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On global existence and attractivity results for nonlinear functional integral equations. (English) Zbl 1197.45005

The authors study the existence and global attractivity of the solutions for the following nonlinear functional integral equation (FIE)

$x\left(t\right)=F\left(t,f\left(t,\alpha \left(t\right)\right),\underset{0}{\overset{\beta \left(t\right)}{\int }}g\left(t,s,x\left(\gamma \left(s\right)\right)\right)ds\right)$

for all $t\in {ℝ}_{+}$, where $f:{ℝ}_{+}×ℝ\to ℝ$, $g:{ℝ}_{+}×{ℝ}_{+}×ℝ\to ℝ$, $F:{ℝ}_{+}×ℝ×ℝ\to ℝ$ and $\alpha ,\beta ,\gamma :{ℝ}_{+}\to {ℝ}_{+}$.

They give two existence and global attractivity results for the solution of the FIE via a measure theoretic fixed point theorem of B. Dhage [Commun. Appl. Nonlinear Anal. 15, No. 2, 89–101 (2008; Zbl 1160.47041)]. These results improve and generalize the attractivity results of J. Banas and B. Rzepka [Appl. Math. Lett. 16, No. 1, 1–6 (2003; Zbl 1015.47034)], J. Banas and B. Dhage [Nonlinear Anal., Theory Methods Appl. 69, No. 7 (A), 1945–1952 (2008; Zbl 1154.45005)], B. Dhage [Nonlinear Anal., Theory Methods Appl. 70, No. 7 (A), 2485–2493 (2009; Zbl 1163.45005)] and X. Hu and J. Yan [J. Math. Anal. Appl. 321, No. 1, 147–156 (2006; Zbl 1108.45006)] under some weaker Lipschitz conditions.

MSC:
 45G10 Nonsingular nonlinear integral equations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 45M05 Asymptotic theory of integral equations 45M10 Stability theory of integral equations 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc.