# A New Topological Degree Theory for Perturbations of Demicontinuous Operators and Applications to Nonlinear Equations with Nonmonotone Nonlinearities

Authors: Teffera M. Asfaw
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

LetX be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X⁎. Let T:X⊇DT→2X⁎ be maximal monotone of type Γdϕ (i.e., there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ with ϕ(0)=0 such that 〈v⁎,x-y〉≥-dx-ϕy for all x∈DT, v⁎∈Tx, and y∈X),L:X⊃D(L)→X⁎ be linear, surjective, and closed such that L-1:X⁎→X is compact, and C:X→X⁎ be a bounded demicontinuous operator. A new degree theory is developed for operators of the type L+T+C. The surjectivity of L can be omitted provided that RL is closed, L is densely defined and self-adjoint, and X=H, a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for L+C, where C is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when L is monotone, a maximality result is included for L and L+T. The theory is applied to prove existence of weak solutions in X=L20,T;H01Ω of the nonlinear equation given by ∂u/∂t-∑i=1N(∂/∂xi)Aix,u,∇u+Hλx,u,∇u=fx,t,  x,t∈QT;  ux,t=0,  x,t∈∂QT; and ux,0=ux,T,  x∈Ω, where λ>0, QT=Ω×0,T, ∂QT=∂Ω×0,T, Aix,u,∇u=∂/∂xiρx,u,∇u+aix,u,∇u(i=1,2,…,N), Hλx,u,∇u=-λΔu+gx,u,∇u, Ω is a nonempty, bounded, and open subset of RN with smooth boundary, and ρ,ai,g:Ω¯×R×RN→R satisfy suitable growth conditions. In addition, a new existence result is given concerning existence of weak solutions for nonlinear wave equation with nonmonotone nonlinearity.


---

## Body

## 1. Introduction and Preliminaries

Throughout the paper,X,· denotes a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X∗. For x∈X and x∗∈X∗, the duality pairing 〈x∗,x〉 denotes the value x∗x. Let J:X→2X∗ be the normalized duality mapping given by (1)Jx=x∗∈X∗:x∗,x=x2,x∗=x. It is well-known that Jx≠∅ for all x∈X because of the Hahn-Banach Theorem. Since X and X∗ are locally uniformly convex reflexive Banach spaces, it is well-known that J is single valued and homeomorphism. For a multivalued operator T from X into X∗, the domain of T denoted by DT is given as DT=x∈X:Tx≠∅. The range of T, denoted by RT, is given by RT=⋃x∈DTTx and graph of T, denoted by GT, is given by GT=x,v∗:x∈DT,v∗∈Tx. The following definition is used in the squeal.Definition 1.
A multivalued operatorT:X⊇DT→2X∗ is called (i)
“monotone” if, for allx∈DT, y∈DT, v∗∈Tx, and u∗∈Ty, we have v∗-u∗,x-y≥0,(ii)
“maximal monotone” ifT is monotone and u∗-u0∗,x-x0≥0 for every x,u∗∈GT implies x0∈DT and u0∗∈Tx0. This is equivalent to saying that T is “maximal monotone” if and only if RT+λJ=X∗ for every λ>0,(iii)
“coercive” if eitherDT is bounded or there exists a function ψ:0,∞→-∞,∞ such that ψt→∞ as t→∞ and (2)y∗,x≥ψxx∀x∈DT,y∗∈Tx.It is the goal of the paper to develop a topological degree theory for classes of operators of the typeL+T+C, where L, T, and C satisfy one of the following conditions:(i)
L:X⊃DL→X∗ is linear, surjective, and closed such that L-1:X∗→X is compact, T:X⊃DT→2X∗ is maximal monotone of type Γdϕ, and C:X→X∗ is bounded demicontinuous operator.(ii)
X=H, a real Hilbert space, L:H⊃DL→H is linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact, T:H⊃DT→2H is maximal monotone of type Γdϕ, and C:H→H is bounded demicontinuous operator. The main reason for the need of such a theory is the existence of nonlinear problems (i.e., nonlinear equations and variational inequality problems) which cannot be addressed by the existing theories under minimal assumptions on L, C, and T. In addition, considering the classes of operators of the type L+T+C, it is an essential contribution to have a theory useful to drive existence theorems to treat larger class of problems. Therefore, Section 2 gives a preliminary lemma, which will be useful to extend the definition of pseudomonotone homotopy of maximal monotone operators initially introduced by Browder [1, 2]. Section 3 deals with the construction of the degree mapping along with basic properties and homotopy invariance results. The main contribution of this work is providing a new degree theory for treating nonlinear problems involving operators of type L+T+C, where L, T, and C satisfy condition (i) or (ii). In this theory, the operator L might not be pseudomonotone type and C is just bounded demicontinuous operator. The well-known degree for monotone type operators, which is attributed to Browder [1, 2], is for operators of type T+C, where T:X⊃DT→2X∗ is maximal monotone and C:G¯→X∗ is bounded demicontinuous operator of type S+. In view of this, the degree mapping constructed herein allows C to be bounded demicontinuous operator not necessarily compact, bounded of type S+, or pseudomonotone. To the best of the author’s knowledge, this degree mapping is new and has the potential to address new classes of problems such as wave equations with nonmonotone nonlinearities. As a consequence of the theory, new existence results are given for the solvability of operator inclusions of the type Lu+Tu+Cu∋f∗, u∈DL∩DT. In the last section, examples are provided proving existence of weak solutions for nonlinear parabolic as well as hyperbolic problems in appropriate Sobolev spaces. For degree theories for bounded demicontinuous S+ perturbations of maximal monotone operators, the reader is referred to the papers of Browder [1, 2], Kobayashi and Ôtani [3], Hu and Papageorgiou [4], Berkovits and Mustonen [5, 6], Berkovits [7], Kartsatos and Skrypnik [8], and Kien et al. [9] and the references therein. For recent topological degree theories for bounded pseudomonotone perturbations of maximal monotone operators, the reader is referred to the recent papers of Asfaw and Kartsatos [10] and Asfaw [11]. Basic definitions, properties, and existence theorems concerning operators of monotone type can be found in the books of Barbu [12, 13], Pascali and Sburlan [14], and Zeidler [15].


