Electronic Journal of Qualitative Theory of Differential Equations 2012, No. 27, 1-10;http://www.math.u-szeged.hu/ejqtde/
EXISTENCE RESULTS FOR SECOND ORDER CONVEX SWEEPING PROCESSES IN p-UNIFORMLY SMOOTH AND
q-UNIFORMLY CONVEX BANACH SPACES
MESSAOUD BOUNKHEL
Abstract. In a previous work the authors proved under a complex assump- tion on the set-valued mapping, the existence of Lipschitz solutions for second order convex sweeping processes inp-uniformly smooth andq-uniformly con- vex Banach spaces. In the present work we prove the same result, under a condition on the distance function to the images of the set-valued mapping.
Our assumption is much simpler than the one used in the former paper.
1. Introduction
In [5], the authors studied the following extensions of convex sweeping processes from Hilbert spacesH to reflexive smooth Banach spacesX:
(SSP) Find T >0,x∗: [0, T]→J(cl(ν0)) andu∗: [0, T]→X∗ such that
u∗(0) =J(u0), J∗(u∗(t))∈K(J∗(x∗(t))), for allt∈[0, T];
x∗(t) =J(x0) + Z t
0
u∗(s)ds, for allt∈[0, T];
(u∗)′(t)∈ −N(K(J∗(x∗(t)));J∗(u∗(t))) a.e. on [0, T],
wherex0∈X,u0∈K(x0),ν0:=J∗(ν0∗),ν0∗ be an open neighborhood ofJ(x0) in X∗,K:cl(ν0)⇉X be a set-valued mapping taking nonempty closed convex values inX, andJ :X →X∗ is the duality mapping defined fromX intoX∗(see Section 2 for the definitions). The mappingx∗is called a solution of (SSP)
Clearly, (SSP) coincides with the well known second order convex sweeping process studied in many works (see for instance [4, 7, 10] and the reference therein) in the Hilbert space setting in whichJ is the identity mapping. The authors in [5] proved the following theorem.
Theorem 1.1. Let p, q >1,X be a separable Banach space which isp-uniformly convex and q-uniformly smooth, and let K : cl(ν0)⇉X be a set-valued mapping taking nonempty closed convex values in X and satisfying: for any x, x′ ∈ cl(ν0) and any ϕ, ϕ′∈X∗,
(1.1) |(dVK(x′))q−1q (ϕ′)−(dVK(x))q−1q (ϕ)| ≤λkJ(x′)−J(x)k+γkϕ′−ϕk.
Assume thatJ(K(x))⊂ L, for some convex compact setL ∈X∗. Then (SSP) has at least one Lipschitz solution.
1991Mathematics Subject Classification. 34A60, 49J53 .
Key words and phrases. Uniformly smooth Banach spaces, Uniformly convex Banach spaces, Sweeping process, Generalized projection, Duality mapping.
THIS PROJECT WAS SUPPORTED BY KING SAUD UNIVERSITY, DEANSHIP OF SCIENTIFIC RESEARCH, COLLEGE OF SCIENCE RESEARCH CENTER.
EJQTDE, 2012 No. 27, p. 1
They proved the existence of solutions under (1.1) the Lipschitz behavior of the function (x, ψ) 7→ (dVK(J∗(x)))q−1q (ψ) defined on X∗×X∗, where dVS(ψ) :=
infx∈SV(ψ, x) and V(ϕ, x) =kϕk2−2hϕ, xi+kxk2. In this paper we prove the previous theorem under the Lipschitz continuity of the functionx7→d
q p
K(x)(u),∀u∈ X, (with constant depending onusee Theorem 3.1), which is defined onX and is easier to handle with, than the function (y, ψ)7→(dVK(J∗(y)))q−1q (ψ) used in (1.1).
Also, in the case of Banach spaces (not necessarily Hilbert) the Lipschitz assumption (3.1) is much easier to be checked than (1.1).
Before proving our main result in Theorem 3.1, we recall from [5] some needed concepts and results and for more details we refer the reader to [5] 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) := infu∈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 [11])
(2.1) N(S; ¯x) ={ϕ∈X∗:hϕ, x−xi ≤¯ 0 for allx∈S}.
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, 14, 15]. Let V :X∗×X →Rbe defined by
V(ϕ, x) =kϕk2−2hϕ, xi+kxk2, for any ϕ∈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 ¯xis called a generalized projection ofϕontoS. The set of all such points is denoted byπS(ϕ). When the spaceXis not reflexiveπS(ϕ) may be empty for some elementsϕ∈X∗ even whenS is closed and convex (see Example 1.4. in [12]).
