Characterization of the Solvability of Generalized Constrained Variational Equations

Abstract

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.

1. Introduction

It is common knowledge that in studying differential problems, variational methods have come to be essential. For instance, in [1], for a certain impulsive differential equation, its variational structure as well as the existence and uniqueness of a weak solution is shown. Specifically, given 𝑇>0, 𝑓∈𝐿²(0,𝑇), 𝜆>−𝜋²/𝑇², 𝑡₀=0<𝑡₁<⋯<𝑡_𝑘<𝑡_𝑘+1=𝑇, and 𝑑₁,…,𝑑_𝑘∈ℝ, the impulsive linear problem−𝑥^′′(𝑡)(𝑡)(𝑡)(),𝑥(0)(𝑇)+𝜆𝑥=𝑓,𝑡∈0,𝑇=𝑥=0,Δ𝑥^′𝑡_𝑗=𝑑_𝑗,𝑗=1,…,𝑘, (1.1) where Δ𝑥^′(𝑡_𝑗)∶=𝑥^′(𝑡⁺_𝑗)−𝑥′(𝑡⁻_𝑗), is considered, and as a direct application of the classical Lax-Milgram theorem ([2, Corollary 5.8]), possibly the most popular variational tool, it is proven that there exists a unique 𝑥₀∈𝐻¹₀(0,𝑇) such that𝑦∈𝐻¹₀(0,𝑇)⟹^𝑇₀𝑥^′₀𝑦^′+𝜆𝑥₀𝑦=^𝑇₀𝑓𝑦−^𝑘_𝑗=1𝑑_𝑗𝑦𝑡_𝑗. (1.2) In this paper we replace the Lax-Milgram theorem with a characterization of the unique solvability of a certain type of variational equation with constraints. Such a constrained variational equation arises naturally; for instance, when in the variational formulation of an elliptic partial differential equation, its essential boundary constraints are treated as constraints. This result allows us to consider problems without data functions in a Hilbert space, which is beyond the control of the classical theory ([3, section II.1 Proposition 1.1], [4, Lemma  4.67]). In particular, it extends the class of the said impulsive linear problems admitting a weak solution, since we prove that for any 𝑓∈𝐿^𝑝(0,𝑇), where 1<𝑝<∞ and not necessarily 𝑝=2, and for all 𝜆>−𝜆_𝑝,𝑇, for some 𝜆_𝑝,𝑇>0 only depending on 𝑝 and 𝑇, the impulsive equation has one and only one solution.

To help understand specifically the sort of constrained variational inequality under consideration, we have selected a simple but illustrative model problem, which will adequately serve our purposes related to linear impulsive problems. For 𝑇>0, 𝑣₀,𝑣_𝑇∈ℝ, and 𝑓∈𝐿²(0,𝑇), let us consider the corresponding Poisson’ equation with nonhomogeneous Dirichlet boundary conditions−𝑥^′′(),𝑥(0)=𝑓in0,𝑇=𝑣₀,𝑥(𝑇)=𝑣_𝑇, (1.3) whose usual weak formulation isfind𝑥₀∈𝐻¹(0,𝑇)suchthat𝑥∈𝐻¹₀(0,𝑇)⟹^𝑇₀𝑥^′₀𝑥^′=^𝑇₀𝑥𝑓𝑥,₀(0)=𝑣₀,𝑥₀(𝑇)=𝑣_𝑇. (1.4) By imposing essential boundary conditions weakly, we can equivalently write that variational formulation as a constrained variational equation. To be more concrete, let 𝑋 be the Sobolev space 𝐻¹(0,𝑇), let 𝑌∶=ℝ², and let 𝑎∶𝑋×𝑋→ℝ and 𝑏∶𝑋×𝑌→ℝ be the continuous bilinear forms given by𝑎(𝑥,𝑥)∶=^𝑇₀𝑥^′𝑥^′,(),𝑏(𝑥(0)(𝑇𝑥,𝑥∈𝑋𝑥,𝑦∶=,𝑥))⋅𝑦,𝑥∈𝑋,𝑦∈𝑌 (1.5) (“⋅” stands for the Euclidean inner product in ℝ²), and let 𝑥^∗₀∶𝑋→ℝ and 𝑦^∗₀∶𝑌→ℝ be the continuous linear functionals defined by𝑥^∗₀(𝑥)∶=_Ω(),𝑦𝑓𝑥,𝑥∈𝑋_∗0𝑦∶=𝑣₀,𝑣_𝑇⋅𝑦,𝑦∈𝑌. (1.6) Since𝐾∶=𝑥∈𝑋∶𝑦∈𝑌⟹𝑏𝑥,𝑦=0=𝐻¹₀(Ω), (1.7) then the weak formulation above leads us to consider the following variational equation with constraints: find 𝑥₀∈𝑋 such that𝑥∈𝐾⟹𝑎𝑥₀,𝑥=𝑥^∗₀(𝑥),𝑦∈𝑌⟹𝑏𝑥₀,𝑦=𝑦^∗₀𝑦. (1.8) With regard to this problem, a more abstract approach has been adopted: let 𝑋 and 𝑌 be the Hilbert spaces, let 𝑥^∗₀∶𝑋→ℝ and 𝑦^∗₀∶𝑌→ℝ be continuous linear functionals, let 𝑎∶𝑋×𝑋→ℝ and 𝑏∶𝑋×𝑌→ℝ be continuous bilinear forms, and let 𝐾∶={𝑥∈𝑋∶𝑦∈𝑌⇒𝑏(𝑥,𝑦)=0}. Under these assumptions,find𝑥₀𝑥∈𝑋suchthat𝑥∈𝐾⟹𝑎₀,𝑥=𝑥^∗₀(𝑥)𝑦∈𝑌⟹𝑏𝑥₀,𝑦=𝑦^∗₀𝑦. (1.9) The known classical results are nothing more than sufficient conditions guaranteeing that such a variational equation with constraints has a solution. However, when the function data do not belong to Hilbert spaces, these results do not apply. For this reason we study a more general type of variational equation with constraints, whose most important particular case relies on this construction: given a reflexive Banach space 𝑋, normed spaces 𝑌, 𝑍, and 𝑊, continuous bilinear forms 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ, and continuous linear functionals 𝑦^∗₀∶𝑌→ℝ and 𝑤^∗₀∶𝑊→ℝ, denoting𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0, (1.10) find, if possible, 𝑥₀∈𝑋 such that𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥₀,𝑤=𝑤^∗₀(𝑤). (1.11) In Theorem 2.2 of Section 2 we characterize when this constrained variational equation, in effect a more general variational inequality with constraints in the framework of locally convex spaces, admits a solution. The particular normed case is discussed in Sections 3 and 4. In the first one, the version for normed spaces of Theorem 2.2 leads to an easier statement and, under hypotheses of uniqueness, it is possible to obtain a stability estimation for the solution. The reflexive case, which immediately follows and extends the classical known results, is illustrated with a non-Hilbertian data example. Section 4 completes the normed space setting, and as an application of Theorem 2.2 we also obtain analogous results for dual normed spaces. Finally, Section 5 is concerned with solving weakly the aforementioned kind of impulsive differential equation, generalizing the linear results in [1] to the reflexive context.

From now on, we assume that all the spaces are real, although our results are equally valid and easily adapted to the complex case.

2. Variational Inequalities with Constraints in Lcs

We first discuss a characterization of the existence of solutions to some constrained variational inequalities in the general setting of locally convex spaces. In order to state our main result, Theorem 2.2, the Hahn-Banach theorem, is required. Although there is a long history of using the Hahn-Banach theorem, recently a fine reformulation of this fundamental result has been developed in [5, 6] (see Proposition 2.1 below). It is known as the Hahn-Banach-Lagrange theorem and has encountered numerous applications in different branches of the mathematical analysis (see [5–8]). Let us recall that if 𝑋 is a real vector space, a function 𝑆∶𝑋→ℝ is sublinear provided that it is subadditive and positively homogeneous. For such an 𝑆, if 𝐶 is a nonempty convex subset of a vector space, then 𝑗∶𝐶→𝑋 is said to be 𝑆-convex if𝑥,𝑦∈𝐶,0<𝑡<1⟹𝑆𝑗𝑡𝑥+(1−𝑡)𝑦−𝑡𝑗(𝑥)−(1−𝑡)𝑗𝑦≤0. (2.1) Finally, a convex function 𝑘∶𝐶→ℝ∪{∞} is proper when there exists 𝑥∈𝐶 with 𝑘(𝑥)<∞.

Proposition 2.1 ([6, Theorem 2.9]). Let 𝑋 be a nontrivial vector space, and let 𝑆∶𝑋→ℝ be a sublinear function. Assume in addition that 𝐶 is a nonempty convex subset of a vector space, 𝑘∶𝐶→ℝ∪{∞} is a proper convex function, and 𝑗∶𝐶→𝑋 is 𝑆-convex. Then there exists a linear functional 𝐿∶𝑋→ℝ such that 𝐿≤𝑆,inf_𝐶𝐿∘𝑗+𝑘=inf_𝐶𝑆∘𝑗+𝑘. (2.2)

Now we state the main result of this section, along the lines of [5–7]. To this end, some notations are required. For two real vector spaces 𝑋 and 𝑌, a bilinear form 𝑎∶𝑋×𝑌→ℝ, and 𝑥∈𝑋, 𝑦∈𝑌, and 𝑎(⋅,𝑦) stands for the linear functional on 𝑋𝑥∈𝑋⟼𝑎𝑥,𝑦∈ℝ (2.3) and 𝑎(𝑥,⋅) for the analogous linear functional on 𝑌. In addition, given a real Hausdorff locally convex space 𝑋, we will write 𝑋^∗ to denote its dual space (continuous linear functionals on 𝑋).

The characterization is stated as follows.

