In this paper, an extension of Darbo fixed point theorem is introduced. By applying our extension, we obtain a coupled fixed point theorem and a solution for an integral equation. The proofs of our results are based on the technique of measure of noncompactness. Definition 1.1. [6] A mapping µ : M E −→ [0, ∞) is called a measure of noncompactness if it satisfies the following conditions: (1) The family Kerµ = {X ∈ M E : µ(X) = 0} is nonempty and Kerµ ⊆ N E. (2) X ⊆ Y =⇒ µ(X) ≤ µ(Y). (3) µ(X) = µ(X). (4) µ(Conv(X)) = µ(X).
Here, some extensions of Darbo fixed point theorem associated with measures of noncompactness are proved. Then, as an application, our attention is focused on the existence of solutions of the integral equationx(t)=F(t,f(t,x(α1(t)), x(α2(t))),((Tx)(t)/Γ(α))×∫0t(u(t,s,max[0,r(s)]|x(γ1(τ))|, max[0,r(s)]|x(γ2(τ))|)/(t-s)1-α)ds, ∫0∞v(t,s,x(t))ds), 0<α≤1,t∈[0,1]in the space of real functions defined and continuous on the interval[0,1].
Using the fixed point method, we investigate the Hyers-Ulam stability of the system of additive-cubic-quartic functional equations with constant coefficients in non-Archimedean 2-normed spaces. Also, we give an example to show that some results in the stability of functional equations in (Archimedean) normed spaces are not valid in non-Archimedean normed spaces. MSC: 39B82; 46S10; 39B52; 47S10; 47H10
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