# Characterization of the Solvability of Generalized Constrained Variational Equations

Authors: Manuel Ruiz Galán
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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<⋯<tk<tk+1=T, and d1,…,dk∈ℝ, the impulsive linear problem(1.1)-x′′(t)+λx(t)=f(t),t∈(0,T),x(0)=x(T)=0,Δx′(tj)=dj,j=1,…,k,
where Δx′(tj):=x′(tj+)-x′(tj-), 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∈H01(0,T) such that(1.2)y∈H01(0,T)⟹∫0T(x0′y′+λx0y)=∫0Tfy-∑j=1kdjy(tj).
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∈Lp(0,T), where 1<p<∞ and not necessarily p=2, and for all λ>-λp,T, for some λp,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. ForT>0, v0,vT∈ℝ, and f∈L2(0,T), let us consider the corresponding Poisson’ equation with nonhomogeneous Dirichlet boundary conditions(1.3)-x′′=fin(0,T),x(0)=v0,x(T)=vT,
whose usual weak formulation is(1.4)findx0∈H1(0,T)suchthat{x∈H01(0,T)⟹∫0Tx′0x′=∫0Tfx,x0(0)=v0,x0(T)=vT.
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 by(1.5)a(x,x̃):=∫0Tx′x̃′,(x,x̃∈X),b(x,y):=(x(0),x(T))⋅y,(x∈X,y∈Y)
(“·” stands for the Euclidean inner product in ℝ2), and let x0*:X→ℝ and y0*:Y→ℝ be the continuous linear functionals defined by(1.6)x0*(x):=∫Ωfx,(x∈X),y0*(y):=(v0,vT)⋅y,(y∈Y).
Since(1.7)K:={x∈X:y∈Y⟹b(x,y)=0}=H01(Ω),
then the weak formulation above leads us to consider the following variational equation with constraints: find x0∈X such that(1.8)x∈K⟹a(x0,x)=x0*(x),y∈Y⟹b(x0,y)=y0*(y).
With regard to this problem, a more abstract approach has been adopted: let X and Y be the Hilbert spaces, let x0*:X→ℝ and y0*: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,(1.9)findx0∈Xsuchthat{x∈K⟹a(x0,x)=x0*(x)y∈Y⟹b(x0,y)=y0*(y).
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 y0*:Y→ℝ and w0*:W→ℝ, denoting(1.10)Kb:={y∈Y:b(y,⋅)=0},
find, if possible, x0∈X such that(1.11)y∈Kb⟹a(x0,y)=y0*(y),w∈W⟹c(x0,w)=w0*(w).
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, Theorem2.2, 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 if(2.1)x,y∈C,0<t<1⟹S(j(tx+(1-t)y)-tj(x)-(1-t)j(y))≤0.
Finally, a convex function k:C→ℝ∪{∞} is proper when there exists x∈C with k(x)<∞.Proposition 2.1 ([6, Theorem 2.9]).
LetX 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
(2.2)L≤S,inf⁡C(L∘j+k)=inf⁡C(S∘j+k).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 X(2.3)x∈X⟼a(x,y)∈R
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.
LetX 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
(2.4)(y,w)∈C×D⟹a(⋅,y),c(⋅,w)∈X*.
Then,
(2.5)thereexistsx0**∈X**suchthat{y∈C⟹Φ(y)≤x0**(a(⋅,y))w∈D⟹Ψ(w)≤x0**(c(⋅,w))
if, and only if, there exists a continuous seminorm p:X*→ℝ so that
(2.6)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤p(a(⋅,y)+c(⋅,w)).
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 thatX 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
(2.7)j(y,w):=a(⋅,y)+c(⋅,w),((y,w)∈C×D),
and the proper convex function k:C×D→ℝ∪{∞} given by
(2.8)k(y,w):=-Φ(y)-Ψ(w),((y,w)∈C×D),
obtaining thus that there exists a linear functional L:X*→ℝ such that, on the one hand,
(2.9)L≤p,
and therefore L=x0**∈X** for some x0**∈X**, and on the other hand, satisfyies
(2.10)inf⁡(y,w)∈C×D(x0**(a(⋅,y)+c(⋅,w))-Φ(y)-Ψ(w))=inf⁡(y,w)∈C×D(p(a(⋅,y)+c(⋅,w))-Φ(y)-Ψ(w)).
But we are assuming that
(2.11)inf⁡(y,w)∈C×D(p(a(⋅,y)+c(⋅,w))-Φ(y)-Ψ(w))≥0.
Hence
(2.12)inf⁡(y,w)∈C×D(x0**(a(⋅,y)+c(⋅,w))-Φ(y)-Ψ(w))≥0,
that is,
(2.13)(y,w)∈C×D⟹Φ(z)+Ψ(w)≤x0**(a(⋅,y)+c(⋅,w)).
Since (0,0)∈C×D and Φ(0)≥0 and Ψ(0)≥0, taking in this last inequality (y,0)∈C×D yields
(2.14)y∈C⟹Φ(y)≤x0**(a(⋅,y)),
while for (0,w)∈C×D it implies
(2.15)w∈D⟹Ψ(w)≤x0**(c(⋅,w)),
and we have proven (2.5) as we wish.
