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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**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.