Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

“…Second, the nonlinear term and boundary conditions of fractional BVP (1.1) depend not only on the unknown functions but also on fractional derivatives that make the problem considered here more generalized and complicated than the problems considered in papers. 37,38 Third, compared with literature, 37,38 we not only discuss the existence and uniqueness of the results but also establish Ulam's type stability results. Moreover, Theorem 3.1 (see Section 3) shows that the problem (1.1) has at least one solution provided, the nonlinear term satisfies a linear growth condition that makes the existence condition weaker than condition (3) used in prior work.…”

“…Next, in 2019, Mehandiratta et al 38 extended the results of Graef et al 37 to the case of n + 1 nodes and n edges. In their study, the investigation of the fractional BVP on G was presented as follows: i (l i t, x, ), i = 1, 2, … , n, t ∈ [0, 1] and f i satisfied the following conditions:…”

“…[33][34][35][36] However, limited work has been conducted on fractional BVPs on graphs. 37,38 First, in 2014, Graef et al 37 considered a fractional BVP on a star graph (a collection of n edges incident to a single node point (see Figure 1)), which is a graph consisting of three nodes and two edges, that is, and…”

Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance

“…In recent years, the discussion of fractional initial value problems (IVPs) and BVPs have attracted the attention of many scholars and valuable results have been obtained (see ). Various methods have been utilized to study fractional IVPs and BVPs such as the Banach contraction map principle (see [8][9][10][11]), fixed point theorems (see [12][13][14][15][16][17][18]), monotone iterative method (see [19][20][21]), variational method (see [22][23][24]), fixed point index theory (see [17][18][19][20][21][22][23][24][25]), coincidence degree theory (see [26][27][28][29]), and numerical methods [30,31]. For instance, Jiang (see [26]) studied the existence of solutions using coincidence degree theory for the following fractional BVP:…”

“…The main difficulties in solving the present BVP are: Constructing suitable Banach spaces for BVP (1); Since [0, +∞) is noncompact, it is difficult to prove that operator N is L-compact; The theory of Mawhin's continuation theorem is characterized by higher dimensions of the kernel space on resonance BVPs, therefore, constructing projections P and Q is difficult; Estimating a priori bounds of the resonance problem on an infinite interval with dim KerL = 2 (see Section 3, Lemmas [11][12][13][14][15][16].…”