And conversely, ifx0**∈X** satisfies (2.5), then
(2.16)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤x0**(a(⋅,y)+c(⋅,w)),
so for the continuous seminorm p on X*(2.17)p(x*):=|x0**(x*)|,(x*∈X*),
we have that
(2.18)Φ(y)+Ψ(w)≤p(a(⋅,y)+c(⋅,w)).
In any case we have stated the inequalityx0**≤p.As we can see, all the topological assumptions fall onX. 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 conditionx0**≤p, which seems to be irrelevant, will entail in the normed case a control of the norm ∥x0**∥.Let us also emphasize that Theorem2.2 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 ofX 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 BX 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 spaceX, 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 thatX 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
(3.1)(y,w)∈C×D⟹a(⋅,y),c(⋅,w)∈X*,
then the constrained variational problem
(3.2)findx0**∈X**suchthat{y∈C⟹Φ(z)≤x0**(a(⋅,y))w∈D⟹Ψ(w)≤x0**(c(⋅,w))
is solvable if, and only if, for some ρ≥0(3.3)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤ρ‖a(⋅,y)+c(⋅,w)‖.
Moreover, when these statements hold and there exists (y,w)∈C×D satisfying a(·,y)+c(·,w)≠0, then
(3.4)min⁡⁡{‖x0**‖:x0**∈X**,{y∈C⟹Φ(y)≤x0**(a(⋅,y))w∈D⟹Ψ(w)≤x0**(c(⋅,w))}=(sup⁡(y,w)∈C×D,a(⋅,y)+c(⋅,w)≠0Φ(y)+Ψ(w)‖a(⋅,y)+c(⋅,w)‖)+.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,
(3.5)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤‖x0**‖‖a(⋅,y)+c(⋅,w)‖.
Hence, if
(3.6)α:=(sup⁡(y,w)∈C×D,a(⋅,y)+c(⋅,w)≠0Φ(y)+Ψ(w)‖a(⋅,y)+c(⋅,w)‖)+
then
(3.7)‖x0**‖≥α,
and so
(3.8)α<∞.
But in addition we have that
(3.9)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤α‖a(⋅,y)+c(⋅,w)‖,
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
(3.10)p(x*):=α‖x*‖
satisfies (2.6); therefore, Theorem 2.2 implies the existence of x0**∈X** for which (3.2) is valid and x0**≤p, that is,
(3.11)‖x0**‖≤α,
which together with (3.7) finally yields (3.4).Of course, when in Corollary3.1 we additionally assume that X is reflexive, then the existence of ρ≥0 such that(3.12)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤ρ‖a(⋅,y)+c(⋅,w)‖
is equivalent to the existence of a solution in X to the constrained variational inequality, that is,(3.13)thereexistsx0∈Xsuchthat{y∈C⟹Φ(y)≤a(x0,y),w∈D⟹Ψ(w)≤c(x0,w).
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 thatX 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
(3.14)Kb:={y∈Y:b(y,⋅)=0},Kc:={x∈X:c(x,⋅)=0}
and supposes that
(3.15)y∈Kb,w∈W⟹a(⋅,z),c(⋅,w)∈X*,
then
(3.16)y∈Kb,w∈W⟹‖a(⋅,y)∣Kc‖≤‖a(⋅,y)+c(⋅,w)‖.Proof.
Lety∈Kb and ε>0. Since a(·,y)∣Kc∈X*, choose x0∈Kc so that
(3.17)‖x0‖=1,a(x0,y)>‖a(⋅,y)∣Kc‖-ε.
Then, given w∈W, we have that
(3.18)‖a(⋅,y)∣Kc‖<a(x0,y)+ε=a(x0,y)+c(x0,w)+ε(x0∈Kc)≤‖x0‖‖a(⋅,y)+c(⋅,w)‖+ε=‖a(⋅,y)+c(⋅,w)‖+ε,
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.
LetX 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  y0*∈Y* and  w0*∈W*, and write
(3.19)Kb:={y∈Y:b(y,⋅)=0},Kc:={x∈X:c(x,⋅)=0},Rw0*:={x∈X:c(x,⋅)=w0*}.
Then, the corresponding constrained variational equation admits a solution; that is,
(3.20)thereexistsx0∈Xsuchthat{y∈Kb⟹a(x0,y)=y0*(y),w∈W⟹c(x0,w)=w0*(w),
if, and only if,
(3.21)Rw0*≠∅,(3.22)∀x∈Rw0*thereexistsδ≥0suchthaty∈Kb⟹y0*(y)-a(x,y)≤δ‖a(⋅,y)∣Kc‖.
In addition, if one of these equivalent conditions is valid and there exists y∈Kb with a(·,y)∣Kc≠0, then one can take x0∈X in (3.20) with
(3.23)‖x0‖=min⁡x∈Rw0*(sup⁡y∈Kb,a(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖).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,
(3.24)Rw0*=x0+Kc.
Thus, given x∈Rw0* there exists x1∈Kc such that x=x0+x1, so for all y∈Kb(3.25)y0*(y)-a(x,y)=-a(x1,y)≤‖x1‖‖a(⋅,y)∣Kc‖,
and we have shown (3.22).
