Stud. Univ. Babe¸ s-Bolyai Math. 66(2021), No. 4, 723–738 DOI: 10.24193/subbmath.2021.4.10 Existence of positive solutions for a class of BVPs in Banach spaces Lyna Benzenati, Svetlin Georgiev Georgiev and Karima Mebarki Abstract. In this work, we use index fixed point theory for perturbation of expan- sive mappings by ℓ-set contractions to study the existence of bounded positive solutions for a class of two-point boundary value problem (BVP) associated to second-order nonlinear differential equation on the positive half-line. The nonlin- earity, which may exhibit a singularity at the origin, is written as a sum of two functions which behave differently. These functions, depend on the solution and its derivative, take values in a general Banach space and have at most polynomial growth. An example to illustrate the main results is given. Mathematics Subject Classification (2010): 34B15, 34B18, 34B40, 47H08, 47H10. Keywords: Boundary value problem, Green’s function, unbounded interval, mea- sure of noncompactness, fixed point index, sum operator. 1. Introduction The theory of ordinary differential equations in Banach space is a rapidly growing area of research, it is developed for example in the books by Guo et al. [12], Guo and Lakshmikantham [11], Lakshmikantham and Leela [14], Deimling [2], and Zeidler [19] or in the papers by P. Li et al. [15] and by Y. Liu [16]. In the past decades, the study of BVPs defined on compact intervals has been considered by many authors with application of a huge variety of methods and tech- niques. However, BVPs defined on unbounded intervals are scarce, as they require other types of techniques to overcome the lack of compactness. Historically, these problems began at the end of nineteenth century with A. Kneser [13]. In this work, the lack of compactness is overcome with some techniques and specific tools. Let P be a cone in some Banach space E, that is a closed convex subset such that α P⊂P for all positive real number α and P∩ (−P )= {0}. Notice that E is partially ordered by cone P , i.e. x ≤ y if and only if y − x ∈P . For details on cone theory see [11].