Sentence examples for methods we prove from inspiring English sources

Using the fixed point and direct methods, we prove the Hyers Ulam stability of the following Cauchy Jensen functional equation.

To design efficient numerical methods, we prove that Helmholtz free energy density is a concave function with respect to the temperature under certain physical conditions.

In this paper, using the fixed point and direct methods, we prove the Hyers-Ulam stability of the following additive functional equation.

Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces.

In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.

Using fixed point methods, we prove the stability and superstability of -ternary additive, quadratic, cubic, and quartic homomorphisms in -ternary rings for the functional equation, for each.

Using the contradiction method, we prove the lemma.

Using the Picard iteration method, we prove the existence of equation (1.1) in Sect.

Using the periodic unfolding method, we prove some convergence results and describe the homogenized problems.

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of the functional equation.

Using fixed point method, we prove the Hyers-Ulam stability of derivations on proper Jordan CQ*-algebras.