CARPATHIAN J. MATH. Volume 37 (2021), No. 3, Pages 407 - 416 Online version at https://www.carpathian.cunbm.utcluj.ro/ Print Edition: ISSN 1584 - 2851; Online Edition: ISSN 1843 - 4401 DOI: https://doi.org/10.37193/CJM.2021.03.04 Measures of noncompactness and infinite systems of integral equations of Urysohn type in L ∞ (G) SHAHRAM BANAEI 1 ,VAHID PARVANEH 2 and MOHAMMAD MURSALEEN 3,4 ABSTRACT. In this article, applying the concept of measure of noncompactness, some fixed point theorems in the Fr´ echet space L ∞ (G) (where G ⊆ R ω ) have been proved. We handle our obtained consequences to inquiry the existence of solutions for infinite systems of Urysohn type integral equations. Our results extend some famous related results in the literature. Finally, to indicate the effectiveness of our results we present a genuine example. 1. I NTRODUCTION AND PRELIMINARIES Measure of noncompactness (MNC) approaches ([8], [17]) have an substantial role in nonlinear functional analysis and fixed point theory. Heretofore, applying MNC ap- proaches many articles have been extracted on the existence and behavior of solutions for nonlinear differential and integral equations. Some of these papers are [2, 3, 6, 7, 11, 14]. In this paper, we extract some fixed point theorems in Fr´ echet spaces with the assis- tance of MNC approaches and the Tychonoff fixed point theorem (TFPT), which are ex- tensions of the results presented in [18, 19, 20, 21]. The conformation of this paper is as follows. In part 1, some preliminaries and concepts are summoned. Part 2 is allocated to stating some fixed point theorems of Darbo-type in the space L ∞ (G). Finally, in part 3, we apply our results to contemplate the existence of solutions for the following infinite system of nonlinear integral equations: (1.1) σ n (ι)= ρ n  ι, σ 1 (ι) ...,σ n (ι),...,  G η n (ι, κ, (σ j (κ)) ∞ j=1 )dκ  where G ⊆ R ω in which R ω denotes the countable cartesian product of R with itself. Note that some classes of infinite system of nonlinear integral equations have been investigated in [10, 12, 15]. All over this paper, B is assumed to be an infinite dimensional Banach space or a Fr´ echet space. As well as, B(x, r) marks the closed ball centered at x with radius r. The symbol B r stands for the ball B(0,r). If Q be a subset of B, then the closure and closed convex hull of Q, are announced by Q and ConvQ, respectively. Furthermore, the family of all nonempty bounded subsets and the collection of all relatively compact subsets of B are indicated by M B and N B , respectively. A vector space Q over the field R which is endowed with a topology such that the maps (ι, κ) → ι + κ and (υ,ι) → υι are continuous from Q×Q and R ×Q to Q is called a topological vector space (TVS). A TVS is called locally convex if the origin has a neighborhood basis (i.e. a local base) consisting of convex sets [16]. Fr´ echet spaces are locally convex and complete with respect to a translation invariant metric. Received: 19.11.2020. In revised form: 16.04.2021. Accepted: 23.04.2021 2010 Mathematics Subject Classification. 47H08, 47H10. Key words and phrases. Measure of noncompactness, Tychonoff fixed point theorem, Fr´ echet space, System of inte- gral equations. Corresponding author: Sh. Banaei; math.sh.banaei@gmail.com. 407