## 2. A Preliminary Lemma

The following lemma is useful towards the extension of the definition of a pseudomonotone homotopy of maximal monotone operators introduced by Browder [1, 2].Definition 2.
A familyTtt∈0,1 is said to be uniformly of type (i)
Γdϕ if there exist d≥0 and a nondecreasing function ϕ:0,∞→0,∞ such that ϕ0=0 and (3)vt,x-y≥-dx-ϕy uniformly for all t∈0,1 and x∈DTt, vt∈Ttx, and y∈X,(ii)
Γd if there exists d≥0 such that vt,x≥-dx uniformly for all t∈0,1 and x∈DTt and vt∈Ttx. If Tt=T for all t∈[0,1], then the operator T is said to be of type Γdϕ or Γd if it satisfies either (i) or (ii), respectively.It is easy to see that a family of monotone operatorsTtt∈0,1 is uniformly of type Γd if 0∈Tt0 for all t∈0,1. It is also true that the class Γd includes the class Γdϕ. The lemma below is used in the construction of the degree.Lemma 3.
LetTtt∈0,1 be a family of maximal monotone operators uniformly of type Γd. Then the following four conditions are equivalent: (i)
For any sequencestn in 0,1,un∈DTtn and wn∗∈Ttnun such that un⇀u in X, and tn→t0∈0,1 and wn∗⇀w0∗ in X∗ as n→∞ with limsupn→∞⁡wn∗,un-u≤0, it follows that u∈DTt0,w0∗∈Tt0u, and wn∗,un→w0∗,u as n→∞.(ii)
For eachε>0, the operator defined by ψt,w=Tt+εJ-1w is continuous from 0,1×X∗ to X.(iii)
For each fixedw∈X∗, the operator defined by ψwt=Tt+εJ-1w is continuous from 0,1 to X.(iv)
For any given pairx,u∈GTt0 and any sequence tn→t0 as n→∞, there exist sequences xn and un such that un∈Ttnxn and xn→x and un→u as n→∞.Proof.
The proof for the implicationsii⇒iii⇒iv⇒i follows from the result attributed to to Browder [1, 2] without requiring the condition 0∈Tt0 for all t∈0,1. Next we give the proof of the implication i⇒ii. Fix ε>0. Let tn,wn∗∈0,1×X∗ such that tn→t0 and wn∗→w0∗ as n→∞. For each n, let un=Ttn+εJ-1wn∗ and u0=Tt0+εJ-1w0∗. It follows that wn∗=zn∗+εJun for some zn∗∈Ttnun, for all n, and w0∗=z0∗+εJu0 for some z0∗∈Tt0u0. We shall show that un is bounded. By applying Γd condition on the family Ttt∈0,1, we see that (4)wn∗-w0∗,un-u0=zn∗+εJun-z0∗-εJu0,un-u0=zn∗-z0∗,un-u0+εJun-Ju0,un-u0≥zn∗-z0∗,un-u0+εun2-2unu0+u02=zn∗-z0∗,un-u0+εun-u02=zn∗,un-zn∗,u0-z0∗,un+z0∗,u0+εun-u02≥-dun-zn∗u0-z0∗un-du0+εun-u02≥-dun+u0-wn∗+εunu0-z0∗un+εun-u02∀n;that is, we get(5)εun-u02≤dun+u0+wn∗+εunu0+z0∗un+wn∗-w0∗un-u0∀n.Since wn∗ is bounded, (83) implies the boundedness of the sequence un. Assume, without loss of generality, that un⇀u,Jun⇀z as n→∞. Since wn∗=zn∗+εJun, it follows that zn∗⇀y0∗=w0∗-εz as n→∞. By the condition in (i) and monotonicity of J and wn∗→w0∗ as n→∞ and boundedness of un, we obtain that (6)limsupn→∞zn∗,un-u=limsupn→∞wn∗-εJun,un-u=limsupn→∞wn∗,un-u-εliminfn→∞Jun-Ju,un-u-εliminfn→∞Ju,un-u≤0.Consequently, we arrive at (7)limsupn→∞zn∗,un≤y0∗,u.Thus, by using conditions in (i), it follows that u∈DTt0, y0∗∈Tt0u, and zn∗,un→y0∗,u as n→∞, which implies (8)limsupn→∞Jun,un-u=0.Since J is of type S+ and continuous, we have un→u and Jun→Ju as n→∞, which implies y0∗=w0∗-εJu; that is, w0∗∈Tt0+εJu, implying in turn that u=Tt0+εJ-1w0∗; that is, ψtn,wn∗→ψt,w0∗ as n→∞. This shows that ψ is continuous from 0,1×X∗ into X. Therefore, the equivalency of the four statements is proved.A larger class of pseudomonotone homotopies of maximal monotone operators is introduced below. The original definition of pseudomonotone homotopy of maximal monotone operatorsTtt∈0,1 is attributed to Browder [2] which requires the family to satisfy 0∈Tt0 for all t∈0,1.Definition 4.
A familyTtt∈0,1 of maximal monotone operators uniformly of type Γdϕ is called a “pseudomonotone homotopy of type Γdϕ” if one of the equivalent conditions of Lemma 3 holds.


