Partial differential hemivariational inequalities

Definition 1.1.

A pair (x,u), with x∈C⁢([0,T];E) and u:[0,T]→K measurable, is said to be a mild solution of problem (PDHVI) in (1.1) if

and

For simplicity, x is called a mild trajectory of problem (PDHVI) and u is called a variational control function.

Definition 2.1.

Let F:X→P⁢(Y) be a set-valued mapping. We note the following definitions:

1. We say that F is upper semicontinuous (u.s.c., for short) at x∈X if, for every open set O⊂Y with F⁢(x)⊂O, there exists a neighborhood N⁢(x) of x such that F⁢(N⁢(x)):=⋃y∈N⁢(x)F⁢(y)⊂O. If this occurs at every x∈X, then F is called upper semicontinuous.

2. We say that F is lower semicontinuous (l.s.c., for short) at x∈X if, for every open set O⊂Y with F⁢(x)∩O≠∅, there exists a neighborhood N⁢(x) of x such that F⁢(y)∩O≠∅ for all y∈N⁢(x). If this is true for every x∈X, then F is called lower semicontinuous.

3. We say that F is continuous if it is simultaneously u.s.c. and l.s.c.

Lemma 2.2.

Assume that X and Y are metric spaces and F:X→K⁢(Y) is a closed and compact set-valued mapping. Then F is u.s.c.

Definition 2.3.

The Hausdorff metric H:C⁢b⁢(Y)×C⁢b⁢(Y)→ℝ+ is defined by

Definition 2.4.

Let E, E1, E2 be Banach spaces and let I⊂ℝ be an interval.

1. F:I→P⁢(E) is said to be measurable if for every open subset O⊂E the set F+⁢(O):={t∈I:F⁢(t)⊂O} is measurable in ℝ,

2. F:I→C⁢b⁢(E) is said to be strongly measurable if there exists a sequence {Fn}n≥1 of step set-valued mappings such that H⁢(F⁢(t),Fn⁢(t))→0 as n→∞ for a.e. t∈I,

3. F:I×E1→P⁢(E2) is said to be superpositionally measurable if for every measurable set-valued mapping Q:I→K⁢(E1) the set-valued mapping Φ:I→P(E2) given by Φ⁢(t):=F⁢(t,Q⁢(t)) is measurable.

Theorem 2.5.

Let E and E0 be Banach spaces, let F:[0,T]×E0→K⁢(E) satisfy the Carathéodory conditions and let U:[0,T]→K⁢(E0) be measurable. Assume that g:[0,T]→E is measurable and g⁢(t)∈F⁢(t,U⁢(t)) for a.e. t∈[0,T]. Then there exists u:[0,T]→E0 measurable such that u⁢(t)∈U⁢(t) and g⁢(t)∈F⁢(t,u⁢(t)) for a.e. t∈[0,T].

Theorem 2.6.

Let E and E0 be Banach spaces and let G:[0,T]×E0→K⁢(E) satisfy the following conditions:

1. For every x∈E0, the mapping G⁢(⋅,x):[0,T]→K⁢(E) has a strongly measurable selection.

2. For a.e. t∈[0,T], the mapping G⁢(t,⋅):E0→K⁢(E) is u.s.c.

Then for every strongly measurable function q:[0,T]→E0 there exists a strongly measurable selection g:[0,T]→E for G⁢(⋅,q⁢(⋅)):[0,T]→K⁢(E) such that g⁢(t)∈G⁢(t,q⁢(t)) for a.e. t∈[0,T].

Definition 2.7.

Let E be a Banach space and let (𝐀,≥) be a partially ordered set. A mapping β:P⁢(E)→𝐀 is called a measure of noncompactness (MNC, for short) in E if β⁢(co¯Ω)=β⁢(Ω) for every Ω∈P⁢(E).

Theorem 2.8.

Let χ1 and χ2 be two real MNCs in E1 and E2, respectively. Suppose that X⊂E1 and that F:X×E1→K⁢(E2) satisfy the following conditions:

1. There is a constant k∈ℝ+ such that for any x∈X, the mapping F⁢(x,⋅):E1→K⁢(E2) is k-Lipschitz with respect to the Hausdorff metric H on K⁢(E2), i.e.,

   H⁢(F⁢(x,y1),F⁢(x,y2))≤k⁢∥y1-y2∥ for all y1,y2∈E1.