To conclude, we prove the converse (3.22)⇒(3.20). Let y∈Kb and w∈W, and let x∈Rw0*, whose existence guarantees (3.22). Then
(3.26)y0*(y)+w0*(w)=y0*(y)-a(x,y)+a(x,y)+c(x,w)(x∈Rw0*)≤y0*(y)-a(x,y)+‖x‖‖a(⋅,y)+c(⋅,w)‖≤δ‖a(⋅,y)∣Kc‖+‖x‖‖a(⋅,y)+c(⋅,w)‖(by(3.22))≤(δ+‖x‖)‖a(⋅,y)+c(⋅,w)‖(byLemma3.2).
Therefore, there exists ρ:=δ+∥x∥≥0 such that
(3.27)y∈Kb,w∈W⟹y0*(y)+w0*(w)≤ρ‖a(⋅,y)+c(⋅,w)‖,
so Corollary 3.1 (in combination with the reflexivity of X) for C:=Kb, D:=W, Φ:=y0*, and Ψ:=w0* 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
(3.28)min⁡{‖x0‖:x0∈Rw0*,y0*∣Kb=a(x0,⋅)∣Kb}=(sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖)+=sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖.
So, for establishing (3.23) and concluding the proof, it suffices to state the equality
(3.29)sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖=min⁡x∈Rw0*(sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖).
Suppose, on the one hand, that x∈Rw0*, y∈Kb, and w∈W, with a(·,y)∣Kc≠0, which in view of Lemma 3.2 implies that a(·,y)+c(·,w)≠0. Then
(3.30)y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖=y0*(y)-a(x,y)‖a(⋅,y)+c(⋅,w)‖+a(x,y)+c(x,w)‖a(⋅,y)+c(⋅,w)‖(x∈Rw0*)≤|y0*(y)-a(x,y)|‖a(⋅,y)+c(⋅,w)‖+‖x‖≤|y0*(y)-a(x,y)|‖a(⋅,y)∣Kc‖+‖x‖(byLemma3.2).
Thus
(3.31)sup⁡(y,w)∈Kb×Wa(⋅,y)∣Kc≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖≤sup⁡y∈Kba(⋅,y)∣Kc≠0|y0*(y)+w0*(w)|‖a(⋅,y)∣Kc‖+‖x‖=sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)+w0*(w)‖a(⋅,y)∣Kc‖+‖x‖,
and therefore, as x∈Rw0* is arbitrary,
(3.32)sup⁡(y,w)∈Kb×W,a(⋅,y)∣Kc≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖≤inf⁡x∈Rw0*(sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖).
And, on the other hand, if x∈Rw0*, y∈Kb, and w∈W satisfy a(·,y)+c(·,w)≠0 but a(·,y)∣Kc=0,
(3.33)y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖=y0*(y)-a(x,y)‖a(⋅,y)+c(⋅,w)‖+a(x,y)+c(x,w)‖a(⋅,y)+c(⋅,w)‖(x∈Rw0*)=a(x,y)+c(x,w)‖a(⋅,y)+c(⋅,w)‖≤‖x‖.
Hence
(3.34)sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0,a(⋅,y)∣Kc=0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖≤‖x‖,
and thus
(3.35)sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0,a(⋅,y)∣Kc=0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖≤inf⁡x∈Rw0*‖x‖≤inf⁡x∈Rw0*(sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖).
Therefore, it follows from (3.32), Lemma 3.2, and (3.35) that
(3.36)sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖≤inf⁡x∈Rw0*(sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖).
But thanks to (3.28) we can choose x0∈Rw0* with y0*∣Kb=a(x0,·)∣Kb in such a way that
(3.37)‖x0‖=sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖,
so finally
(3.38)‖x0‖=min⁡x∈Rw0*(sup⁡y∈Kba(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖)=sup⁡(y,w)∈Kb×W,a(⋅,y)+c(⋅,w)≠0y0*(y)+w0*(w)‖a(⋅,y)+c(⋅,w)‖,
and (3.29) is proven.Remark 3.4.
In Theorem3.3 we do not need to suppose that a and c are continuous: it suffices to impose
(3.39)(y,w)∈Kb×W⟹a(⋅,y),c(⋅,w)∈X*,
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 Corollary3.1 (or [8, Corollary  1.3]) we have, with X being reflexive,(3.40)Rw0*≠∅
if, and only if, there exists μ≥0 such that(3.41)w∈W⟹w0*(w)≤μ‖c(⋅,w)‖.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 Theorem3.3. If the variational equation with constraints (3.20) has a solution, then it is unique if, and only if,
(3.42)x∈Kc,a(x,⋅)∣Kb=0⟹x=0.Proof.
Ifa satisfies the nondegeneracy condition (3.42), then for any x∈Rw0* we have Rw0*=x+Kc, so if (3.20) has two solutions x1 and x̃1, then for some x2,x̃2∈Kc, x1=x+x2 and x̃1=x+x̃2; hence
(3.43)y∈Kb⟹a(x2-x̃2,y)=0.
By hypothesis it follows that x2=x̃2; that is, x1=x̃1.
And conversely, if there existsx0∈Kc, x0≠0, such that
(3.44)y∈Kb⟹a(x0,y)=0,
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 Theorem3.3 imply uniqueness of the solution and simplify the control of the norm of the solution.Corollary 3.6.
