Sentence examples for utilizing the fixed point from inspiring English sources

They dealt with many boundary conditions given in the literature in a unified way by utilizing the fixed point index theory in cones.

Finally, by utilizing the fixed point index theory, we establish the multiplicity of positive solutions for the case 0 < λ < λ ∗.

By utilizing the fixed point theorems, we derive many existence results concerning the mild solutions for problem (2) under different assumptions on the nonlinear term and nonlocal term.

Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type.

Furthermore, we utilize the fixed point theorem and Leray-Schauder degree theory to study the existence of solutions for boundary value problem (1.3) (in Section 3).

Very recently, Burton has successfully utilized the fixed point theorem to investigate the stability for deterministic systems in [22]; Luo in [23] and Appleby in [22] have applied this valuable method into dealing with the stability for stochastic differential equations.

By virtue of some inequalities associated with Green's function, without the assumption of the nonnegativity of f i, we utilize the fixed point index theory to establish our main results.

To our knowledge, there are still very few to utilize the fixed point results on mixed monotone operators with perturbation to study the existence and uniqueness of a positive solution for nonlinear fractional differential equation boundary value problems.

The proof of Lemma 3 follows by utilizing the fixed-point principle of Schauder-Tychonoff.

In this section, we first formulate and prove sufficient conditions for the existence and uniqueness of a mild solution of system (1)–(3) by utilizing the fixed-point theory.

The method we utilized for the analysis is the sequential technique and regularization, and the existence of positive solutions is obtained by the fixed point theorem.