Neutral set differential equations

Stability of Set Differential Equations in Fréchet Spaces

“…Proof. From the previous section, we know that equation (12) can be reduced to a system of SDEs on the Banach spaces E i by considering the mapping F as a projective limit.…”

“…Nowadays, the theory of SDEs has been developed into an independent subject area. There are a few results on the existence, stability, and other properties of solutions for various equations, such as SDEs [2][3][4][5][6][7][8][9][10][11], set functional differential equations [12][13][14][15][16], set integrodifferential equations [17][18][19][20], SDEs on time scales [21,22], SDEs with causal operators [23][24][25][26][27], and others [15,[28][29][30][31], and references are given therein. Systematic development of set differential equations has been provided by Lakshmikantham et al [32] and Martynyuk [33].…”

Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

“…e stability results of various set-valued differential equations were obtained by applying Lyapunov's direct method. e results can be found in the monograph [15], the papers for the set-valued differential equation [16][17][18][19][20][21][22][23][24], setvalued functional differential equation [25][26][27][28][29], and other equations [6,9,[30][31][32]. However, according to what we know so far, there are few stability results for the set-valued differential equation via the fixed point method.…”

Higher order convergence for a class of set differential equations with initial conditions

“…Some basic results have been established on existence, uniqueness, continuous dependence of solutions and the corresponding stability properties [20,21,3,16,29,14,4,30,5,28]. Meanwhile, some generalized results have also been established for various types of SDEs, such as set functional differential equations [1,2], impulsive set differential equations [7,25,24,11], set differential equations involving causal operators [8,12], set differential equations in Banach spaces [22,23], set dynamic equations on time scales [15], to name a few.…”

“…For the IVP(1), assume that(H 13 ): V, W ∈ C 1 [I, K c (R n )]are coupled lower and upper solutions of type II of (1), andV (t) ≤ W (t), (H 14 ): F (t, U ) is hyperconvex in U of order 2k, G(t, U ) is hyperconcave in U of order 2k, k ≥ 1, where F , G ∈ C 2k+1 [Ω, K c (R n )]. i.e., D H 2k+1 U (t, U ) ≥ θ, D H 2k+1 U G(t, U ) ≤ θ, (H 15 ): D H U F (t, U ) ≤ − max Ω [D H 2k+1 U F (t, U )] (W − V ) 2k (2k − 1)!…”