The following proposition is needed in the proof of the main theorem. For its proof we refer the reader to [13].
Proposition 2.2. 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 of uontoS, that isx¯∈PS(u);
ii) hJ(u−x), x¯ −xi ≤¯ 0 for allx∈S;
iii) J(u−x)¯ ∈N(S; ¯x).
EJQTDE, 2012 No. 27, p. 2
Assume now thatX isp-uniformly convex andq-uniformly smooth Banach space (for their definitions we refer the reader to the reference [5] and the references therein) and letSbe closed nonempty set inX. Recall the definition of the function dVS :X∗ →[0,∞[, given bydVS(ϕ) = infx∈SV(ϕ, x). Clearly, in Hilbert spacesdVS coincide withd2S. We need the two following lemma proved in [5] respectively.
Lemma 2.3. 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−ykp≤V(J(x), y)≤βkx−ykq, for allx, y ∈S.
Proposition 2.4. IfSis a bounded set inX, thendVS(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 dVS(ϕ)≤β(dS(J∗(ϕ)))q, for allϕ∈ S2, whereβ depends onl1 andl2.
The following lemma is taken from [1].
Proposition 2.5. Let p ≥ 2, q > 1 and let X be a p-uniformly convex and q- uniformly smooth Banach space. The duality mappingJ :X →X∗ is Lipschitz on bounded sets, that is,
kJ(x)−J(y)k ≤C(R)kx−yk, for allkxk ≤R,kyk ≤R.
Here C(R) := 32Lc22(q−1)−1 andc2= max{1, R} and1< L <1.7.
Let us mention that the Lipschitz continuity on bounded sets of the duality mapping J∗ on X∗, is not ensured in general by Proposition 2.5 because X∗ is p′-uniformly convex andq′-uniformly smooth Banach space withp′= p−1p ,q′= q−1q and by the fact thatp′ ∈[1,2] wheneverp≥2. However,J∗is uniformly continuous on bounded sets.
The following proposition summarizes two important results proved respectively in [12, 6]
Proposition 2.6. LetX be a reflexive Banach space with dual spaceX∗ andS be a nonempty, closed and convex subset of X. The following properties hold:
(π1) πS(ϕ)6=∅, for anyϕ∈X∗;
(π2) If X is also smooth, thenϕ∈N(S,x), if and only if,¯ ∃α >0 so thatx¯ ∈ πS(J(¯x) +αϕ).
We end this section with the following lemma needed in our proofs (for the proof we refer the reader for instance to [9]).
Lemma 2.7. LetX be a reflexive Banach space and letC:I→X be a set-valued mapping with nonempty closed convex values. Then the functional I:v7→I(v) :=
RT
0 δ∗C(t)(v(t))dt from X∗ to R is weakly lower semi-continuous in the following sense: for any (vn) a sequence of mappingsvn :I →X∗ such that vn →v∗ in the weak star topology ofL∞(I, X∗), we have
Z T 0
δ∗C(t)(v∗(t))dt≤lim inf
n
Z T 0
δ∗C(t)(vn(t))dt.
EJQTDE, 2012 No. 27, p. 3
3. Main result.
Now, we are ready to prove the main result in the following theorem.
Theorem 3.1. Instead of (1.1) in Theorem 1.1, assume that
(3.1) |d
q p
K(x′)(u)−d
q p
K(x)(u)| ≤λ(u)kx′−xk, for allu∈X,
with λ : X → [0,∞) is bounded on bounded sets. Then (SSP) has at least one Lipschitz solution.
Proof. We give the proof in four steps.
Step 1. Construction of approximants.Letµ >0 such thatJ(x0) +µB∗⊂ν0∗ and letl >0 such thatL⊂lB∗. LetT ∈(0,µl) and putI:= [0, T]. For eachn∈N, we consider the partition ofI given byIn,i:= [tn,i, tn,i+1), for alli= 0, . . . , n−1, withtn,i=iµn,µn:= Tn, andIn,n:={T}.
For everyn∈Nwe define the following approximating mappings on each interval In,ias follows
(3.2)
u∗n(t) :=J(un,i), un(t) =J∗(u∗n(t)) =un,i, x∗n(t) =J(x0) +
Z t 0
u∗n(s)ds, xn(t) =J∗(x∗n(t)), whereun,0=u0 and for alli= 0, . . . , n−1 the point un,i+1 is given by (3.3) un,i+1=πK(xn(tn,i+1))(J(un,i)).