LetX be a real reflexive Banach space, let  Y,  Z, and  W be real normed spaces, let  w0*∈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
(3.45)Kb:={y∈Y:b(y,⋅)=0},Kc:={x∈X:c(x,⋅)=0}
and suppose that
(3.46)x∈Kc,a(x,⋅)∣Kb=0⟹x=0
and that there exist constants α,μ>0 with
(3.47)y∈Kb⟹α‖y‖≤‖a(⋅,y)∣Kc‖,(3.48)w∈W⟹w0*(w)≤μ‖c(⋅,w)‖.
Then, for each y0*∈Y*, the corresponding variational equation with constraints admits one and only one solution; that is,
(3.49)thereexistsauniquex0∈Xsuchthat{y∈Kb⟹a(x0,y)=y0*(y),w∈W⟹c(x0,w)=w0*(w).
Furthermore, if one defines
(3.50)Rw0*:={x∈X:c(x,⋅)=w0*},
then the a priori estimate
(3.51)∥x0∥≤‖y0*‖α+(1+‖a‖α)min⁡x∈Rw0*‖x‖≤‖y0*‖α+(1+‖a‖α)μ
is valid for the norm of the solution x0.Proof.
Corollary3.1 (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
(3.52)‖x0‖=min⁡x∈Rw0*(sup⁡y∈Kb,a(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖)
holds. Finally, we have by condition (3.47) that
(3.53)min⁡x∈Rw0*(sup⁡y∈Kb,a(⋅,y)∣Kc≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖)=min⁡x∈Rw0*(sup⁡y∈Kb,y≠0y0*(y)-a(x,y)‖a(⋅,y)∣Kc‖+‖x‖)≤inf⁡x∈Rw0*(sup⁡y∈Kb,y≠0y0*(y)+‖a‖‖x‖‖y‖α‖y‖+‖x‖)=‖y0*‖α+(1+‖a‖α)inf⁡x∈Rw0*‖x‖,
and by Corollary 3.1 (or [8, Corollary  1.3]) and (3.48) that
(3.54)inf⁡x∈Rw0*‖x‖=min⁡x∈R‖x‖=sup⁡w∈W,c(⋅,w)≠0w0*(w)‖c(⋅,w)‖≤μ,
and thus we have the announced bound.If we assume a condition onc 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 that(3.55)w∈W⟹γ‖w‖≤‖c(⋅,w)‖,
then we have the stability estimate(3.56)‖x0‖≤‖y0*‖α+(1+‖a‖α)‖w0*‖γ
for the solution.Taking into account that whenX=Y, Z=W, and b=c conditions (3.42) and (3.47) are satisfied if a is coercive on Kb×Kb, 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 thatX and  Z are reflexive real Banach spaces,  x0*∈X*,  z0*∈Z*, and  a:X×X→ℝ and  b:X×Z→ℝ are continuous bilinear forms. Suppose in addition that, taking
(3.57)Kb:={x∈X:b(x,⋅)=0},Rz0*:={x∈X:b(x,⋅)=z0*},
there exists α>0 such that
(3.58)x∈Kb⟹α‖x‖2≤a(x,x)
and that Rz0*≠∅. Then
(3.59)thereexistsauniquex0∈Xsuchthat{x∈Kb⟹a(x0,x)=x0*(x),z∈Z⟹b(x0,z)=z0*(z).
Besides, the solution x0 satisfies the a priori estimate:
(3.60)‖x0‖≤‖x0*‖α+(1+‖a‖α)min⁡x∈Rz0*‖x‖.
In particular, if there exists γ>0 such that
(3.61)z∈Z⟹γ‖z‖≤‖b(⋅,z)‖,
then for the norm of x0 the following estimation:
(3.62)‖x0‖≤‖x0*‖α+(1+‖a‖α)‖z0*‖γ
holds.In Theorem3.3 and Lemma 3.5 we drive a characterization when the variational equation with constraints (3.20) admits a unique solution, for two fixed functionals y0*∈Y* and w0*∈W*. The known particular cases in the literature are stated for arbitrary functionals y0*∈Y* and w0*∈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.
LetX 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
(3.63)Kb:={y∈Y:b(y,⋅)=0},Kc:={x∈X:c(x,⋅)=0}.
Then, for all y0*∈Y* and w0*∈W*,
(3.64)thereexistsoneandonlyonex0∈Xsuchthat{y∈Kb⟹a(x0,y)=y0*(y),w∈W⟹c(x0,w)=w0*(w)
if, and only if,
(3.65)x∈Kc,a(x,⋅)∣Kb=0⟹x=0
and there exist α,γ>0 so that
(3.66)y∈Kb⟹α‖y‖≤‖a(⋅,y)∣Kc‖,(3.67)w∈W⟹γ‖w‖≤‖c(⋅,w)‖.
In addition, if one of these equivalent conditions is satisfied, one has the following stability estimate:
(3.68)‖x0‖≤‖y0*‖α+(1+‖a‖α)‖w0*‖γ.Proof.
In view of Corollary3.6 and Lemma 3.5 we deduce, provided that (3.65), (3.66), and (3.67) hold, that for all y0*∈Y* and w0*∈W* the constrained variational problem (3.64) admits a unique solution, whose norm satisfies estimation (3.68).