## 3. Degree Theory in Reflexive Banach Space withRL=X∗

The section deals with the main contribution of the paper. A new topological degree mapping is constructed for operators of typeT+C+L, where T:X⊃DT→2X∗ is maximal monotone of type Γdϕ, C:X→X∗ is bounded demicontinuous operator, and L:X⊃DL→X∗ is linear, surjective, and closed such that L-1:X∗→X is compact. The construction is based on the Leray-Schauder degree mapping for the operator I+L-1Tε+C, where Tε:X→X∗ is the Yosida approximant of T. Since L is surjective, L-1 is compact, and Tε is bounded continuous operator, it follows that L-1Tε+C:X→X∗ is a well-defined compact operator. Next we prove the following theorem.Theorem 5.
LetG be a nonempty, bounded, and open subset of X. Let Ttt∈0,1 be a pseudomonotone homotopy of maximal monotone operators uniformly of type Γdϕ and Ct=tC1+1-tC2t∈0,1 with Cii=1,2:X→X∗ is bounded demicontinuous operator and let L:X⊃DL→X∗ be linear, surjective, and closed such that L-1:X∗→X is compact. Assume, further, that 0∉L+Tt+CtDL∩DTt∩∂G for all t∈0,1. Then there exists ε0>0 such that (9)dLSI+L-1Tεt+Ct,G,0is well-defined and independent of ε∈0,ε0 and t∈0,1, where dLS denotes the Leray-Schauder degree mapping for compact displacement of the identity and Tεt:X→DTt is the Yosida approximant of Tt.Proof.
Suppose the hypotheses hold. Assume to the contrary that there existεn↓0+, xn∈∂G, and tn∈0,1 such that(10)xn+L-1Tεntnxn+Ctnxn=0∀n.Since L is surjective, it follows that xn=-L-1Tεntnxn+Ctnxn∈DL and Lxn+Tεntnxn+Ctnxn=0 for all n. The uniform boundedness of the family Ctt∈0,1 implies the boundedness of Ctnxn. Since Ttn is of type Γdϕ, let d≥0 and ϕ be as in Definition 2. Let vn∗=Tεntnxn. It is well-known that Jεntnxn∈DT, vn∗∈TtnJεntnxn, and Jεntnxn=xn-εnJ-1vn∗ for all n. For each x∈X, we see that(11)Lxn,xn-x=-vn∗,xn-x-Ctnxn,xn-x=-vn∗,xn-Jεntnxn+Jεntnxn-x-Ctnxn,xn-x≤-vn∗,εnJ-1vn∗-vn∗,Jεntnxn-x+Ctnxnxn-x≤-εnvn∗2+dJεntnxn+ϕx+Ctnxnxn≤-εnvn∗2+dxn+dεnvn∗+ϕx+Ctnxnxn=-εnvn∗2+dεnvn∗+dxn+ϕx+Ctnxnxn=-εnzn2+dεnzn+dxn+ϕx+Ctnxnxn≤d2εn4+dxn+ϕx+Ctnxnxn≤K1x, where zn=vn∗ for all n and K1 is an upper bound for d2+ϕx+dxn+Ctnxnxn. Now, setting xn-x in place of x, we obtain that(12)Lxn,x=Lxn,xn-xn-x≤k+ϕxn-xfor all n. Since ϕ is nondecreasing and xn is bounded, we see that (13)Lxn,x≤K2x,where K2x is an upper bound for k+ϕxn-x. By similar argument, setting xn+x in place of x, we get (14)Lxn,-x=Lxn,xn-xn+x≤k+ϕxn+xfor all n; that is, Lxn,x≥-k-ϕxn+x≥-K3x, where K3x is an upper bound for k+ϕxn+x. For each x∈X, combining these two inequalities shows that there exists Nx≥0 such that Lxn,x≤Nx for all n. By applying the well-known uniform boundedness principle, we conclude that Lxn is bounded. Consequently, we obtain the boundedness of vn∗. Since Ctnxn is bounded and L-1 is compact, we assume without loss of generality that there exists a subsequence, denoted again by L-1vn∗+Ctnxn, such that L-1vn∗+Ctnxn→x0 as n→∞; that is, xn→x0∈∂G as n→∞. Assume without loss of generality that tn→t0∈0,1, vn∗⇀v0∗, Lxn⇀y0∗, and Ctnxn⇀Ct0x0 as n→∞. Since xn→x0, it follows that Jεntnxn=xn-εnJ-1vn∗→x0 as n→∞; that is, (15)limsupn→∞vn∗,xn-x0=0.Since Ttt∈0,1 is a pseudomonotone homotopy of type Γdϕ, (iv) of Lemma 3 implies that x0∈DTt0, v0∗∈Tt0x0, and vn∗,xn→v0∗,x0 as n→∞. Since L is closed, we conclude that x0∈DL and Lx0=y0∗, which implies that 0∈L+T+CDL∩DT∩∂G. However, this is impossible. Therefore, there exists ε0>0 such that dLSI+L-1Tεt+Ct,G,0 is well-defined for all ε∈0,ε0 and t∈0,1. Next we prove that there exists ε1∈0,ε0 such that dLSI+L-1Tεt+Ct,G,0 is independent of ε∈0,ε1 and t∈0,1. Suppose this is false; that is, there exist εn↓0+, δn↓0+, and tn∈0,1 such that(16)dLSI+L-1Tεntn+Ctn,G,0≠dLSL-1Tδntn+Ctn,G,0for all n. For each n, we consider the homotopy (17)Hns,x=x+L-1sTεntnx+1-sTδntnx+L-1Ctnx,s,x∈0,1×G¯.Since L-1 is compact and Tεntn, Tδntn, and Ctn are bounded continuous operators, we observe that Hns,·s∈0,1 is Leray-Schauder type homotopy. We shall show that Hns,·s∈0,1 is an admissible homotopy for all large n; that is, for all large n, we have 0∉Hns,∂G for all s∈0,1. Suppose there exists a subsequence of n, denoted again by n, such that there exist xn∈∂G, tn∈0,1, and sn∈0,1 such that (18)xn+L-1snTεntnxn+1-snTδntnxn+Ctnxn=0for all n; that is, we have (19)Lxn+snTεntnxn+1-snTδntnxn+Ctnxn=0for all n. Assume without loss of generality that sn→s0∈0,1 and tn→t0∈0,1 as n→∞. For each n, let vn∗=Tεntnxn, un∗=Tδntnxn, and zn∗=snvn∗+1-snun∗. It is well-known that vn∗∈TtnJεntnxn, un∗∈TtnJδntnxn, Jεntnxn=xn-εnJ-1vn∗, and Jδntnxn=xn-δnJ-1un∗ for all n. Let x∈X. By the definition of pseudomonotone homotopy of type Γdϕ and uniform boundedness condition on Ctt∈0,1, we see that (20)Lxn,xn-x=-snvn∗,xn-x-1-snun∗,xn-x-Ctnxn,xn-x=-snvn∗,xn-Jεntnxn+Jεntnxn-x-1-snun∗,xn-Jδntnxn+Jδntnxn-x+Ctnxnxn=-snvn∗,εnJ-1vn∗-snvn∗,Jεntnxn-x-1-snun∗,δnJ-1un∗-1-snun∗,Jδntnxn-x+Ctnxn≤-snεnvn∗2+sndJεntnxn+ϕx+-1-snδnun∗2+1-sndJδntnxn+ϕx+Ctnxn=-snεnvn∗2+sndεnvn∗+-1-snδnun∗2+1-sndδnun∗+2ϕx+dxn+Ctnxn≤-εnsnvn∗2+dεnsnvn∗+-δn1-snun∗2+dδn1-snun∗+2ϕx+dxn+Ctnxn=-εnτn2+dεnτn+-δnλn2+dδnλn+2ϕx+dxn+Ctnxn≤d2εn4+d2δn4+2ϕx+dxn+Ctnxn≤21+ϕx+dxn+Ctnxn=K4x,where τn=snvn∗, λn=1-snun∗, and K4x is upper bound for 21+ϕx+dxn+Ctnxn. By following the argument used in the first part of this proof, it follows that Lxn is bounded; that is, zn∗ is bounded. By the compactness of L-1, there exists a subsequence, denoted again by xn, such that xn→x0 as n→∞. Assume without loss of generality that Lxn⇀h0∗ and zn∗⇀z0∗ as n→∞. Since L is closed, we have x0∈DL and h0∗=Lx0. Since {zn∗} and Ctnxn are bounded and xn→x0 as n→∞, we get zn∗,xn→〈z0∗,x0〉 as n→∞. To complete the proof, we consider the following cases.
Case I. snvn∗ is bounded. Since zn∗ is bounded, 1-snun∗ is also bounded. Since Ttt∈0,1 is of type Γdϕ (i.e., of type Γd), it follows that vt,x≥-dx for all x∈DTt and vt∈Ttx for all t∈0,1. Let x,y∈GTt0. Since Ttt∈0,1 is a pseudomonotone homotopy of type Γdϕ, by (iv) of Lemma 3, there exists a sequence yn,yn∗∈GTtn such that yn→x and yn∗→y as n→∞. On the other hand, the monotonicity of Ttn implies (21)vn∗-yn∗,xn-εnJ-1vn∗-yn≥0∀n;that is,(22)vn∗,xn≥vn∗,yn+yn∗,xn-yn+εnvn∗2-εnyn∗vn∗∀n.In a similar manner, we get(23)un∗,xn≥un∗,yn+yn∗,xn-yn+δnvn∗2-δnyn∗un∗for all n. Multiplying (22) and (23) by sn and 1-sn, respectively, and adding the resulting inequalities, we get(24)zn∗,xn≥zn∗,yn+yn∗,xn-yn+snεnvn∗2-yn∗vn∗+1-snδnun∗2-yn∗un∗∀n.Since zn∗ is bounded and xn→x0, it follows that zn∗,xn→z0∗,x0 as n→∞. Consequently, using (24), we obtain(25)z0∗,x0=limn→∞zn∗,xn≥liminfn→∞zn∗,yn+yn∗,xn-yn-limsupn→∞snεnvn∗yn∗+1-snδnyn∗un∗=z0∗,x+y,x0-xfor all x,y∈GTt0, which yields z0∗-y,x0-x≥0. By the maximal monotonicity of Tt0, we conclude that x0∈DTt0 and z0∗∈Tt0x0. Therefore, we obtain that x0∈DL∩DTt0∩∂G and z0∗∈Tt0x0 such that Lx0+z0∗+t0C1x0+1-t0C2x0=0. However, this is a contradiction.
Case II. Suppose snvn∗ is unbounded. Then there exists a subsequence, denoted again by snvn∗, such that snvn∗→+∞ as n→∞. Then 1-snun∗,vn∗ and un∗ being unbounded. Assume without loss of generality that vn∗→∞ and un∗→∞ as n→∞. If either εnsnvn∗2 or δn1-snun∗2 is unbounded, (24) implies(26)zn∗,xn≥zn∗,yn+yn∗,xn-yn+snεnvn∗21-yn∗vn∗+1-snδnun∗21-yn∗un∗.Assuming εnsnvn∗2→∞ or δn1-snun∗2→∞ and taking limits in (26) imply that(27)z0∗-y,x0-x≥∞,which is impossible. Thus, εnsnvn∗2 and δn1-snun∗2 are bounded. Consequently, we get (28)snεnvn∗=snεnvn∗2vn∗⟶0as n→∞. Similarly, we have 1-snδnun∗→0 as n→∞. In all cases, (24) and (25) yield a contradiction. Therefore, by using the compactness of L-1 and boundedness of Tεntn and Tδntn, we proved that the family Hnt,·s∈0,1 is an admissible homotopy of Leray-Schauder type; that is, dHns,·,G,0 is independent of s∈0,1 for all large n; that is, (29)dLSI+L-1Tεntn+Ctn,G,0=dLSI+L-1Tδntn+Ctn,G,0.However, this is impossible. Therefore, there exists ε1>0 such that dLSI+L-1Tεt+Ct,G,0 is well-defined and independent of ε∈0,ε1 and t∈0,1. The proof is completed.Next we give the definition of the required degree mapping.Definition 6.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃DT→2X∗ be maximal monotone of type Γdϕ, C:X→X∗ be bounded demicontinuous operator, and L:X⊃D(L)→X∗ be linear, surjective, and closed such that L-1:X∗→X is compact. Assume, further, that f∗∉L+T+CDT∩DL∩∂G. Then the degree of L+T+C at f∗∈X∗ with respect to G is given by (30)dL+T+C,G,f∗=limε↓0+dLSI+L-1Tε+C-f∗,G,0,where dLS denotes the Leray-Schauder degree mapping for compact perturbations of the identity and Tε is the Yosida approximant of T.The degreed satisfies the following basic properties and homotopy invariance result.Theorem 7.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃DT→2X∗ be maximal monotone of type Γdϕ, L:X⊃DL→X∗ be linear, surjective, and closed such that L-1:X∗→X is compact, and C:X→X∗ be bounded demicontinuous operator. Then (i)
(normalization) there existsα>0 such that dL+αJ,G,0=1 if 0∈G and dL+αJ,G,0=0 if 0∉G¯. If L is monotone, then dL+J,G,0=1 if 0∈G and dL+J,G,0=0 if 0∉G¯;(ii)
(existence) if0∉L+T+CDL∩DT∩∂G and dL+T+C,G,0≠0, then 0∈L+T+CDL∩DT∩G;(iii)
(decomposition) letG1 and G2 be nonempty and disjoint open subsets of  G such that f∗∉L+T+CG¯∖G1∪G2. Then (31)dL+T+C,G,f∗=dL+T+C,G1,f∗+dL+T+C,G2,f∗;(iv)
(translation invariance) letf∗∉L+T+CDL∩DT∩∂G. Then we have (32)dL+T+C-f∗,G,0=dL+T+C,G,f∗;(v)
