JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 301319 10.1155/2013/301319 301319 Research Article Eigenvalue for Densely Defined (S+)L Perturbations of Multivalued Maximal Monotone Operators in Reflexive Banach Spaces 0000-0002-0320-0599 Ibrahimou Boubakari Yao Jen-Chih Department of Public Health Western Kentucky University 1906 College Heights Boulevard Bowling Green, KY 42104 USA wku.edu 2013 3 2 2013 2013 07 11 2012 08 01 2013 2013 Copyright © 2013 Boubakari Ibrahimou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let X be a real reflexive Banach space and let X be its dual. Let X be open and bounded such that 0. Let T:XD(T)2X be maximal monotone with 0D(T) and 0T(0). Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem Tx+λPx0, where the operator P:XD(P)X is a single-valued of class (S+)L. The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator P:X2X is multivalued and is investigated.

1. Preliminaries

In what follows we assume that X is a real or complex Banach space and has been renormed such that it and its dual X* are locally uniformly convex. The normalized duality mapping is defined by (1)J(x)={x*X:x*=x,x,x*=x2}. The mapping T:XD(T)2X* is said to be “monotone” if for every x,yD(T), uTx, and vTy we have (2)u-v,x-y0. A monotone operator T is “maximal monotone” if G(T) is maximal in X×X*, when X×X* is partially ordered by inclusion. In our setting, a monotone operator T is maximal if and only if R(T+λJ)=X* for every λ>0. If T:XD(T)2X* is maximal monotone operator, the operator Tt(T-1+tJ-1)-1:XX* is called the Yosida approximant of T and the following, which can be found in [1, page 102] is true.

Lemma 1.

Let T:XD(T)2X* be a maximal monotone operator with 0D(T) and 0T(0). Then

Tt is a bounded maximal monotone mapping with 0=Tt(0) for each t>0;

TtxT{0}x in X as t0 for all xD(T), where T{0}x denotes the element y*Tx of minimum norm;

||Ttx|| as t0 for all xD(T)¯.

Also TtxTJtx, where Jt=I-tJ-1Tt:XX and satisfies: if xconvD(T)¯, then Jtxx in X as t0, where convM denote the convex hull of the set M.

The proof of the next lemma can be found in Kartsatos and Skrypnik ,

Lemma 2.

Let T:XD(T)2X* be maximal monotone such that 0D(T) and 0T(0). Then the mapping (t,x)Ttx is continuous on the set (0,)×X.

An operator T:XD(T)Y, with Y another Banach space, is bounded if it maps bounded subsets of D(T) onto bounded sets. It is compact if it is continuous and maps bounded subsets of D(T) onto relatively compact subsets of Y. It is demicontinuous if for every sequence {xn} such that xnx0 we have TxnTx0.

We say that the operator T:XD(T)2X* satisfies condition (Sq) on a set AD(T) if for every sequence {xn}A such that xnx0X and any yn*Txn, with yn* (some)y*X*, we have xnx0. If A=D(T), then we say that T satisfies (Sq).

Definition 3.

Let X be a separable reflexive Banach space and let L be a dense subspace of X. A mapping P:D(P)XX* is said to be of class (S+)L if, for any sequence of finite dimensional subspaces Fn of L with Un=1Fn¯=X, hX*, {xn}n=1D(P) with xnx0 and (3)limsupnPxn-h,xn0,limnPxn-h,v=0 for all vUn=1Fn, we have xnx0, xD(P) and Px0=h.

If h=0, then we call P a mapping of class (S+)0,L.

Definition 4.

Let X be a separable reflexive Banach space and let L be a dense subspace of X. A multivalued mapping P:D(P)X2X* is said to be of class (S+)L if it satisfies the following conditions:

Px is bounded closed and convex for each xD(P),

P is weakly upper semicontinuous in each finite dimensional space, that is, for each finite dimensional space F of L, FD(P), P:FD(P)2X* is upper semicontinuous in the weak topology,

if for any sequence of finite dimensional subspaces Fn of L with LUn=1Fn¯, hX*, {xn}n=1D(P)L and xnx0 such that (4)limsupnpn-h,xn0,limnpn-h,v=0

for all vUn=1Fn and some pnPxn, then we have xnx0, xD(P) and hPx0.

If h=0, then we call P a mapping of class (S+)0,L.

