We show the existence of at least one Lipschitz solution for ex- tensions of convex sweeping processes in reflexive smooth Banach spaces

Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 168, pp. 1–6.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS FOR CONVEX SWEEPING PROCESSES IN p-UNIFORMLY SMOOTH AND q-UNIFORMLY

CONVEX BANACH SPACES

MESSAOUD BOUNKHEL

Abstract. We show the existence of at least one Lipschitz solution for ex- tensions of convex sweeping processes in reflexive smooth Banach spaces. Our result is proved under a weaker assumption on the moving set than those in [3], and using a different discretization.

1. Main result

Bounkhel and Al-yusof [3] studied the following extension of the convex sweeping processes from Hilbert spacesH to reflexive smooth Banach spacesX:

(SP) Findu: [0, T]→X such that u(t) =u₀+Rt 0u(s)ds,˙

−d

dt(J(u(t)))∈N(C(t);u(t)) a.e. in [0, T] andu(t)∈C(t),∀t∈[0, T], where J :X →X^∗ is the duality mapping defined fromX into X^∗ (see Section 2 for the definition).

Clearly, (SP) coincides with the well known convex sweeping process introduced and studied in [8] in the Hilbert space setting in whichJ is the identity mapping.

The authors in [3] proved the following theorem.

Theorem 1.1. Let p, q >1, X be ap-uniformly convex and q-uniformly smooth Banach space, T > 0, I = [0, T] and C : I ⇒ X be a set-valued mapping closed convex values satisfying for any t, t⁰ ∈I and anyx∈X

|(d^V_C(t0))^1/q0(ψ)−(d^V_C(t))^1/q0(φ)| ≤λ|t⁰−t|+γkψ−φk, (1.1) whereλ, γ >0, andq⁰= _q−1^q . Assume that

J(C(t))⊂K,∀t∈I for some convex compact setK inX^∗. (1.2) Then(SP)has at least one Lipschitz solution

They proved the existence of solutions under the Lipschitz continuity of the function (t, ψ) 7→ (d^V_C(t))^1/q0(ψ) defined on I×X^∗, and under the compactness assumption (1.2). Using a different discretization we prove the previous theorem

2000Mathematics Subject Classification. 34A60, 49J53.

Key words and phrases. Uniformly smooth and uniformly convex Banach spaces;

state dependent sweeping process; generalized projection; duality mapping.

c

2012 Texas State University - San Marcos.

Submitted February 14, 2012. Published October 4, 2012.

1

under the boundedness of C and the compactness of their values which is clearly weaker than the compactness assumption (1.2), and under the Lipschitz continuity of the usual distance functiont7→(d_C(t))^1/q0(u), for allu∈X, defined on Iwhich is easier to handle with, than the function used in (1.1). Although, both Lipschitz assumptions coincide in the Hilbert space setting, in the case of Banach spaces the Lipschitz continuity of the distance function is easier to be checked than (1.1).

Before proving our main result in Theorem 3.1, we recall from [3] some needed concepts and results and for more details we refer the reader to [3] and the references therein.

2. Preliminaries

LetX be a Banach space with topological dual spaceX^∗. We denote bydS the usual distance function toS; i.e., dS(x) := inf_u∈Skx−uk. Let S be a nonempty closed convex set ofX and ¯xbe a point inS. The convex normal cone ofS at ¯xis defined by (see for instance [6])

N(S; ¯x) ={ϕ∈X^∗:hϕ, x−¯xi ≤0 for allx∈S}. (2.1) The normalized duality mappingJ :X ⇒X^∗ is defined by

J(x) ={j(x)∈X^∗:hj(x), xi=kxk²=kj(x)k²}.

Many properties of the normalized duality mapping J have been studied. For the details, one may see the books [1, 10, 11]. LetV :X^∗×X→Rbe defined by

V(ϕ, x) =kϕk²−2hϕ, xi+kxk², for anyϕ∈X^∗ andx∈X.

Based on the functionalV, a setπS(ϕ) of generalized projections ofϕ∈X^∗ onto S is defined as follows (see [2]).

