Ω) dt. (3.4) Proof. Let us first observe that by (3.1) it follows that functional F is well defined. We shall first show that for every direction h ∈ W there exists a directional