We will need the following two conditions:

there exist a subspace L of X such that LD(P), L¯=X and that the operator P satisfies condition (S+)L;

there exist a function ϕ:++ which is nondecreasing such that (5)p,x-ϕ(x),pPx,xD(P).

The following lemma can be found in Zeidler [3, page 915].

Lemma 5.

Let T:XD(T)2X* be maximal monotone. Thus the following are true:

{xn}D(T)xnx0 and Txnyny0 imply x0D(T), and y0Tx0;

{xn}D(T)xnx0 and Txnyny0 imply x0D(T), and y0Tx0.

We will need the following lemma from Adhikari and Kartsatos .

Lemma 6.

Assume that the operators T:XD(T)2X* and T0:XD(T0)X* are maximal monotone, with 0D(T)D(T0) and 0T(0)T0(0). Assume further that T+T0 is maximal monotone. Assume that there is a positive sequence {tn} such that tn0, a sequence {xn}D(T0) and a sequence wnT0xn such that xnx0 and Ttnxn+wny0*X*. Then the following are true:

the inequality (6)limnTtnxn+wn,xn-x0<0 is impossible;

if (7)limnTtnxn+wn,xn-x0=0, then x0D(T+T0) and y*(T+T0)x0.

The proof of the following lemma is given in  proof of Theorem 7 but we will repeat it here for completeness and future reference.

Lemma 7.

Let T:XD(T)2X* be a quasibounded maximal monotone operator such that 0T(0). Let {tn}(0,) and {tn}X be such that (8)unS,Ttnun,unS1, where S,S1 are positive constants. Then there exists a number K>0 such that ||Ttnun||K for all n=1,2,.

Proof.

Let (9)wn=Ttnun=(T-1+tnJ-1)-1un. We have that (10)wnTJtnun=Txn,      tnwn=J(un-xn), where xn=Jtnun. Thus, (11)wn,xn=wn,un-tnJ-1wn=wn,un-tnwn,J-1wn=wn,un-tnwn2Ttnun,unS1. From (11) we obtain (12)tnwn2=wn,un-wn,xn. Now since 0T(0) and wnTxn, we have that wn,xn0, which implies tn||wn||2S1. We claim that {wn} is bounded, if not we may assume that ||wn|| and ||wn||||wn||2 for all n. Thus tn||wn||S1 and (13)tnwn=J(un-xn)=un-xn which implies that {xn} is bounded. Now since T is strongly quasibounded, the boundedness of {xn} and {wn,xn} imply the boundedness of {wn}, that is, a contradiction. It follows that {Ttnun} is bounded and the proof is complete.

We denote by Jψ the duality mapping with gauge function ψ. The function ψ:++ is continuous, strictly increasing such that ψ(0)=0 and ψ(r) as r. This mapping Jψ is continuous, bounded, surjective, strictly and maximal monotone, and satisfies condition (S+). Also, (14)Jψx,x=ψ(x)x,Jψx=ψ(x),xX for these facts we refer to Petryshyn [6, pages 32-33 and 132].

2. The Eigenvalue Result

In this section we are using the topological degree developed by Kartsatos and Quarcoo in . The following result will also improve Theorem 4 of Kartsatos and Skrypnik in  since we are no longer assuming that the perturbation is quasibounded.

Theorem 8.

Let Ω be open and bounded with 0Ω. Assume that the operator T:XD(T)2X* is maximal monotone with 0D(T) and 0T(0). Assume that the operator P:XD(P)X*, with LD(P), satisfies condition (S+)L and (P2). Let ϵ, ϵ0, and Λ be positive numbers. Assume that

there exists λ(0,Λ] such that (15)Tx+λPx+ϵJψx0 has no solution in D(T+P)Ω.

Then

there exists (λ0,x0)(0,Λ]×(D(T+P)Ω) such that (16)Tx0+λ0Px0+ϵJψx00;

if 0T(D(T)Ω), T satisfies (Sq) on Ω, and property (𝒫) is satisfied for every ϵ(0,ϵ0], then there exists (λ0,x0)(0,Λ]×(D(T+P)Ω) such that Tx0+λ0Px00.

Proof.