(homotopy invariance) letTtt∈0,1 be a pseudomonotone homotopy of maximal monotone operators uniformly of type Γdϕ and Ci:X→X∗i=1,2 is bounded demicontinuous operator. Let (33)Ht,x=Lx+Ttx+tC1x+1-tC2x,t,x∈0,1×DL∩DTt∩G¯. Then dHt,·,G,0 is independent of t∈0,1 provided that 0∉Ht,DL∩DTt∩∂G for all t∈0,1.Proof.
(i) Suppose the hypotheses hold. SinceL-1:X∗→X is continuous, there exists d>0 such that Lx≥dx for all x∈DL. Let α∈0,d and Ht,x=x+tL-1αJx, t,x∈0,1×DL∩G¯. If there exist t0∈0,1 and x0∈∂G such that 0=x0+t0L-1Jx0, then it follows that x0∈DL and Lx0+t0αJx0=0. Since dx0≤Lx0≤t0αx0≤αx0 (i.e., d-αx0≤0), this gives d≤α or x0=0. But these are impossible because 0∈G and d>α. Since tL-1αJt∈0,1 is a family of compact operators from G¯ into X such that H~t,x=tL-1αJx,  t,x∈0,1×G¯, is uniformly continuous in t∈0,1 uniformly for all x∈G¯, it follows that I+tL-1αJt∈0,1 is an admissible homotopy of Leray-Schauder type; that is, (34)dHt,·,G,0=dLSI+tL-1αJ,G,0is independent of t∈0,1. Therefore, we obtain that dL+αJ,G,0=dH1,·,G,0=dI,G,0=1 if 0∈G and dL+αJ,G,0=0 if 0∉G¯.
(ii) Suppose0∉L+T+CDL∩DT∩∂G and dL+T+C,G,0≠0. By the definition of the degree mapping d, we see that dLSI+L-1Tε+C,G,0≠0 for all sufficiently small ε>0; that is, for each εn↓0+, there exists xn∈DL∩DT∩G such that (35)xn+L-1Tεnxn+Cxn=0∀n.By the arguments used in the proof of Theorem 5, one can easily show that 0∈L+T+CDL∩DT∩G. The details are omitted here.
(iii) Suppose the hypotheses hold. The definition of the degree mappingd and decomposition property of the Leray-Schauder degree dLS imply (36)dL+T+C,G,f∗=dLSI+L-1Tε+C-f∗,G,0=dLSI+L-1Tε+C-f∗,G1,0+dLSI+L-1Tε+C-f∗,G2,0=dL+T+C,G1,f∗+dL+T+C,G2,f∗ for all sufficiently small ε>0. This completes the proof of (iii). The proof of (iv) follows from the translation invariance property of Leray-Schauder degree.
(v) LetCtx=tC1x+1-tC2x,t,x∈0,1×G¯. The proof of Theorem 5 confirms the existence of ε1>0 such that dLSI+L-1Tεt+Ct,G,0 is well-defined and independent of t∈0,1 and ε∈0,ε1; that is, by the definition of the degree, we get that(37)dHt,·,G,0=limε↓0+dLSI+L-1Tεt+Ct,G,0is well-defined and independent of t∈0,1. The details are omitted here.Consequently, we prove the following new existence result.Theorem 8.
LetT:X⊃DT→2X∗ be densely defined maximal monotone with 0∈T(0) and of type Γdϕ, L:X⊃DL→X∗ be linear monotone, surjective, and closed such that L-1:X∗→X is compact, and C:X→X∗ be bounded demicontinuous operator. Let f∗∈X∗. Assume, further, that there exists R>0 such that (38)Lx+v∗+Cx-f∗,x>0for all x∈DL∩DT∩∂BR0 and v∗∈Tx. Then f∗∈L+T+CDL∩DT∩BR0. In addition, L+T+C is surjective if L+T+C is coercive.Proof.
By the continuity ofL-1, there exists d>0 such that dx≤Lxfor all x∈DL. Let ε∈0,d and (39)Ht,x=Lx+Tx+tCx-f∗+εJx,t,x∈0,1×DL∩DT.Since 0∈T0 and L0=0, we see that Ht,x can be rewritten as (40)Ht,x=tLx+Tx+Cx+εJx-f∗+1-tLx+Tx+εJx,t,x∈0,1×DL∩DT. Since L and T are monotone with 0∈T0, we get (41)Lx+v∗+tCx-f∗+εJx,x=tLx+v∗+Cx-f∗,x+1-tLx+v∗+εJx,x≥1-tLx+v∗+εJx,x≥εx2>0for all t∈0,1, x∈DL∩DT∩∂BR0, and v∗∈Tx; that is, we have (42)tLx+v∗+Cx+εJx-f∗+1-tv∗+Lx+εJx,x>0for all x∈DL∩DT∩∂BR0, t∈0,1, and v∗∈Tx; that is, 0∉Ht,DL∩DT∩∂BR0 for all t∈0,1. Since Tt=Tt∈0,1 is a pseudomonotone homotopy of maximal monotone operators of type Γdϕ, (v) of Theorem 7 implies that (43)dL+T+C+εJ,BR0,f∗=dL+T+εJ,BR0,0.Next we show that dL+T+εJ,BR0,0=1. We consider (44)Kt,x=Lx+tTx+εJx,t,x∈0,1×DL∩DT∩B¯R0.Following the above arguments, it is not difficult to show that 0∉Kt,DL∩DT∩∂BR0 for all t∈0,1. Since T is densely defined, it is well-known that Tt=tTt∈0,1 is a pseudomonotone homotopy of type Γdϕ; that is, (v) of Theorem 7 gives (45)dL+T+εJ,BR0,0=dL+εJ,BR0,0=1.Consequently, we get (46)dL+T+C+εJ,BR0,f∗=dL+T+εJ,BR0,0=dL+εJ,BR0,0=1;that is, for each εn↓0+, there exist xn∈DL∩DT∩BR0 and vn∗∈Txn such that (47)Lxn+vn∗+Cxn+εnJxn=f∗∀n.By using Γdϕ condition on T and boundedness of C, we can follow the arguments used in the proof of Theorem 5 to conclude that f∗∈L+T+CDL∩DT∩BR0. Furthermore, L+T+C is surjective provided that L+T+C is coercive. The proof is completed.Next we give the following important theorem on maximality ofL and L+T without requiring (i) T or L to be quasibounded and 0∈DL∩DT and (ii) DL∩DT∘≠∅. The maximality condition (i) and (ii) are attributed to Browder and Hess [16] and Rockafellar [17], respectively.Theorem 9.
LetT:X⊃DT→2X∗ be densely defined maximal monotone of type Γdϕ and 0∈T0 and L:X⊃DL→X∗ be linear monotone, surjective, and closed such that L-1:X∗→X is compact. Then L+T is maximal monotone.Proof.
Letf∗∈X∗. By the boundedness and continuity of J and monotonicity of L and T with 0∈T0, it follows that there exists R>0 such that Lx+v∗+Jx-f∗,x≥x2-f∗x for all x∈D(L)∩DT and v∗∈Tx; that is, L+T+J is coercive. By Theorem 8, we conclude that RL+T+J=X∗; that is, L+T is maximal monotone. The maximality of L follows by setting T=0.