And conversely, suppose that for arbitraryy0*∈Y* and w0*∈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 w0*∈W* there exists x0∈X with w0*=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 w0*=0∈W* we have that for all y0*∈Y* there exists x0∈Kc such that a(x0,·)|Kb=y0*|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 Remark3.4, that in this proper extension of the Lax-Milgram theorem we just need to assume that
(3.69)(y,w)∈Kb×W⟹a(⋅,y),c(⋅,w)∈X*,
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 Theorem3.8 (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 Lp(0,T). In addition, the standard norm in the Sobolev space W1,p(0,T) is given by(3.70)‖x‖1,p:=(‖x‖pp+‖x′‖pp)1/p,(x∈W1,p(0,T)),
and the inherited norm on the subspace W01,p(0,T) is equivalent to the norm |·|1,p defined as (3.71)|x|1,p:=‖x′‖p,(x∈W01,p(0,T)),
thanks to the well-known Poincaré inequality ([16, Theorem  6.30]), which asserts that there exists a constant cp,T>0, depending only on T and p, in such a way that(3.72)x∈W01,p(0,T)⟹‖x‖p≤cp,T|x|1,p.
In fact, it is easy to check that we can take(3.73)cp,T=Tp1/p,
and for p=2 the optimal c2,T is given by(3.74)c2,T=Tπ
(see [17, Section  1.1.3]). As usual, we denote by W-1,p′(0,T) the dual space of W01,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 W01,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.
LetT>0 and  1<p<∞, with conjugate exponent  p′, and consider the continuous bilinear form  a0:W01,p(0,T)×W01,p′(0,T)→ℝ given for each  x∈W01,p(0,T) and  y∈W01,p′(0,T) by
(3.75)a0(x,y):=∫0Tx′y′.
Then a0 satisfies the inf-sup condition. More specifically,
(3.76)y∈W01,p′(0,T)⟹11+T-1/p′|y|1,p′≤|a0(⋅,y)|-1,p′.Proof.
It is sufficient to deal withy∈C0∞(0,T). The description of the dual space W-1,p′(0,T) of W01,p(0,1) (see for instance [2, Proposition  8.14]) guarantees that for some y0∈Lp′(0,T),
(3.77)x∈W01,p(0,T)⟹a0(x,y)=∫0Tx′y0|a0(⋅,y)|-1,p′=‖y0‖p′.
Then
(3.78)x∈W01,p(0,T)⟹∫0Tx′(y′-y0)=0,
and so
(3.79)x∈C0∞(0,T)⟹∫0Tx′(y′-y0)=0.
But taking into account that
(3.80){x′:x∈C0∞(0,T)}={z∈C0∞(0,T):∫0Tz=0},
if x0∈C0∞(0,T) with ∫0Tx0=1, then
(3.81){x′:x∈C0∞(0,T)}={w-(∫0Tw)x0:w∈C0∞(0,T)},
so (3.79) is equivalent to
(3.82)w∈C0∞(0,T)⟹∫0T(w-(∫0Tx)x0)(y′-y0)=0,
or in other words,
(3.83)w∈C0∞(0,T)⟹∫0Tw(y′-y0)=∫0Tw∫0Tx0(y′-y0),
that is,
(3.84)y′-y0=∫0Tx0(y′-y0).
Therefore, we have that for some λ∈ℝ(3.85)y′-y0=λ.
Hence, integrating and noting that y∈C0∞(0,T),
(3.86)λ=-1T∫0Ty0,
and thus, as a consequence of the triangular and the Holdër inequalities,
(3.87)|y|1,p′≤‖y0‖p′+|λ|T≤(1+T-1/p′)‖y0‖p′,
so
(3.88)11+T-1/p′|y|1,p′≤‖y0‖p′=|a0(⋅,y)|-1,p′.The corresponding bilinear forma:W01,p(Ω)×W01,p′(Ω)→ℝ defined for each (x,y) in W01,p(Ω)×W01,p′(Ω) as(3.89)a(x,y):=∫Ω∇x⋅∇y,
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 thatT>0, v0,vT∈ℝ, 1<p<∞, f∈Lp(Ω), and consider the elliptic boundary value problem
(3.90)-x′′=fin(0,T),x(0)=v0,x(T)=vT,
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∈W01,p′(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:
(3.91)findx0∈W1,p(0,T)suchthat{y∈W01,p′(0,T)⟹∫0Tx0′y′=∫0Tfyx0(0)=v0,x0(T)=vT.
Let us consider the real reflexive Banach space
(3.92)X:=W1,p(0,T)
and the normed spaces
(3.93)Y:=W1,p′(0,T),Z:=W:=R2,
with ℝ2 endowed with its ∥·∥1 norm, the continuous linear functionals y0*∈Y* and w0*∈W* defined for each y∈Y and w∈W, respectively, as
(3.94)y0*(y):=∫0Tfy,(3.95)w0*(w1,w2):=(v0,vT)⋅w,
and the continuous bilinear forms a:X×Y→ℝ, b:Y×Z→ℝ, and c:X×W→ℝ given by
(3.96)a(x,y):=∫0Tx′y′,(x∈X,y∈Y),b(y,(z1,z2)):=(y(0),y(T))⋅z,(y∈Y,z∈Z),c(x,(w1,w2)):=(x(0),x(T))⋅w,(x∈X,w∈W).
Now Kb=W01,p′(0,T), so the variational formulation above is nothing more than the variational equation with the following constraints:
(3.97)findx0∈Xsuchthat{y∈Kb⟹y0*(y)=a(x0,y),w∈W⟹w0*(w)=c(x0,w).
