By S. Elaydi, J. Cushing, R. Lasser, V. Papageorgiou, A. Ruffing

This quantity includes talks given at a joint assembly of 3 groups operating within the fields of distinction equations, specified services and functions (ISDE, OPSFA, and SIDE). The articles mirror the range of the themes within the assembly yet have distinction equations as universal thread. Articles conceal subject matters in distinction equations, discrete dynamical platforms, precise features, orthogonal polynomials, symmetries, and integrable distinction equations.

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**Example text**

If cpu,$u are strictly increasing maps for all u , then card(3) 5 1. 45 Proof. Reasoning by contradiction, suppose that there exist u , w E 7 , u < w. 5, Fix(cp,) = Fix($,) = { w } . This implies that either w > $,(x) > x for all x E (0, w ) or 0 < $,(x) < x for all x E (0, w ) . Suppose that w = $,(w) > $,(z) > x for all x E (0, w ) . Since cp, o I), = Idl(O,oo)(see (7)) and cp, is increasing, it follows x > cpw(x) for all x E (0, w ) . 5 it is Fix(cp,) = Fix($,) = {u}. In particular, $,(u) = f ( u , w ) = cp,(w) > w = $,(w), in contradiction with the fact that $ , is increasing.

88262 N ~ for 38 Table 3. Errors No = 16, N = 20. and convergence rates ~ Table 4. Errors and convergence rates 2 on ~WN" "for on W N ~for 1 1 Figure 1. Surface plot solution for NO = 32, N = 20, E = In Fig. 1 we present a surface plot of the numerical approximation generated by the numerical method (14)-(16) applied to problem (1)-(3) with data (38) for No = 32, N = 20, E = 39 6. Conclusion An initial-boundary value problem for a quasilinear singularly perturbed Sobolev equation having a initial jump was considered.

1. The maps CJ and p are strictly decreasing, with lim ~ ( w=)lim p ( w ) = +co, w+o W+O lim ~ ( w=) lim p ( w ) = 0. 2. For all u > 0 and n = 0 , 1 , 2 , . . , it is Proof. By (4)and f ( x , y ) = o(x)p(y) it is straightforward to check u = u ( u ) . p ( a ( u ) p ( u ) )and u = p ( u ) . These equalities give = $$#, % u > 0. Moreover since a, = f(u,a,) = f ( u , h - l ( u ) ) = u(u)p(h-'(u)),and also a, = f ( a z L , u= ) f ( h - l ( u ) ,u)= c ~ ( h - l ( u ) ) p ( uIt) remains . to extend the property t o any positive integer n.