Existence and properties of nontrivial solutions for novel fractional differential equations with ϕ-Hilfer operators and boundary conditions

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 79 - 18
• Main Authors: Salemi, Zahra, Afshari, Hojjat, Ahmadkhanlu, Asghar
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Boundary conditions; Boundary value problems; Differential equations; Fixed point theory; Fixed points (mathematics); Fractional calculus; Green's functions; Integral boundary conditions; Mathematics; Mathematics and Statistics; Non-trivial solution; Operators (mathematics); Theorems; ϕ-Hilfer operator; Non-trivial solution; Fixed point theory; Integral boundary conditions; Hilfer operator
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03331-5

This study investigates a class of fractional differential equations of the form D 0 + μ , ϕ q ( n ) + f ( n , q ( n ) ) = 0 , 0 < n < 1 , 1 < μ ≤ 2 , where D 0 + μ , ϕ represents the Caputo fractional derivative. The nonlinear function f : [ 0 , 1 ] × [ 0 , ∞ ) → [ 0 , ∞ ) is assumed to be continuous, and ϕ ∈ C 2 [ 0 , 1 ] satisfies ϕ ′ ( n ) > 0 . The problem is supplemented with nonlocal boundary conditions: q ( 0 ) = 0 , q ( 1 ) = ρ ∫ 0 1 p ( r ) q ( r ) ϕ ′ ( r ) d r , 0 < ρ < 1 . By constructing an equivalent integral representation using Green’s function, the existence of nontrivial positive solutions is established through the application of fixed point theorems. The analysis provides new insights into the solvability and properties of solutions for this class of fractional boundary value problems.