Generalizations of the Lax-Milgram Theorem

Abstract

We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations.

Existence Theorem for Noninstantaneous Impulsive Evolution Equations

“…In [13], Drivaliaris and Yannakakis proved the following variant form of generalized Lax-Milgram eorem [14]. Theorem 1.…”

Existence of Solutions of a Class of Nonlinear Singular Equations in Lorentz Spaces

“…First part is mainly using ideas of An et al [3] and Drivaliaris and Yannakakis [10]. The proof of the estimate is inspired by Napoli and Mariani [18].…”

Extensions of the Lax–Milgram theorem to Hilbert $$C^*$$-modules

“…It has several applications in various disciplines such as partial differential equations, integral operators, fractional differential equations, differential operators, system of boundary value problems, linear system of equations, etc; see for example [9,13]. This theorem has been generalized by several mathematicians in linear and nonlinear forms; see for example [17,19,5,21]. A version of the Lax-Milgram theorem for Hilbert C-modules and C-sesquilinear forms is given in [8].…”

“…A version of the Lax-Milgram theorem for Hilbert C-modules and C-sesquilinear forms is given in [8]. In this note, we adopt the version of the Lax-Milgram theorem presented in [5] and generalize the theorem in the setting of Hilbert C * -modules over W * -modules and over C * -algebras of compact operators.…”