Theorem 2.2. Let 𝑋 be a real Hausdorff locally convex space such that its dual space 𝑋^∗ is also a real Hausdorff locally convex space. Suppose that 𝑌 and 𝑊 are real vector spaces, 𝐶 and 𝐷 are convex subsets of 𝑌 and 𝑊, respectively, with (0,0)∈𝐶×𝐷, Φ∶𝐶→ℝ∪{−∞} and Ψ∶𝐷→ℝ∪{−∞} are concave functions such that Φ(0)≥0 and Ψ(0)≥0, and 𝑎∶𝑋×𝑌→ℝ and 𝑐∶𝑋×𝑊→ℝ are bilinear forms satisfying that 𝑦,𝑤∈𝐶×𝐷⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋^∗. (2.4) Then, thereexists𝑥₀^∗∗∈𝑋^∗∗𝑦suchthat𝑦∈𝐶⟹Φ≤𝑥₀^∗∗𝑎(𝑤)⋅,𝑦𝑤∈𝐷⟹Ψ≤𝑥₀^∗∗(𝑐(⋅,𝑤)) (2.5) if, and only if, there exists a continuous seminorm 𝑝∶𝑋^∗→ℝ so that 𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝑝𝑎⋅,𝑦+𝑐(⋅,𝑤). (2.6) Furthermore, if one of these equivalent conditions holds, then it is possible to choose 𝑥₀^∗∗ and 𝑝 with 𝑥₀^∗∗≤𝑝.

Proof. We can assume without loss of generality that 𝑋 is nontrivial, which is exactly the same as 𝑋^∗ being nontrivial, thanks to the Hahn-Banach theorem.
Let us first assume that (2.6) is true for some continuous seminorm 𝑝∶𝑋^∗→ℝ. The Hahn-Banach-Lagrange theorem (Proposition 2.1) applies, with the sublinear function 𝑆=𝑝, the 𝑆-convex mapping 𝑗∶𝐶×𝐷→𝑋^∗ defined as 𝑗𝑦,𝑤∶=𝑎⋅,𝑦+𝑐(⋅,𝑤),𝑦,𝑤∈𝐶×𝐷, (2.7) and the proper convex function 𝑘∶𝐶×𝐷→ℝ∪{∞} given by 𝑘𝑦,𝑤∶=−Φ𝑦−Ψ(𝑤),𝑦,𝑤∈𝐶×𝐷, (2.8) obtaining thus that there exists a linear functional 𝐿∶𝑋^∗→ℝ such that, on the one hand, 𝐿≤𝑝, (2.9) and therefore 𝐿=𝑥₀^∗∗∈𝑋^∗∗ for some 𝑥₀^∗∗∈𝑋^∗∗, and on the other hand, satisfyies inf_{𝑦,𝑤∈𝐶×𝐷}𝑥₀^∗∗𝑎⋅,𝑦+𝑐(⋅,𝑤)−Φ𝑦−Ψ(𝑤)=inf_{(𝑦,𝑤)∈𝐶×𝐷}𝑝𝑎⋅,𝑦+𝑐(⋅,𝑤)−Φ𝑦−Ψ(𝑤). (2.10) But we are assuming that inf_{(𝑦,𝑤)∈𝐶×𝐷}𝑝𝑎⋅,𝑦+𝑐(⋅,𝑤)−Φ𝑦−Ψ(𝑤)≥0. (2.11) Hence inf_{(𝑦,𝑤)∈𝐶×𝐷}𝑥₀^∗∗𝑎⋅,𝑦+𝑐(⋅,𝑤)−Φ𝑦−Ψ(𝑤)≥0, (2.12) that is, 𝑦,𝑤∈𝐶×𝐷⟹Φ(𝑧)+Ψ(𝑤)≤𝑥₀^∗∗𝑎⋅,𝑦+𝑐(⋅,𝑤). (2.13) Since (0,0)∈𝐶×𝐷 and Φ(0)≥0 and Ψ(0)≥0, taking in this last inequality (𝑦,0)∈𝐶×𝐷 yields 𝑦∈𝐶⟹Φ𝑦≤𝑥₀^∗∗𝑎⋅,𝑦, (2.14) while for (0,𝑤)∈𝐶×𝐷 it implies 𝑤∈𝐷⟹Ψ(𝑤)≤𝑥₀^∗∗(𝑐(⋅,𝑤)), (2.15) and we have proven (2.5) as we wish.
And conversely, if 𝑥₀^∗∗∈𝑋^∗∗ satisfies (2.5), then 𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝑥₀^∗∗𝑎⋅,𝑦+𝑐(⋅,𝑤), (2.16) so for the continuous seminorm 𝑝 on 𝑋^∗𝑝𝑥^∗∶=𝑥₀^∗∗𝑥^∗,𝑥^∗∈𝑋^∗, (2.17) we have that Φ𝑦+Ψ(𝑤)≤𝑝𝑎⋅,𝑦+𝑐(⋅,𝑤). (2.18)
In any case we have stated the inequality 𝑥₀^∗∗≤𝑝.

As we can see, all the topological assumptions fall on 𝑋. Thus, 𝐶 and 𝐷 are nothing more than convex sets, and there is no topological assumption on them, not even that they are closed. In particular, no continuity is supposed for Φ or Ψ.

Let us also note that the condition 𝑥₀^∗∗≤𝑝, which seems to be irrelevant, will entail in the normed case a control of the norm ‖𝑥₀^∗∗‖.

Let us also emphasize that Theorem 2.2 captures the essence of the Hahn-Banach theorem from a variational standpoint. Theorem 2.2 extends the Lax-Milgram-type result given in [8, Theorem  1.2]. But the latter in turn is an equivalent reformulation of the Hahn-Banach theorem (Proposition 2.1), as shown in [8, Theorem  3.1]. Because the Hahn-Banach theorem and the Hahn-Banach-Lagrange theorem are equivalent results, Theorem 2.2 is nothing more than an equivalent version of the Hahn-Banach theorem.

3. Constrained Variational Equations in Reflexive Banach Spaces

Both in this section and the next one we turn to the study of constrained variational equations, only in the case of 𝑋 being a normed space and considering a kind of locally convex topology that, in some sense, satisfies a compactness property, which actually ensures that the solutions 𝑥₀^∗∗∈𝑋^∗∗ of the constrained variational inequality (2.5) belong to 𝑋. To be more precise, in this section we fix the norm topology in 𝑋 and deduce that 𝑥₀^∗∗∈𝑋 in an obvious way when 𝑋 is reflexive; that is, its closed unit ball 𝐵_𝑋 is weakly compact ([2, Theorem  3.17]), which suffices for the applications in Section 5. In the next section we assume that 𝑋 is a dual Banach space, that is, 𝑋=𝐸^∗ for some normed space 𝐸, endowed with its weak-star topology 𝑤(𝐸^∗,𝐸), in which, by the way, its closed unit ball is closed ([2, Theorem 3.16]). Continuing with the contents of this section, we provide an estimation of the norm of the solution only in terms of the data. We also generalize to the reflexive framework the classical Hilbertian characterization [3, 4] of those constrained problems (1.9) that admit a solution, indeed obtaining a proper extension of a result in the reflexive context that follows from [9, Theorem 2.1], developed for mixed variational formulations of elliptic boundary value problems.

Thus it is that we consider a real normed space 𝑋, equipped with its norm topology, and the topology in 𝑋^∗ is taken to be that associated with the canonical norm of 𝑋^∗. In this way we obtain the said estimation of the norm of a solution to the variational inequality with constraints (2.5). Moreover, we replace the existence of the continuous seminorm 𝑝 in Theorem 2.2 with that of a nonnegative constant.

Corollary 3.1. Suppose that 𝑋 is a real normed space, 𝑌 and 𝑊 are real vector spaces, 𝐶 and 𝐷 are convex subsets of 𝑌 and 𝑊, respectively, (0,0)∈𝐶×𝐷, Φ∶𝐶→ℝ∪{−∞} and Ψ∶𝐷→ℝ∪{−∞} are concave functions such that Φ(0)≥0 and Ψ(0)≥0. If 𝑎∶𝑋×𝑌→ℝ and 𝑐∶𝑋×𝑊→ℝ are bilinear forms so that 𝑦,𝑤∈𝐶×𝐷⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋^∗, (3.1) then the constrained variational problem find𝑥₀^∗∗∈𝑋^∗∗(𝑧)suchthat𝑦∈𝐶⟹Φ≤𝑥₀^∗∗𝑎(𝑤)⋅,𝑦𝑤∈𝐷⟹Ψ≤𝑥₀^∗∗(𝑐(⋅,𝑤)) (3.2) is solvable if, and only if, for some 𝜌≥0𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝜌𝑎⋅,𝑦+𝑐(⋅,𝑤). (3.3) Moreover, when these statements hold and there exists (𝑦,𝑤)∈𝐶×𝐷 satisfying 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0, then min𝑥₀^∗∗∶𝑥₀^∗∗∈𝑋^∗∗,𝑦∈𝐶⟹Φ𝑦≤𝑥₀^∗∗𝑎(𝑤)⋅,𝑦𝑤∈𝐷⟹Ψ≤𝑥₀^∗∗(𝑐(=⋅,𝑤))sup_{(𝑦,𝑤)∈𝐶×𝐷,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}Φ𝑦+Ψ(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)₊. (3.4)

Proof. The equivalence between (3.2) and (3.3) clearly follows from Theorem 2.2 for the real Hausdorff locally convex space 𝑋 endowed with its norm topology and considering in its dual space 𝑋^∗ the dual norm topology, and from the fact that if 𝑝∶𝑋^∗→ℝ is a continuous seminorm, then it is bounded above by a suitable positive multiple of the norm.
Let us finally suppose that (3.2) or equivalently (3.3) holds. In order to prove (3.4) provided that there exists (𝑦,𝑤)∈𝐶×𝐷 such that 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0, let us start by fixing an arbitrary element 𝑥₀^∗∗ in 𝑋^∗∗ for which (3.2) is valid. Then, 𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝑥₀^∗∗𝑎⋅,𝑦+𝑐(⋅,𝑤). (3.5) Hence, if 𝛼∶=sup_{(𝑦,𝑤)∈𝐶×𝐷,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}Φ(𝑦)+Ψ(𝑤)𝑎(⋅,𝑦)+𝑐(⋅,𝑤)₊ (3.6) then 𝑥₀^∗∗≥𝛼, (3.7) and so 𝛼<∞. (3.8) But in addition we have that 𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝛼𝑎⋅,𝑦+𝑐(⋅,𝑤), (3.9) since for (𝑦,𝑤)∈𝐶×𝐷 with 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0 it clearly holds, and when 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)=0, it suffices to make use of the fact that (3.2) is valid to arrive at the same conclusion. In summary, the continuous seminorm 𝑝∶𝑋^∗→ℝ given for each 𝑥^∗∈𝑋^∗ by 𝑝𝑥^∗∶=𝛼𝑥^∗ (3.10) satisfies (2.6); therefore, Theorem 2.2 implies the existence of 𝑥₀^∗∗∈𝑋^∗∗ for which (3.2) is valid and 𝑥₀^∗∗≤𝑝, that is, 𝑥₀^∗∗≤𝛼, (3.11) which together with (3.7) finally yields (3.4).