2. F⁢(Ω×{y}) is relatively compact in E2 for every bounded set Ω⊂X and y∈E1.

Then G:X→K⁢(E2) defined as G⁢(x)=F⁢(x,x) is (k,χ1,χ2)-bounded.

Definition 2.9.

Let E be a Banach space, X⊂E a closed subset, β a real MNC in E and a constant 0≤k<1. A set-valued mapping F:X→K⁢(E) is said to be (k,β)-condensing if β⁢(F⁢(Ω))≤k⁢β⁢(Ω) for every Ω⊆X.

Proposition 2.10.

Assume that Ω⊂L1⁢([0,T];E) is integrably bounded and the sets Ω⁢(t):={x⁢(t)∈E:x∈Ω} are relatively compact for a.e. t∈[0,T]. Then Ω is weakly compact in L1⁢([0,T];E).

Theorem 2.11.

Let X be a Banach space, C a closed convex subset of X, D an open subset of C (relative to C) and 0∈D. Suppose that the set-valued mapping Γ:D¯→K⁢v⁢(C) is compact, u.s.c. and does not possess fixed points on ∂⁡D. Then one of the following two cases holds:

1. Γ has a fixed point in D,

2. there are x∈∂⁡D and λ∈(0,1) with x∈λ⁢Γ⁢x.

Proposition 2.12.

Let C be a closed subset of a Banach space E and let F:C→K⁢(E) be a closed set-valued mapping, which is β-condensing on every bounded subset of C, where β is a monotone MNC in E. If Fix⁢F is bounded, then it is compact.

Theorem 2.13.

If G:L1⁢([0,T];E)→C⁢([0,T];E) is the Cauchy operator

then for every sequence {fn}⊂L1⁢([0,T];E) which is integrably bounded and with the sets {fn⁢(t)} relatively compact for a.e. t∈[0,T], one has that the sequence {G⁢fn}n=1∞ is relatively compact in C⁢([0,T];E). Moreover, if fn⇀f0 then G⁢fn→G⁢f0.

Proposition 2.14.

Let κ:E1→R be locally Lipschitz of rank Lu>0 near the point u∈E1. Then there hold:

1. The function v↦κ∘⁢(u;v) is finite, positively homogeneous, subadditive and satisfies

   |κ∘⁢(u;v)|≤Lu⁢∥v∥E1.

2. κ∘⁢(u;v) is upper semicontinuous as a function of (u,v).

3. ∂⁡κ⁢(u) is a nonempty, convex, weak* compact subset of E1* with

   ∥ξ∥E1*≤Lu for all ⁢ξ∈∂⁡κ⁢(u).

4. For each v∈E1, we have

   κ∘⁢(u;v)=max⁡{〈ξ,v〉:ξ∈∂⁡κ⁢(u)}.

Definition 3.1.

We say that {un}⊂K is an approximating sequence for (QVH) if

with some un*∈Q⁢(un) for all n, and εn→0+ as n→∞.

Definition 3.2.

Problem (QVH) is said to be strongly well-posed if the solution set Π of (QVH) is nonempty and every approximating sequence for (QVH) contains a subsequence strongly converging to a point of Π.

Theorem 3.3.

Suppose that ϕ:E1→R is a property convex lower semicontinuous functional, B:E1→E1* is a linear continuous operator, Q:K→P⁢(E1*) is a lower semicontinuous set-valued operator and J:E1→R is a locally Lipschitz function. If Q-B+∂⁡J:K→P⁢(E1*) is monotone, then problem (QVH) is strongly well-posed if and only if

where

Proof.

We rely on the following two claims.

Claim 1. Problem (QVH) is strongly well-posed if and only if its solution set Π is nonempty, compact, and

where e⁢(A1,A2):=supa∈A1⁡d⁢(a,A2) with d⁢(a,A2):=infb∈A2⁡∥a-b∥E1 for A1,A2⊂E1.

