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My research interests are topological degree theories and their applications to eigenvalues, invairiance of domain theorems and nonlinear operator equations in Banach spaces. This area of mathematics has applications in differential equations and is an active area of current research. A great number of problems in applications involve nonlinearity in their interpretation. My Ph.D. advisor is Professor Athanassios G. Kartsatos. He has done a lot in topological degree theories and applications to various problems involving so-called monotone type operators in Banach spaces. I was inspired by him and Professor Gianni Dal Maso to work in the area of Nonlinear Analysis.

RESEARCH INTERESTS
Dhruba R. Adhikari


Nonlinear analysis has been proven to be a very powerful tool in solving vast number of nonlinear problems in the fields of all sciences, engineering, and economics. Motivated from the works of Professor Athanassios G. Kartsatos and Professor Gianni Dal Maso, I have been working in the area of nonlinear operators and applications of various degree theories to operator equations involving monotone and accretive operators in Banach spaces (cf., [2], [3], [4], [5]).

My previous research work focuses on problems of the existence of nonzero solutions, invariance of domain results, and eigenvalues for operators acting from subsets of reflexive Banach space X to its dual space X*. More precisely, let X be a real reflexive Banach space and G1, G2 two nonempty, open and bounded subsets of X such that 0 Î G2 and [`(G2)] Ì G1. Consider the problem Tx+Cx = 0, where T:X É D(T)® X is an accretive or monotone operator with 0 Î D(T) and T(0)=0, while C:X É D(C)® X can be, e.g., one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type (S+) with T positively homogeneous of degree  £ 1; (d) quasi-bounded and satisfies a generalized (S+)-condition w.r.t. the operator T. Solutions are sought for the problem lying in the set D(T+C)Ç(G1\G2). The degree theories developed by Leray and Schauder, Browder [7], [8] and Skrypnik [13] are used. This is an existence theorem for nonzero solutions to operator equations (possibly nonlinear) of the form Tx+ Cx = 0. Such a problem has also been considered for T:X É D(T)® X* and C:[`D(T)]® X* where T is maximal monotone and C is bounded demicontinuous and of type (S+). Also, the degree theory by Kartsatos and Skrypnik [10] for densely defined operators T, C has been utilized. Here, the operators T and C have a dense linear subspace in their domains. Moreover, the new degree theory developed by Kartsatos and Skrypnik [12] has been used to prove a similar result for the operators of type T+C, where T:X É D(T)® 2X* is a multi-valued maximal monotone operator with 0 Î D(T) and 0 Î T(0) and C: X É D(C)® X* a densely defined quasi-bounded and finitely continuous operator of type ([S\tilde]+). In this case the operator T need not be densely defined and the operator C need not be quasibounded w.r.t T. An invariance of domain problem for the combination T+f+G, where T:X É D(T)® X* is maximal monotone, f:[`D(T)]® X* bounded demicontinuous of class (S+) and G:[`D(T)]® X* a compact multifunction of class (P), has been established. Kartsatos and Skrypnik in [9] have considered this combination without the operator G. The sum L+T+C for the domain invariance results and eigenvalues is considered in the spirit of the degree theory developed by Addou and Mermri [1]. This degree theory generalizes the Berkovits-Mustonen theory [6] for the combination L+C. Here, L: X É D(L)® X* is a densely defined linear maximal monotone operator, T:X® X* bounded maximal monotone and C:[`D]® X* is bounded demicontinuous and of type (S+) w.r.t. D(L), where D Ì X is open and bounded.

My recent research paper [2] in collaboration with Professor A.G. Kartsatos is on the construction of a more generalized degree theory for the combination L+T+C, where L,  C are as above, and T:X É D(T)® 2X* is a strongly quasibounded maximal monotone operator. This development generalizes the Berkovits-Mustonen degree theory [6] and improves the Addou-Mermri degree theory [1] to possibly unbounded maximal monotone operators. As generalizations of results in [9] and [11], invariance of domain theorems have been established and an eigenvalue problem has been solved for this generalized combination. These results are the subjects of the paper [4].

My current research work with Professor A.G. Kartsatos in [5] is the introduction of a new topological degree theory for the sum T+S+C, where T:X É D(T)® 2X* is strongly quasibounded maximal monotone, S:X É D(S)® X* maximal monotone, and C:X É D(C)® X* quasibounded and such that it satisfies a generalized (S+)-condition w.r.t. the operator S. We assume that D(S)=L Ì D(T)ÇD(C), where L is a dense subspace of X, and 0 Î T(0), S(0)=0.  The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators.  This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik [10] just for the single-valued sum S+C, as above.  Applications of the new degree theory will be given in the field of partial differential equations on cylindrical domains.

My future research plan is to give a continuation of the recent works and work on eigenvalues and critical point theory by applying topological degree theories to operators of monotone type and accretive operators. It is relevant to mention at this point that the topological degree theories can be applied to various surjectivity results, invariance of domain results, eigenvalues, critical points, ranges of sums and balls in the ranges of nonlinear operators. One of the basic ideas about how we can apply degree theories is that we construct certain admissible homotopies so that one can efficiently apply, among other properties, the homotopy invariance property of the underlying degree theory.

References

[1]
A. Addou and B. Mermri, Topological degree and application to a parabolic variational inequality problem, Int. J. Math. Math. Sci. 25 (2001), 273-287.
[2]
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.
[3]
D.R. Adhikari, A.G. Kartsatos, Topological Degree Theories and Nonlinear Operator Equations in Banach Spaces, Nonlinear Analysis 69 (2008) 1235-1255.
[4]
D.R. Adhikari, A.G. Kartsatos, Invariance of Domain and Eigenvalues for Perturbations of Densely Defined Linear Maximal Monotone Operators (submitted to Set-valued Analysis).
[5]
D.R. Adhikari, A.G. Kartsatos, A New Topological Degree Theory for Perturbations of the Sum of Two Maximal Monotone Operators (preprint).
[6]
J. Berkovits, V. Mustonen, On the topological degree for perturbations of linear maximal monotone mappings and applications to a class of parabolic problems, Rend. Mat. Appl. 12 (1992), 597-621.
[7]
F.E. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Appl. Math. 18 Part 2, Providence, 1976.
[8]
F.E. Browder, Fixed Point Theory and Nonlinear Problems, Bull. Amer. Math. Soc. 9 (1983), 1-39.
[9]
A. G. Kartsatos, I. V. Skrypnik, Invariance of Domain for Perturbations of Maximal Monotone Operators in Banach Spaces, (to appear).
[10]
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.
[11]
A.G. Kartsatos, I.V. Skrypnik, On the Eigenvalue Problem for Perturbed Nonlinear Maximal Monotone Operators in Reflexive Banach Spaces, Trans. Amer. Math. Soc. 358 (2005), 3851-3881.
[12]
A. G. Kartsatos, I. V. Skrypnik, A New Topological Degree Theory for Densely Defined Quasibounded ([S\tilde]+)-Perturbations of Multivalued Maximal Monotone Operators in Reflexive Banach Spaces, Abstract and Applied Analysis (2005), 121-158
[13]
I. V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Probelms, Amer. Math. Soc. Transl., Ser. II, 139, AMS, Rhode Island, 1994.

 

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