Of course, when in Corollary 3.1 we additionally assume that 𝑋 is reflexive, then the existence of 𝜌≥0 such that𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝜌𝑎⋅,𝑦+𝑐(⋅,𝑤) (3.12) is equivalent to the existence of a solution in 𝑋 to the constrained variational inequality, that is,thereexists𝑥₀𝑦𝑥∈𝑋suchthat𝑦∈𝐶⟹Φ≤𝑎₀,(𝑤)𝑥,𝑦𝑤∈𝐷⟹Ψ≤𝑐₀,𝑤. (3.13) Next we focus our effort on proving that Corollary 3.1, with the additional hypothesis of the reflexivity of 𝑋, provides a result, Theorem 3.8, generalizing the classical Hilbertian theory.

Lemma 3.2. Assume that 𝑋 is a real normed space, 𝑌 and 𝑊 are real vector spaces, and 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ are bilinear forms. If one writes 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐∶={𝑥∈𝑋∶𝑐(𝑥,⋅)=0} (3.14) and supposes that 𝑦∈𝐾_𝑏,𝑤∈𝑊⟹𝑎(⋅,𝑧),𝑐(⋅,𝑤)∈𝑋^∗, (3.15) then 𝑦∈𝐾_𝑏,𝑤∈𝑊⟹𝑎⋅,𝑦_∣𝐾𝑐≤𝑎⋅,𝑦+𝑐(⋅,𝑤). (3.16)

Proof. Let 𝑦∈𝐾_𝑏 and 𝜀>0. Since 𝑎(⋅,𝑦)_∣𝐾𝑐∈𝑋^∗, choose 𝑥₀∈𝐾_𝑐 so that 𝑥₀𝑎𝑥=1,₀,𝑦>𝑎⋅,𝑦_∣𝐾𝑐−𝜀. (3.17) Then, given 𝑤∈𝑊, we have that 𝑎⋅,𝑦_∣𝐾𝑐<𝑎𝑥₀𝑥,𝑦+𝜀=𝑎₀,𝑦+𝑐𝑥₀,𝑤+𝜀𝑥₀∈𝐾_𝑐≤𝑥₀𝑎(=𝑎(⋅,𝑦+𝑐⋅,𝑤)+𝜀⋅,𝑦+𝑐⋅,𝑤)+𝜀, (3.18) and the announced inequality follows from the arbitrariness of 𝜀>0.

In the next result we establish the first characterization of the solvability of a variational equation with constraints.

Theorem 3.3. Let 𝑋 be a real reflexive Banach space, let 𝑌,𝑍, and 𝑊 be real normed spaces, and let 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ be bilinear forms with 𝑎 and 𝑐 being continuous. Let 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗, and write 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐{()},𝑅∶=𝑥∈𝑋∶𝑐𝑥,⋅=0_𝑤∗0∶=𝑥∈𝑋∶𝑐(𝑥,⋅)=𝑤^∗₀. (3.19) Then, the corresponding constrained variational equation admits a solution; that is, thereexists𝑥₀∈𝑋suchthat𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥₀,𝑤=𝑤^∗₀(𝑤), (3.20) if, and only if, 𝑅_𝑤^∗₀≠∅,∀𝑥∈𝑅_𝑤^∗₀thereexists𝛿≥0suchthat𝑦∈𝐾_𝑏⟹𝑦^∗₀𝑦−𝑎𝑥,𝑦≤𝛿𝑎⋅,𝑦_∣𝐾𝑐. (3.21)(3.22) In addition, if one of these equivalent conditions is valid and there exists 𝑦∈𝐾_𝑏 with 𝑎(⋅,𝑦)_∣𝐾𝑐≠0, then one can take 𝑥₀∈𝑋 in (3.20) with 𝑥₀=min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖. (3.23)

Proof. Let us begin by stating (3.20)⇒(3.22). Let 𝑥₀ be a solution to the variational equation with constraints (3.20). Then, in particular, 𝑅_𝑤^∗₀=𝑥₀+𝐾_𝑐. (3.24) Thus, given 𝑥∈𝑅_𝑤^∗₀ there exists 𝑥₁∈𝐾_𝑐 such that 𝑥=𝑥₀+𝑥₁, so for all 𝑦∈𝐾_𝑏𝑦^∗₀𝑦−𝑎𝑥,𝑦=−𝑎𝑥₁≤𝑥,𝑦₁𝑎⋅,𝑦_∣𝐾𝑐, (3.25) and we have shown (3.22).
To conclude, we prove the converse (3.22)⇒(3.20). Let 𝑦∈𝐾_𝑏 and 𝑤∈𝑊, and let 𝑥∈𝑅_𝑤^∗₀, whose existence guarantees (3.22). Then 𝑦^∗₀𝑦+𝑤^∗₀(𝑤)=𝑦^∗₀𝑦−𝑎𝑥,𝑦+𝑎𝑥,𝑦+𝑐(𝑥,𝑤)𝑥∈𝑅_𝑤^∗₀≤𝑦^∗₀𝑦+‖𝑥𝑎(𝑎−𝑎𝑥,𝑦‖⋅,𝑦+𝑐⋅,𝑤)≤𝛿⋅,𝑦_∣𝐾𝑐+‖𝑥𝑎((≤(‖𝑥𝑎(.‖⋅,𝑦+𝑐⋅,𝑤)by3.22)𝛿+‖)⋅,𝑦+𝑐⋅,𝑤)byLemma3.2 (3.26) Therefore, there exists 𝜌∶=𝛿+‖𝑥‖≥0 such that 𝑦∈𝐾_𝑏,𝑤∈𝑊⟹𝑦^∗₀𝑦+𝑤^∗₀(𝑤)≤𝜌𝑎⋅,𝑦+𝑐(⋅,𝑤), (3.27) so Corollary 3.1 (in combination with the reflexivity of 𝑋) for 𝐶∶=𝐾_𝑏, 𝐷∶=𝑊, Φ∶=𝑦^∗₀, and Ψ∶=𝑤^∗₀ ensures, on the one hand, that the variational system (3.20) has a solution, hence stating the equivalence between (3.20) and (3.22), and, on the other hand, that if one of these conditions holds, then min𝑥₀∶𝑥₀∈𝑅_𝑤^∗₀,𝑦^∗_0∣𝐾𝑏=𝑎𝑥₀,⋅_∣𝐾𝑏=sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀(𝑦)+𝑤^∗₀(𝑤)𝑎(⋅,𝑦)+𝑐(⋅,𝑤)₊=sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤). (3.28) So, for establishing (3.23) and concluding the proof, it suffices to state the equality sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)=min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖. (3.29) Suppose, on the one hand, that 𝑥∈𝑅_𝑤^∗₀, 𝑦∈𝐾_𝑏, and 𝑤∈𝑊, with 𝑎(⋅,𝑦)_∣𝐾𝑐≠0, which in view of Lemma 3.2 implies that 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0. Then 𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(=𝑦⋅,𝑦+𝑐⋅,𝑤)^∗0𝑦−𝑎𝑥,𝑦𝑎(+𝑎()⋅,𝑦+𝑐⋅,𝑤)𝑥,𝑦+𝑐𝑥,𝑤𝑎⋅,𝑦+𝑐(⋅,𝑤)𝑥∈𝑅_𝑤^∗₀≤𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎(+‖𝑥‖≤𝑦⋅,𝑦+𝑐⋅,𝑤)_∗0𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖byLemma3.2. (3.30) Thus sup_{𝑦,𝑤∈𝐾𝑏×𝑊𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)≤sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖=sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖, (3.31) and therefore, as 𝑥∈𝑅_𝑤^∗₀ is arbitrary, sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)≤inf_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖. (3.32) And, on the other hand, if 𝑥∈𝑅_𝑤^∗₀, 𝑦∈𝐾_𝑏, and 𝑤∈𝑊 satisfy 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0 but 𝑎(⋅,𝑦)_∣𝐾𝑐=0, 𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(=𝑦⋅,𝑦+𝑐⋅,𝑤)^∗0𝑦−𝑎𝑥,𝑦𝑎(+𝑎()⋅,𝑦+𝑐⋅,𝑤)𝑥,𝑦+𝑐𝑥,𝑤𝑎⋅,𝑦+𝑐(⋅,𝑤)𝑥∈𝑅_𝑤^∗₀=𝑎𝑥,𝑦+𝑐(𝑥,𝑤)𝑎(≤‖𝑥‖.⋅,𝑦+𝑐⋅,𝑤) (3.33) Hence sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0,𝑎(⋅,𝑦)𝑐∣𝐾=0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(≤‖𝑥‖,⋅,𝑦+𝑐⋅,𝑤) (3.34) and thus sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0,𝑎(⋅,𝑦)𝑐∣𝐾=0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)≤inf_{𝑥∈𝑅𝑤∗0}‖𝑥‖≤inf_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖. (3.35) Therefore, it follows from (3.32), Lemma 3.2, and (3.35) that sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎⋅,𝑦+𝑐(⋅,𝑤)≤inf_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖. (3.36) But thanks to (3.28) we can choose 𝑥₀∈𝑅_𝑤^∗₀ with 𝑦^∗_0∣𝐾𝑏=𝑎(𝑥₀,⋅)_∣𝐾𝑏 in such a way that 𝑥₀=sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(,⋅,𝑦+𝑐⋅,𝑤) (3.37) so finally 𝑥₀=min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎(⋅,𝑦)_∣𝐾𝑐+‖𝑥‖=sup_{(𝑦,𝑤)∈𝐾𝑏×𝑊,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}𝑦^∗₀𝑦+𝑤^∗₀(𝑤)𝑎(,⋅,𝑦+𝑐⋅,𝑤) (3.38) and (3.29) is proven.

Remark 3.4. In Theorem 3.3 we do not need to suppose that 𝑎 and 𝑐 are continuous: it suffices to impose 𝑦,𝑤∈𝐾_𝑏×𝑊⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋^∗, (3.39) as seen in its proof. Since for our applications the bilinear forms 𝑎 and 𝑐 are continuous, for the sake of simplicity, we have assumed this hypothesis. However, in Theorem 4.2 we impose these less restrictive assumptions, in a more general setting.

