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.Advanced Nonlinear Studies 2 (2002), 325-356 Multiplicity results for systems of asymptotically linear second order equations* Anna Capietto, Francesca Dalbono Dipartimento di Matematica, Universitri di Torino, Via Carlo Alberto 10, 10123 Torino, Italia e-mail: capietto@dm.unito.it, dalbono@dm.unito.it Received 10 September 2001 Communicated by Fabio Zanolin Abstract We prove the existence and multiplicity of solutions, with prescribed nodal prop- erties, for a BVP associated with a system of asymptotically linear second order equations. The

Advanced Nonlinear Studies 14 (2014) 159–182 Nonlocal Robin Problem for Elliptic Quasilinear Second Order Equations Mikhail Borsuk, Krzysztof Żyjewski Department of Mathematics and Informatics University of Warmia and Mazury in Olsztyn, Sloneczna 54, Olsztyn-Kortowo, Poland e-mail: borsuk@uwm.edu.pl, krzysztof.zyjewski@uwm.edu.pl Received 29 January 2013 Communicated by Laurent Véron Abstract We investigate the behavior of weak solutions to the nonlocal Robin problem for quasi- linear elliptic divergence second order equations in a neighborhood of the boundary

Advanced Nonlinear Studies 11 (2011), 675–694 Dynamics of Periodic Second-Order Equations Between an Ordered Pair of Lower and Upper Solutions Antonio J. Ureña∗ Departmento de Matemática Aplicada Facultad de Ciencias, Universidad de Granada, 18071, Granada e-mail: ajurena@ugr.es Received 21 August 2010 Communicated by Rafael Ortega Abstract We consider periodic second-order equations having an ordered pair of lower and upper solutions and show the existence of asymptotic trajectories heading towards the maximal and minimal periodic solutions which lie between

References [1] Borsuk M.V., Transmission problems for elliptic second-order equations in non-smooth domains , Birkhäuser, Basel, 2010. [2] Hernández J., Mancebo F.J., Vega J.M., On the linearization of some singular, nonlinear elliptic problems and applications , Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 777–813. [3] Lazer A.C., McKenna P.J., On a singular nonlinear elliptic boundary-value problem , Proc. Amer. Math. Soc. 111 (1991), 721–730. [4] Mityushev V., Adler P., Darcy flow around a two-dimensional lens , I. Phys. A: Math. Gen. 39

[1] Borsuk M.V., A priori estimates and solvability of second order quasilinear elliptic equations in a composite domain with nonlinear boundary condition and conjugacy condition, Trudy Mat. Inst. Steklov., 1968, 103, 15–50 (in Russian) [2] Borsuk M., Transmission Problems for Elliptic Second-Order Equations in Non-Smooth Domains, Front. Math., Birkhäuser/Springer, Basel, 2010 [3] Borsuk M., Kondratiev V., Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North-Holland Math. Library, 69, Elsevier, Amsterdam, 2006 [4] Furusho Ya

15 A primer on linear PDE in 2D II: second order equations A second order partial differential equation is an equation in which the higher par- tial derivatives are of second order. We will deal with linear second order PDE in two dimensions, whose general form is auxx + buxy + cuyy + dux + euy + fu = h(x, y), (x, y) ∈ ℝ2 with a, b, c not all zero. As usual the independent variables x, y can be labeled differ- ently, such as t, x or t, y. The features of this equation essentially depend on the co- efficients a, b, c of the second order partial derivatives. It is

§3. A MODIFIED THEORY FOR SECOND ORDER EQUATIONS WITH AN INDEFINITE ENERGY FORM In this section we show how the theory developed in Section 2 can be applied to motions governed by equations which are second order in tim e , that is of the form (3.1) utt - Lu , where u is an element of a Hilbert space £ and L is a se lf-a d jo in t operator on £ . Assuming that incoming and outgoing subspaces exist, the theory applies directly when L is negative; if L has a finite number of positive eigenvalues, some modifications of the theory are required. One can

RUSS. J. Numer. Anal Math. Modelling, Vol.13, No. 4, pp.335-344 (1998) © VSP 1998 Compact schemes for systems of second-order equations without mixed derivatives V.I. PAASONEN* Abstract — In this paper we carry out a formalized construction of compact difference schemes of fourth- order accuracy in space for multidimensional systems of differential second-order equations of the main types (elliptic, parabolic, and hyperbolic ones). They have variable coefficients, which depend on the sought solution, an arbitrary elliptic operator without mixed derivatives, and

DEMONSTRATIO MATHEMATICA Vol. XXXIV No 3 2001 Gerd Herzog ON POSITIVE SOLUTIONS OF LINEAR SECOND ORDER EQUATIONS IN BANACH SPACES Abstract. In an ordered Banach space we consider initial value problems of the form u"(t) = C(t)u(t) + f(t), u(0) = uq, u'(0) = u\. Involving quasimonotonicity methods we give conditions which imply that u(t) > 0. 1. Introduction Let (E, || • ||) be a real Banach space ordered by a cone K. A cone if is a closed convex subset of E such that \K C K (A > 0) and K i l ( - K ) = {0}. Then E is ordered by x<y<$y — X€K. We will

Chapter 6 Inverse problems for parabolic and elliptical type second-order equations In determining thermophysical and filtration parameters of a medium and density of distribution of heat sources there arise inverse problems for parabolic type equations. This problem has been investigated by a great number of researchers; mainly during the 1970s. Here, first of all, the works by Beznosh- chenko [295-301], Iskenderov [302-304] (also with Budak [305-307]), Kliba- nov [181,308-314], Cannon [315-321], should be recalled and it should be noted that they have