(i) Assume that (𝒫) is true and that (16) is false. We consider the homotopy inclusion (17)H(t,x)Tx+tΛPx+ϵJψx0,t[0,1]. It is clear that this inclusion has no solution xD(H(t,·))Ω for t(0,1], because tΛ(0,Λ] and our assumption about (16). It is also true for t=0 because 0(T+ϵJψ)(0) and the operator T+ϵJψ is strictly monotone, hence one-to-one. We are going to show that H(t,x) is an admissible homotopy for the degree in . To this end, we set Tt=T, PttΛP+Jψ and we recall the operators Tt,s=Ts(T-1+sJ-1)-1:XX*, s>0, and we set Jt,s=Js=I-sJ-1Tt,s=I-sJ-1Ts:XX. We have D(H(0,·))=D(T) and D(H(t,·))=D(T+P), t(0,1]. We also set Dt=D(tΛP)=D(tP). We have D0=X and Dt=D(P) for t(0,1]. Let Ω be an open and bounded subset of X. We know that the equation (18)Ttx+Ptx0 has no solution xD(H(t,·))Ω for any t[0,1]. Now we consider the equation (19)Tsx+tΛPx+ϵJψ=0, and we show that there exists s1>0 such that (20)0(Ts+tΛP+ϵJψ)(DtΩ),(s,t)(0,s1]×[0,1]. Assume that this is not true, then there exist {sn}(0,) with sn0, {tn}[0,1], with tnt0, {xn}Ω with xnx0, and (21)Tsnxn+tnΛPxn+ϵJψxn=0. Clearly we cannot have tn=0 for any n, since (Tsn+Jψ)(0)=0 and the operator Tsn+Jψ is strictly monotone, hence one-to-one. Thus tn>0, for all n. From (21) we have that (22)Tsnxn,xn=-tnΛPxn,xn-ϵJψxn,xntnΛϕ(xn)-ϵψ(xn)Λϕ(S1)+ϵψ(S1), where S1 is the bound of {xn}. Thus we have the boundedness of Tsnxn,xn. Using Lemma 7, we have the boundedness of Tsnxn. We may thus assume that Tsnxnh1*. From (21) we also have that the sequence {Pxn} is bounded and we may assume that Pxnh0*.

If t0=0, then from (23)ϵψ(xn)xn=ϵJψxn,xn-tnΛPxn,xn0 we obtain xn0Ω, which is a contradiction to 0Ω. Hence it follows that t0>0. Since {Jψxn} is bounded, we may assume that Jψxnh2* and we have t0Λh0*=-h1*-ϵh2*.

From (21) it follows that (24)Tsnxn+tnΛPxn+ϵJψxn,x=0,xUn=1Fn. Thus (25)Tsnxn+tnΛPxn+ϵJψxn,xn=0,limnTsnxn+tnΛPxn+ϵJψxn,v=0,vUn=1Fn. Now (26)tnΛPxn+h1*+ϵh2*,xn=Tsnxn+tnΛPxn+ϵJψxn,xn-Tsnxn+ϵJψxn,xn-x0-Tsnxn+ϵJψxn,x0+h1*+ϵh2*,xn, hence (27)limsupntnΛPxn+h1*+ϵh2*,xnlimsupnTsnxn+ϵJψxn+tnΛPxn,xn--liminfnTsnxn+ϵJψxn,xn-x0--h1*+ϵh2*,x0+h1*+ϵh2*,x0-liminfnTsnxn+ϵJψxn,xn-x00, where the last inequality follows from Lemma 6. Now since P is of class (S+)L, it follows that xnx0, x0D(P) and t0ΛPx0=-h1*-ϵh2*. Also we have that (28)Tsnxn,xn-x0=0 and using Lemma 6 we get that x0D(T) and Tx0h1*=-t0ΛPx0-ϵJψx0. This is a contradiction since xnx0, we have x0Ω. We have shown that H(t,x) is an admissible homotopy for our degree. We can now work as in Theorem 3 in  in order to show that d(H(t,·),Ω,0)=const.

Thus (29)d(H(t,·),Ω,0)=d(Ts+ϵJψ,Ω,0)=1, where the last equality follows from Theorem 3, (i) of .

Consequently, the inclusion H(t,x)0 has a solution in Ω for each t[0,1]. In particular, this says that Tx+λPx+ϵJψx0 has a solution in Ω for every λ>0. This of course contradicts condition (𝒫) and finishes the proof of (i).

(ii) Let λn(0,Λ], xnD(P)Ω be such that, for some un*Txn, (30)un*+λnPxn+(1n)Jψxn=0 again, λn=0 for any n is not possible. Since λn>0, we have by property (P2) that (31)un*,xn=-λnPxn,xn-1nJψxn,xnλnϕ(xn)-1nψ(xn)xnΛϕ(S1)+ψ(S1)S1, where S1 is the bound of {xn}. This show that un*,xn is bounded and further we obtain the boundedness of {un*} by Lemma 2. We may thus assume that λnλ0, xnx0, un*h0*, then λnPxn-h0*.

