Solvability of mixed type operator equations

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Abstract

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.

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 equationsx=g+BFϕ,fUAxin the space L1(0,1), where gL1(0,1), B, A are bounded linear operators on L1(0,1), ϕ is a function from (0,1) to (0,1). Furthermore, the operatorFϕ,f:L1(0,1)×L1(0,1)L1(0,1)is the superposition operator generated by the functions ϕ, f and is defined by(Fϕ,f(Ax))(t)=f(t,Ax(ϕ(t))),t(0,1).Finally, the nonlinear Urysohn integral operator(Ux)(t)=01u(t,s,x(s))ds.Clearly 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 L1=L1(0,1), the class of Lebesgue integrable functions on the interval (0, 1) with the usual normx=01x(t)dt,xL1(0,1).

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), namelyHx=g+BFϕ,fUAx.Therefore Eq. (6) can be written asx=Hx=g+BFϕ,fUAx.

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 gL1(0,1) and is a.e. nondecreasing on the interval (0,1).

  • (ii)

    f:(0,1)×RR satisfies the Carathéodory conditions. There exists a

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