Coupled Fixed Point Theorem for the Generalized Langevin Equation with Four-Point and Strip Conditions

Abstract

By considering a metric space with partially ordered sets, we employ the coupled fixed point type to scrutinize the uniqueness theory for the Langevin equation that included two generalized orders. We analyze our problem with four-point and strip conditions. The description of the rigid plate bounded by a Newtonian fluid is provided as an application of our results. The exact solution of this problem and approximate solutions are compared.

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

“…Let δ(t) ∈ C ν and f (t) ∈ C µ . Then, there exists a unique solution x(t) ∈ C α 1 ,β 1 1−γ 1 to Problems ( 9) and (10) given by…”

“…where a is the particle's radius, ν is the fluid's viscosity, 1 σ is the friction coefficient for unit mass, F(t, x(t)) = − 1 σ x ′ (t) + 1 m R(t) and R(t) is a random force. FLEs have attracted many scholars to study properties of solutions for FLEs-for instance, the existence and uniqueness of solutions for FLEs with Caputo or Riemann-Liouville fractional derivatives [7,8], boundary value problems for FLEs [9][10][11][12], etc. Baghani and Nieto [13] studied the following Langevin differential equation with two different fractional orders: c D ξ ( c D ν + λ)x(t) = h(t, x(t)).…”

Finite-Time Stability to Fractional Delay Cauchy Problem of Hilfer Type

“…Ordinary differentiation and integration of arbitrary order, which may be non-integer, are generalized in fractional calculus. Many studies have focused on differential equations of fractional order [1][2][3]. Many definitions of fractional integrals and derivatives may be found in the literature, ranging from the most well-known Riemann-Liouville (R-L) and Caputo-type fractional derivatives to others like the Hadamard fractional derivative, the Katugampola fractional derivative, the Hilfer fractional derivative, and so on.…”

Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

“…Interest in fractional differential equations has risen significantly in recent years [1][2][3]. Numerous writers have investigated fractional equations with varying conditions and fractional evolution equations; for instance, see [4][5][6][7][8]. Fractional differential equations with non-local conditions are commonly seen in real-life scenarios.…”