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
(3.98)x∈W01,p(0,T)⟹11+T-1/p|x|1,p≤|a0(x,⋅)|-1,p,
which, in view of the fact that Kc=W01,p(0,T), is equivalent to
(3.99)x∈Kc⟹11+T-1/p|x|1,p≤|a(x,⋅)∣Kb|-1,p,
and the equivalence of the norm |·|1,p and the usual one in W01,p(0,T) yields (3.65).
To prove that condition (3.66) is satisfied, we fix y∈Kb=W01,p′(0,T) and apply Proposition 3.10 using equivalence of the norm |·|1,p′ and the usual one in W01,p′(0,T).
In order to conclude, let us deduce the inf-sup condition (3.67), more specifically, that there exists γp,T>0, depending only on p and T, such that
(3.100)w∈R2⟹γp,T‖w‖1≤‖c(⋅,w)‖W1,p(0,T)*,
where ∥·∥W1,p(0,T)* denotes the dual norm of ∥·∥1,p in W1,p(0,T)*, equivalently,
(3.101)inf⁡w∈R2,‖w‖1=1‖c(⋅,w)‖W1,p(0,T)*≥γp,T.
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∈W1,p(0,T)(3.102)x0(t):=1,(t∈[0,T]),
for which
(3.103)‖x0‖1,p=1T1/p.
Then
(3.104)‖c(⋅,w)‖W1,p(0,T)*≥c(x0,w)T1/p=w1+w2T1/p=1T1/p.
On the other hand, if w1<0<w2, for
(3.105)x0(t):=-1+2Tt,(t∈[0,T]),
we have that x0∈W1,p(0,T) and
(3.106)‖x0‖1,p≤T1/p(3p+(2T)p)1/p,
and so,
(3.107)‖c(⋅,w)‖W1,p(0,T)*≥c(x0,w)‖x0‖1,p≥-w1+w2T1/p(3p+(2/T)p)1/p=1T1/p(3p+(2/T)p)1/p.
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
(3.108)γp,T:=min⁡{1T1/p,T-1/p(3p+(2T)p)-1/p}>0.
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∈W1,p(0,T), for whose norm the stability estimation
(3.109)‖x0‖≤θp,T(‖f‖p+‖(v0,vT)‖∞)
is valid, where θp,T>0 depends only on p and T.With the applications of Section5 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(3.110)-▵x=finΩ,x=gon⁡Γ
admits a constrained variational formulation similar to that of Example 3.11, where Γ is the topological boundary of Ω, 1<p<∞, f∈Lp(Ω), and g∈W1/p′,p(Γ), the space of traces on Γ of functions in the Sobolev space W1,p(Ω).Next we show that the reflexivity ofX is essential in Theorem 3.8.Proposition 3.12.
IfX 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 y0*∈Y* and w0*∈W* satisfying the assumptions in Theorem 3.8, except obviously the reflexivity of X,  but for any x0∈X the constrained variational equation
(3.111)y∈Kb⟹a(x0,y)=y0*(y),w∈W⟹c(x0,w)=w0*(w)
does not hold.Proof.
It suffices to takeZ:=X, Y:=W:=X*, a, b and c the corresponding duality pairings, y0*∈Y* any continuous linear functional on Y and w0*:=x0**∈X**∖X. Then, since Kb=0 and Kc=0, the first condition in the characterization holds. The second one is also satisfied, because
(3.112)w∈W⟹‖w‖=‖c(⋅,w)‖.
However, although for all x0∈X(3.113)y∈Kb⟹a(x0,y)=y0*(y),
for each x0∈X,
(3.114)c(x0,⋅)≠w0*,
since c(·,x0)=x0∈X but w0*=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 write(3.115)Kb:={y∈Y:b(y,⋅)=0},Kc:={x∈X:c(x,⋅)=0}.
Then, for all y0*∈Y*, w0*∈W* there exists a unique (x0,z0)∈X×Z such that(3.116)y∈Y⟹y0*(y)=a(x0,y)+b(y,z0),w∈W⟹w0*(w)=c(x0,w)
if, and only if,(3.117)x∈Kc,a(x,⋅)∣Kb=0⟹x=0,
and there exist α,β,γ>0 such that(3.118)y∈Kb⟹α‖y‖≤‖a(⋅,y)∣Kc‖,z∈Z⟹β‖z‖≤‖b(⋅,z)‖,w∈W⟹γ‖w‖≤‖c(⋅,w)‖.
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 {en}n≥1 be the usual basis of ℓ2, and let Z:=span{e2n: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,(3.119)a(x,y):=∑n=1∞x(n)y(2n-1),
so a is continuous and ∥a∥=1; for y∈Y and n≥1,(3.120)b(y,e2n):=ny(2n)
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. Now(3.121)Kb={y∈l2:n≥1⟹y(2n)=0},Kc=l2.
Let us check condition (3.65). If x∈Kc and a(x,·)|Kb=0, then for n≥1, e2n-1∈Kb, so(3.122)x(n)=a(x,e2n-1)=0,
and therefore x=0. Finally, condition (3.66) is also satisfied, since for each y∈Kb,(3.123)y=∑n=1∞y(2n-1)e2n-1,
and thus defining(3.124)ỹ:=∑n=1∞y(2n-1)en,
then(3.125)‖y‖=‖ỹ‖,a(ỹ,y)=‖y‖2,
and since ∥a∥=1, we arrive at(3.126)‖y‖=‖a(⋅,y)‖.