Let us note that according to Corollary 3.1 (or [8, Corollary  1.3]) we have, with 𝑋 being reflexive,𝑅_𝑤^∗₀≠∅ (3.40) if, and only if, there exists 𝜇≥0 such that𝑤∈𝑊⟹𝑤^∗₀(𝑤)≤𝜇‖𝑐(⋅,𝑤)‖. (3.41)

In connection with uniqueness we have the following elementary characterization, which establishes the equivalence of such uniqueness with that of the corresponding homogeneous variational equation with constraints.

Lemma 3.5. Let one make the same assumptions and use the same notations as in Theorem 3.3. If the variational equation with constraints (3.20) has a solution, then it is unique if, and only if, 𝑥∈𝐾_𝑐,𝑎(𝑥,⋅)_∣𝐾𝑏=0⟹𝑥=0. (3.42)

Proof. If 𝑎 satisfies the nondegeneracy condition (3.42), then for any 𝑥∈𝑅_𝑤^∗₀ we have 𝑅_𝑤^∗₀=𝑥+𝐾_𝑐, so if (3.20) has two solutions 𝑥₁ and 𝑥₁, then for some 𝑥₂,𝑥₂∈𝐾_𝑐, 𝑥₁=𝑥+𝑥₂ and 𝑥₁=𝑥+𝑥₂; hence 𝑦∈𝐾_𝑏⟹𝑎𝑥₂−𝑥₂,𝑦=0. (3.43) By hypothesis it follows that 𝑥₂=𝑥₂; that is, 𝑥₁=𝑥₁.
And conversely, if there exists 𝑥₀∈𝐾_𝑐, 𝑥₀≠0, such that 𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=0, (3.44) then given a solution 𝑥 of (3.20), 𝑥+𝑥₀ is also a solution, which is different than 𝑥.

Hypotheses more restrictive than those of Theorem 3.3 imply uniqueness of the solution and simplify the control of the norm of the solution.

Corollary 3.6. Let 𝑋 be a real reflexive Banach space, let 𝑌, 𝑍, and 𝑊 be real normed spaces, let 𝑤^∗₀∈𝑊^∗, and let 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ be bilinear forms such that 𝑎 and 𝑐 are continuous. Let one take 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐∶={𝑥∈𝑋∶𝑐(𝑥,⋅)=0} (3.45) and suppose that 𝑥∈𝐾_𝑐,𝑎(𝑥,⋅)_∣𝐾𝑏=0⟹𝑥=0 (3.46) and that there exist constants 𝛼,𝜇>0 with 𝑦∈𝐾_𝑏⟹𝛼𝑦≤𝑎⋅,𝑦_∣𝐾𝑐,𝑤∈𝑊⟹𝑤^∗₀(𝑤)≤𝜇‖𝑐(⋅,𝑤)‖. (3.47)(3.48) Then, for each 𝑦^∗₀∈𝑌^∗, the corresponding variational equation with constraints admits one and only one solution; that is, thereexistsaunique𝑥₀∈𝑋suchthat𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥₀,𝑤=𝑤^∗₀(𝑤). (3.49) Furthermore, if one defines 𝑅_𝑤^∗₀∶=𝑥∈𝑋∶𝑐(𝑥,⋅)=𝑤^∗₀, (3.50) then the a priori estimate ‖𝑥₀𝑦‖≤^∗0𝛼+‖𝑎‖1+𝛼min_{𝑥∈𝑅𝑤∗0}‖𝑥‖≤𝑦^∗₀𝛼+‖𝑎‖1+𝛼𝜇 (3.51) is valid for the norm of the solution 𝑥₀.

Proof. Corollary 3.1 (or [8, Corollary  1.3]) and Theorem 3.3, together with conditions (3.47) and (3.48), imply the existence of a solution whose uniqueness follows from Lemma 3.5. Besides, we deduce from Theorem 3.3 that for the solution 𝑥₀, the identity 𝑥₀=min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖ (3.52) holds. Finally, we have by condition (3.47) that min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖=min_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑦≠0}𝑦^∗₀𝑦−𝑎𝑥,𝑦𝑎⋅,𝑦_∣𝐾𝑐+‖𝑥‖≤inf_{𝑥∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑦≠0}𝑦^∗₀𝑦+‖𝑎‖‖𝑥‖𝑦𝛼𝑦+‖𝑥‖=𝑦^∗₀𝛼+‖𝑎‖1+𝛼inf_{𝑥∈𝑅𝑤∗0}‖𝑥‖, (3.53) and by Corollary 3.1 (or [8, Corollary  1.3]) and (3.48) that inf_{𝑥∈𝑅𝑤∗0}‖𝑥‖=min_𝑥∈𝑅‖𝑥‖=sup_{𝑤∈𝑊,𝑐(⋅,𝑤)≠0}𝑤^∗₀(𝑤)‖𝑐(⋅,𝑤)‖≤𝜇, (3.54) and thus we have the announced bound.

If we assume a condition on 𝑐 stronger than (3.48), the so-called inf-sup or Babuška-Brezzi condition (see [10–14] for some recent developments); that is, there exists 𝛾>0 such that𝑤∈𝑊⟹𝛾‖𝑤‖≤‖𝑐(⋅,𝑤)‖, (3.55) then we have the stability estimate𝑥₀≤𝑦^∗₀𝛼+‖𝑎‖1+𝛼𝑤^∗₀𝛾 (3.56) for the solution.

Taking into account that when 𝑋=𝑌, 𝑍=𝑊, and 𝑏=𝑐 conditions (3.42) and (3.47) are satisfied if 𝑎 is coercive on 𝐾_𝑏×𝐾_𝑏, we deduce the following immediate consequence, which is well known (see [3, section II.1 Proposition 1.1] and [4, Lemma  4.67]) in the particular case of 𝑋 and 𝑍 being Hilbert spaces.

Corollary 3.7. Suppose that 𝑋 and 𝑍 are reflexive real Banach spaces, 𝑥^∗₀∈𝑋^∗, 𝑧^∗₀∈𝑍^∗, and 𝑎∶𝑋×𝑋→ℝ and 𝑏∶𝑋×𝑍→ℝ are continuous bilinear forms. Suppose in addition that, taking 𝐾_𝑏∶={𝑥∈𝑋∶𝑏(𝑥,⋅)=0},𝑅_𝑧^∗₀∶=𝑥∈𝑋∶𝑏(𝑥,⋅)=𝑧^∗₀, (3.57) there exists 𝛼>0 such that 𝑥∈𝐾_𝑏⟹𝛼‖𝑥‖²≤𝑎(𝑥,𝑥) (3.58) and that 𝑅_𝑧^∗₀≠∅. Then thereexistsaunique𝑥₀∈𝑋suchthat𝑥∈𝐾_𝑏⟹𝑎𝑥₀,𝑥=𝑥^∗₀(𝑥),𝑧∈𝑍⟹𝑏𝑥₀,𝑧=𝑧^∗₀(𝑧). (3.59) Besides, the solution 𝑥₀ satisfies the a priori estimate: 𝑥₀≤𝑥^∗₀𝛼+‖𝑎‖1+𝛼min_{𝑥∈𝑅𝑧∗0}‖𝑥‖. (3.60) In particular, if there exists 𝛾>0 such that 𝑧∈𝑍⟹𝛾‖𝑧‖≤‖𝑏(⋅,𝑧)‖, (3.61) then for the norm of 𝑥₀ the following estimation: 𝑥₀≤𝑥^∗₀𝛼+‖𝑎‖1+𝛼𝑧^∗₀𝛾 (3.62) holds.

In Theorem 3.3 and Lemma 3.5 we drive a characterization when the variational equation with constraints (3.20) admits a unique solution, for two fixed functionals 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗. The known particular cases in the literature are stated for arbitrary functionals 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗. Now, we derive a characterization along those lines. The particular case 𝑋=𝑌, 𝑍=𝑊, and 𝑏=𝑐 was stated in [15, Corollary 2.7].

Theorem 3.8. Let 𝑋 be a real reflexive Banach space, let 𝑌, 𝑍, and 𝑊 be real normed spaces, and let 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ be bilinear forms with 𝑎 and 𝑐 being continuous. Let 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐∶={𝑥∈𝑋∶𝑐(𝑥,⋅)=0}. (3.63) Then, for all 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗, thereexistsoneandonlyone𝑥₀∈𝑋suchthat𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥₀,𝑤=𝑤^∗₀(𝑤) (3.64) if, and only if, 𝑥∈𝐾_𝑐,𝑎(𝑥,⋅)_∣𝐾𝑏=0⟹𝑥=0 (3.65) and there exist 𝛼,𝛾>0 so that 𝑦∈𝐾_𝑏⟹𝛼𝑦≤𝑎⋅,𝑦_∣𝐾𝑐‖𝑤‖≤‖𝑐(.,𝑤∈𝑊⟹𝛾⋅,𝑤)‖ (3.66)(3.67) In addition, if one of these equivalent conditions is satisfied, one has the following stability estimate: 𝑥₀≤𝑦^∗₀𝛼+‖𝑎‖1+𝛼𝑤^∗₀𝛾. (3.68)

Proof. In view of Corollary 3.6 and Lemma 3.5 we deduce, provided that (3.65), (3.66), and (3.67) hold, that for all 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ the constrained variational problem (3.64) admits a unique solution, whose norm satisfies estimation (3.68).
And conversely, suppose that for arbitrary 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ there exists a unique solution of (3.64). Then obviously we have that (3.65) holds, from Lemma 3.5. Moreover, since, in particular, for all 𝑤^∗₀∈𝑊^∗ there exists 𝑥₀∈𝑋 with 𝑤^∗₀=𝑐(𝑥₀,⋅); then, the uniform boundedness theorem and Corollary 3.1 (or [8, Corollary 1.3]) imply (3.67). In a similar way we can arrive at (3.66): taking 𝑤^∗₀=0∈𝑊^∗ we have that for all 𝑦^∗₀∈𝑌^∗ there exists 𝑥₀∈𝐾_𝑐 such that 𝑎(𝑥₀,⋅)_|𝐾𝑏=𝑦^∗_0|𝐾𝑏, which according to the Hahn-Banach theorem, the uniform boundedness theorem and Corollary 3.1 (or [8, Corollary 1.3]) are exactly (3.66).

