Sabbavarapu Nageswara Rao, Mahammad Khuddush, Mutum Zico Meetei, Merdi Ahmed Orsud, Boundary value problems of fractional order: Analyzing iterative systems with positive solutions, Vol. 2025 (2025), No. 16, pp. 1-18

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DOI: 10.23952/jnfa.2025.16

Received December 2, 2024; Accepted April 11, 2025; Published May 26, 2025

Abstract. This study addresses a fractional-order iterative system

$D_{0^+}^{i_1}\Big[D_{0^+}^{i_2}r_t(q)\Big]+\sigma(q)\hslash_t(r_{t+1}(q))=0,\ 0<q<1,\,1<i_1,i_2<2,$

satisfying two-point integral boundary conditions

$r_t(0)=D_{0^+}^{i_2}r_t(0)=0,\ r_t(1)=\int_0^1\Lambda(z)r_t(z)dz$

and

$D_{0^+}^{i_2}r_t(1)=\int_0^1\Lambda(z)\big(D_{0^+}^{i_2}r_t(z)\big)dz.$

Here, we consider the system where $t$ is an integer from $1$ to $w,$ and $r_{w+1}$ equals $r_1$ and $\sigma(q)$ belongs to the Lebesgue space $L^q[0,1]$ for $1\le p\le\infty$ and may exhibit singularities within $(0, 1/2).$ The study employs Riemann-Liouville fractional derivatives of orders $i_1$ and $i_2,$ denoted as $D_{0^+}^{i}.$ By leveraging fixed-point index theory within a Banach space cone, the existence of infinitely many positive solutions to this problem is established. Furthermore, in the specific case where $w$ equals $1,$ sufficient conditions for a unique solution are derived by using Rus's theorem within a complete metric space.

How to Cite this Article:
S.N. Rao, M. Khuddush, M.Z. Meetei, M.A. Orsud, Boundary value problems of fractional order: Analyzing iterative systems with positive solutions, J. Nonlinear Funct. Anal. 2025 (2025) 16.