Some Solvability Theorems for Nonlinear Equations with Applications to Projected Dynamical Systems

The study of solvability of nonlinear equations defined in topological vector spaces is a fundamental problem considered in Nonlinear Analysis. This problem is so important because it is related to the study of solvability of differential or integral equations. The literature related to this subject is huge. See [1]–[4], [7], [10], [11], [17], [23]–[25] among the other papers and books published until now. The solvability of nonlinear equations has been studied by many authors using several kinds of mathematical tools and related to this problem several chapters of Nonlinear Analysis have been created as for example, the fixed point theory, the theory of monotone operators, the theory of accretive operators, the theory of normal solvable operator, etc. Several solvability theorems have been obtained using as assumptions the monotonicity [17], [19], [24], [25] or some geometrical assumptions, [1], or topological degrees. We present in this paper some simple solvability theorems applicable to the study of some problems considered in the theory of projected dynamical systems [5], [6], [8], [16], [21], [22]. The theory of projected dynamical systems is a new chapter in the theory of dynamical systems. In our solvability results some inequalities are essential. The inequalities are so important in mathematics [20].