Remark 3.9. We emphasize, in view of Remark 3.4, that in this proper extension of the Lax-Milgram theorem we just need to assume that 𝑦,𝑤∈𝐾_𝑏×𝑊⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋^∗, (3.69) and not necessarily that 𝑎 and 𝑐 are continuous. Theorem 4.3 is stated in these terms, and in a more general framework.

Let us again take up the elliptic boundary value problem considered in Introduction, in this case a more general one with non-Hilbertian data. We make use of our results with that elliptic boundary value problem for which the classical theory in the Hilbert framework does not apply. Thus we show how Theorem 3.8 (or Theorem 3.3) increases the class of elliptic boundary value problems for which it is known that the corresponding constrained variational equation derived from weakly imposing boundary conditions has a unique solution. But before doing so, we give a technical result, interesting in itself, and recall some common notations. For 𝑇>0 and 1<𝑝<∞,‖⋅‖_𝑝 stands for the usual norm in the Lebesgue space 𝐿^𝑝(0,𝑇). In addition, the standard norm in the Sobolev space 𝑊^1,𝑝(0,𝑇) is given by‖𝑥‖_1,𝑝∶=‖𝑥‖^𝑝_𝑝+𝑥^′𝑝_𝑝^1/𝑝,𝑥∈𝑊^1,𝑝(0,𝑇), (3.70) and the inherited norm on the subspace 𝑊₀^1,𝑝(0,𝑇) is equivalent to the norm |⋅|_1,𝑝 defined as |𝑥|_1,𝑝∶=𝑥^′_𝑝,𝑥∈𝑊₀^1,𝑝(0,𝑇), (3.71) thanks to the well-known Poincaré inequality ([16, Theorem  6.30]), which asserts that there exists a constant 𝑐_𝑝,𝑇>0, depending only on 𝑇 and 𝑝, in such a way that𝑥∈𝑊₀^1,𝑝(0,𝑇)⟹‖𝑥‖_𝑝≤𝑐_𝑝,𝑇|𝑥|_1,𝑝. (3.72) In fact, it is easy to check that we can take𝑐_𝑝,𝑇=𝑇𝑝^1/𝑝, (3.73) and for 𝑝=2 the optimal 𝑐_2,𝑇 is given by𝑐_2,𝑇=𝑇𝜋 (3.74) (see [17, Section  1.1.3]). As usual, we denote by 𝑊^−1,𝑝′(0,𝑇) the dual space of 𝑊₀^1,𝑝(0,𝑇), where 𝑝^′ is the conjugate exponent of 𝑝 defined through the relation 1/𝑝+1/𝑝^′=1. The dual norm of ‖⋅‖_1,𝑝, when restricted to 𝑊₀^1,𝑝(0,𝑇), will be denoted by ‖⋅‖_−1,𝑝′ and that of |⋅|_1,𝑝 by |⋅|_−1,𝑝′.

The following result is a particular case of [18, Theorem  4], although we include it because we provide an explicit inf-sup constant, which will allow us to improve such a result in the concrete case to be used in Section 5.

Proposition 3.10. Let 𝑇>0 and 1<𝑝<∞, with conjugate exponent 𝑝^′, and consider the continuous bilinear form 𝑎₀∶𝑊₀^1,𝑝(0,𝑇)×𝑊^1,𝑝′₀(0,𝑇)→ℝ given for each 𝑥∈𝑊₀^1,𝑝(0,𝑇) and 𝑦∈𝑊^1,𝑝′₀(0,𝑇) by 𝑎₀𝑥,𝑦∶=^𝑇₀𝑥^′𝑦^′. (3.75) Then 𝑎₀ satisfies the inf-sup condition. More specifically, 𝑦∈𝑊^1,𝑝′₀()⟹10,𝑇1+𝑇^−1/𝑝′𝑦_1,𝑝′≤𝑎₀⋅,𝑦_−1,𝑝′. (3.76)

Proof. It is sufficient to deal with 𝑦∈𝐶^∞₀(0,𝑇). The description of the dual space 𝑊^−1,𝑝′(0,𝑇) of 𝑊₀^1,𝑝(0,1) (see for instance [2, Proposition  8.14]) guarantees that for some 𝑦₀∈𝐿^𝑝′(0,𝑇), 𝑥∈𝑊₀^1,𝑝(0,𝑇)⟹𝑎₀𝑥,𝑦=^𝑇₀𝑥^′𝑦₀𝑎₀⋅,𝑦_−1,𝑝′=𝑦_0𝑝′. (3.77) Then 𝑥∈𝑊₀^1,𝑝(0,𝑇)⟹^𝑇₀𝑥^′𝑦^′−𝑦₀=0, (3.78) and so 𝑥∈𝐶^∞₀(0,𝑇)⟹^𝑇₀𝑥^′𝑦^′−𝑦₀=0. (3.79) But taking into account that 𝑥^′∶𝑥∈𝐶^∞₀(0,𝑇)=𝑧∈𝐶^∞₀(0,𝑇)∶^𝑇₀𝑧=0, (3.80) if 𝑥₀∈𝐶^∞₀(0,𝑇) with ∫^𝑇₀𝑥₀=1, then 𝑥^′∶𝑥∈𝐶^∞₀(0,𝑇)=𝑤−^𝑇₀𝑤𝑥₀∶𝑤∈𝐶^∞₀(0,𝑇), (3.81) so (3.79) is equivalent to 𝑤∈𝐶^∞₀(0,𝑇)⟹^𝑇₀𝑤−^𝑇₀𝑥𝑥₀𝑦^′−𝑦₀=0, (3.82) or in other words, 𝑤∈𝐶^∞₀(0,𝑇)⟹^𝑇₀𝑤𝑦^′−𝑦₀=^𝑇₀𝑤^𝑇₀𝑥₀𝑦^′−𝑦₀, (3.83) that is, 𝑦′−𝑦₀=^𝑇₀𝑥₀𝑦^′−𝑦₀. (3.84) Therefore, we have that for some 𝜆∈ℝ𝑦^′−𝑦₀=𝜆. (3.85) Hence, integrating and noting that 𝑦∈𝐶^∞₀(0,𝑇), 1𝜆=−𝑇^𝑇₀𝑦₀, (3.86) and thus, as a consequence of the triangular and the Holdër inequalities, 𝑦_1,𝑝′≤𝑦_0𝑝′+|𝜆|𝑇≤1+𝑇^−1/𝑝′𝑦_0𝑝′, (3.87) so 11+𝑇^−1/𝑝′𝑦_1,𝑝′≤𝑦_0𝑝′=𝑎₀⋅,𝑦_−1,𝑝′. (3.88)

The corresponding bilinear form 𝑎∶𝑊₀^1,𝑝(Ω)×𝑊^1,𝑝′₀(Ω)→ℝ defined for each (𝑥,𝑦) in 𝑊₀^1,𝑝(Ω)×𝑊^1,𝑝′₀(Ω) as𝑎𝑥,𝑦∶=_Ω∇𝑥⋅∇𝑦, (3.89) when Ω is a convex bounded plane polygon domain also satisfies the inf-sup condition (see [19, Theorem  2.1]). However we are just interested in the 1D case, since it is the one to be used in the applications of Section 5.

Now we are in a position to return to the mentioned example.