Definition 2.1. LetS be a nonempty subset ofX andϕ∈X^∗. If there exists a point ¯x∈S satisfying

V(ϕ,x) = inf¯

x∈SV(ϕ, x),

then ¯x is called a generalized projection of ϕ onto S. The set of all such points is denoted by πS(ϕ). When the spaceX is not reflexive πS(ϕ) may be empty for some elementsϕ∈X^∗ even whenS is closed and convex (see [7, Example 1.4]).

The two following propositions are needed in the proof of the main theorem. For their proofs we refer the reader to [5, 9] respectively.

Proposition 2.2. LetSbe a nonempty closed convex subset ofXandx∈S. Then

∂d_S(x) =N_S(x)∩B.

Proposition 2.3. For a nonempty closed convex subset S of a reflexive smooth Banach spaceX andu∈S, the following assertions are equivalent:

(i) ¯x∈S is a projection ofuontoS, that is ¯x∈P_S(u);

(ii) hJ(u−x), x¯ −xi ≤¯ 0 for allx∈S;

(iii) J(u−x)¯ ∈N(S; ¯x).

Assume now thatX isp-uniformly convex andq-uniformly smooth Banach space and let S be closed nonempty set in X. Recall the definition of the function d^V_S :X^∗→[0,∞[, given by d^V_S(ϕ) = inf_x∈SV(ϕ, x). Clearly, in Hilbert spaces,d^V_S coincides withd²_S. We need the two following lemmas proved in [3].

Lemma 2.4. Let p, q > 1, X be a p-uniformly convex and q-uniformly smooth Banach space, and let S be a bounded set. Then there exist two constants α > 0 andβ >0 so that αkx−yk^p≤V(J(x), y)≤βkx−yk^q, for allx, y∈S.

Proposition 2.5. IfSis a bounded set inX, thend^V_S(ϕ)≤β(dS(J^∗(ϕ)))^q, whereβ depends on the bound ofSand onϕ. As a consequence, for setsS1andS2inXand X^∗ bounded byl1 andl2 respectively, we have d^V_S(ϕ)≤β(dS(J^∗(ϕ)))^q, for allϕ∈ S2, whereβ depends onl1 andl2.

The following proposition is taken from [1].

Proposition 2.6. Let p≥2 and let X be ap-uniformly convex and q-uniformly smooth Banach space. The duality mapping J :X →X^∗ is Lipschitz on bounded sets; that is,

kJ(x)−J(y)k ≤C(R)kx−yk, for allkx≤R,kyk ≤R.

Here C(R) := 32Lc²₂(q−1)⁻¹ andc2= max{1, R} and1< L <1.7. The Lipschitz continuity on bounded sets of the duality mappingJ_∗ onX^∗, follows from the fact that X^∗ isp⁰-uniformly convex andq⁰-uniformly smooth Banach space withp⁰ and q⁰ are the conjugate numbers ofpandq respectively; i.e.,p⁰=_p−1^p ,q⁰= _q−1^q .

The following proposition summarizes some results proved in [4, 7].

Proposition 2.7. LetX be a reflexive Banach space with dual spaceX^∗ andS be a nonempty, closed and convex subset ofX. The following properties hold:

(π1) π_S(ϕ)6=∅, for anyϕ∈X^∗;

(π2) If X is also smooth, then ϕ ∈N(S,x), if and only if, there exists¯ α > 0 such that ¯x∈π_S(J(¯x) +αϕ).

3. Main result

Now, we are ready to prove our main result in the following theorem.

Theorem 3.1. Instead of (1.1) and (1.2) in Theorem 1.1, assume that C is bounded with compact values and that

|(dC(t⁰))^p/q(u)−(d_C(t))^p/q(u)| ≤λ|t⁰−t|. (3.1) Then(SP)has at least one Lipschitz solution.

Proof. Assume thatT = 1. Consider∀n∈N the following partition ofI I_n,i= (t_n,i, t_n,i+1], t_n,i= i

n, 0≤i≤n−1, I_n,0={0}.