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 y0*∈Y* and w0*∈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 y0*∈Y* and w0*∈W*.


## 4. A Word on Dual Banach Spaces

As we commented in Section3, 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 Section3, we merely enunciate them.We begin with the analogue to Corollary3.1, which also follows from Theorem 2.2, but with a different locally convex space.Corollary 4.1.
Assume thatX 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
(4.1)(y,w)∈C×D⟹a(⋅,y),c(⋅,w)∈X,
then
(4.2)thereexistsx0*∈X*suchthat{y∈C⟹Φ(z)≤a(x0*,y),w∈D⟹Ψ(w)≤c(x0*,w),
if, and only if, there exists ρ≥0 such that
(4.3)(y,w)∈C×D⟹Φ(y)+Ψ(w)≤ρ‖a(⋅,y)+c(⋅,w)‖.
Furthermore, if one of these equivalent conditions holds and there exists (y,w)∈C×D so that a(·,y)+c(·,w)≠0, then
(4.4)min⁡{‖x0*‖:x0*∈X*,{y∈C⟹Φ(y)≤a(x0*,y)w∈D⟹Ψ(w)≤c(x0*,w)}=(sup⁡(y,w)∈C×D,a(⋅,y)+c(⋅,w)≠0Φ(y)+Ψ(w)‖a(⋅,y)+c(⋅,w)‖)+.Corollaries3.1 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.
LetX,  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
(4.5)Kb:={y∈Y:b(y,⋅)=0},Kc:={x*∈X*:c(x*,⋅)=0},
and assume that
(4.6)(y,w)∈Kb×W⟹a(⋅,y),c(⋅,w)∈X.
If in addition y0*∈Y* and w0*∈W* and
(4.7)Rw0*:={x*∈X*:c(x*,⋅)=w0*},
then
(4.8)thereexistsx0*∈X*suchthat{y∈Kb⟹a(x0*,y)=y0*(y),w∈W⟹c(x0*,w)=w0*(w),
if, and only if,
(4.9)Rw0*≠∅∀x*∈Rw0*thereexistsδ≥0suchthaty∈Kb⟹y0*(y)-a(x*,y)≤δ‖a(⋅,y)∣Kc‖.
Moreover, when these conditions are satisfied and there exists y∈Kb such that a(·,y)∣Kc≠0, then one can take a solution x0*∈X* with
(4.10)‖x0*‖=min⁡x*∈Rw0*(sup⁡y∈Kb,a(⋅,y)∣Kc≠0y0*(y)-a(x*,y)‖a(⋅,y)∣Kc‖+‖x*‖).The version for anyy0*∈Y* and w0*∈W*, with the ingredient of uniqueness, is stated in these terms.Theorem 4.3.
LetX,  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
(4.11)Kb:={y∈Y:b(y,⋅)=0},Kc:={x*∈X*:c(x*,⋅)=0},
and assume that
(4.12)y∈Kb,w∈W⟹a(⋅,y),c(⋅,w)∈X.
Then, for all y0*∈Y* and w0*∈W*, the corresponding constrained variational equation is uniquely solvable, that is,
(4.13)thereexistsoneandonlyonex0*∈X*suchthat{y∈Kb⟹a(x0*,y)=y0*(y),w∈W⟹c(x0*,w)=w0*(w)
if, and only if,
(4.14)x*∈Kc,a(x*,⋅)∣Kb=0⟹x*=0
and there exist α,γ>0 such that
(4.15)y∈Kb⟹α‖y‖≤‖a(⋅,y)∣Kc‖,w∈W⟹γ‖w‖≤‖c(⋅,w)‖.
In addition, if one of these equivalent conditions holds, then
(4.16)‖x0*‖≤‖y0*‖α+(1+‖a‖α)‖w0*‖γ.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 Example3.11; 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∈Lp(0,T), with 1<p<∞, λ∈ℝ, t0=0<t1<⋯<tk<tk+1=T, and v0,vT,d1,…,dk∈ℝ, the impulsive linear problem in question is(5.1)-x′′(t)+λx(t)=f(t),t∈(0,T),x(0)=v0,x(T)=vT,Δx′(tj)=dj,j=1,…,k,
where Δx′(tj)=x′(tj+)-x′(tj-). Now the notion of a weak solution, along the lines of [1] (multiply by a test function y∈W01,p′(0,T), with 1/p+1/p′=1, integrate by parts and take into account the impulsive conditions Δx′(tj)=dj), is defined as(5.2)x0∈W1,p(0,T)suchthat{y∈W01,p′(0,T)⟹∫0T(x0′y′+λx0y)=∫0Tfy-∑j=1kdjy(tj),x0(0)=v0,x0(T)=vT.
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.
LetT>0,  1<p<∞ with conjugate exponent  p′ and  λ∈ℝ, and consider the continuous bilinear form  aλ:W01,p(0,T)×W01,p′(0,T)→ℝ given for each  x∈W01,p(0,T) and  y∈W01,p′(0,T) by
(5.3)aλ(x,y):=∫0T(x′y′+λxy).