Example 3.11. Assume that 𝑇>0, 𝑣₀,𝑣_𝑇∈ℝ, 1<𝑝<∞, 𝑓∈𝐿^𝑝(Ω), and consider the elliptic boundary value problem −𝑥^′′(),𝑥(0)=𝑓in0,𝑇=𝑣₀,𝑥(𝑇)=𝑣_𝑇, (3.90) which does not admit the classical treatment ([3, section II.1 Proposition  1.1], [4, Lemma  4.67]), since the involved spaces are not Hilbert, except for 𝑝=2. However, Theorems 3.3 and 3.8 apply. Indeed, if we multiply equation −𝑥^′′=𝑓 in (0,𝑇) by a test function 𝑦∈𝑊^1,𝑝′₀(0,𝑇)(1/𝑝+1/𝑝^′=1), and integrate by parts, then we obtain the weak formulation of problem (3.90), that in the particular case 𝑝=2 coincides with the classical one: find𝑥₀∈𝑊^1,𝑝(0,𝑇)suchthat𝑦∈𝑊^1,𝑝′₀(0,𝑇)⟹^𝑇₀𝑥^′₀𝑦^′=^𝑇₀𝑥𝑓𝑦₀(0)=𝑣₀,𝑥₀(𝑇)=𝑣_𝑇. (3.91) Let us consider the real reflexive Banach space 𝑋∶=𝑊^1,𝑝(0,𝑇) (3.92) and the normed spaces 𝑌∶=𝑊^1,𝑝′(0,𝑇),𝑍∶=𝑊∶=ℝ², (3.93) with ℝ² endowed with its ‖⋅‖₁ norm, the continuous linear functionals 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ defined for each 𝑦∈𝑌 and 𝑤∈𝑊, respectively, as 𝑦^∗₀𝑦∶=^𝑇₀𝑤𝑓𝑦,_∗0𝑤₁,𝑤₂∶=𝑣₀,𝑣_𝑇⋅𝑤, (3.94)(3.95) and the continuous bilinear forms 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ given by 𝑎𝑥,𝑦∶=^𝑇₀𝑥^′𝑦^′,,𝑏𝑧𝑥∈𝑋,𝑦∈𝑌𝑦,₁,𝑧₂𝑦(0)(𝑇,𝑐𝑤∶=,𝑦)⋅𝑧,𝑦∈𝑌,𝑧∈𝑍𝑥,₁,𝑤₂∶=(𝑥(0),𝑥(𝑇))⋅𝑤,(𝑥∈𝑋,𝑤∈𝑊). (3.96) Now 𝐾_𝑏=𝑊^1,𝑝′₀(0,𝑇), so the variational formulation above is nothing more than the variational equation with the following constraints: find𝑥₀∈𝑋suchthat𝑦∈𝐾_𝑏⟹𝑦^∗₀𝑦=𝑎𝑥₀,𝑦,𝑤∈𝑊⟹𝑤^∗₀(𝑤)=𝑐𝑥₀,𝑤. (3.97) In order to prove that this problem has a unique solution, let us check (3.65), (3.66), and (3.67). The first of these conditions follows from the duality 1/𝑝+1/𝑝^′=1 and Proposition 3.10, which imply 𝑥∈𝑊₀^1,𝑝()⟹10,𝑇1+𝑇^−1/𝑝|𝑥|_1,𝑝≤𝑎₀(𝑥,⋅)_−1,𝑝, (3.98) which, in view of the fact that 𝐾_𝑐=𝑊₀^1,𝑝(0,𝑇), is equivalent to 𝑥∈𝐾_𝑐⟹11+𝑇^−1/𝑝|𝑥|_1,𝑝≤𝑎(𝑥,⋅)_{∣𝐾𝑏−1,𝑝}, (3.99) and the equivalence of the norm |⋅|_1,𝑝 and the usual one in 𝑊₀^1,𝑝(0,𝑇) yields (3.65).
To prove that condition (3.66) is satisfied, we fix 𝑦∈𝐾_𝑏=𝑊^1,𝑝′₀(0,𝑇) and apply Proposition 3.10 using equivalence of the norm |⋅|_1,𝑝′ and the usual one in 𝑊^1,𝑝′₀(0,𝑇).
In order to conclude, let us deduce the inf-sup condition (3.67), more specifically, that there exists 𝛾_𝑝,𝑇>0, depending only on 𝑝 and 𝑇, such that 𝑤∈ℝ²⟹𝛾_𝑝,𝑇‖𝑤‖₁≤‖𝑐(⋅,𝑤)‖_{𝑊1,𝑝(0,𝑇)}^∗, (3.100) where ‖⋅‖_{𝑊1,𝑝(0,𝑇)}^∗ denotes the dual norm of ‖⋅‖_1,𝑝 in 𝑊^1,𝑝(0,𝑇)^∗, equivalently, inf_{𝑤∈ℝ2,‖𝑤‖1=1}‖𝑐(⋅,𝑤)‖_{𝑊1,𝑝(0,𝑇)}^∗≥𝛾_𝑝,𝑇. (3.101) Thus, let 𝑤=(𝑤₁,𝑤₂)∈ℝ² with ‖𝑤‖₁=1. On the one hand, let us assume that 𝑤₁,𝑤₂≥0. Then we define the function 𝑥₀∈𝑊^1,𝑝(0,𝑇)𝑥₀(𝑡)∶=1,(𝑡∈[0,𝑇]), (3.102) for which 𝑥_01,𝑝=1𝑇^1/𝑝. (3.103) Then ‖𝑐(⋅,𝑤)‖_{𝑊1,𝑝(0,𝑇)}^∗≥𝑐𝑥₀,𝑤𝑇^1/𝑝=𝑤₁+𝑤₂𝑇^1/𝑝=1𝑇^1/𝑝. (3.104) On the other hand, if 𝑤₁<0<𝑤₂, for 𝑥₀(𝑡)2∶=−1+𝑇𝑡,(𝑡∈[0,𝑇]), (3.105) we have that 𝑥₀∈𝑊^1,𝑝(0,𝑇) and 𝑥_01,𝑝≤𝑇^1/𝑝3^𝑝+2𝑇^𝑝1/𝑝, (3.106) and so, ‖𝑐(⋅,𝑤)‖_{𝑊1,𝑝(0,𝑇)}^∗≥𝑐𝑥₀,𝑤𝑥_01,𝑝≥−𝑤₁+𝑤₂𝑇^1/𝑝(3^𝑝+(2/𝑇)^𝑝)^1/𝑝=1𝑇^1/𝑝(3^𝑝+(2/𝑇)^𝑝)^1/𝑝. (3.107) As the cases 𝑤₁,𝑤₂≤0 and 𝑤₂<0<𝑤₁ follow from the preceding ones, we arrive at the inf-sup condition for 𝑐 (3.101) with 𝛾_𝑝,𝑇1∶=min𝑇^1/𝑝,𝑇^−1/𝑝3^𝑝+2𝑇^{𝑝−1/𝑝}>0. (3.108)
Therefore the variational equation with constraints (3.97), that is, the weak formulation obtained by imposing weakly the boundary conditions of (3.90), admits a unique solution 𝑥₀∈𝑊^1,𝑝(0,𝑇), for whose norm the stability estimation 𝑥₀≤𝜃_𝑝,𝑇𝑓_𝑝+𝑣₀,𝑣_𝑇∞ (3.109) is valid, where 𝜃_𝑝,𝑇>0 depends only on 𝑝 and 𝑇.

With the applications of Section 5 in mind, we are just interested in the 1D case, yet as we commented previously, in [19, Theorem  2.1] an analogue of Proposition 3.10 is established for a convex bounded polygon Ω⊂ℝ². In such a case, it is possible to prove that the boundary value problem−▵𝑥=𝑓inΩ,𝑥=𝑔onΓ (3.110) admits a constrained variational formulation similar to that of Example 3.11, where Γ is the topological boundary of Ω, 1<𝑝<∞, 𝑓∈𝐿^𝑝(Ω), and 𝑔∈𝑊^{1/𝑝′,𝑝}(Γ), the space of traces on Γ of functions in the Sobolev space 𝑊^1,𝑝(Ω).

Next we show that the reflexivity of 𝑋 is essential in Theorem 3.8.

Proposition 3.12. If 𝑋 is a nonreflexive real normed space, then there exist real normed spaces 𝑌, 𝑍, and 𝑊, continuous bilinear forms 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ, and 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ satisfying the assumptions in Theorem 3.8, except obviously the reflexivity of 𝑋, but for any 𝑥₀∈𝑋 the constrained variational equation 𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥₀,𝑤=𝑤^∗₀(𝑤) (3.111) does not hold.

Proof. It suffices to take 𝑍∶=𝑋, 𝑌∶=𝑊∶=𝑋^∗, 𝑎, 𝑏 and 𝑐 the corresponding duality pairings, 𝑦^∗₀∈𝑌^∗ any continuous linear functional on 𝑌 and 𝑤^∗₀∶=𝑥₀^∗∗∈𝑋^∗∗⧵𝑋. Then, since 𝐾_𝑏=0 and 𝐾_𝑐=0, the first condition in the characterization holds. The second one is also satisfied, because 𝑤∈𝑊⟹‖𝑤‖=‖𝑐(⋅,𝑤)‖. (3.112) However, although for all 𝑥₀∈𝑋𝑦∈𝐾_𝑏⟹𝑎𝑥₀,𝑦=𝑦^∗₀𝑦, (3.113) for each 𝑥₀∈𝑋, 𝑐𝑥₀,⋅≠𝑤^∗₀, (3.114) since 𝑐(⋅,𝑥₀)=𝑥₀∈𝑋 but 𝑤^∗₀=𝑥₀^∗∗∉𝑋.

The following result, stated in [9, Theorem 2.1], has a certain relation with Theorem 3.8 and ensures that the mixed variational formulation of some boundary value problems is uniquely solvable, when the data functions belong to reflexive spaces, unlike the classical Babuška-Brezzi theory ([3, 11, 20, 21]), developed only for the Hilbert framework: if 𝑋, 𝑌, 𝑍, and 𝑊 are real reflexive Banach spaces and 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ are continuous bilinear forms, and we write𝐾_𝑏,𝐾∶=𝑦∈𝑌∶𝑏𝑦,⋅=0_𝑐∶={𝑥∈𝑋∶𝑐(𝑥,⋅)=0}. (3.115) Then, for all 𝑦^∗₀∈𝑌^∗, 𝑤^∗₀∈𝑊^∗ there exists a unique (𝑥₀,𝑧₀)∈𝑋×𝑍 such that𝑦∈𝑌⟹𝑦^∗₀𝑦=𝑎𝑥₀,𝑦+𝑏𝑦,𝑧₀,𝑤∈𝑊⟹𝑤^∗₀(𝑤)=𝑐𝑥₀,𝑤 (3.116) if, and only if,𝑥∈𝐾_𝑐,𝑎(𝑥,⋅)_∣𝐾𝑏=0⟹𝑥=0, (3.117) and there exist 𝛼,𝛽,𝛾>0 such that𝑦∈𝐾_𝑏⟹𝛼𝑦≤𝑎(⋅,𝑦)_∣𝐾𝑐,‖𝑧‖≤‖𝑏(,‖𝑤‖≤‖𝑐(.𝑧∈𝑍⟹𝛽⋅,𝑧)‖𝑤∈𝑊⟹𝛾⋅,𝑤)‖ (3.118) This result establishes the existence of one and only one solution of the constrained variational equation (3.64) under consideration in Theorem 3.8: if the required assumptions in [9, Theorem  2.1] hold, then in particular we deduce the existence of a unique solution for the variational equation with constraints (3.64), since if (𝑥₀,𝑧₀)∈𝑋×𝑍 is the unique solution of the mixed problem (3.116), then 𝑥₀ is the unique solution of the constrained variational equation (3.64). However, [9, Theorem  2.1] has some additional hypotheses (continuity of 𝑏 and its inf-sup condition), in addition to the reflexivity of 𝑌, 𝑍, and 𝑊, so they are independent statements. Indeed, let 𝑋 and 𝑌 be the sequential space ℓ₂, endowed with its usual Hilbertian norm, let {𝑒_𝑛}_𝑛≥1 be the usual basis of ℓ₂, and let 𝑍∶=span{𝑒_2𝑛∶𝑛≥1}, the subspace of ℓ₂ endowed with its inherited norm, and 𝑊∶=0. We define the bilinear forms 𝑎∶𝑋×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ as follows: given (𝑥,𝑦)∈𝑋×𝑌,𝑎𝑥,𝑦∶=^∞_𝑛=1𝑥(𝑛)𝑦(2𝑛−1), (3.119) so 𝑎 is continuous and ‖𝑎‖=1; for 𝑦∈𝑌 and 𝑛≥1,𝑏𝑦,𝑒_2𝑛∶=𝑛𝑦(2𝑛) (3.120) and it is extended to 𝑌×𝑍 by linearity, so 𝑏 is not continuous; obviously, since 𝑊=0, then 𝑐=0. In particular (3.67) holds. Now𝐾_𝑏=𝑦∈ℓ₂(),𝐾∶𝑛≥1⟹𝑦2𝑛=0_𝑐=ℓ₂. (3.121) Let us check condition (3.65). If 𝑥∈𝐾_𝑐 and 𝑎(𝑥,⋅)_|𝐾𝑏=0, then for 𝑛≥1, 𝑒_2𝑛−1∈𝐾_𝑏, so𝑥(𝑛)=𝑎𝑥,𝑒_2𝑛−1=0, (3.122) and therefore 𝑥=0. Finally, condition (3.66) is also satisfied, since for each 𝑦∈𝐾_𝑏,𝑦=^∞_𝑛=1𝑦(2𝑛−1)𝑒_2𝑛−1, (3.123) and thus defining𝑦∶=^∞_𝑛=1𝑦(2𝑛−1)𝑒_𝑛, (3.124) then𝑦=,𝑎=𝑦𝑦𝑦,𝑦², (3.125) and since ‖𝑎‖=1, we arrive at𝑦=𝑎⋅,𝑦. (3.126) Thus Theorem 3.8 guarantees that the corresponding constrained variational equation (3.64) has a unique solution, unlike [9, Theorem  2.1], which does not apply.

