Abstract
In this paper, a generalization of Diaz-Margolis’s fixed point theorem is established. As applications of the generalized Diaz-Margolis’s fixed point theorem, we present some existence theorems of the Hyers-Ulam stability for a general class of the nonlinear Volterra integral equations in Banach spaces.
Highlights
Introduction and preliminariesThe stability of functional equations was originally raised in a famous talk given by Ulam [ ] at Wisconsin University in
Since the rapid growth of the study of stability of functional equations has been developed at a high rate by several authors in the last decades; for more details, we refer the readers to [ – ] and references therein
Du Journal of Inequalities and Applications (2015) 2015:407 we study the nonlinear generalized Volterra integral equation given by x y(x) = κ(x) + s(x) G x, τ, y(τ ) dτ, ∀x ∈ I, ( . )
Summary
Since γ ∈ [ , ), we obtain limn→∞ λn = and limn→∞ sup{p(wn, wm) : m > n} =. Wn ∈ L implies wn ∈ W for all n ∈ N. For each n ∈ N, we obtain p(v, Tv) ≤ p(v, wn+ ) + p(wn+ , Tv) ≤ p(v, wn+ ) + α p(wn, v) p(wn, v) < p(v, wn+ ) + p(wn, v). Again, we know that the function x → p(v, x) is continuous on the set W. By taking the limit at both sides of the previous inequality and applying Theorem . V ∈ L, we know p(z, v) < ∞ from Theorem. ), we obtain p(z, v) = p(Tz, Tv) ≤ α p(z, v) p(z, v), which implies.
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