Putµn = 1/n. Fixn≥2. Define by induction u_n,0=u₀∈C(0);

un,i+1∈π(C(tn,i+1);un,i), for 0≤i≤n−1, and

un(t) :=J^∗(u^∗_n(t)) u^∗_n(t) :=J(u_n,i) +(t−t_n,i)

µn

(J(u_n,i+1)−J(u_n,i)), for allt∈I_n,i

andu^∗_n(0) =J(u0). The construction is well defined since the generalized projection π exits by Proposition 2.7. Clearly u^∗_n and un are continuous on all I and u^∗_n is differentiable onI\ {t_n,i} and ˙u^∗_n(t) =^J(un,i+1_µ^)−J(un,i)

n , for allt∈I\ {t_n,i}.

Let us find an upper bound estimate for the expression kJ(un,i+1)−J(un,i)k.

First, we have to point out that the sequence uⁿ_i is bounded by some l because the set-valued mapping C is bounded. Now, since X is q-uniformly smooth and p-uniformly convex and the sequenceuⁿ_i is bounded byl, there exist some constants αandβ depending onl such that

αkun,i+1−un,ik^p≤V(J(un,i), un,i+1)≤βkun,i+1−un,ik^q, and so by the construction of the sequenceuⁿ_i and Proposition 2.5 we obtain

αku_n,i+1)−u_n,ik^p≤d^V_C(t

n,i+1)(J(u_n,i))≤βd^q_C(t

n,i+1)(u_n,i) and so by the Lipschitz continuity in (3.1) we obtain

(α

β)^1pkun,i+1)−u_n,ik ≤d^q/p_C(t

n,i+1)(u_n,i)−d^q/p_C(t

n,i)(u_n,i)

≤λ|tn,i+1−tn,i|=λµn, and so

kun,i+1)−un,ik ≤¯λµn,

where ¯λ= (_α^β)^1pλ. Using now the Lipschitz property of the duality mappingJ in Proposition 2.6, we can write

kJ(un,i+1)−J(un,i)k ≤C(l)kun,i+1−un,ik ≤C(l)¯λµn.

This inequality ensures the Lipschitz continuity ofu^∗_non allIwith ratioδ:=C(l)¯λ.

Using the characterization of the normal cone, in terms of the generalized projection πprojection operator stated in Proposition 2.7, we can write for a.e. t∈I

J(un,i+1)−J(un,i)∈ −N(C(tn,i+1);un,i+1), which ensures together with Proposition 2.2 that

−J(u_n,i+1)−J(u_n,i) µn

∈δ∂d_C(tn,i+1)(u_n,i+1).

Define now onIn,i the functionsθn:I→I byθn(0) = 0, and θn(t) =tn,i+1, for allt∈In,i. Then the above inclusion becomes

−u˙^∗_n(t)∈δ∂d_C(θn(t))(u_n(θ_n(t))). (3.2) Now, let us prove that the sequence (un) has a convergent subsequence. Clearly, we have B ={un;n≥2} is equi-Lipschitz and bounded. So it remains to prove that B(t) ={un(t);n≥2} is relatively compact inX, for allt∈I. By construction we have

u_n(θ_n(t))∈C(θ_n(t)), ∀t∈I and alln≥2, (3.3) and hence by the Lipschitz property ofd^p/q_C and the equi-Lipschitz property ofu_n we can write

d^p/q_C(t)(u_n(t)) =d^p/q_C(t)(u_n(t))−d^p/q_C(θ

n(t))(u_n(θ_n(t))≤λµ_n+kun(θ_n(t))−u_n(t)k

≤(λ+δ)µ_n.

Assume by contradiction thatB(t0) is not relatively compact inX for somet0∈I.

So,γ(B(t0))≥2¯δ >0, for some ¯δ∈(0,1]. Fix nown0 ∈Nsuch that µn ≤µn₀ <

(^δ₂^¯)^p/q

λ+δ , for alln≥n₀. So

un(t)∈C(t) + (λ+δ)^q/pµ^q/p_n0 B, for alln≥n0and allt∈I, which implies

B(t)⊂C(t) + (λ+δ)^q/pµ^q/p_n0 B, for allt∈I.