Let us point out another difference between [9, Theorem  2.1] and our results: that theorem gives a characterization for all 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗, whereas in Theorem 3.3 we have the advantage that we drive a characterization when the constrained variational problem (3.64) is uniquely solvable, but for two fixed functionals 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗.

4. A Word on Dual Banach Spaces

As we commented in Section 3, we now consider a real dual normed space, endowed with its weak-star topology, and its dual space with its norm topology. The main results, Theorems 4.2 and 4.3, extend the corresponding ones in the reflexive case, Theorems 3.3 and 3.8. Yet these results allow us to obtain our applications in the next section, so we have decided to introduce them first. Furthermore, for the sake of the simplicity in the exposition, we consider it appropriate.

Since the results are analogously stated as those in Section 3, we merely enunciate them.

We begin with the analogue to Corollary 3.1, which also follows from Theorem 2.2, but with a different locally convex space.

Corollary 4.1. Assume that 𝑋 is a real normed space, 𝑌 and 𝑊 are real vector spaces, 𝐶 and 𝐷 are convex subsets of 𝑌 and 𝑊, respectively, with (0,0)∈𝐶×𝐷, and Φ∶𝐶→ℝ∪{−∞} and Ψ∶𝐷→ℝ∪{−∞} are concave functions such that Φ(0)≥0 and Ψ(0)≥0. If in addition 𝑎∶𝑋^∗×𝑌→ℝ and 𝑐∶𝑋^∗×𝑊→ℝ are bilinear forms satisfying 𝑦,𝑤∈𝐶×𝐷⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋, (4.1) then thereexists𝑥^∗₀∈𝑋^∗(𝑧)𝑥suchthat𝑦∈𝐶⟹Φ≤𝑎^∗₀,(𝑤)𝑥,𝑦𝑤∈𝐷⟹Ψ≤𝑐^∗₀,𝑤, (4.2) if, and only if, there exists 𝜌≥0 such that 𝑦,𝑤∈𝐶×𝐷⟹Φ𝑦+Ψ(𝑤)≤𝜌𝑎⋅,𝑦+𝑐(⋅,𝑤). (4.3) Furthermore, if one of these equivalent conditions holds and there exists (𝑦,𝑤)∈𝐶×𝐷 so that 𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0, then min𝑥^∗₀∶𝑥^∗₀∈𝑋^∗,𝑦∈𝐶⟹Φ𝑦≤𝑎𝑥^∗₀(𝑤)𝑥,𝑦𝑤∈𝐷⟹Ψ≤𝑐^∗₀=,𝑤sup_{(𝑦,𝑤)∈𝐶×𝐷,𝑎(⋅,𝑦)+𝑐(⋅,𝑤)≠0}Φ(𝑦)+Ψ(𝑤)𝑎(⋅,𝑦)+𝑐(⋅,𝑤)₊. (4.4)

Corollaries 3.1 and 4.1 are different results, although the latter is more general than the particular case of the former when 𝑋 is reflexive. For this reason, the next two statements are extensions of Theorems 3.3 and 3.8.

Theorem 4.2. Let 𝑋, 𝑌, 𝑍, and 𝑊 be real normed spaces, and let 𝑎∶𝑋^∗×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋^∗×𝑊→ℝ be bilinear forms. Let one write 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐∶=𝑥^∗∈𝑋^∗∶𝑐𝑥^∗,⋅=0, (4.5) and assume that 𝑦,𝑤∈𝐾_𝑏×𝑊⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋. (4.6) If in addition 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ and 𝑅_𝑤^∗₀∶=𝑥^∗∈𝑋^∗∶𝑐𝑥^∗,⋅=𝑤^∗₀, (4.7) then thereexists𝑥^∗₀∈𝑋^∗suchthat𝑦∈𝐾_𝑏⟹𝑎𝑥^∗₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥^∗₀,𝑤=𝑤^∗₀(𝑤), (4.8) if, and only if, 𝑅_𝑤^∗₀≠∅∀𝑥^∗∈𝑅_𝑤^∗₀thereexists𝛿≥0suchthat𝑦∈𝐾_𝑏⟹𝑦^∗₀𝑦−𝑎𝑥^∗,𝑦≤𝛿𝑎⋅,𝑦_∣𝐾𝑐. (4.9) Moreover, when these conditions are satisfied and there exists 𝑦∈𝐾_𝑏 such that 𝑎(⋅,𝑦)_∣𝐾𝑐≠0, then one can take a solution 𝑥^∗₀∈𝑋^∗ with 𝑥^∗₀=min_{𝑥∗∈𝑅𝑤∗0}sup_{𝑦∈𝐾𝑏,𝑎(⋅,𝑦)𝑐∣𝐾≠0}𝑦^∗₀𝑦−𝑎𝑥^∗,𝑦𝑎(⋅,𝑦)_∣𝐾𝑐+𝑥^∗. (4.10)

The version for any 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗, with the ingredient of uniqueness, is stated in these terms.

Theorem 4.3. Let 𝑋, 𝑌, 𝑍, and 𝑊 be real normed spaces, let 𝑎∶𝑋^∗×𝑌→ℝ, 𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋^∗×𝑊→ℝ be bilinear forms, and let 𝐾_𝑏∶=𝑦∈𝑌∶𝑏𝑦,⋅=0,𝐾_𝑐∶=𝑥^∗∈𝑋^∗∶𝑐𝑥^∗,⋅=0, (4.11) and assume that 𝑦∈𝐾_𝑏,𝑤∈𝑊⟹𝑎⋅,𝑦,𝑐(⋅,𝑤)∈𝑋. (4.12) Then, for all 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗, the corresponding constrained variational equation is uniquely solvable, that is, thereexistsoneandonlyone𝑥^∗₀∈𝑋^∗suchthat𝑦∈𝐾_𝑏⟹𝑎𝑥^∗₀,𝑦=𝑦^∗₀𝑦,𝑤∈𝑊⟹𝑐𝑥^∗₀,𝑤=𝑤^∗₀(𝑤) (4.13) if, and only if, 𝑥^∗∈𝐾_𝑐,𝑎𝑥^∗,⋅_∣𝐾𝑏=0⟹𝑥^∗=0 (4.14) and there exist 𝛼,𝛾>0 such that 𝑦∈𝐾_𝑏⟹𝛼𝑦≤𝑎⋅,𝑦_∣𝐾𝑐,𝑤∈𝑊⟹𝛾‖𝑤‖≤‖𝑐(⋅,𝑤)‖. (4.15) In addition, if one of these equivalent conditions holds, then 𝑥^∗₀≤𝑦^∗₀𝛼+‖𝑎‖1+𝛼𝑤^∗₀𝛾. (4.16)

As in the reflexive case, [22, Theorem  2.2] provides conditions that imply that the constrained variational equation (4.13) is uniquely solvable, but that result is more restrictive than Theorem 4.3, which is shown similarly as in Section 3 with [9, Theorem  2.1] and Theorem 3.8.

5. Application to Linear Impulsive Differential Equations

We now apply the same technique that motivated our results, as in Example 3.11; namely, the boundary conditions are weakly imposed. To be more concrete, we consider the impulsive differential problem in Section 1, previously studied in [1], but with nonhomogenous Dirichlet conditions and a non-Hilbertian data function: given 𝑓∈𝐿^𝑝(0,𝑇), with 1<𝑝<∞, 𝜆∈ℝ, 𝑡₀=0<𝑡₁<⋯<𝑡_𝑘<𝑡_𝑘+1=𝑇, and 𝑣₀,𝑣_𝑇,𝑑₁,…,𝑑_𝑘∈ℝ, the impulsive linear problem in question is−𝑥^′′(𝑡)(𝑡)(𝑡)(),𝑥(0)+𝜆𝑥=𝑓,𝑡∈0,𝑇=𝑣₀,𝑥(𝑇)=𝑣_𝑇,Δ𝑥^′𝑡_𝑗=𝑑_𝑗,𝑗=1,…,𝑘, (5.1) where Δ𝑥^′(𝑡_𝑗)=𝑥^′(𝑡⁺_𝑗)−𝑥^′(𝑡⁻_𝑗). Now the notion of a weak solution, along the lines of [1] (multiply by a test function 𝑦∈𝑊₀^1,𝑝′(0,𝑇), with 1/𝑝+1/𝑝′=1, integrate by parts and take into account the impulsive conditions Δ𝑥′(𝑡_𝑗)=𝑑_𝑗), is defined as𝑥₀∈𝑊^1,𝑝(0,𝑇)suchthat𝑦∈𝑊^1,𝑝′₀(0,𝑇)⟹^𝑇₀𝑥^′₀𝑦^′+𝜆𝑥₀𝑦=^𝑇₀𝑓𝑦−^𝑘_𝑗=1𝑑_𝑗𝑦𝑡_𝑗,𝑥₀(0)=𝑣₀,𝑥₀(𝑇)=𝑣_𝑇. (5.2) The classical Lax-Milgram theorem obviously does not apply in this context. For this very reason we express equivalently this variational formulation as a variational equation with constraints. To this end, two technical results are required. The first of them generalizes Proposition 3.10. Moreover, for the bilinear form under consideration we obtain something better than [18, Theorem  4], since 𝜆 admits certain negative values.

