Abstract:In this paper, we investigate the finite approximate controllability of fractional semilinear differential equations involving the Hilfer derivative. We show that if the linear part is approximate controllable, then under suitable conditions the semilinear system is finite approximate controllable.
“…Definition 5 (Approximate Controllability [39]). System ( 1) is called approximately controllable on [0, τ] if, for every φ ∈ C([−h, 0]; Z), ς 1 ∈ P C(I, Z), and ≥ 0, there exists ṽ ∈ P C(I, U) such that the corresponding solution ς(µ) of (1) satisfies ς(0) = φ(0) and ς(τ) − ς 1 Z ≤ .…”
In this paper, we investigate the exact and approximate controllability, finite time stability, and β–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with the help of a fixed point approach. The famous generalized Grönwall inequality is used to study the finite time stability and β–Hyers–Ulam–Rassias stability. Finally, the main results are verified with the help of an example.
“…Definition 5 (Approximate Controllability [39]). System ( 1) is called approximately controllable on [0, τ] if, for every φ ∈ C([−h, 0]; Z), ς 1 ∈ P C(I, Z), and ≥ 0, there exists ṽ ∈ P C(I, U) such that the corresponding solution ς(µ) of (1) satisfies ς(0) = φ(0) and ς(τ) − ς 1 Z ≤ .…”
In this paper, we investigate the exact and approximate controllability, finite time stability, and β–Hyers–Ulam–Rassias stability of a fractional order neutral impulsive differential system. The controllability criteria is incorporated with the help of a fixed point approach. The famous generalized Grönwall inequality is used to study the finite time stability and β–Hyers–Ulam–Rassias stability. Finally, the main results are verified with the help of an example.
“…where Ω ⊆ R N is a bounded domain containing the origin, p ∈ (1, ∞), s ∈ (0, 1), 1 < r < q < p, 0 ≤ θ < sp < N, α+β = p * s,θ and λ, µ > 0 are two parameters, p * s,θ = (N−θ)p N−ps is the fractional Hardy-Sobolev exponent, the fractional p-Laplacian operator (−△) s p is the nonlocal operator defined on smooth functions by (−△) s p u(x) = 2 lim are widely studied. We refer the readers to [2,5,6,10,13,[18][19][20][21][22][23][24] and references therein.…”
In this paper, we prove the existence of infinitely many solutions for a fractional p&q-Laplacian system involving Hardy-Sobolev exponents and obtain new conclusion under different conditions. The methods used here are based on variational methods and LjusternikSchnirelmann theory.
“…Fractional differential equations play a significant role in various fields of science and engineering such as fluid mechanics, mechanics of materials, biology, plasma physics, finance, chemistry, image processing (see, for example, [1][2][3][4]14,[18][19][20]26]). Variable-order fractional calculus is an extension of the constant-order fractional calculus.…”
A numerical approach for solving the multi-term variable-order space fractional nonlinear partial differential equations is proposed. The fractional derivatives are described in the Caputo sense. The numerical approach is based on generalized Laguerre polynomials and finite difference method. The proposed scheme transforms the main problem to a system of nonlinear algebraic equations. The nonlinear system is solved by using Newton's method. The validity and the applicability of the proposed technique are shown by numerical examples.
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