Monotonicity methods for nonlinear evolution equations. (English) Zbl 0894.34055

The paper deals with evolution problems of the form \[ u'(t)+ A(t)u(t)= f(t),\quad 0<t<T,\quad u(0)= 0, \] where \(A(t): X\to X^*\) (\(X\) is a Banach space and \(X^*\) is its dual space), \(f\in V= L^p(0,T;X)^*\), \(u_0\in X\). The authors show that under suitable conditions the map \(\widetilde A:V\to V^*\) defined by \(\widetilde A(u)(t)= A(t)u(t)\) inherits the monotonicity properties of \(A(t)\). The results are applied to generalize previous papers by N. Hirano [Nonlinear Anal., Theory Methods Appl. 13, No. 6, 599-609 (1989; Zbl 0682.34010)] and N. U. Ahmed and X. Xiang [Nonlinear Anal., Theory Methods Appl. 22, No. 1, 81-89 (1994; Zbl 0806.34051)].


34G20 Nonlinear differential equations in abstract spaces
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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[1] Berkovits, J.; Mustonen, V., Topological degree for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems, Rc. mat., serie VII, 12, 597-621, (1992) · Zbl 0806.47055
[2] Hirano, N., Nonlinear evolution equations with nonmonotonic perturbations, Nonlinear analysis, 13, 599-609, (1989) · Zbl 0682.34010
[3] Ahmed, N.U.; Xiang, X., Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations, Nonlinear analysis, 22, 81-89, (1994) · Zbl 0806.34051
[4] Zeidler, E., Nonlinear functional analysis and its applications, IIA and IIB, (1990), Springer Philadelphia
[5] Lions, J.-L., Quelques méthodes de resolution des problémes aux limites non linéaires, (1969), Dunod, Gauthier-Villars New York · Zbl 0189.40603
[6] Browder, F.E., Fixed point theory and nonlinear problems, Bull. am. math. soc., 9, 1-39, (1983) · Zbl 0533.47053
[7] Landes, R.; Mustonen, V., On pseudo-monotone operators and nonlinear noncoercive variational problems on unbounded domains, Math. annln, 248, 241-246, (1980) · Zbl 0416.35072
[8] Gossez, J.P.; Mustonen, V., Pseudomonotonicity and the Leray-Lions condition, Diff. integral eqns, 6, 37-45, (1993) · Zbl 0786.35055
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