As
x∗n(tn,1) =J(x0) + Z tn,1
0
u∗n(s)ds⊂J(x0) +ltn,1B∗⊂J(x0) +µB∗⊂ν0∗, so
xn(tn,1) =J∗(x∗n(tn,1))⊂J∗(ν0∗) =ν0,
and as K has nonempty closed convex values, by Proposition 2.6 one can choose a pointun,1∈πK(xn(tn,1))(J(un,0)). Similarly, we can define, by induction, all the points (un,i)i
Let us defineθn(t) :=tn,i, andρn(t) :=tn,i+1ift∈In,i. Then, the definition of xn(·) andun(·) yield for allt∈I,
(3.4) un(t)∈K(xn(θn(t)))⊂lB.
So, the mappingsx∗n(·) are Lipschitz with ratio l and they are also equibounded, withkx∗nk∞≤ kx0k+lT. Hence the mappingsxn(·) are continuous.
Observe also that for alln∈Nandt∈I one has
(3.5) xn(t)∈ν0.
Indeed, the definition ofxn(·) andun(·) ensure that, for allt∈I, x∗n(t) =J(x0) +
Z t 0
u∗n(s)ds⊂J(x0) +ltB∗⊂J(x0) +µB∗⊂ν0∗, and so
xn(t) =J∗(x∗n(t))⊂J∗(ν0∗) =ν0,
EJQTDE, 2012 No. 27, p. 4
and henceK(xn(t)) is well defined for allt∈I.
Now we define the piecewise affine approximants fromI toX∗ as follows (3.6) vn∗(t) :=J(un,i) +µ−n1(t−tn,i)(J(un,i+1)−J(un,i)), ift∈In,i. Define the mappingv from ItoX by
(3.7) vn(t) =J∗(vn∗(t)), for allt∈I.
Observe that vn∗(θn(t)) = J(un,i) and vn(θn(t)) = un,i, for all i= 0, . . . , n and so by (3.3),(3.5), and (3.6) one has
vn(θn(t))∈K(xn(tn,i)) =K(xn(θn(t)))⊂lB.
Now, we check that the mappingsvn∗ are equi-Lipschitz. Let us first find an upper bound estimate for the expressionkJ(un,i+1)−J(un,i)k. Clearly, the sequence (uni) is bounded byl. Consequently,λ(uni) is bounded for anyi, n. Let ¯λbe its bound, that is,λ(uni)≤¯λ, for anyi, n. Now, sinceXisq-uniformly smooth andp-uniformly convex and the sequence (uni) is bounded byl, there exists some constants αand β depending onl such that
αkun,i+1−un,ikp≤V(J(un,i), un,i+1)≤βkun,i+1−un,ikq, and so by the construction of the sequenceuni and Proposition 2.4 we get
αkun,i+1−un,ikp≤dVK(xn(tn,i+1))(J(un,i))≤βdqK(x
n(tn,i+1))(un,i) and so by the Lipschitz property in (3.1) we obtain
(α
β)1pkun,i+1−un,ik ≤ d
q p
K(xn(tn,i+1))(un,i)−d
q p
K(xn(tn,i))(un,i)
≤ λ(un,i)|xn(tn,i+1)−xn(tn,i)| ≤λl|t¯ n,i+1−tn,i| ≤lλµ¯ n, and so
kun,i+1−un,ik ≤ λµˆ n,
where ˆλ=l(βα)1pλ. Using now the Lipschitz property of the duality mapping¯ J in Proposition 2.5, we can write
(3.8) kJ(un,i+1)−J(un,i)k ≤C(l)kun,i+1−un,ik ≤C(l)ˆλµn. So, for anyt, t′∈In,i one has
kvn∗(t′)−vn∗(t)k=µ−1n |t′−t|kJ(un,i+1)−J(un,i)k ≤C(l)ˆλ|t′−t|.
This inequality, with the continuity ofv∗non (tn,i)i, shows that the mappingsv∗nare equi-Lipschitz on allI with ratioδ:=C(l)ˆλand hence the mappings vn=J∗(v∗n) are uniformly continuous onIbecausevn∗is bounded andJ∗is uniformly continuous on bounded sets. By the definition ofu∗n(·) andvn∗(·) one has
kvn∗(t)−u∗n(t)k ≤ µ−n1|t−tn,i|kJ(un,i+1)−J(un,i)k ≤δµn, and hence
(3.9) kvn∗−u∗nk∞→0.