## 4. Degree Theory in a Real Hilbert Space withRL≠H

The content of this section outlines the construction of the degree mapping for operators of the typeL+T+C in the setting of a real Hilbert space, where T and C are as in Section 3 and L:H⊇DL→H is linear densely defined, self-adjoint, closed, and range closed. The closedness of RL is achieved if we assume RL=NL⊥, where NL denotes the null space of L. Under this condition, one can easily see that the restriction of L to DL∩RL is one to one and onto RL. Let P:H→NL be the orthogonal projection onto NL. In addition, it is well-known that H=NL⊕RL. For each ε>0, it follows that Lε=εP+L:H⊃DL→H is linear and surjective. For further properties of operators of type L, the reader is referred to the paper by Brézis and Nirenberg [18]. In the following lemma, we shall show that Lε-1:H→H is compact for suitable ε>0.Lemma 10.
LetL:H⊇DL→H be linear, densely defined, and self-adjoint and L-1:RL→H be compact. Then there exists α>0 such that, for each ε∈0,α, the operator Lε:H⊃DL→H is surjective and Lε-1:H→H is compact.Proof.
By the property of the orthogonal projectionP onto NL, it is well-known that P is nonexpansive; that is, Px-Py≤x-y for all x∈H and y∈H. Since L-1:RL→H is compact (i.e., it is continuous and linear), there exists α>0 such that (48)Lx-Ly≥αx-y∀x∈DL,y∈DL.For each ε∈0,α, we see that(49)Lεx-Lεy=Lx+εPx-Ly+εPy=Lx-Ly+εPx-Py≥Lx-Ly-εPx-Py≥αx-y-εx-y=α-εx-yfor all x∈DL and y∈DL, which implies that Lε is expansive; that is, Lε-1:H→H is continuous. Next we show that Lε-1 is compact. Let xn be bounded in H and yn=εP+L-1xn for all n; that is, εP+Lyn=xn for all n; that is, yn=L-1xn-εPyn for all n. The boundedness of yn follows because of the expansiveness of εP+L-1. Since L-1 is compact, we assume by passing to a subsequence that yn=L-1xn-εPyn→y0 as n→∞; that is, the compactness of Lε-1 is proved.As a consequence of Lemma10, it follows that Lε-1B is a compact operator provided that B:H→H is a bounded operator. Theorem 11 gives analogous result like that of Theorem 5.Theorem 11.
LetG be a nonempty, bounded, and open subset of H. Let Ctx=tC1x+1-tC2x, t,x∈0,1×∈G¯, where Cii=1,2:H→H is bounded demicontinuous operator. Let L:H⊃DL→H be linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact. Suppose Ttt∈0,1 is a pseudomonotone homotopy of maximal monotone operators uniformly of type Γdϕ. Assume, further, that 0∉L+Tt+CtDL∩DTt∩∂G for all t∈0,1. Then degree dLSI+Lε-1Tεt+Ct,G,0 is well-defined and independent of sufficiently small ε>0 and t∈0,1.Proof.
SinceLε-1 is compact and C and Tε are bounded demicontinuous operators, it follows that Lε-1Tε+C:H→H is a compact operator. Suppose there exist xn∈∂G and tn∈0,1 such that xn+εnP+L-1Tεnxn+Ctnxn=0 for all n; that is, εnPxn+Lxn+Tεntnxn+Ctnxn=0 for all n. Since εnP+L is surjective, it follows that xn∈DL, Lxn∈RL, and εnPxn+Tεntnx+Ctnxn∈RL for all n; that is, we get (50)xn+L-1εnPxn+Tεntnx+Ctnxn=0∀n.Since C1, C2, and P are bounded and Ttt∈0,1 is uniformly of type Γdϕ, we can follow the arguments in the proof of Theorem 5 to conclude that Lxn and Tεntnxn are bounded. As a result, the compactness of L-1 implies the existence of a subsequence, denoted again by L-1εnPxn+Tεntnx+Ctnxn, such that (51)xn=-L-1εnPxn+Tεntnx+Ctnxn⟶x0∈∂Gasn⟶∞.
SinceC and P are bounded and the family Tt is uniformly of type Γdϕ, the proof can be completed by following exactly similar arguments as in the proof of Theorem 5. The details are omitted here.Based on Theorem11, the definition of the degree mapping is given below.Definition 12.
LetG be a nonempty, bounded, and open subset of H. Let T:H⊃DT→2H be maximal monotone of type Γdϕ, C:H→H be bounded demicontinuous operator, and L:H⊃D(L)→H be linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact. Assume, further, that f∗∉L+T+C(DT∩DL∩∂G). Then the degree of L+T+C at f∗∈H with respect to G is defined by (52)dL+T+C,G,f∗=limε↓0+dLSI+Lε-1Tε+C-f∗,G,0,where Lε=εP+L, P:H→NL is the orthogonal projection onto NL, dLS denotes the Leray-Schauder degree mapping for compact perturbations of the identity, and Tε is the Yosida approximant of T.The basic properties and homotopy invariance results like that of Theorem5 and existence theorems analogous to Theorems 8 and 9 can be proved in Hilbert space setting by using the surjectivity of εP+L instead of the surjectivity of L. The degree satisfies the following properties.Theorem 13.
LetG be a nonempty, bounded, and open subset of H. Let T:H⊃DT→2H be maximal monotone of type Γdϕ, L:H⊃DL→H be linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact, and C:H→H be bounded demicontinuous operator. Then (i)
(normalization) there existsα>0 such that dL+αJ,G,0=1 if 0∈G and dL+αJ,G,0=0 if 0∉G¯. If L is monotone, then dL+J,G,0=1 if 0∈G and dL+J,G,0=0 if 0∉G¯;(ii)
(existence) if0∉L+T+C(DL∩DT∩∂G) and dL+T+C,G,0≠0, then 0∈L+T+CDL∩DT∩G;(iii)
(decomposition) letG1 and G2 be nonempty and disjoint open subsets of G such that f∗∉L+T+CG¯∖G1∪G2. Then (53)dL+T+C,G,f∗=dL+T+C,G1,f∗+dL+T+C,G2,f∗;(iv)
(translation invariance) letf∗∉L+T+CDL∩DT∩∂G. Then we have (54)dL+T+C-f∗,G,0=dL+T+C,G,f∗;(v)
(homotopy invariance) letTtt∈0,1 be a pseudomonotone homotopy of maximal monotone operators uniformly of type Γdϕ and Ci:H→Hi=1,2 is bounded demicontinuous operator. Let (55)Ht,x=Lx+Ttx+tC1x+1-tC2x,t,x∈0,1×(DL∩DTt∩G¯). Then dHt,·,G,0 is independent of t∈0,1 provided that 0∉H(t,DL∩DTt∩∂G) for all t∈0,1.Proof.
