On the existence of a three point boundary value problem at resonance inRn

Existence of solutions to fractional differential equations with three-point boundary conditions at resonance with general conditions

“…where f : [0, 1] × R n × R n → R n is a Carathéodory function and the square matrix A satisfies certain conditions. These results for second order differential equations in [14] and [13] were generalized to fractional order case α ∈ (1, 2] in [4] and [18]. It should be highlighted that, in [13], the authors successfully removed the constricted conditions used in [14] by making use of the property of Moore-Penrose pseudoinverse matrix technique.…”

“…Recently, the authors in [14,13] investigated the following second order differential system u ′′ (t) = f (t, u(t), u ′ (t)), 0 < t < 1,…”

Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

“…In [1], Gupta studied the solvability of three-point BVPs for nonlinear second-order ordinary differential equations. Since then, many authors have investigated the existence and multiplicity of solutions for three-point boundary value problems for nonlinear integer-order ordinary differential equations and nonlinear fractional-order differential equations, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.…”

Solvability of integral boundary value problems at resonance in $R^{n}$

“…Let A = (a ij ) n×n be a square matrix of order n. α : [0, 1] → R is a bounded variation function, and 1 0 u(t) dα(t) = ( under the following assumptions: (H1) B is a diagonalization matrix, and det(I -B) = 0; (H2) 1 0 t(1t) dα(t) = 0; (H3) f : [0, 1] × R 2n → R n satisfies the Carathéodory conditions. If the condition (H 1 ) is considered, the associated linear problem -u (t) = 0, u(0) = 0, u(1) = A 1 that, when n = 1, the existence theory of integral boundary value problems for ordinary differential equations or fractional differential equations has been well studied; we refer the reader to [4,10,17,20,21,24,25,[29][30][31][32][33][34][35]37] for some recent results at non-resonance and to [2,3,16,18,22,23,27,36] for results at resonance. When n ≥ 2 and A is not a diagonal matrix, IBVP (1.1) becomes a system of ordinary differential equations with coupled boundary conditions.…”