Abstract
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations.
Highlights
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces
The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations
In this paper we study integral inequalities of the form y x ≤fxgxq ◦ y dμ, x ∈ X, 1.1
Summary
In this paper we study integral inequalities of the form y x ≤fxgxq ◦ y dμ, x ∈ X, Sx and the corresponding integral equations yx fx gx q ◦ y dμ, x ∈ X, Sx where. It turns out to be useful to study Bihari type inequalities with abstract Lebesgue integral. It is motivated proceeding in this direction as follows. In the second main result we test the scope of the previous theorem by applying it to prove the existence of a maximal and a minimal solution of the integral equation 1.2. Consider the integral inequality y x ≤1 q ◦ ydε[0 1] q y 0 , x ∈ 0, ∞ , 0,x and the corresponding integral equation yx 1 q ◦ ydε[0 1] q y 0 , x ∈ 0, ∞ , 0,x where ε0 is the unit mass at 0 defined on the σ-algebra of Borel subsets of 0, ∞ , and q : 0, ∞ −→ 0, ∞ , q t : t. 1, Since 1.19 has the unique solution y x 0, x ∈ X, the successive approximations yn
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