The proofs can be easily completed as in the arguments used in the proofs of Theorems5 and 7.This part of the theory improves the degree theory developed by Berkovits and Mustonen [19] for operators of the type L+C, where C is bounded demicontinuous pseudomonotone. In the present paper, we only assumed that C is bounded demicontinuous operator. Berkovits and Mustonen [19, Theorem 10, p. 959], gave an existence result for solvability of operator equations of the type Lx+cx+Cx=h, where C is bounded demicontinuous operator with bounded range, c>0 such that -c∈σL (where σL denotes the set of all eigenvalues of L), cI+C is pseudomonotone, and the recession function (cf. Brézis and Nirenberg [18]) corresponding to C given by JC(u)=liminft→∞,v→uCtv,v satisfies h,v<JCv for all v∈Ker⁡L+cI with v=1. However, for c=0, these conditions on C and L exclude the possibility that Ker⁡L={0}. If R(C) is bounded and c is any constant, we can easily see that cI+C is sublinear for all x satisfying x≥1 (i.e., cx+Cx≤cx+d/xx≤c+dx=τx for all x≥1, where τ=c+d and d=supCx:x∈H<∞). As a result of Theorem 14 below, the surjectivity of L+cI+C follows under mild assumption on the constant c omitting both conditions such that cI+C is pseudomonotone and -c∈σL.Theorem 14.
LetL:H⊇DL→H be linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact and C:H→H be bounded demicontinuous operator. Assume, further, that there exist nonnegative constants τ∈0,α and θ such that Cx≤τx+θ for all x∈DL with sufficiently large x, where α is the largest positive constant satisfying αx≤Lx for all x∈DL. Then L+C is surjective. If a reflexive Banach space X is used instead of a real Hilbert space H, then the same conclusion holds provided that L is surjective.Proof.
Letf∈H and ε∈0,α-τ (i.e., ε∈0,α). Consider the homotopy equation given by (56)Ht,x=Lx+tCx-f+ε1-tJx,t,x∈0,1×DL.Consequently, we have (57)Ht,x=Lx+tCx-f+ε1-tJx≥Lx-tCx-f-1-tεJx≥Lx-Cx-f-εx≥Lx-τx-θ-f-εx=Lx1-τ+εxLx-f+θLx=Lx1-τ+εα-f+θLx for all x∈DL∖0 and t∈0,1. Since ε∈0,α-τ (i.e., 1>τ+ε/α) and the right hand side of this inequality is independent of t∈0,1, letting x→∞ implies that there exists R=Rf>0 such that Ht,x>0 for all x∈DL∩∂BR0 and t∈0,1. Consequently, by using (i) and (v) of Theorem 13, we conclude that (58)dHt,·,BR0,0=dH1,·,BR0,0=dL+C-f,BR0,0=dH0,·,BR0,0=dL+εJ,BR0,0=1; that is, f=Lx+Cx is solvable in DL. Since f∈H is arbitrary, we conclude that L+C is surjective. The proof is completed.In the case whenRC is bounded, we can apply Theorem 14 to conclude that L+cI+C is surjective because of the sublinearity of cI+C with τ=c+d∈0,α and C is demicontinuous operator. As a result, it follows that c can be zero and L+C is surjective if α>d without C being pseudomonotone. In [19], Berkovits and Mustonen gave an existence theorem for the surjectivity of operators of the type L+C, where L satisfies conditions of Theorem 14 and C:H→H is bounded demicontinuous pseudomonotone satisfying the following: (i) Lx,x≥-1/αLx2 for all x∈DL, where α is the largest positive constant, (ii) there exist c1>0 and η>0 such that Cx≥ηx-c1 for all x∈DL, and (iii) there exist γ∈0,α and c2>0 such that Cx,x≥1/γCx2-c2 for all x∈DL. By combining (ii) and (iii), we can easily see that (59)Cx≤γx+γc2for all x satisfying x≥1+c1/η, which is a sublinearity condition used in Theorem 14. This shows that conditions (ii) and (iii) used by Berkovits and Mustonen [19, Theorem 8, p. 957] give stronger conditions on C as compared with the sublinearity of C. However, Theorem 14 does not need condition (ii) or (iii). The last but the main improvement of Theorem 14 over that of Berkovits and Mustonen [19] is dropping the requirement of C to be pseudomonotone. It is worth mentioning here that the same conclusion holds in Theorem 14 if the sublinearity condition on C holds for all x∈DL with sufficiently large x. As a result, we get the following corollary.Corollary 15.
LetL:H⊇DL→H be linear, densely defined, self-adjoint, closed, and range closed such that L-1:RL→H is compact and C:H→H be bounded demicontinuous operator. Assume, further, that there exist nonnegative constants μ and β such that 0≤μ≤βα-1 and (60)Cx,x≥βCx2-μx∀x∈H,where αβ≥1 and α is the largest positive constant such that αx≤Lx for all x∈DL. If Cx→∞ as x→∞, then L+C is surjective. If a reflexive Banach space X is used instead of a real Hilbert space H, then the same conclusion holds provided that L is surjective.Proof.
By the side condition onC, we see that(61)βCx2≤μx+Cx,x=μx+Cxx∀x∈H.Consequently, we get (62)Cx≤μxβCx+1βx∀x∈H∖0.Since Cx→∞ as x→∞, there exists R>0 such that Cx≥1 for all x≥R; that is, we have (63)Cx≤τ~x∀xwithx≥R,where τ~=μ+1/β. Since τ~∈0,α by the hypotheses, we can apply Theorem 14 with θ=0 and τ=τ~ to conclude that L+C is surjective. The proof is completed.We can observe that the largest positive constantα used in the hypotheses of Theorem 14 satisfies the condition (64)Lx,x≥-1αLx2∀x∈DL.In [18], Brézis and Nirenberg [18, Theorem III.2, p. 270] proved that f∈RL+C∘ provided that the following conditions hold: (i)
dim⁡NL<∞ and there are positive constants γ<α, μ, and λ such that (65)Cx-f,x≥1γCx2-μx-λu1+1,where u=u1+u2, u1∈NL, and u2∈RL.(ii)
JCv>f,v+c0v for all v∈NL with v≠0, where c0>0, γ<α1+μ/c0-1, and JC is the recession function of C. In view of these, Corollary 15 does not require dim⁡NL<∞ or (ii). In conclusion, Theorem 14 and its corollary gave new surjectivity results with weaker assumptions on L and C. In addition, we note here that Berkovits and Mustonen [19] proved surjectivity of L+C under weak coercivity condition of the type Cx≥ηx-c1 for all x∈DL, for some η>0 and c1≥0, and condition of type (i) with the possibility of having infinite dimensional null space of L.