EJQTDE, 2012 No. 27, p. 5
The definition ofvn(·) given by (3.7) and the relation (3.3) yield
(3.10) vn(θn(t))∈K(xn(θn(t))), for allt∈In,i,(i= 0, . . . , n−1), and by the definition ofvn∗(·), one has for a.e. t∈In,i
(3.11) (vn∗)′(t) =µ−n1(J(un,i+1)−J(un,i)).
So, by the characterization of the convex normal cones stated in Proposition 2.6, we get for a.e. t∈I
(3.12) (v∗n)′(t)∈ −N(K(xn(ρn(t)));vn(ρn(t))).
Indeed, by construction
un,i+1 ∈ πK(xn(tn,i+1))(J(un,i))
= πK(xn(tn,i+1))(J(un,i+1)−[J(un,i+1)−J(un,i)])
⇔ J(un,i+1)−J(un,i)∈ −N(K(xn(tn,i+1));un,i+1)
⇔ µ−1n (J(un,i+1)−J(un,i))∈ −N(K(xn(ρn(t)));vn(ρn(t))).
and hence (3.12) holds.
Step 2. Uniform convergence of the sequences xn(·) and vn(·). Since µ−1n (t−tn,i)≤1, for allt∈In,i andJ(un,i+1), J(un,i)∈L, andLis a convex set inX∗ one gets for allt∈I,
v∗n(t) = J(un,i) +µ−1n (t−tn,i)[J(un,i+1)−J(un,i)]
=
1−t−tn,i µn
J(un,i) +t−tn,i µn
J(un,i+1)∈L.
Thus for every t∈I, the set {v∗n(t) :n∈N} is relatively compact inX∗. On the other hand, it is clear by (3.8) and (3.11) that
k(v∗n)′(t)k ≤δ.
(3.13)
Therefore, this estimate and Theorem 0.3.4 in [3] ensure the existence of a Lipschitz mappingu∗:I→X∗ such that:
• (v∗n) converges uniformly tou∗ onI.
Clearly, we have the weak star convergence of ((v∗n)′) to some limitω inL∞(I, X∗) and easily, we can check thatω= (u∗)′a.e. onI. Indeed, the weak star convergence of ((v∗n)′) toω inL∞(I, X∗) ensures for anyt∈I and anyy∈L1(I, X)
limn h(vn∗)′−ω, yiL∞(I,X∗),L1(I,X)= 0, that is,
limn
Z T 0
h(v∗n)′(s)−ω(s), y(s)iX∗,Xds= 0.
Hereh·,·iL∞(I,X∗),L1(I,X)denotes the dual pairing between the spacesL1(I, X) and L∞(I, X∗), and h·,·iX∗,X denotes the dual pairing between the spacesX andX∗. Fix now anyt∈[0, T] and defineyk :I→X byyk ≡ψ[0,t](·)·ek, where (ek)⊂X EJQTDE, 2012 No. 27, p. 6
is a sequence separating points in X∗ (such sequences exist in reflexive separable Banach spaces). Then for anyk∈Nwe have
hlimn
Z t 0
(v∗n)′(s)ds, ekiX∗,X =h Z t
0
ω(s)ds, ekiX∗,X. This ensures
limn
Z t 0
(vn∗)′(s)ds= Z t
0
ω(s)ds.
Consequently, u∗(t) = lim
n vn∗(t) = lim
n [J(u0) + Z t
0
(vn∗)′(s)ds] =J(u0) + Z T
0
w(s)ds,
and sinceu∗ is absolutely continuous, we deduce thatω= (u∗)′ a.e. onI.
We define now the continuous mappingu:I→X by (3.14) u(t) =J∗(u∗(t)), for allt∈I.
Then, it is clear that (vn) converges uniformly to u, because J∗ is uniformly con- tinuous on bounded sets.
Now, we define the Lipschitz mappingx∗:I→X by (3.15) x∗(t) =J(x0) +
Z t 0
u∗(s)ds, for allt∈I, and the continuous mappingx:I→X is given by
(3.16) x(t) =J∗(x∗(t)), for allt∈I.
Then by the definition ofx∗n one obtains for allt∈I, kx∗n(t)−x∗(t)k=k
Z t 0
(u∗n(s)−u∗(s))dsk ≤Tku∗n−u∗k∞, and by (3.9) we get
(3.17) kx∗n−x∗k∞ ≤Tku∗n−v∗nk∞+Tkvn∗−u∗k∞→0 asn→ ∞.