If problem (QVH) is strongly well-posed, then a fortiori Π≠∅. Since Π⊆Ω⁢(ε) for all ε>0, any sequence {un}⊂Π satisfies un∈Ω⁢(εn) with εn→0+, so according to Definition 3.1, the sequence {un} is an approximating sequence of (QVH). Therefore, by Definition 3.2, there exists a subsequence of {un} converging to a point in Π, which expresses that Π is compact in E1. It remains to check the equality in (3.3). If there were β>0, εn→0+ and vn∈Ω⁢(εn) so that dE1⁢(vn,Π)>β for all n, then it would contradict the strong well-posedness in the sense of Definition 3.2 because {vn} is an approximating sequence of (QVH), so the claim is verified.

For the converse assertion in the claim let {un}⊂K be an approximating sequence of (QVH). Then there exist εn→0+ as n→∞ and un*∈Q⁢(un) such that

which reads as un∈Ω⁢(εn) for all n. By (3.3) there exists a sequence {ωn} in Π satisfying ∥un-ωn∥E1→0 as n→∞. The compactness of Π ensures that a subsequence of {ωn} strongly converges to some u0∈Π, therefore the same occurs for a subsequence of {un}. This enables us to conclude that problem (QVH) is strongly well-posed.

Claim 2. We have that u∈K is a solution of (QVH) if and only if

Indeed, if u∈K is a solution of (QVH), then there exists u*∈Q⁢(u) such that

where J∘⁢(u;v-u)=〈ξu,v-u〉, with ξu∈∂⁡J⁢(u) (see Proposition 2.14). Since Q-B+∂⁡J:K→P⁢(E1*) is monotone, one has

Therefore, using again Proposition 2.14, we may write

Conversely, suppose that u∈K solves (3.4). If v∈K and λ∈(0,1), then the convexity of K implies uλ:=(1-λ)⁢u+λ⁢v∈K, hence (3.4) and the convexity of ϕ yield

Using that Q is l.s.c., we have that for any u*∈Q⁢(u) there exists a sequence uλ*→u* as λ→0 with uλ*∈Q⁢(uλ). Then passing to the inferior limit as λ→0+ results in

which completes the proof of Claim 2.

Now we proceed to prove the theorem. Suppose that problem (QVH) is strongly well-posed. By Claim 1, we know that Π is nonempty and compact. Then, since according to (3.2) we have Π⊂Ω⁢(ε), it turns out that

By (3.5) we obtain

Taking into account (3.3), it follows that limε→0⁡χE1⁢(Ω⁢(ε))=0.

Conversely, assume that condition (3.1) holds. Then the closure Ω⁢(ε)¯ of Ω⁢(ε) is nonempty, bounded, closed, increasing with ε>0, and satisfies

Setting Ω=⋂ε>0Ω⁢(ε)¯, the generalized Cantor theorem (see [15, p. 412]) entails that Ω is nonempty and compact in E1 with

Let us prove that Π=Ω. Observing that Π⊆Ω, we only need to show that Ω⊆Π. For each u∈Ω, from dE1⁢(u,Ω⁢(ε)¯)=0, we infer that corresponding to any εn→0+ there exist sequences un∈K and un*∈Q⁢(un) satisfying

Hence un→u∈K and, by the monotonicity of Q-B+∂⁡J, we get

By Proposition 2.14 we obtain

Invoking Claim 2, we derive that u∈Π, so Ω=Π.

From (3.6) it is seen that limε→0⁡e⁢(Ω⁢(ε),Π)=0. In view of Claim 1 and the compactness of Π, we conclude that problem (QVH) is strongly well-posed in the sense of Definition 3.2, which completes the proof. ∎

Theorem 3.4.

Assume that problem (QVH) is strongly well-posed in the sense of Definition 3.2 and

is a monotone operator. Then the solution set Π of (QVH) is nonempty, compact and convex in E1.

Proof.

It follows from Claim 1 in the proof of Theorem 3.3 that the solution set Π of problem (QVH) is nonempty and compact. It remains to prove the convexity of Π.

Let u1,u2∈Π and λ∈(0,1), so we have the following:

Since Q-B+∂⁡J is monotone, for each v*∈Q⁢(v) we may write that

Setting uλ:=λ⁢u1+(1-λ)⁢u2∈K and J∘⁢(v;v-uλ)=〈ξv,v-uλ〉 with ξv∈∂⁡J⁢(v), by the convexity of ϕ and the monotonicity of Q-B+∂⁡J, we obtain