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3, 329-353. [Barbu1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [BarPr] V. Barbu & Th. Precupanu, Convexity and Optimization in Banach Spaces, Editura Academiei and Reidel, Bucharest and Dordrecht, 1986. [Br´ezis1] H. Br´ezis, Op´erateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. [Br´ezis2] H. Br´ezis, Analyse fonctionelle, Masson, Paris, 1983. [Dieudo] J. Dieudonn´e, El´ements d’analyse.

1, we present an auxiliary result. 1 ([HokMo]). a. r ∈ (0, 1). 9) Then, the operator F˜ is maximal monotone in H. PROOF Obviously, F˜ is monotone. a. r ∈ (0, 1), u (0), u (1) ∈ γ u(0), −u(1) . 10). 11) whenever γz + γˆ z = z1 , where z1 ∈ IR2 depends only on u1 and γˆ = 1 e2 − 1 1 + e2 2e 2e 1 + e2 , z= 1 −e 1 −e −1 c1 c2 . As γˆ is a strongly positive matrix, it follows that γ + γˆ is a surjective mapping. Therefore, the equation γz + γˆ z = z1 has a solution z ∈ IR2 . 1. 1) we can see that G(0, ·) and G(1, ·) are surjective mappings.

We shall say that this semigroup is generated by −A. 11) on [0, T ], for each T > 0). 1) considered on [0, ∞). 4 Let A: D(A) ⊂ H → H be a maximal monotone operator and let the family {S(t): D(A) → D(A) | t ≥ 0} be the semigroup generated by −A. 2) converges strongly, as t → ∞, for every u0 ∈ D(A), f ∈ L1 (0, ∞; H), and the limit of u(t), if it exists, is an element of F : = A−1 0. 3 The above result reduces the study of asymptotic behavior of solutions u(t) in the case f ∈ L1 (0, ∞; H) to the asymptotic behavior of S(t)x, x ∈ D(A).

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