Hence (x∗n) converges uniformly to x∗ on I and so (xn) = (J∗(x∗n)) converges uniformly toJ∗(x∗) =xonIbecauseJ∗ is uniformly continuous on bounded sets.
This completes the second step.
Step 3. Existence of a solution.First observe that (x∗n◦θn), (x∗n◦ρn) and (vn∗◦θn), (v∗n◦ρn) converge uniformly on Ito x∗ and u∗ respectively. Recall now that vn(ρn(t)) ∈ K(xn(ρn(t))), for all t ∈ I and n ∈ N. It follows then by our assumptions that
d
p q
K(x(t))(vn(ρn(t))) = d
p q
K(x(t))(vn(ρn(t)))−d
p q
K(xn(ρn(t)))(vn(ρn(t)))
≤ λkx(t)¯ −xn(ρn(t))k
≤ λkx(t)¯ −xn(t)k+ ¯λkxn(t)−xn(ρn(t))k
≤ λkJ¯ ∗(x∗(t))−j∗(x∗n(t))k+ ¯λkj∗(x∗n(t))−j∗(x∗n(ρn(t)))k.
EJQTDE, 2012 No. 27, p. 7
Hence, by the fact thatkx∗n−x∗k∞→0 andkx∗n(t)−x∗(ρn(t))k ≤lµn→0, and the unifrom continuity ofJ∗, we obtain
dK(x(t))(vn(ρn(t)))→0, and so
dK(x(t))(u(t)) ≤ dK(x(t))(vn(ρn(t))) +kvn(ρn(t))−u(t)k
≤ dK(x(t))(vn(ρn(t))) +kvn(ρn(t))−vn(t)k+kvn(t)−u(t)k →0, which ensures by the closedness of the values of K, that u(t) ∈ K(x(t)), for all t∈I.
Now, let us prove that the mappingx∗is a solution of our problem (SSP). Using the weak star convergence of ((v∗n)′) to (u∗)′inL∞(I, X∗) and Lemma 2.7 we obtain
lim inf
n
Z T 0
δ∗K(x(t))(−(v∗n)′(t))dt≥ Z T
0
δK(x(t))∗ (−(u∗)′(t))dt.
Again, we use the weak star convergence of (v∗n)′ to (u∗)′ in L∞(I, X∗) with the uniform convergence ofvn◦ρn touto get
limn
Z T 0
h(vn∗)′(t), vn(ρn(t))idt= Z T
0
h(u∗)′(t), u(t)idt.
Therefore,
Z T 0
hδ∗K(x(t))(−(u∗)′(t)) +h(u∗)′(t), u(t)ii dt≤ lim inf
n
Z T 0
hδK(x(t))∗ (−(vn∗)′(t)) +h(vn∗)′(t), vn(ρn(t))ii (3.18) dt.
By (3.12) and the definition of the normal cone we have
h−(vn∗)′(t);y−vn(ρn(t))i ≤0, ∀y∈K(xn(ρn(t))), a.e. t∈I.
(3.19)
Fix anytin Ifor which (3.19) holds and let anyv∈K(x(t)). By (3.1) we have d
p q
K(xn(ρn(t)))(v) = |d
p q
K(xn(ρn(t)))(v)−d
p q
K(x(t))(v)|
≤ λkx¯ n(ρn(t))−x(t)k
≤ λ¯[kxn(ρn(t))−xn(t)k+kxn(t)−x(t)k]
≤ λ¯[kxn◦ρn−xnk∞+kxn−xk∞]. (3.20)
Putλn:= ¯λ[kxn◦ρn−xnk∞+kxn−xk∞]
q
p. Clearlyλn→0 asn→ ∞by the uniform convergence of the sequence (xn) tox. Thus,
dK(xn(ρn(t)))(v) ≤ λn, (3.21)
which ensures the existence ofwn ∈K(xn(ρn(t))) with kv−wnk ≤λn. Hence by (3.19) we have
h−(vn∗)′(t);wn−vn(ρn(t))i ≤0 (3.22)
EJQTDE, 2012 No. 27, p. 8
and so by using (3.13) we obtain
h−(vn∗)′(t);v−vn(ρn(t))i = h−(vn∗)′(t);wn−vn(ρn(t))i+h−(vn∗)′(t);v−wni
≤ k(vn∗)′(t)kkv−wnk ≤δλn. Thus
h−(vn∗)′(t);vi+h(v∗n)′(t);vn(ρn(t))i ≤δλn,∀v∈K(x(t)), a.e. t∈I.
