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An existence result for nonlinear evolution equations of second order

100%

Kandilakis D.

nr 2

153-160

In this paper we consider a second order differential equation involving the difference of two monotone operators. Using an auxiliary equation, a priori bounds and a compactness argument we show that the differential equation has a local solution. An example is also presented in detail.

Quasilinear vector differential equations with maximal monotone terms and nonlinear boundary conditions

63%

Bader R. , Papageorgiou N. S.

2000

tom 73

nr 1

69-92

We consider a quasilinear vector differential equation which involves the p-Laplacian and a maximal monotone map. The boundary conditions are nonlinear and are determined by a generally multivalued, maximal monotone map. We prove two existence theorems. The first assumes that the maximal monotone map involved is everywhere defined and in the second we drop this requirement at the expense of strengthening the growth hypothesis on the vector field. The proofs are based on the theory of operators of monotone type and on the Leray-Schauder fixed point theorem. At the end we present some special cases (including the classical Dirichlet, Neumann and periodic problems), which illustrate the general and unifying features of our work.

Periodic solutions for quasilinear vector differential equations with maximal monotone terms

63%

Kourogenis N. , Papageorgiou N.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1997

tom 17

nr 1-2

67-81

We consider a quasilinear vector differential equation with maximal monotone term and periodic boundary conditions. Approximating the maximal monotone operator with its Yosida approximation, we introduce an auxiliary problem which we solve using techniques from the theory of nonlinear monotone operators and the Leray-Schauder principle. To obtain a solution of the original problem we pass to the limit as the parameter λ > 0 of the Yosida approximation tends to zero.

Extremal solutions for nonlinear neumann problems

63%

Fiacca A. , Servadei R.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2001

tom 21

nr 2

191-206

In this paper, we study a nonlinear Neumann problem. Assuming the existence of an upper and a lower solution, we prove the existence of a least and a greatest solution between them. Our approach uses the theory of operators of monotone type together with truncation and penalization techniques.