A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations

Abstract

We present some fixed point results in Banach algebras based on the so called degree of nondensifiability φ d . It is shown that φ d is an alternative method to measures of noncompactnes to obtain fixed point result. As an application of the usefulness of φ d it is proved the existence of solution for some quadratic integral equations.

A quantitative version of the Arzelà-Ascoli theorem based on the degree of nondensifiability and applications

“…we find χ i (B) = 0 < 1 = φ d (B). Another example that shows that φ d and χ i are essentially different can be found in [12,Example 3.4]. Despite its name, χ i is not a MNC (see, for instance, [1, p. 9]).…”

“…• Unlike many of the results based in MNCs for integral equations of type (4.1), we do not require the uniform continuity of the family of functions (f (s, x(s)) s∈I (see the above cited references). • Similar conditions to (C1)-(C4) are assumed in [12] to prove the existence of solutions of certain integral equations. In fact, condition (C3) is required in [12] Proof.…”

“…• Similar conditions to (C1)-(C4) are assumed in [12] to prove the existence of solutions of certain integral equations. In fact, condition (C3) is required in [12] Proof. Clearly, the existence of solutions of (4.1) is equivalent to the existence of fixed points of the map which, from condition (C1), satisfies T (x) ∈ C(I, X) for every x ∈ C(I, X).…”

“…As it is shown in [10,11,12], we can use φ d as an alternative to the MNCs in certain fixed point problems, because in some cases it seems to work out under more general conditions than the MNCs. So, with a suitable combination of a Sadovskiȋ fixed point theorem type for φ d (see Theorem 2.11) and Theorem 3.2, we will prove in Section 4 a result regarding the existence of solutions of certain Volterra integral equations.…”

The ε-approximated complete invariance property

“…On the other hand, the so called degree of nondensifiability (DND), explained in detail in Section 2, has been used to prove, under suitable conditions, the existence of fixed points of continuous self mappings defined into a non-empty, bounded, closed and convex subset of a Banach space (see [3] and references therein). In the present paper, for a given metric space (X, d), we introduce the concept of ε-approximated complete invariance property (ε-ACIP), which generalizes the CIP one and, by using the DND, we relate in our main result (see Theorem 3.2) this novel concept with the DND of a bounded metric space.…”