An Extension of Darbo’s Theorem and Its Application

Abstract

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].

Positive solutions of fractional integral equations by the technique of measure of noncompactness

“…We are going to show that equation ( 1.1 ) has a positive solution that belongs to space . The obtained results extend several papers (see [ 1 – 5 , 8 ], for example). Finally, an example is presented to show the efficiency of our results.…”

“…Following the trend of Lemma 2.2 , various authors proved several DFPTs and their coupled versions by using different types of contractive conditions in the sense of the MNC (for example, see [ 1 – 5 , 8 ]).…”

Common fixed point theorems for two and three mappings

“…A contractive condition in terms of a measure of noncompactness, firstly used by Darbo [1], is one of the fruitful tools to obtain fixed point and common fixed point theorems. There are many extensions of this condition which are known as generalizations of Darbo's fixed point theorem; see for example [2][3][4][5][6][7][8][9][10][11] and the references therein.…”

F ( ψ , φ ) $F(\psi,\varphi)$ -Contraction in terms of measure of noncompactness with application for nonlinear integral equations

“…In this article the measure of noncompactness is used instead of the metric d , which is used in [ 4 ] and [ 11 ]. Also, contractions associated with a measure of noncompactness in two linear and integral types and the application in solving integral equations are considered, while in the two mentioned article just the linear contractions in terms of the metric d is used for achieving a fixed point.…”