Then the properties ofγ and the compactness of the values ofC imply γ(B(t0)) =γ({un(t0) :n≥n0})≤γ((C(t0)) +γ((λ+δ)^q/pµ^q/p_n0 B)

≤2(λ+δ)^q/pµ^q/p_n0 <δ,¯

which is a contradiction. Therefore, the setB(t) is relatively compact inX for any t ∈ I. Thus, Arzela-Ascoli theorem concludes that (u_n) has a subsequence (still denoted u_n) converging uniformly to some u. Since lim_nθ_n(t) = t, we can write lim_nu_n(θ_n(t)) = lim_nu_n(t) =u(t) uniformly onI. So the sequenceu^∗_n=J(u_n) will converge uniformly tou^∗=J(u) onI, sinceJ is uniformly continuous on bounded sets. We also have ( ˙u^∗_n) converges weakly star inL^∞(I, X^∗) to somew. So, by the reflexivity and the separability of the spaceX, we can write

u^∗(t) =J(u(t)) = lim

n u^∗_n(t) = lim

n

u^∗_n(0) + Z t

0

˙ u^∗_n(s)ds

=u₀+ Z t

0

w(s)ds.

Hence ˙u^∗(t) = _dt^dJ(u(t)) =w(t) a.e. onI. Let us prove thatuis the solution of our problem. First, we have to prove thatu(t)∈C(t), for allt∈I. Using now the Lipschitz property of the functiont7→d^q/p_C(t) to write for allt∈I

d^q/p_C(t)(u_n(θ_n(t)) =d^q/p_C(t)(u_n(θ_n(t))−d^q/p_C(θ

n(t))(u_n(θ_n(t))

≤λ|θn(t)−t| ≤λµn, and so

dC(t)(u(t)) =dC(t)(un(θn(t)) +kun(θn(t)−u(t)k

≤(λµn)^p/q+kun(θn(t))−u(t)k →0,

asn→ ∞, by the fact that lim_nu_n(θ_n(t)) =u(t) uniformly onI. So the closedness of the setC(t) ensuresu(t)∈C(t), for allt∈I. Going back to (3.2) we have

−u˙^∗_n(t))∈N(C(θn(t));un(θn(t))), a.e. onI.

So, Proposition 2.2 ensures for a.e. t∈I,

h−u˙^∗_n(t));x−u_n(θ_n(t))i ≤0, ∀x∈C(θ_n(t)). (3.4) Using the fact that ˙u^∗_nconverges to_dt^dJ(u(·)) in the weak star topology ofL^∞(I, X^∗), we can pass to the limit in (3.4) to obtain

h−d

dtJ(u(t));x−u(t)i ≤0, ∀x∈C(t), a.e. onI. (3.5) Indeed, fixt∈I, for which ˙u^∗_n(t) exists and converges weakly to _dt^dJ(u(t)), and let xbe any element inC(t). Then, we have

x∈C(θ_n(t)) + (λµ_n)^q/pB;

that is, x =yn(t) + (λµn)^q/pbn, with bn ∈B and yn(t) ∈ C(θn(t)). Hence (3.4) yields

h−d

dtJ(u(t)), x−u(t)i

=h−d

dtJ(u(t)) + ˙u^∗_n(t)), x−u(t)i+h−u˙^∗_n(t)), x−u(t)i

=h−d

dtJ(u(t)) + ˙u^∗_n(t)), x−u(t)i+h−u˙^∗_n(t)), u_n(θ_n(t))−u(t)i +h−u˙^∗_n(t), y_n(t)−u_n(θ_n(t))i+h−u˙^∗_n(t)),(λµ_n)^q/pb_ni

≤ hu˙^∗_n(t))− d

dtJ(u(t)), x−u(t)i+λ(λµn)^q/p+λkun θn(t)−u(t) k →0 asn→ ∞. So,

h−d

dtJ(u(t)), x−u(t)i ≤0, for allx∈C(t), (3.6) which by Proposition 2.2 gives

− d

dtJ(u(t))∈N(C(t);u(t)), a.e. onI (3.7)

and hence the proof is complete.

Acknowledgments. The author extends his appreciation to the Deanship of Sci- entific Research at King Saud University for funding the work through the research group project No. RGP-VPP-024.

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Messaoud Bounkhel

King Saud University, Department of Mathematics, P.O. Box 2455, Riyadh 11451, Riyadh, Saudi-Arabia

E-mail address:[email protected], bounkhel [email protected]