Non-linear boundary value problems with generalized \(p\)-Laplacian, ranges of \(m\)-accretive mappings and iterative schemes. (English) Zbl 1352.47033
Summary: In this paper, we first prove some perturbation results on the ranges of maximal monotone operators, one of which is then used to show that the non-linear elliptic equation involving the generalized \(p\)-Laplacian operator with Neumann boundary conditions has a unique solution in \(L^2(\Omega)\). This unique solution is shown to be the zero point of a suitably defined nonlinear \(m\)-accretive mapping. Finally, two kinds of iterative sequences are constructed and proved to converge strongly and weakly to the unique solution, respectively. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.
MSC:
| 47H14 | Perturbations of nonlinear operators |
| 47H05 | Monotone operators and generalizations |
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
| 47J25 | Iterative procedures involving nonlinear operators |
| 47N20 | Applications of operator theory to differential and integral equations |
| 35J66 | Nonlinear boundary value problems for nonlinear elliptic equations |
Keywords:
monotone operator; accretive mapping; zero point; generalized \(p\)-Laplacian operator; non-linear elliptic boundary value problem; iterative schemeReferences:
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