If λ0=0, then (30) implies that (32)limnun*=limn[-λnPxn-(1n)Jψxn]=0. Since T satisfies (Sq), it follows that xnx0Ω. Now, by the demiclosedness of T (see Lemma 6) we obtain that x0D(T), 0Tx0 and this contradicts 0T(D(T)Ω). Hence, λ0>0. Repeating the proof of (i) starting from (21), we get again that (33)limsupnλnPxn+h0*,xn0 and since P is of class (S+)L, we have xnx0, -h0*=λ0Px0. Using the demiclosedness of T, we obtain x0D(T)Ω, h0Tx0 and Tx0+λ0Px00. The proof is now complete.

3. Continuous Branches of Eigenvectors

In this section we are interested in showing that the results obtained in previous sections could be used in order to obtain the existence of continuous branches of eigenvectors. We need the following definition.

Definition 9.

Let T:XD(T)2X*, P:XD(P)X* be given and consider the problem (34)Tx+λPx0. An eigenvector x is solution of (34) for some eigenvalue λ with x(D(T)D(P)). We say that the nonzero eigenvectors of the problem (34) form a continuous branch of infinite length if there exists r0>0 such that, for every rr0, the sphere Br(0) contains at least one nonzero eigenvector of (34).

Theorem 10.

Assume that the operator T:XD(T)2X* is maximal monotone with 0D(T) and 0T(0). Assume that the operator P:XD(P)X* with LD(P) is of class (S+)L and satisfies (P2). Let Λ be a positive number. Assume that Tx0 implies x=0, T satisfies (Sq) and

(𝒫1) there exists α>0 and λ(0,Λ] such that (35)|Tx+λPx|α,xD(T+P).

Then the nonzero eigenvectors of problem (34) form a continuous branch of infinite length with corresponding eigenvalues λ(0,Λ].

Proof.

Let r0>0 be given. Let ϵ0>0 be so small that ϵ0r0<α. Then (36)|Tx+λPx+ϵJx||Tx+λPx|-|ϵJx|α-ϵxα-ϵ0r0,xD(T+P), which implies that the inclusion (37)Tx+λPx+ϵJx0 has no solution xD(T+P)Br(0) for any ϵ(0,ϵ0]. Since 0T(Br0(0)) and T is of type (Sq), Theorem 8 implies the existence of a solution xλ0D(T+P)Br0(0), for some λ0(0,Λ]. The same argument can be repeated for any r>r0. This complete the proof.

Remark 11.

This result is also true when P:XD(P)2X* is assumed to be multivalued. We have the following theorem without proof.

Theorem 12.

Assume that the operator T:XD(T)2X* is maximal monotone with 0D(T) and 0T(0). Assume that the operator P:XD(P)2X* with LD(P) is of class (S+)L and satisfies (P2). Let Λ be a positive number. Assume that Tx0 implies x=0, T satisfies (Sq), and

( 𝒫 1 ) there exists α>0 and λ(0,Λ] such that (38)|Tx+λPx|α,xD(T+P). Then the nonzero eigenvectors of problem (34) form a continuous branch of infinite length with corresponding eigenvalues λ(0,Λ].

In the following result, we assume that the operator T is defined and bounded on all of X. In this case we demonstrate the fact that the assumption |Tx+Px|0 may be replaced by the assumption ||Px||0 on D(P).

Theorem 13.

Assume that the operator T:XD(T)2X* is maximal monotone and bounded with 0D(T) and 0T(0). Assume that the operator P:XD(P)X*, with LD(P) satisfies condition (S+)L and (P2). Assume that Tx0 implies x=0, T satisfies (Sq) and there exist α>0 such that (39)Pxα,xD(P). Then the nonzero eigenvectors of the problem (34) form a continuous branch of infinite length.

Proof.

We show that the problem (34) has eigenvectors on the set Br(0) for every r>0. To this end, let fix r>0, ϵ>0 and show the existence of λ~>0 such that (40)d(T+λ~P+ϵJ,Br(0),0)=0. If this is not true, then there exist a sequence of {λn}(0,) such that λn and one of the following is true:

the degree d(T+λnP+ϵJ,Br(0),0) is not well-defined;

d(T+λnP+ϵJ,Br(0),0)0.

