Sentence examples for a solution to the integral from inspiring English sources

In fact, a solution to the integral equations in (12) is a continuous function satisfying the conditions in (12).

Based on the Fredholm theory, we show the existence and uniqueness of a solution to the integral system (2.56).

Clearly, every solution to the problem is a solution to the integral matrix equation Y t) = exp bigl t-t_{j} mu A(t_{j}) bigl t-t_{j} mut^{t}_{t_{j}} exp bigl((t-s)mu A(t_{j}) bigr) mubigl(A(s)-A(t_{j})bigr) Y_{j, ds.

Now, all the assumptions of Theorem 2.1 are satisfied with ψ ( t ) = t, for all t ∈ R +, u ( x, y ) = d ( x, y ), and L = 0. Therefore, f and g have a common fixed point, that is, a solution to the integral equation (3.1).

Then Ψ u is a solution to the integral equation ∫ Ω ϕ 0 Ψ u u 2 d x = ∫ Ω ( ϕ 0 + φ u ) φ u u 2 d x. and Ψ u ≤ 0. Since the map Φ : u ∈ H 0 1 → φ u ∈ H 0 1 is continuously differentiable, we define the reduced C 1 functional K ( u ) : = I ( u, φ u ).

We say that a continuous interval function (X: [a,b] to K_{C}(mathbb {R})) is a solution to the interval fractional integral equation (3.1) if it satisfies equation (3.1).

In this paper, we establish a solution to the following integral equation: u ( t ) = ∫ 0 T G ( t, s ) f ( s, u ( s ) ) d s for all  t ∈ [ 0, T ], (1).

Indeed, if u ∈ C ( [ 0, a ], R F ) is a solution to the fuzzy integral equation u ( t ) = I q f ( t, u ( t ) ). and f ( t, u ( t ) ) ∈ C ( ( 0, a ], R F ) ∩ L 1 ( ( 0, a ), R F ), then u is also a solution to Eq. (1).

Conditions (3.1) and (3.12) are more general than the analogues in the previous literature, and theorems related to those conditions have a more general character because of the parameter s and arbitrary coefficients q, p. In this section, we will use Corollary 3.23 to show that there is a solution to the following integral equation: x ( t ) = int_{0}^{T} G bigl( t,r,x ( r ) bigr),dr.

Next we shall establish an important result related to the existence of a solution to the interval-valued integral equation (4), and then we can obtain the existence of a solution to the Cauchy type problem (1 - 2) under certain conditions.

The existence and uniqueness of a solution to the corresponding boundary integral system will be given in Section 3. The potential theory and Fredholm theory will be used to prove our main results.