Proposition 5.1. Let 𝑇>0, 1<𝑝<∞ with conjugate exponent 𝑝^′ and 𝜆∈ℝ, and consider the continuous bilinear form 𝑎_𝜆∶𝑊₀^1,𝑝(0,𝑇)×𝑊^1,𝑝′₀(0,𝑇)→ℝ given for each 𝑥∈𝑊₀^1,𝑝(0,𝑇) and 𝑦∈𝑊^1,𝑝′₀(0,𝑇) by 𝑎_𝜆𝑥,𝑦∶=^𝑇₀𝑥^′𝑦^′+𝜆𝑥𝑦. (5.3) Then there exists 𝛿_𝑝,𝑇>0 such that if 𝜆>−𝛿_𝑝,𝑇, then 𝑎_𝜆 satisfies the inf-sup condition. More precisely, for all 𝜆>−𝛿_𝑝,𝑇, where 𝛿_𝑝,𝑇𝑝∶=^1/𝑝𝑝^′1/𝑝′𝑇²1+𝑇^−1/𝑝′, (5.4) there exists 𝛼_{𝑝,𝑇,𝜆}>0, depending only on 𝑝, 𝑇, and 𝜆, such that 𝑦∈𝑊^1,𝑝′₀(0,𝑇)⟹𝛼_{𝑝,𝑇,𝜆}𝑦_1,𝑝′≤𝑎_𝜆⋅,𝑦_−1,𝑝. (5.5)

Proof. The mentioned result in [18] gives us proof for the case 𝜆≥0. On the other hand, in Proposition 3.10 we have shown that 𝑎₀ (the bilinear form 𝑎_𝜆 with 𝜆=0) satisfies the inf-sup condition, obtaining a concise inf-sup constant. In fact we are going to make use of that result. To do this, let us note that 𝑎_𝜆=𝑎₀+𝜆𝑎, where 𝑎∶𝑊₀^1,𝑝(0,𝑇)×𝑊^1,𝑝′₀(0,𝑇)→ℝ is the continuous bilinear form defined for each (𝑥,𝑦)∈𝑊₀^1,𝑝(0,𝑇)×𝑊^1,𝑝′₀(0,𝑇) as 𝑎𝑥,𝑦∶=^𝑇₀𝑥𝑦. (5.6) But for all (𝑥,𝑦)∈𝑊₀^1,𝑝(0,𝑇)×𝑊^1,𝑝′₀(0,𝑇) we have, in view of (3.72), that 𝑎𝑥,𝑦≤‖𝑥‖_𝑝𝑦_𝑝′≤𝑐_𝑝,𝑇𝑐_𝑞,𝑇|𝑥|_1,𝑝𝑦_1,𝑝′≤𝑇²𝑝^1/𝑝𝑝^′1/𝑝′|𝑥|_1,𝑝𝑦_1,𝑝′, (5.7) so 𝑎⋅,𝑦_−1,𝑝≤𝑇²𝑝^1/𝑝𝑝^′1/𝑝′𝑦_1,𝑝′. (5.8) Taking into account this last inequality, Proposition 3.10, and the fact that 𝑎_𝜆=𝑎₀+𝜆𝑎, we conclude that for 𝑝0>𝜆>−^1/𝑝𝑝^′1/𝑝′𝑇²1+𝑇^−1/𝑝′ (5.9) it follows that 𝑦∈𝑊^1,𝑝′₀()⟹10,𝑇1+𝑇^−1/𝑝′𝑇+𝜆²𝑝^1/𝑝𝑝^′1/𝑝′𝑦_1,𝑝′≤𝑎₀⋅,𝑦_−1,𝑝′, (5.10) and the proof is complete.

The second technical result is very simple but will be useful to establish the stability of the solution.

Lemma 5.2. Let 𝜆∈ℝ, 𝑇>0, 𝑑₁,…,𝑑_𝑘∈ℝ, 1<𝑝<∞, 1/𝑝+1/𝑝^′=1, and suppose that 𝑓∈𝐿^𝑝(0,𝑇). Then the linear functional 𝑦^∗₀∶𝑊^1,𝑝′(0,𝑇)→ℝ given for each 𝑦∈𝑊^1,𝑝′(0,𝑇) by 𝑦^∗₀𝑦∶=^𝑇₀𝑓𝑦−^𝑘_𝑗=1𝑑_𝑗𝑦𝑡_𝑗 (5.11) is continuous, and in addition 𝑦^∗_{0−1,𝑝′}≤𝑇𝑝^′1/𝑝′𝑓_𝑝+𝑇^𝑘1/𝑝_𝑗=1𝑑_𝑗. (5.12)

Proof. Assume without loss of generality that 𝑦∈𝐶^∞₀(0,𝑇). Given 𝑗=1,…,𝑘 it follows from the Hölder inequality that 𝑦𝑡_𝑗=^𝑡𝑗₀𝑦^′≤𝑦_1,1≤𝑇^1/𝑝𝑦_1,𝑝′, (5.13) from which the announced inequality clearly follows.

Now we can show that the impulsive linear problem (5.1) admits a unique weak solution.

Theorem 5.3. Let 𝑇>0, 1<𝑝<∞, whose conjugate exponent is 𝑝^′, 𝑓∈𝐿^𝑝(0,𝑇), 𝑡₀=0<𝑡₁<⋯<𝑡_𝑘<𝑡_𝑘+1=𝑇, and 𝑣₀,𝑣_𝑇,𝑑₁,…,𝑑_𝑘∈ℝ. Then there exists 𝜆_𝑝,𝑇>0, specifically, 𝜆_𝑝,𝑇𝑝∶=^1/𝑝𝑝^′1/𝑝′𝑇²1min1+𝑇^−1/𝑝,11+𝑇^−1/𝑝′, (5.14) in such a way that if 𝜆>−𝜆_𝑝,𝑇, then the corresponding impulsive linear problem −𝑥^′′(𝑡)(𝑡)(𝑡)(),𝑥(0)+𝜆𝑥=𝑓,𝑡∈0,𝑇=𝑣₀,𝑥(𝑇)=𝑣_𝑇,Δ𝑥^′𝑡_𝑗=𝑑_𝑗,𝑗=1,…,𝑘, (5.15) admits a unique weak solution. Furthermore, there exists 𝜒_{𝑝,𝑇,𝜆}>0, depending only on 𝑝, 𝑇, and 𝜆, such that 𝑥_0𝑊0^1,𝑝_(0,𝑇)≤𝜒_{𝑝,𝑇,𝜆}𝑓_𝑝+^𝑘_𝑗=1𝑑_𝑗+𝑣₀,𝑣_𝑇∞. (5.16)

Proof. Our aim is to prove the existence of one and only one 𝑥₀∈𝑊^1,𝑝(0,𝑇) such that 𝑦∈𝑊^1,𝑝′₀(0,𝑇)⟹^𝑇₀𝑥^′₀𝑦^′+𝜆𝑥₀𝑦=^𝑇₀𝑓𝑦−^𝑘_𝑗=1𝑑_𝑗𝑦𝑡_𝑗,𝑥₀(0)=𝑣₀,𝑥₀(𝑇)=𝑣_𝑇 (5.17) which moreover depends continuously on the initial data. Yet this variational problem admits an equivalent reformulation as the variational problem with constraints find𝑥₀∈𝑋suchthat𝑦∈𝐾_𝑏⟹𝑦^∗₀𝑦=𝑎𝑥₀,𝑦,𝑤∈𝑊⟹𝑤^∗₀(𝑤)=𝑐𝑥₀,𝑤, (5.18) where 𝑋 is now the real reflexive Banach space 𝑋∶=𝑊^1,𝑝(0,𝑇) (5.19) and 𝑌, 𝑍, and 𝑊 are the real normed spaces 𝑌∶=𝑊^1,𝑝′(0,𝑇),𝑍∶=𝑊∶=ℝ², (5.20)ℝ² is equipped with its ‖⋅‖₁ norm, 𝑎∶𝑋×𝑌→ℝ,𝑏∶𝑌×𝑍→ℝ, and 𝑐∶𝑋×𝑊→ℝ are the continuous bilinear forms defined by 𝑎𝑥,𝑦∶=^𝑇₀𝑥^′𝑦^′,,𝑏𝑦(0)(𝑇,𝑐()(𝑥(0)(𝑇(),+𝜆𝑥𝑦𝑥∈𝑋,𝑦∈𝑌𝑦,𝑧∶=,𝑦)⋅𝑧,𝑦∈𝑌,𝑧∈𝑍𝑥,𝑤∶=,𝑥))⋅𝑤,𝑥∈𝑋,𝑤∈𝑊 (5.21) and 𝑦^∗₀∈𝑌^∗ and 𝑤^∗₀∈𝑊^∗ are the continuous linear functionals given as 𝑦^∗₀𝑦∶=^𝑇₀𝑓𝑦−^𝑘_𝑗=1𝑑_𝑗𝑦𝑡_𝑗,,𝑤𝑦∈𝑌_∗0(𝑤)∶=𝑣₀,𝑣_𝑇⋅𝑤,(𝑤∈𝑊). (5.22) The fact that the variational problem coincides with the constrained variational equation follows from the fact that 𝐾_𝑏=𝑊^1,𝑝′₀(0,𝑇).
Let us finally check (3.65), (3.66), and (3.67) in order to apply Theorem 3.8 and conclude the proof.
Since 𝐾_𝑏=𝑊^1,𝑝′₀(0,𝑇) and 𝐾_𝑐=𝑊₀^1,𝑝(0,𝑇), (3.65) follows from Proposition 5.1 and (3.66) from this equivalent reformulation of such a result: there exists 𝛼_{𝑝,𝑇,𝜆}>0, depending only on 𝑝, 𝑇, and 𝜆, such that 𝑥∈𝑊₀^1,𝑝(0,𝑇)⟹𝛼_{𝑝,𝑇,𝜆}|𝑥|_1,𝑝≤𝑎_𝜆(𝑥,⋅)_−1,𝑝′, (5.23) provided that 𝑝𝜆>−^1/𝑝𝑝^′1/𝑝′𝑇²1+𝑇^−1/𝑝′. (5.24) On the other hand, (3.67) was proven in Example 3.11.
Finally, the continuous dependence property follows from (3.68) and Lemma 5.2.

Theorem 5.3 generalizes [1, Theorem  3.3] to the reflexive framework, although a constant better than𝜆_2,𝑇=2𝑇²11+𝑇^−1/2 (5.25) is obtained, to be more precise,𝜋²𝑇². (5.26)

In [23, 24] the damped case is also considered, but only in the Hilbertian framework. For some recent developments of the theory of impulsive differential equations with a variational approach, we also refer to [25–27] and their extension [28], and to [29]. Sea also [30] for impulsive differential equations in abstract Banach spaces.

Acknowledgment

This research is partially supported by the Junta de Andaluca Grant FQM359.