Taking the supremum onv overK(x(t)) and integrating overIwe get Z T
0
h
δ∗K(x(t))(−(v∗n)′(t)) +h(v∗n)′(t);vn(ρn(t))ii
dt≤δT λn. Hence
lim inf
n
Z T 0
h
δK(x(t))∗ (−(vn∗)′(t)) +h(vn∗)′(t);vn(ρn(t))ii dt≤0, and so, combining with (3.18) we obtain
Z T 0
h
δ∗K(x(t))(−(u∗)′(t)) +h(u∗)′(t), u(t)ii dt≤0, that is,
Z T 0
δK(x(t))∗ (−(u∗)′(t))dt≤ Z T
0
h−(u∗)′(t), u(t)idt.
Sinceu(t)∈K(x(t)), the last inequality becomes equality and we write Z T
0
δK(x(t))∗ (−(u∗)′(t))dt= Z T
0
h−(u∗)′(t), u(t)idt, and hence for a.e. t∈I we have
δK(x(t))∗ (−(u∗)′(t)) =h−(u∗)′(t), u(t)i, that is,
h−(u∗)′(t), wi ≤ h−(u∗)′(t), u(t)i,∀w∈K(x(t)), a.e. t∈I.
Thus,
(u∗)′(t)∈ −N(K(J∗(x∗(t)));J∗(u∗(t))), a.e. onI,
that is, x∗ is a solution of (SSP) and so the proof of the theorem is complete.
Acknowledgments.This Project was supported by King Saud Univer- sity, Deanship of Scientific Research, College of Science Research Center. The author would like to thank the referee for (her or his) helpful sug- gestions and for the pertinent and serious remarks improving the final version of the paper.
EJQTDE, 2012 No. 27, p. 9
References
1. Y. Alber and I. Ryazantseva,Nonlinear ill-posed problems of monotone type. Springer, Dor- drecht, 2006.
2. Y. Alber,Generalized projection operators in Banach spaces: properties and applications, Funct. Different. Equations 1 (1), 1-21 (1994).
3. J-P. Aubin and A. Cellina,Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.
4. M. Bounkhel, General existence results for second order nonconvex sweeping process with unbounded perturbations. Port. Math. (N.S.) 60 (2003), no. 3, 269-304.
5. M. Bounkhel and R. Al-yusof,First and second order convex sweeping processes in reflexive smooth Banach spaces, Set-Valued Var. Anal. 18 (2010), no. 2, 151–182.
6. M. Bounkhel and R. Al-yusof,Proximal analysis in reflexive smooth Banach spaces, Nonlin- ear Analysis: Theory, Methods & Applications, Volume 73, Issue 7, 1 October 2010, Pages 1921-1939.
7. M. Bounkhel and D. Azzam,Existence results for second order nonconvex sweeping processes, Set-Valued Anal. 12 (2004), no. 3, 291-318.
8. M. Bounkhel and L. Thibault,On various notions of regularity of sets, Nonlinear Anal.:
Theory, Methods and Applications, Vol.48(2002), No. 2, 223-246.
9. C. Castaing,Equations differentielles multivoques avec contraintes sur l’´etat dans les espaces de Banach. S´eminaire d’analyse convexe, expos´e No. 13, Montpellier, 1978.
10. C. Castaing,Quelques probl`emes d’´evolution du second ordre, S´eminaire d’analyse convexe, expos´e No. 5, Montpellier, 1988.
11. F. H. Clarke,Optimization and Nonsmooth Analysis, John Wiley & Sons Inc., New York, 1983.
12. J. Li,The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl. Vol. 306. 2005, 55-71.
13. J. P. Penot and R. Ratsimahalo,Characterizations of metric projections in Banach spaces and applications,Abstr. Appl. 3(1998), no. 1-2, 85-103.
14. W. Takahashi,Nonlinear Functional Analysis, 2000. Yokohama Publishers.
15. M. M. Vainberg,Variational Methods and Method of Monotone Operators in the Theory of Nonlinear Equations, John Wiley, New York, 1973.
(Received March 17, 2011)
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Riyadh, Saudi Arabia
E-mail address: [email protected]
EJQTDE, 2012 No. 27, p. 10
