Solvability of mixed type operator equations
Introduction
The class of functional operator equations of various types plays very important role in numerous mathematical research areas. In this work we study the solvability of the functional operator equationsin the space , where , B, A are bounded linear operators on , ϕ is a function from to . Furthermore, the operatoris the superposition operator generated by the functions ϕ, f and is defined byFinally, the nonlinear Urysohn integral operatorClearly this equation is general than that studied in [5], [13], [14].
We present in Section 2 the relevant results needed in our work. The main result will be presented in Section 3. We apply this result with particular choices to the operators involved and we are able to draw several conclusions.
Section snippets
Preliminaries
This section is devoted to recall some notations and results that will be needed in the sequel. Denote by , the class of Lebesgue integrable functions on the interval (0, 1) with the usual norm
The superposition operator is one of the simplest and most important operators that is investigated in nonlinear functional analysis and in the theories of differential integral and functional equations [3], [4], [5], [6], [7], [8], [15], [4], [20]. This operator is
Main result
Consider the operator equation (1). Let us denote by H the operator determined by the right hand side of Eq. (1), namelyTherefore Eq. (6) can be written as
To establish our main result concerning existence of a monotone solution of Eq. (7) we impose suitable conditions on the functions involved in that equation. Namely we assume
- (i)
The function and is a.e. nondecreasing on the interval .
- (ii)
satisfies the Carathéodory conditions. There exists a
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