Maximality theorems on the sum of two maximal monotone operators and application to variational inequality problems. (English) Zbl 1470.47036

Summary: Let \(X\) be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space \(X^\ast\). Let \(T : X \supseteq D(T) \rightarrow 2^{X^\ast}\) and \(A : X \supseteq D(A) \rightarrow 2^{X^\ast}\) be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for \(T + A\) under weaker sufficient conditions. These theorems improve the well-known maximality results of R. T. Rockafellar [Trans. Am. Math. Soc. 149, 75–88 (1970; Zbl 0222.47017)] who used condition \(\overset\circ{D(T)} \cap D(A) \neq \emptyset\) and F. E. Browder and P. Hess [J. Funct. Anal. 11, 251–294 (1972; Zbl 0249.47044)] who used the quasiboundedness of \(T\) and condition \(0 \in D(T) \cap D(A)\). In particular, the maximality of \(T + \partial \phi\) is proved provided that \(\overset\circ{D(T)} \cap D(\phi) \neq \emptyset\), where \(\phi : X \rightarrow(- \infty, \infty]\) is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator.


47H05 Monotone operators and generalizations
49J40 Variational inequalities
Full Text: DOI


[1] Brézis, H.; Crandall, M. G.; Pazy, A., Perturbations of nonlinear maximal monotone sets in Banach space, Communications on Pure and Applied Mathematics, 23, 123-144, (1970) · Zbl 0182.47501
[2] Browder, F. E.; Hess, P., Nonlinear mappings of monotone type in Banach spaces, Journal of Functional Analysis, 11, 3, 251-294, (1972) · Zbl 0249.47044
[3] Brézis, H., Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Université de Grenoble. Annales de l’Institut Fourier, 18, 115-175, (1968) · Zbl 0169.18602
[4] Zeidler, E., Nonlinear Functional Analysis and Its Applications, (1990), New York, NY, USA: Springer, New York, NY, USA
[5] Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 75-88, (1970) · Zbl 0222.47017
[6] Asfaw, T. M.; Kartsatos, A. G., Variational inequalities for perturbations of maximal monotone operators in reflexive Banach spaces, The Tohoku Mathematical Journal. Second Series, 66, 2, 171-203, (2014) · Zbl 1318.47082
[7] Chen, Y.; Cho, Y.; Kumam, P., On the maximality of sums of two maximal monotone operators, Journal of Mathematical Analysis, 7, 2, 24-30, (2016) · Zbl 1362.47031
[8] Asfaw, T. M., A new topological degree theory for pseudomonotone perturbations of the sum of two maximal monotone operators and applications, Journal of Mathematical Analysis and Applications, 434, 1, 967-1006, (2016) · Zbl 1344.47041
[9] Carl, S.; Le, V. K., Quasilinear parabolic variational inequalities with multi-valued lower-order terms, Zeitschrift für Angewandte Mathematik und Physik, 65, 5, 845-864, (2014) · Zbl 1317.35137
[10] Carl, S.; Le, V. K.; Motreanu, D., Existence, comparison, and compactness results for quasilinear variational-hemivariational inequalities, International Journal of Mathematics and Mathematical Sciences, 3, 401-417, (2005) · Zbl 1085.49009
[11] Carl, S., Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems, Nonlinear Analysis: Theory, Methods & Applications, 65, 8, 1532-1546, (2006) · Zbl 1233.35090
[12] Carl, S.; Motreanu, D., General comparison principle for quasilinear elliptic inclusions, Nonlinear Analysis: Theory, Methods & Applications, 70, 2, 1105-1112, (2009) · Zbl 1152.35306
[13] Kenmochi, N., Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Mathematical Journal, 4, 229-263, (1974) · Zbl 0284.47030
[14] Kenmochi, N., Pseudomonotone operators and nonlinear elliptic boundary value problems, Journal of the Mathematical Society of Japan, 27, 121-149, (1975) · Zbl 0292.35034
[15] Kenmochi, N., Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of Differential Equations, IV, 203-298, (2007), Amsterdam, The Netherlands: Elsevier/North-Holland, Amsterdam, The Netherlands · Zbl 1192.35083
[16] Asfaw, T. M., New surjectivity results for perturbed weakly coercive operators of monotone type in reflexive Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 113, 209-229, (2015) · Zbl 1315.47050
[17] Asfaw, T. M., New variational inequality and surjectivity theories for perturbed noncoercive operators and application to nonlinear problems, Advances in Mathematical Sciences and Applications, 24, 2, 611-668, (2014) · Zbl 1343.47057
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.