By Donald W. Hight

An exploration of conceptual foundations and the sensible purposes of limits in arithmetic, this article deals a concise advent to the theoretical examine of calculus. It analyzes the assumption of a generalized restrict and explains sequences and features to these for whom instinct can't suffice. Many routines with strategies. 1966 variation.

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Viable Solutions to Fractional Difference and Differential Equations 23 Remark 7. If the fractional differential equation RL α 0 D x(t) = f (t, x(t)), 0 < α < 1, t ∈ (0, T ], (14) has a viable solution in K, then for every ε > 0 there exists h > 0 such that there exists a solution y to system (a Δα h y) (nh) = f (nh, y(nh + a)) (15) that is viable in Kε , Remark 8. Let us notice that if K = Rn+ one can consider all above results as positivity ones. 5 Conclusions The problem of viability that consists in finding conditions under which there exists at least one solution to a system, which maintains in a certain set of constrains is considered.

The work was supported by Bialystok University of Technology grant G/WM/3/12. References 1. : Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011) 2. : Viability Theory. Birkh¨ auser, Berlin (1991) 3. : Set-Valued Analysis. , Boston (1990) 4. : Positive solutions for boundary value problems of nonlinear fractional differential equations. Journal of Mathematical Analysis and Applications 311(2), 495–505 (2005) 5.

Remark 6. Note that: a Δα h,∗ : F(hN)a → F(hN)a+(1−α)h , where α ∈ (0, 1]. The property of the composition of h-sums was proved in [14]. Proposition 2. Let x be a real valued function defined on (hN)a , where a, h ∈ R, h > 0. For α, β > 0 the following equalities hold: −α a+βh Δh −β a Δh x (t) = −(α+β) x a Δh (t) = −β a+αh Δh −α a Δh x (t) , where t ∈ (hN)a+(α+β)h . The next proposition gives a useful identity of transforming Caputo fractional difference equations into fractional summations for the case when an order is from the interval (0, 1].

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