## 5. Applications

In this section, we shall apply the abstract existence results to prove existence of weak solutions for nonlinear parabolic and hyperbolic problems such as wave and minimal surface equations. In these examples, the main contribution is that the Leray-Lion condition which guarantees pseudomonotonicity of the nonlinear term(s) is dropped. This will help to treat larger class of nonlinear equations and inequalities in appropriate Sobolev spaces. In the following examples, the norm ofu∈H, where H=L20,T;V, where V=H01Ω, is denoted by u.Example 16 (nonlinear parabolic equation).
LetH=L20,T;V, where V=H01(Ω). We prove the existence of weak solution for the nonlinear parabolic problem given by(66)∂u∂t-∑i=1N∂∂xiAix,u,∇u+Hλx,u,∇u=fx,tinQT,ux,t=0x,t∈∂QT,ux,0=ux,Tx∈Ω,where QT=Ω×0,T, Ω is a nonempty, bounded, and open subset of RN with smooth boundary, ∂QT=∂Ω×0,T, λ>0, Aix,u,∇u=∂/∂xiρx,u,∇u+aix,u,∇u, Hλx,u,∇u=-λΔu+gx,u,∇u, and bi=∂ρ(x,η,ξ)/∂xi (i=1,2,3,…,N) for all x,η,ξ∈Ω×R×RN. Suppose the following hypotheses are satisfied: (i)
aii=1,…,N, bi(i = 1,2,…,N),andg:Ω×R×RN↦R are Carathéodory functions; that is, x↦aix,η,ξ, x↦bix,η,ξ, and x↦gx,η,ξ are measurable functions for almost all η,ξ∈R×RN and η,ξ↦aix,η,ξ, η,ξ↦bix,η,ξ, and η,ξ↦gx,η,ξ are continuous for almost all x∈Ω.(ii)
There existk1∈L2Q and c1≥0 such that (67)max⁡ρx,η,ξ,∇ρx,η,ξ≤k1x,t+c1η+ξ,∑i=1N∂∂xiρx,η,ξξi≥η2for all x,η,ξ∈Ω×R, where x=x1,x2,…,xN∈RN and (68)∇xρx,η,ξ=∂ρx,η,ξ∂xii=1N.(iii)
There existk2∈L2Q and c2≥0 such that (69)aix,η,ξ≤k2x,t+c2η+ξfor all i=1,…,N and x,η,ξ∈Ω×R×RN.(iv)
There existk3∈L2Q and c3≥0 such that (70)gx,η,ξ≤k3x,t+c3η+ξfor all x,η,ξ∈Ω×R×RN.(v)
There existsk0∈L2Q such that (71)∑i=1Naix,η,ξξi≥-k0ξfor all x,η,ξ∈Ω×R×RN, where ξ=ξii=1N. Let L:H⊇DL→H be given by Lu=u′-λΔu, u∈DL, where λ>0 and (72)DL=u∈L20,T;H2Ω∩H01Ω:u′∈L2Q,u0=uT;that is, we have(73)Lu,ϕ=u′,ϕ+λ∫Q∇ux,t∇ϕx,tdxdt,ϕ∈H, u∈DL, where u′ is understood in the sense that (74)∫0Tu′tϕtdt=-∫0Tutϕ′tdtfor all ϕ∈C0∞0,T and <,> is the inner product in L2Q. The norm of u in H is given by (75)u=∫0TutV2dt1/2,where utV=utL2Ω+∇utL2Ω1/2,t∈0,1. It is well-known that L is surjective maximal monotone and L-1:H→H exists and is continuous. As a consequence of the compact embedding of H01Ω into L2Ω, it is not difficult to show that L-1 is compact. In addition, the maximal monotonicity of L implies the closedness of graph of L; that is, L is closed. Let A:H→H be given by(76)Au,ϕ=∫Q∇ρx,u,∇u∇ϕx,tdxdt,ϕ∈H, u∈H and B:H→H be given by(77)Bu,ϕ=∑i=1N∫Qaix,u,∇u∂ϕx,t∂xidxdt,ϕ∈H, u∈H. By conditions (i) through (iv), it well-known that A and B are bounded continuous operators; that is, C:H→H given by C=A+B is bounded continuous operator. A weak solution in H of (66) is understood in the sense of the following definition.Definition 17.
Letf∈L2Q. An element u∈H is called a “weak solution” of (66) if (78)Lu+Cu,ϕ=f,ϕ∀ϕ∈H;that is, u satisfies the functional equation (79)u′,ϕ+∫Q∇ux,t∇ϕx,t+∑NNaix,u,∇u∂ϕx,t∂xidxdt=∫Qfx,t-gx,u,∇uϕx,tdxdtfor all ϕ∈H. The following existence result holds.Theorem 18.
Suppose conditions (i) through (v) are satisfied. Then, for eachf∈L2Q, (66) admits at least one weak solution in H.Proof.
LetL, A, and B be as defined in (73), (76), and (77), respectively. It is well-known that L:H⊇DL→H is surjective maximal monotone (i.e., graph closed) and L-1:H→H is compact and C=A+B is bounded continuous operator. By applying Hólder’s inequalities along with conditions (i) through (v), monotonicity of u′, and coercivity of L we can see that (80)Lu+Cu-f,u=Lu,u+Cu-f,u=u′,u-∫QΔux,tux,tdxdt+∫Q∇ρx,u,∇u∇ux,tdxdt+∑i=1N∫Qaix,u,∇u∂u∂xidxdt-fL2QuL2Q≥∫Q∇ux,t2dxdt+∫Qux,t2dxdt-∫Qk0∇udxdt-fL2QuL2Q≥u2-k0L2Q∇uL2Q-fL2QuL2Q≥u2-k0L2Qu-fL2Qu=u21-fL2Q+k0L2Qu for all u∈DL∖0. Since the right side of the above inequality tends to ∞ as u→∞, there exists R=Rf>0 such that (81)Lu+Cu-f,u>0for all u∈DL∩∂BR0; that is, the boundary condition in Theorem 8 is satisfied with linear operator L, maximal monotone operator T=0, and bounded continuous operator C. Therefore, we conclude that the problem f∗=Lu+Cu is solvable in DL, where f∗ is a functional on H generated by f∈L2Q. Since f∈L2Q is arbitrary, there exists u∈DL such that (82)Lu+Cu,ϕ=f,ϕ∀ϕ∈H,that is, (66) admits at least one weak solution in DL. The proof is completed.One of the main advantages of this theory concerning parabolic problems of type (66) is dropping the requirement for C to be pseudomonotone type, that is, dropping the condition(83)∑i=1Naix,η,ξ′-aix,η,ξξi′-ξi≥0;for all x,η,ξ′ and x,η,ξ in Ω×R×RN.Example 19 (minimal surface equation).
LetH, Ω, QT=Ω×0,T, and ∂QT=∂Ω×0,T be as in Example 16. Let βi:Ω×R×RN→R be given by (84)βix,η,ξ=ξi1+ξ2∀x,η,ξ∈Ω×R×RNfor all i=1,2,…,N. It follows that βi (i=1,2,…,N) satisfies (i) and |βix,η,ξ|≤|ξ|≤|η|+ξ, that is, condition (iv) of Example 16 with k3x,t=0 for all x,t∈Q and c3=1. We notice here that (85)-div⁡∇u1+∇u2=-∑i=1N∂∂xi11+∇u2∂u∂xi,u∈H.Let L=∂/∂t-λΔ, A~:H→H, B~:H→H, and C=A~+B~ be defined by (86)A~u,ϕ=∑i=1N∫Qβix,u,∇u∂ϕx,t∂xidxdt,B~u,ϕ=∫Qgx,u,∇uϕx,tdxdt.u∈H and ϕ∈H. By using the operators L and C and following analogous arguments used in the proof of Theorem 18, for each f∗∈H, we can show that f∗=Lu+Cu is solvable in DL; that is, u∈DL satisfies Lu+Cu,ϕ=f,ϕ for allϕ∈H provided that f∈L2Q. Equivalently, we conclude that the minimal surface equation, given by (87)∂u∂t-λΔu-div⁡∇u1+∇u2+gx,u,∇u=fx,tinQT,ux,t=0x,t∈∂QT,ux,0=ux,Tx∈Ω, admits at least one weak solution in H provided that λ>0, f∈L2Q and g satisfies conditions (i) and (iv) of Example 16, and gx,η,ξη≥η2 for all (x,η,ξ)∈Ω×R×RN.Example 20 (nonlinear wave equation).
LetH=L2Ω, Ω=0,2π×0,π. We shall show existence of weak solutions in H of the wave equation given by(88)utt-uxx+gt,x,u=fx,tt,x∈Ω,ut,0=ut,π=0t∈0,2π,ux,0=ux,2πx∈0,π,utx,0=utx,2πx∈0,π, where f∈L2Ω and (a)
g:Ω×R↦R is Carathéodory function;(b)
there existc0>0 and k0∈L2Ω such that (89)gt,x,u≤k0t,x+c0u∀t,x∈Ω.A weak solution u∈H of (88) is understood in the sense of the following definition.Definition 21.
Letf∈L2Ω. An element u∈H is called a “weak solution of (88)” if u satisfies(90)u,ϕtt-ϕxx+Cu,ϕ=f,ϕ∀ϕ∈Cπ2Ω¯,where (91)Cπ2Ω¯=ϕ∈C2Ω¯:ϕ0,t=ϕπ,t=0,ϕx,0=ϕx,2π,ϕtx,0=ϕtx,2π;that is, (92)∫02π∫0πuϕtt-ϕxxdxdt+∫02π∫0πgt,x,uϕdxdt=∫02π∫0πfx,tϕdxdtfor all ϕ∈Cπ2Ω¯.Next we prove the following existence theorem.Theorem 22.
Supposeg satisfies conditions (a) and (b), and let C:H→H be given by Cu=g·,·,u, u∈H. Then, for each f∈H, (88) admits at least one weak solution in H.Proof.
By the sublinearity ofg, we can easily see that C:H→H is bounded continuous operator and Cu≤k0L2Ω+c0u for all u∈H. The abstract representation of the wave operator ∂2/∂t2-∂2/∂x2 in H=L2Ω is the linear operator L:H⊇DL→H given by (93)Lu=∑m∈Z,n∈Z+n2-m2umnφmn,u=∑m∈Z,n∈Z+umnφmn,where umn=u,φmn,  φmnt,x=π-1exp⁡(imt)sin⁡(nx), and (94)DL=u∈H:∑m∈Z,n∈Z+n2-m22umn2<∞.If u∈H2Ω, then, by using successive integration by parts, we see that u∈DL is a solution of Lu+Cu=f if and only if u satisfies (90). It is well-known that L is linear densely defined, self-adjoint, and closed and RL=NL⊥ (i.e., RL is closed), the restriction of L to DL∩RL, denoted again by L, is one to one onto RL, and L-1:RL→H is compact. Since L-1 is continuous, there exists d>0 such that du≤Lu for all u∈DL. Let α>0 be the largest positive constant to satisfy this condition. If c0∈0,α, then, by applying Theorem 14 with θ=k0L2Ω and τ=c0, we conclude that there exists u∈DL such that (95)Lu+Cu,ϕ=f,ϕ∀ϕ∈H;that is, (88) admits a weak solution in DL. The proof is completed.It is important to notice that the monotonicity assumption ong is not required to establish existence of weak solutions for (88). For existence of weak solutions under the requirement that g is monotone, the reader is referred to the papers by Rabinowitz [20], Brézis and Nirenberg [18, 21], Brezis [22], and Barbu and Pavel [23] and the references therein. In an attempt to remove the monotonicity assumption on g, Coron [24] used additional assumption on NL; that is, he assumed the existence of a closed subspace H1 of H such that NL∩H1=0 and H1 is invariant under L and g. For further results on nonmonotone g, the reader is referred to the papers by Coron [24] and Hofer [25]. Consequently, Theorem 22 provides a new result concerning existence of weak solution for the nonlinear wave equation with lower order nonlinear part g satisfying only continuity and sublinearity conditions. In conclusion, we like to mention that various examples of pseudomonotone type operators under Leray-Lion type growth conditions along with (83) can be found in the papers of Landes and Mustonen [26], Mustonen [27], and Mustonen and Tienari [28] and the references therein. Existence results for perturbations of maximal monotone operator T by bounded demicontinuous operator of type S+ or bounded pseudomonotone can be found in the papers of Browder and Hess [16], Kenmochi [29], Kartsatos [30], Asfaw [31–33], and Le [34] and the references therein. For detailed study of ranges of sums of perturbed operators in Hilbert space and more examples and properties of elliptic, parabolic, and hyperbolic linear operators, we mention the papers by Brézis and Nirenberg [18] and Berkovits and Mustonen [19] and the references therein.



---

*Source: 3970621-2016-12-05.xml*