# Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications

Volume 13, Issue 5, pp 239--257
Publication Date: March 03, 2020 Submission Date: September 06, 2019 Revision Date: January 10, 2020 Accteptance Date: January 16, 2020
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### Authors

Teffera M. Asfaw - Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA.

### Abstract

Let $X$ be a real locally uniformly convex reflexive Banach space. Let $T: X\supseteq D(T)\to 2^{X^*}$ and $A:X\supseteq D(A)\to 2^{X^*}$ be maximal monotone operators such that $T$ is of compact resolvents and $A$ is strongly quasibounded, and $C: X\supseteq D(C)\to X^*$ be a bounded and continuous operator with $D(A)\subseteq D(C)$ or $D(C)=\overline{U}$. The set $U$ is a nonempty and open (possibly unbounded) subset of $X$. New degree mappings are constructed for operators of the type $T+A+C$. The operator $C$ is neither pseudomonotone type nor defined everywhere. The theory for the case $D(C)=\overline{U}$ presents a new degree mapping for possibly unbounded $U$ and both of these theories are new even when $A$ is identically zero. New existence theorems are derived. The existence theorems are applied to prove the existence of a solution for a nonlinear variational inequality problem.

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##### ISRP Style

Teffera M. Asfaw, Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 5, 239--257

##### AMA Style

Asfaw Teffera M., Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications. J. Nonlinear Sci. Appl. (2020); 13(5):239--257

##### Chicago/Turabian Style

Asfaw, Teffera M.. "Topological degree theories for continuous perturbations of resolvent compact maximal monotone operators, existence theorems and applications." Journal of Nonlinear Sciences and Applications, 13, no. 5 (2020): 239--257

### Keywords

• Compact resolvents
• continuous operator
• degree theory
• variational inequality
• homotopy invariance
• maximal monotone

•  47H11
•  47H14
•  47H07

### References

• [1] D. R. Adhikari, A. G. Kartsatos, Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory, J. Math. Anal. Appl., 348 (2008), 122--136

• [2] T. M. Asfaw, New variational inequality and surjectivity theories for perturbed noncoercive operators and application to nonlinear problems, Adv. Math. Sci. Appl., 24 (2014), 611--668

• [3] T. M. Asfaw, A new topological degree theory for pseudomonotone perturbations of the sum of two maximal monotone operators and applications, J. Math. Anal. Appl., 434 (2016), 967--1006

• [4] T. M. Asfaw, A new topological degree theory for perturbations of demicontinuous operators and applications to nonlinear equations with nonmonotone nonlinearities, J. Funct. Spaces, 2016 (2016), 15 pages

• [5] T. M. Asfaw, A degree theory for compact perturbations of monotone type operators in reflexive Banach spaces, Abstr. Appl. Anal., 2017 (2017), 13 pages

• [6] T. M. Asfaw, A variational inequality theory for constrained problems in reflexive Banach spaces, Adv. Oper. Theory, 4 (2019), 462--480

• [7] T. M. Asfaw, A. G. Kartsatos, A Browder topological degree theory for multivalued pseudomonotone perturbations of maximal monotone operators in reflexive Banach spaces, Adv. Math. Sci. Appl., 22 (2012), 91--148

• [8] T. M. Asfaw, A. G. Kartsatos, Variational inequalities for perturbations of maximal monotone operators in reflexive Banach spaces, Tohoku Math. J. (2), 66 (2014), 171--203

• [9] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York (2010)

• [10] J. Berkovits, V. Mustonen, On the topological degree for mappings of monotone type, Nonlinear Anal., 10 (1986), 1373--1383

• [11] H. Brezis, M. G. Crandall, A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. Pure Appl. Math., 23 (1970), 123--144

• [12] L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann., 71 (1912), 97--115

• [13] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., 18 (1976), 1--308

• [14] F. E. Browder, Degree of mapping for nonlinear mappings of monotone type, Proc. Nat. Acad. Sci., 80 (1983), 1771--1773

• [15] F. E. Browder, Degree of mapping for nonlinear mappings of monotone type; Strongly nonlinear mapping, Proc. Nat. Acad. Sci. U.S.A., 80 (1983), 2408--2409

• [16] F. E. Browder, Degree of mapping for nonlinear mappings of monotone type: densely defined mapping, Proc. Nat. Acad. Sci. U.S.A., 80 (1983), 2405--2407

• [17] F. E. Browder, P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal., 11 (1972), 251--294

• [18] S. C. Hu, N. S. Papageorgiou, Generalization of Browder's degree theory, Trans. Amer. Math. Soc., 347 (1995), 233--259

• [19] A. G. Kartsatos, I. V. Skrypnik, Topological degree theories for densely defined mappings involving operators of type $(S_+)$, Adv. Differential Equations, 4 (1999), 413--456

• [20] N. Kenmochi, Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations, Hiroshima Math. J., 4 (1974), 229--263

• [21] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in: Handbook of differential equations: stationary partial differential equations, 2007 (2007), 203--298

• [22] J. Kobayashi, M. Otani, Topological degree for $({\rm S})\sb +$-mappings with maximal monotone perturbations and its applications to variational inequalities, Nonlinear Anal., 59 (2004), 147--172

• [23] R. Landes, V. Mustonen, On pseudomonotone operators and nonlinear noncoercive variational problems on unbounded domain, Math. Ann., 248 (1980), 241--246

• [24] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer. Math. Soc., 139 (2011), 1645--1658

• [25] J. Leray, J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3), 51 (1934), 45--78

• [26] N. G. Lloyd, Degree Theory, Cambridge University Press, New York (1978)

• [27] J. Mawhin, Leray-Schauder degree: a half a century of extensions and applications, Topol. Methods Nonlinear Anal., 14 (1999), 195--228

• [28] M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math., 73 (1951), 497--511

• [29] D. O'Regan, Y. J. Cho, Y.-Q. Chen, Topological Degree Theory and Applications, Chapman and Hall/CRC, Boca Raton (2006)

• [30] D. Pascali, S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff & Noordhoff International Publishers, Bucharest (1978)

• [31] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149 (1970), 75--88

• [32] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Amer. Math. Soc., Providence (1997)

• [33] I. I. Vrabie, Compactness methods for nonlinear evolutions, Longman Scientific & Technical, Harlow (1995)

• [34] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York (1990)