Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations. (English) Zbl 0806.34051

The authors study the initial value problem (1) \(u'(t) + Au(t) + G(u(t)) = f(t)\) \((0 \leq t \leq T)\), \(u(0) = u_ 0\), in a Hilbert space \(H\); \(A\) is monotone and hemicontinuous in \(H\) and \(G:V \to V^*\), where \(V\) is reflexive Banach space with \(V \hookrightarrow H \hookrightarrow V^*\); the function \(f(\cdot)\) belongs to \(L^ q (0,T;V^*)\) for some \(q>1\). The main result (under several additional hypotheses) is an existence theorem where the solution \(u(\cdot)\) belongs to \(C(0,T;H) \cap L^ p (0,T;V)\) with \(1/p + 1/q = 1\). This result generalizes previous work of N. Hirano [Nonlinear Anal., Theory Methods Appl. 13, No. 6, 599-609 (1989; Zbl 0682.34010)], where the range of \(G\) belongs to \(H\).


34G20 Nonlinear differential equations in abstract spaces


Zbl 0682.34010
Full Text: DOI


[1] Hirano, N., Nonlinear evolution equations with nonmonotone perturbations, Nonlinear Analysis, 13, 6, 599-609 (1989) · Zbl 0682.34010
[2] Barbu, V.; Precupanu, T., Convexity and Optimization in Banach space (1975), Editura Academiei R.S.R: Editura Academiei R.S.R Bucharest · Zbl 0317.49011
[3] Barbu, V., Nonlinear Semigroups and Evolution Equations in Banach Spaces (1976), Noorhoff: Noorhoff Leyden · Zbl 0328.47035
[4] Ahmed, N. U.; Teo, K. L., Optimal Control of Distributed Parameter Systems (1981), North-Holland: North-Holland New York · Zbl 0472.49001
[5] Browder, F. E., Nonlinear operators and nonlinear equations of evolution in Banach space, Proc. Symp. Pure Math., XVIII, 2 (1976) · Zbl 0176.45301
[6] Brezis, H., Equations et inequations non lineaires dans les espaces vectoriels en dualite, Ann. Inst. Fourier. Univ. Grenoble, 18, 115-175 (1968) · Zbl 0169.18602
[7] Ahmed, N. U., Nonlinear integral equations on reflexive Banach space with application to stochastic integral equations and abstract evolution equations, J. Integral Eqns, 1, 1-15 (1979) · Zbl 0443.45023
[8] Vrabie, I. I., An existence result for a class of nonlinear evolution equations in Banach spaces, Nonlinear Analysis T.M.A., 7, 711-722 (1982) · Zbl 0493.34050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.