Download Differential and Difference Equations with Applications: by Ondřej Došlý, Simona Fišnarová (auth.), Sandra Pinelas, PDF

By Ondřej Došlý, Simona Fišnarová (auth.), Sandra Pinelas, Michel Chipot, Zuzana Dosla (eds.)

The quantity comprises rigorously chosen papers awarded on the overseas convention on Differential & distinction Equations and purposes held in Ponta Delgada – Azores, from July 4-8, 2011 in honor of Professor Ravi P. Agarwal. the target of the collection used to be to collect researchers within the fields of differential & distinction equations and to advertise the alternate of rules and learn. The papers disguise all components of differential and distinction equations with a distinct emphasis on applications.

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Additional info for Differential and Difference Equations with Applications: Contributions from the International Conference on Differential & Difference Equations and Applications

Example text

The other assertions are proved analogously. 36 T. Belkina et al. 1 Reduction of the Second-Order IDE to a Third-Order ODE The known possibility of reducing the second-order IDE (6) to a third-order ODE is important for further exposition. First, we note that (Jm ϕ ) (u) = 1 exp(−u/m) m u 0 ϕ (x) exp (x/m)dx = [ϕ (u) − (Jm ϕ )(u)]/m. (36) Then differentiating IDE (6) in view of (36) gives a linear third-order IDE (b2 /2)u2ϕ (u) + [(b2 + a)u + c]ϕ (u) + (a − λ )ϕ (u) +(λ /m)[ϕ (u) − (Jmϕ )(u)] = 0, u ∈ R+ , (37) which also implies the limit condition lim [cϕ (u) + (a − λ )ϕ (u) + (λ /m)ϕ (u)] = 0.

For a detailed proof of Lemma 3, see [4]. Corollary 1. Under the assumptions of Lemma 3, all solutions of ODE (10) have finite limits as u → ∞ iff condition (16) is fulfilled. Summarizing all results, we obtain the proof of Theorem 1. 5 The Accompanying Singular Problem for Capital Stock Model (The Third “Degenerate” Case: c = 0, b = 0, a > 0, λ > 0, m > 0) For this case, the input singular IDE problem has the form: (b2 /2)u2 ϕ (u) + auϕ (u) − λ [ϕ (u) − (Jmϕ )(u)] = 0, u ∈ R+ , (45) lim ϕ (u) = lim [uϕ (u)] = 0, (46) lim ϕ (u) = 1, (47) u→+0 u→∞ u→+0 lim ϕ (u) = 0, u→∞ and restrictions (3) are needed for the solution.

For this model, the positiveness condition for the net expected income (“safety loading”) has the form (17). Denote by τ = inf{t : Rt < 0} the time of ruin, then P(τ < ∞) is the probability of ruin at the infinite time interval. -L. risk theory [8]: under condition (17) and assuming existence of a constant RL > 0 (“the Lundberg coefficient”) such that equality 0∞ [1 − F(x)] exp (RL x)dx = c/λ > 0 holds, the probability of ruin ξ (u) as a function of the initial surplus admits the estimate ξ (u) = P(τ < ∞) ≤ exp (−RL u), u ≥ 0.

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