Then there exists δp,T>0 such that if λ>-δp,T, then aλ satisfies the inf-sup condition. More precisely, for all λ>-δp,T, where
(5.4)δp,T:=p1/pp′1/p′T2(1+T-1/p′),
there exists αp,T,λ>0, depending only on p, T, and λ, such that
(5.5)y∈W01,p′(0,T)⟹αp,T,λ|y|1,p′≤|aλ(⋅,y)|-1,p.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 aλ 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 aλ=a0+λã, where ã:W01,p(0,T)×W01,p′(0,T)→ℝ is the continuous bilinear form defined for each (x,y)∈W01,p(0,T)×W01,p′(0,T) as
(5.6)ã(x,y):=∫0Txy.
But for all (x,y)∈W01,p(0,T)×W01,p′(0,T) we have, in view of (3.72), that
(5.7)ã(x,y)≤‖x‖p‖y‖p′≤cp,Tcq,T|x|1,p|y|1,p′≤T2p1/pp′1/p′|x|1,p|y|1,p′,
so
(5.8)|ã(⋅,y)|-1,p≤T2p1/pp′1/p′|y|1,p′.
Taking into account this last inequality, Proposition 3.10, and the fact that aλ=a0+λã, we conclude that for
(5.9)0>λ>-p1/pp′1/p′T2(1+T-1/p′)
it follows that
(5.10)y∈W01,p′(0,T)⟹(11+T-1/p′+λT2p1/pp′1/p′)|y|1,p′≤|a0(⋅,y)|-1,p′,
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.
Letλ∈ℝ,  T>0,  d1,…,dk∈ℝ,  1<p<∞,  1/p+1/p′=1, and suppose that  f∈Lp(0,T). Then the linear functional  y0*:W1,p′(0,T)→ℝ given for each  y∈W1,p′(0,T) by
(5.11)y0*(y):=∫0Tfy-∑j=1kdjy(tj)
is continuous, and in addition
(5.12)|y0*|-1,p′≤(Tp′1/p′‖f‖p+T1/p∑j=1k|dj|).Proof.
Assume without loss of generality thaty∈C0∞(0,T). Given j=1,…,k it follows from the Hölder inequality that
(5.13)|y(tj)|=|∫0tjy′|≤|y|1,1≤T1/p|y|1,p′,
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.
LetT>0,  1<p<∞, whose conjugate exponent is  p′,  f∈Lp(0,T),  t0=0<t1<⋯<tk<tk+1=T, and  v0,vT,d1,…,dk∈ℝ. Then there exists  λp,T>0, specifically,
(5.14)λp,T:=p1/pp′1/p′T2min⁡{11+T-1/p,11+T-1/p′},
in such a way that if λ>-λp,T, then the corresponding impulsive linear problem
(5.15)-x′′(t)+λx(t)=f(t),t∈(0,T),x(0)=v0,x(T)=vT,Δx′(tj)=dj,j=1,…,k,
admits a unique weak solution. Furthermore, there exists χp,T,λ>0, depending only on p, T, and λ, such that
(5.16)‖x0‖W01,p(0,T)≤χp,T,λ(‖f‖p+∑j=1k|dj|+‖(v0,vT)‖∞).Proof.
Our aim is to prove the existence of one and only onex0∈W1,p(0,T) such that
(5.17)y∈W01,p′(0,T)⟹∫0T(x0′y′+λx0y)=∫0Tfy-∑j=1kdjy(tj),x0(0)=v0,x0(T)=vT
which moreover depends continuously on the initial data. Yet this variational problem admits an equivalent reformulation as the variational problem with constraints
(5.18)findx0∈Xsuchthat{y∈Kb⟹y0*(y)=a(x0,y),w∈W⟹w0*(w)=c(x0,w),
where X is now the real reflexive Banach space
(5.19)X:=W1,p(0,T)
and Y, Z, and W are the real normed spaces
(5.20)Y:=W1,p′(0,T),Z:=W:=R2,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
(5.21)a(x,y):=∫0T(x′y′+λxy),(x∈X,y∈Y),b(y,z):=(y(0),y(T))⋅z,(y∈Y,z∈Z),c(x,w):=(x(0),x(T))⋅w,(x∈X,w∈W),
and y0*∈Y* and w0*∈W* are the continuous linear functionals given as
(5.22)y0*(y):=∫0Tfy-∑j=1kdjy(tj),(y∈Y),w0*(w):=(v0,vT)⋅w,(w∈W).
The fact that the variational problem coincides with the constrained variational equation follows from the fact that Kb=W01,p′(0,T).
Let us finally check (3.65), (3.66), and (3.67) in order to apply Theorem 3.8 and conclude the proof.
SinceKb=W01,p′(0,T) and Kc=W01,p(0,T), (3.65) follows from Proposition 5.1 and (3.66) from this equivalent reformulation of such a result: there exists α̃p,T,λ>0, depending only on p, T, and λ, such that
(5.23)x∈W01,p(0,T)⟹α̃p,T,λ|x|1,p≤|aλ(x,⋅)|-1,p′,
provided that
(5.24)λ>-p1/pp′1/p′T2(1+T-1/p′).
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.Theorem5.3 generalizes [1, Theorem  3.3] to the reflexive framework, although a constant better than(5.25)λ2,T=2T211+T-1/2
is obtained, to be more precise,(5.26)π2T2.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.



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