In the case (i) there exist eigenvectors xnBr(0) such that (41)Txn+λnPxn+ϵJxn0. In the case (ii) there exist eigenvectors xnBr(0) such that (41) holds. Thus in either case, there exist a sequence {xn}Br(0)¯ such that (41) is true. But this leads to a contradiction because ||λnPxn+ϵJxn||αλn-ϵr, while the set Txn lie in a bounded set. Thus, (40) is true for some λ~>0. We consider the homotopy (42)H(t,x)Tx+tλ~Px+ϵJx,t[0,1],xD(H(t,·)), either there exist t0[0,1] and xt0Br(0) such that (43)Txt0+t0λ~Pxt0+ϵJxt00, or (44)d(H(t,·))=d(H(1,·))=0=d(H(0,·))=1,t[0,1] that is a contradiction. The last equality in (44) follows from Theorem 3, (i) in . It follows that (43) is true. Naturally, we must have t00 in (43) because otherwise 0(T+ϵJ)(Br(0)). This cannot happen because we already have 0(T+ϵJ)(0) and T+ϵJ is one-to-one. From (43) we obtain sequences λn(0,), {xn}Br(0) such that (45)Txn+λnPxn+(1n)Jxn0. We may assume that xnx0. Again, {λn} cannot contain a subsequence {λnk} such that λnk as k because the sequence Txnk+(1/nk)Jxnk lies in a bounded set and λnk||Pxnk|| as k. Also we have that the sequence {Pxn} is bounded. Thus we may assume that Pxnp*X*. Since the sequence {λn} is bounded, we may assume that λnλ0. Again, λ00 otherwise λnPxn+(1/n)Jxn0 and the (Sq) property of T would imply that xnx0Ω. Since T is demiclosed, Lemma 6 would imply Tx00. This is a contradiction to our assumption that Tx0 implies x=0. It follows that λ0>0 and we may also assume that λn>0 for all n.

Now proceeding as in the proof of Theorem 8, it is easy to see that (46)limsupnλnPxn+λ0p*,xn0,limnλnPxn+λ0p*,v=0,vUn=1Fn. Now since P is of class (S+)L, we conclude that xnx0, x0D(P) and Px0=p*. Thus since also yn*-λ0Px0, xnx0 and the demiclosedness of T, we obtain x0D(P)Br(0) and Tx0+λ0Px00. Since r>0 is arbitrary, the nonzero eigenvectors of problem (34) form a continuous branch of infinite length.

Pascali D. Sburlan S. Nonlinear Mappings of Monotone Type 1978 The Hague, The Netherlands Martinus Nijhoff Publishers x+341 MR531036 Kartsatos A. G. Skrypnik I. V. A new topological degree theory for densely defined quasibounded (S˜+)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces Abstract and Applied Analysis 2005 2005 2 121 158 10.1155/AAA.2005.121 MR2179439 Zeidler E. Functional Analysis and Its Applications 1990 II/B New York, NY, USA Springer Adhikari D. R. Kartsatos A. G. Topological degree theories and nonlinear operator equations in Banach spaces Nonlinear Analysis: Theory, Methods & Applications 2008 69 4 1235 1255 10.1016/j.na.2007.06.026 MR2426688 ZBL1142.47346 Browder F. E. Hess P. Nonlinear mappings of monotone type in Banach spaces Journal of Functional Analysis 1972 11 251 294 MR0365242 ZBL0249.47044 Petryshyn W. V. Approximation-Solvability of Nonlinear Functional and Differential Equations 1993 171 New York, NY, USA Marcel Dekker xii+372 Monographs and Textbooks in Pure and Applied Mathematics MR1200455 Kartsatos A. G. Quarcoo J. A new topological degree theory for densely defined (S+)L-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces Nonlinear Analysis: Theory, Methods & Applications 2008 69 8 2339 2354 10.1016/j.na.2007.08.017 MR2446332 Kartsatos A. G. Skrypnik I. V. On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces Transactions of the American Mathematical Society 2006 358 9 3851 3881 10.1090/S0002-9947-05-03761-X MR2219002 ZBL1100.47050 Berkovits J. On the degree theory for densely defined mappings of class (S+)L Abstract and Applied Analysis 1999 4 3 141 152 10.1155/S1085337599000111 MR1811232 ZBL0991.47049