On solvability and approximating the solutions for nonlinear infinite system of fractional functional integral equations in the sequence space \(\ell_p\), \(p > 1\). (English) Zbl 1543.45004
The following infinite system of nonlinear functional integral equations with Riemann-Liouville type fractional operators in the sequence space \(l_p\), \(p>1\), is considered:
\[
x_n(\sigma)=f_n\left(\sigma,x(\sigma),\frac{h_n(\sigma,x(\sigma))}{\Gamma(\alpha)}\int\limits_{0}^{\sigma}\frac{g_n(\sigma,s,x(s))}{(\sigma-s)^{1-\alpha}}ds\right),\] with \( x(\sigma)=(x_j(\sigma))_{j=1}^{\infty}, \; j,n\in\mathbb{N}.
\)
The authors verify and prove the existence of a solution of this generalized system. The result is established by the applications of the Meir-Keeler condensing operator and Hausdorff measure of noncompactness in the sequence space. An iterative numerical algorithm based on the homotopy perturbation approach along with the Adomian decomposition method to find the approximate solution is also suggested. A numerical example is provided.
Reviewer: Alexander N. Tynda (Penza)
MSC:
| 45G15 | Systems of nonlinear integral equations |
| 45L05 | Theoretical approximation of solutions to integral equations |
| 65R20 | Numerical methods for integral equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 46B45 | Banach sequence spaces |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fixed point theorem; fractional system of integral equations; measure of noncompactness; Meir-Keeler condensing operator; sequence spaceReferences:
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