Positive solutions for a class of singular fractional boundary value problem with p-Laplacian

Saroj Panigrahi; Raghvendra Kumar

doi:10.1515/jaa-2024-0104

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Positive solutions for a class of singular fractional boundary value problem with p-Laplacian

Saroj Panigrahi and Raghvendra Kumar

Published/Copyright: November 14, 2024

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From the journal Journal of Applied Analysis

Abstract

In this paper, an attempt has been made to establish the sufficient conditions for the existence and multiplicity of positive solutions by using the fixed point index theory and the Avery–Peterson fixed point theorem respectively for the following class of nonlinear singular fractional differential equation

D 0 + β ⁢ ( ϕ p ⁢ ( D 0 + α ⁢ u ⁢ ( t ) ) ) + f ⁢ ( t , u ⁢ ( t ) , … , u ( n - 2 ) ⁢ ( t ) ) = 0 , t ∈ ( 0 , 1 ) ,

with the boundary conditions

u ( k ) ⁢ ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , D 0 + α ⁢ u ⁢ ( 0 ) = 0 , D 0 + α - 1 ⁢ u ⁢ ( 1 ) = ∫ 0 1 D 0 + α - 1 ⁢ u ⁢ ( t ) ⁢ 𝑑 A ⁢ ( t ) ,

where 0 < β ≤ 1 , n - 1 < α ≤ n , n ≥ 3 , A : [ 0 , 1 ] → ℝ is a function of bounded variation, ϕ p ⁢ ( s ) = | s | p - 2 ⁢ s for p > 1 and ϕ q ⁢ ( s ) is the inverse of ϕ p ⁢ ( s ) , where p , q satisfies the relation 1 p + 1 q = 1 , f may have singularity at t = 0 .

Keywords: Continuous fractional calculus; boundary value problem; positive solutions; fixed point index theory; Avery–Peterson fixed point theorem; Banach space

MSC 2020: 34A08; 45G05

Funding statement: The second author was supported by University Grant Commission Fellowship, India through ref. no. 201610152817, dated Feb 04, 2021.

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Received: 2024-06-26

Revised: 2024-10-04

Accepted: 2024-10-06

Published Online: 2024-11-14

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https://doi.org/10.1515/jaa-2024-0104

Keywords for this article

Continuous fractional calculus; boundary value problem; positive solutions; fixed point index theory; Avery–Peterson fixed point theorem; Banach space