Sentence examples for a unique solution of the nonlinear from inspiring English sources
This implies that there exists a unique solution of the nonlinear Fredholm integral equation (4.1).
This implies that there exists a unique solution of the nonlinear fractional differential equation of Caputo type (4.3).
In order to assure the existence of a unique solution of the nonlinear operator equation (2.15) in, some conditions must be imposed on the parameters appearing in the equation.
Moreover, for each (x_{0} in C[a,b]), the Picard iteration ({x_{n}}), which is defined by (x_{n}) (t)= phi(t)+ int_{a}^{b}K bigl t,s,x_{n-1}(s) bigr),ds for all (n inmathbb{N}), converges to a unique solution of the nonlinear integral equation (4.1).
Moreover, for each (x_{0} in C[a,b]), the Picard iteration ({x_{n}}), which is defined by (x_{n}) (t)= phi(t)+ int_{a}^{t} K bigl t,s,x_{n-1}(s) bigr),ds for all (n inmathbb{N}), converges to a unique solution of the nonlinear integral equation (4.2).
From (3.10), we know that and so has a unique fixed point, which is a unique solution of the generalized strongly nonlinear implicit quasi-variational inequality (2.1).
The purpose of this paper is to employ numerical methods to approximate the solution of the linear integral equations (2) in a piecewise continuous polynomial space and then generate a sequence of approximation solutions where converge to the unique solution of the nonlinear integral equation (1) under some conditions on k(t s u) as mentioned above.
A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions.
The highlight of the viscosity approximation method is that the desired limit point is not only a solution of a nonlinear problem but a unique solution of a classical monotone variational inequality.
Thus given a finite-Einstein-energy solution of the equation □ƒ + λƒ3 = 0 on M0 (λ ⩾ 0) there exist unique solutions ƒ± of the free wave equation which approach ƒ in the Minkowski energy norm as x0 → ± ∞, and every finite-Einstein-energy solution of the free wave equation is of the form ƒ+ (or ƒ−) for a unique solution ƒ of the nonlinear equation.
In this technique, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and furthermore these sequences converge uniformly and quadratically to the unique solution of the given nonlinear differential equations [1].
