By C. De Coster, P. Habets (auth.), M. R. Grossinho, M. Ramos, C. Rebelo, L. Sanchez (eds.)
This paintings, inclusive of expository articles in addition to examine papers, highlights contemporary advancements in nonlinear research and differential equations. the cloth is essentially an outgrowth of autumn college classes and seminars held on the collage of Lisbon and has been completely refereed.
Several issues in traditional differential equations and partial differential equations are the point of interest of key articles, including:
* periodic suggestions of structures with p-Laplacian variety operators (J. Mawhin)
* bifurcation in variational inequalities (K. Schmitt)
* a geometrical method of dynamical platforms within the aircraft through twist theorems (R. Ortega)
* asymptotic habit and periodic options for Navier--Stokes equations (E. Feireisl)
* mechanics on Riemannian manifolds (W. Oliva)
* concepts of decrease and top ideas for ODEs (C. De Coster and P. Habets)
A variety of similar topics facing houses of ideas, e.g., bifurcations, symmetries, nonlinear oscillations, are taken care of in different articles.
This quantity displays wealthy and sundry fields of study and may be an invaluable source for mathematicians and graduate scholars within the ODE and PDE community.
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Additional info for Nonlinear Analysis and its Applications to Differential Equations
In Russian). cz Periodic Solutions of Systems with p-Laplacian-like Operators Jean Mawhin 1 Introduction A great deal of attention has recently been given to extending spectral, bifurcation or existence results for semilinear equations of the second order, in both ordinary and partial differential cases, to the case of nonlinear perturbations of the so-called p-Laplacian operator u f---t 6. pu = div (Igrad ul P- 2grad u) if N ::::: 2, or of some suitable generalization. 2) where u= (U1,U2, ...
6) . e. 6), loT I (¢(u'(t)))'1 dt :::; C3(R2)Nllu'll£l + IAIR2T + Nllell£l :::; C 3 (R2)NR l + IAIR2T + Nllell£! 8) 50 J. 1 that Ihl = Ih(b) I :::; R4. Consequently, and hence for some R6 > 0 which is independent of u and A. 2). 1 are satisfied. 1 is the following corollary. 2. 9) has a unique solution and, for each sufficiently large R > 0, the LeraySchauder degree over B(R) of the associated fixed point operator in Cj, is equal to ±1. Proof. Only the uniqueness has to be proved. 9). Then we have u(O) (¢(u'))' - (¢(v'))' + A(u - v) = 0, = u(T), u'(O) = u'(T), v(O) = v(T), v'(O) = v'(T), Periodic Solutions of Systems with p-Laplacian-like Operators 51 and hence, after scalar multiplication by u - v, and integration by parts over [0, T], we get 10 T (¢(u'(t»-¢(v'(t», u'(t)-v'(t») dt- (T io (A(u(t)-v(t», u(t)-v(t») dt = o.
Further, Ct ::; f3 ::; Ct + 2n ::; f3 + 2n. 5 and deduce the existence of a second solution v =I u mod 2n. 3 Non-well-ordered lower and upper solutions The existence of lower and upper solutions such that Ct 2: f3 is not sufficient to guarantee the solvability of this problem. This is clear from the example u" + u = sint, u'(O) = 0, u'(n) = 0, 20 C. De Coster, P. Habets which has no solution although 0:( t) = 1 and (3 = -1 are respectively lower and upper solutions. The difficulty comes here from the interference of the nonlinearity with the second eigenvalue of the problem '\2 = 1.