On a Resonant Fractional Order Multipoint and Riemann-Stieltjes Integral Boundary Value Problems on the Half-line with Two-dimensional Kernel Ezekiel K. Ojo † , Samuel A. Iyase, and Timothy A. Anake, Abstract—This paper investigates existence of solutions of a resonant fractional order boundary value problem with mul- tipoint and Riemann-Stieltjes integral boundary conditions on the half-line with two-dimensional kernel. We utilised Mawhin’s coincidence degree theory to derive our results. The results obtained are validated with examples. Index Terms—Banach spaces, coincidence degree theory, half- line, resonance, Riemann-Stieltjes integral, two-dimensional kernel. I. I NTRODUCTION F RACTIONAL differential equation serves as a powerful tool for mathematical modelling of complex phenom- ena, such as; viscoelastic media, epidemics, electromagnet- ics, acoustics, control theory, electrochemistry, finance, and materials science found in science and engineering (see [5], [16], [21], [24], [26]. The interest of researchers and scientists have significantly shifted to fractional-order models because, they are more accurate and provide more degrees of freedom than integer-order models.Valuable results have been obtained in the literature on the existence of solutions of fractional order boundary value problems (BVPs) by using different methods.These methods include; coincidence degree theory of Mawhin (see [2], [8], [10], [12], [16], [18], [22], [27], [28], [29], hybrid fixed point theorem [6], Ge and Ren extension of Mawhin coincidence degree theory [9],extension of continuation theorem [25], monotone iter- ative technique [11] and the references therein. A fractional order BVP is at resonance if the corresponding homogeneous equation has non-trivial solution. Some scholars have studied resonant fractional order BVPs on finite interval [0, 1] with finite point or integral boundary conditions in which the dim ker L =1 and the order 1 < α ≤ 2 (see [1], [7], [12], [15], [23], [31]). Recently, Zhang and Liu [30] studied the following class of fractional multipoint boundary value problem at resonance with dim ker L =2 on an infinite interval, and established Manuscript received December 22, 2021; revised February 10, 2023. This work is supported by Covenant University, Nigeria. † Corresponding author: Ezekiel K. Ojo is a research PhD student at the Department of Mathematics, Covenant University, Ota, Ogun State, Nigeria (E-mail: kadjoj@yahoo.com, kadejo.ojopgs@stu.cu.edu.ng) Samuel A. Iyase is a Professor of Mathematics Department, Covenant University, Ota, Ogun State, Nigeria (E-mail:samuel. iyase@covenantuniversity.edu.ng) Timothy A. Anake is a Professor of Mathematics Department, Covenant University, Ota, Ogun State, Nigeria (E-mail: timothy. anake@covenantuniversity.edu.ng) that solution exists by using coincidence degree theory D α 0 +u(t)= f ( t, u(t),D α-2 0 + u(t),D α-1 0 + u(t) ) , 0 <t< +∞, subject to; u(0) = 0, ,D α-2 0 + u(0) = m X i=1 α i D α-2 0 + u(ξ i ), D α-1 0 + u(+∞)= n X j=1 β j D α-1 0 + u(η j ), ∑ m i=1 α i =1= ∑ n j=1 β j η j , ∑ m i=1 α i ξ i =0= ∑ n j=1 β j are critical for resonance; where D α 0 + is the standard Riemann-Liouville fractional derivative of order α, 2 <α ≤ 3, 0 <ξ 1 <ξ 2 < ··· <ξ m < +∞, and 0 <η 1 <η 2 < ··· <η n < +∞. However, the existence of solutions for a resonant frac- tional order boundary value problems on the half-line with multipoint and Riemann-Stieltjes integral boundary condi- tions where dim ker L =2 and 3 <α ≤ 4 have not been widely reported in the literature. We are motivated by this, to focus on investigating existence of solution for the following resonant fractional order boundary value problem D α 0 +x(t)= f  t, x(t),D α-3 0 + x(t),D α-2 0 + x(t),D α-1 0 + x(t)  , x(0) = 0 = D α-3 0 + x(0),D α-2 0 + x(0) = m X i=1 μ i D α-2 0 + x(ξ i ), D α-1 0 + x(+∞)= Z η 0 D α-2 0 + x(t)dA(t), (1) where t ∈ (0, +∞), 3 <α ≤ 4, dim ker L =2, 0 <ξ 1 < ξ 2 <ξ 3 < ··· <ξ m < ∞, η ∈ (0, +∞) and A(t) is a continuous and bounded variation function on (0, +∞). Throughout this investigation, the following assumptions are made: (H 1 ) m X i=1 μ i =1, m X i=1 μ i ξ i =0, Z η 0 tdA(t)=1, and Z η 0 dA(t)=0. (H 2 ) Δ=  1 - m X i=1 μ i e -ξi  Z η 0 (2 + t)e -t dA(t)  +  R η 0 e -t dA(t)  ∑ m j=1 μ i (2 + ξ i )e -ξi - 2  6=0 (H 3 ) There exist nonnegative functions ρ 1 (t),ρ 2 (t),ρ 3 (t),ρ 4 (t),ρ 5 (t) ∈ L 1 (0, +∞) such that Engineering Letters, 31:1, EL_31_1_14 Volume 31, Issue 1: March 2023 ______________________________________________________________________________________