EXISTENCE OF SOLUTIONS OF SEMILINEAR SYSTEMS IN
Let Q : <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img03.JPG" alt="http:/fbpe/img/proy/v27n2/img03.JPG">be a symmetric and positive semi-definite linear operator and f j : <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img04.JPG...
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| Autores principales: | , |
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| Lenguaje: | English |
| Publicado: |
Universidad Católica del Norte, Departamento de Matemáticas
2008
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| Materias: | |
| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172008000200004 |
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| Sumario: | Let Q : <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img03.JPG" alt="http:/fbpe/img/proy/v27n2/img03.JPG">be a symmetric and positive semi-definite linear operator and f j : <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img04.JPG" alt="http:/fbpe/img/proy/v27n2/img04.JPG">(j = 1, 2, ...) be real functions so that, f j(0) = 0 and, for every x = (x1, x2, ....) <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img05.JPG" alt="http:/fbpe/img/proy/v27n2/img05.JPG"><img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img02.JPG" alt="http:/fbpe/img/proy/v27n2/img02.JPG">, it holds that f (x) := (f1(x1), f2(x2), ...) <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img05.JPG" alt="http:/fbpe/img/proy/v27n2/img05.JPG"><img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img02.JPG" alt="http:/fbpe/img/proy/v27n2/img02.JPG">. Sufficient conditions for the existence of non-trivial solutions to the semilinear problem Qx = f (x) are provided. Moreover, if G is a group of orthogonal linear automorphisms of <img border=0 width=32 height=32 src="http:/fbpe/img/proy/v27n2/img02.JPG" alt="http:/fbpe/img/proy/v27n2/img02.JPG">which commute with Q, then such sufficient conditions ensure the existence of non-trivial solutions which are invariant under G. As a consequence, sufficient conditions to ensure solutions of nonlinear partial difference equations on finite degree graphs with vertex set being either finite or infinitely countable are obtained. We consider adaptations to graphs of both Matukuma type equations and Helmholtz equations and study the existence of their solutions. |
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