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Mathematics, Volume 1, Issue 2 (June 2013), Pages 46-75

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Research

Open AccessArticle Stability of Solutions to Evolution Problems
Mathematics 2013, 1(2), 46-64; doi:10.3390/math1020046
Received: 26 February 2013 / Revised: 25 April 2013 / Accepted: 25 April 2013 / Published: 13 May 2013
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Abstract
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have [...] Read more.
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t , in particular, sufficient conditions for this limit to be zero. The evolution problem is: u ˙ = A(t)u + F(t, u) + b(t), t 0; u(0) = u 0 . (*) Here u ˙ := du dt , u = u(t) H, H is a Hilbert space, t R + := [0,), A(t) is a linear dissipative operator: Re(A(t)u,u) γ(t)(u, u) where F(t, u) is a nonlinear operator, F(t, u) c 0 u p , p > 1, c 0 and p are positive constants, b(t) β(t) , and β(t)0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case γ(t) 0 is also treated. Full article
Open AccessArticle On the Class of Dominant and Subordinate Products
Mathematics 2013, 1(2), 65-75; doi:10.3390/math1020065
Received: 25 March 2013 / Revised: 12 April 2013 / Accepted: 22 April 2013 / Published: 15 May 2013
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Abstract
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts [...] Read more.
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients. Full article

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