# A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem

Authors: Teffera M. Asfaw
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

LetX be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). A new topological degree theory is developed for operators of the type T+S+C. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type T+S+C, where C is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.


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## Body

## 1. Introduction: Preliminaries

In what follows, the norm of the spacesX and X∗ will be denoted by ·. For x∈X and x∗∈X∗, the pairing 〈x∗,x〉 denotes the value x∗(x). Let X and Y be real Banach spaces. For an operator T:X→2Y, we define the domain D(T) of T by D(T)={x∈X:Tx≠∅}, and the range R(T) of T by R(T)=⋃x∈D(T)Tx. We also use the symbol G(T) for the graph of T: G(T)={(x,x∗):x∈D(T),x∗∈Tx}. An operator T:X⊃D(T)→Y is “demicontinuous" if it is continuous from the strong topology of D(T) to the weak topology of Y. It is “compact" if it is strongly continuous and maps bounded subsets of D(T) to relatively compact subsets of Y. An operator T:X⊃D(T)→2Y is “bounded" if it maps each bounded subset of D(T) into a bounded subset of Y. It is “finitely continuous" if it is upper semicontinuous from each finite dimensional subspace F of X to the weak topology of Y. Let ϕ:[0,∞)→(-∞,∞) be a continuous and strictly increasing function such that ϕ(t)→∞ as t→∞. The mapping Jϕ:X→2X∗ defined by(1)Jϕx=x∗∈X∗:x∗,x=ϕxx,x∗=ϕxis called the “duality mapping" associated with ϕ. As a consequence of the Hahn-Banach theorem, it is well-known that Jϕ(x)≠∅ for all x∈X. Since X and X∗ are locally uniformly convex, Jϕ is single valued, bounded, monotone, and bicontinuous. The following definitions are needed throughout the paper.Definition 1.
An operatorT:X⊃D(T)→2X∗ is said to be (i)
“monotone" if for everyx∈D(T), y∈D(T), u∗∈Tx, and v∗∈Ty, we have u∗-v∗,x-y≥0;(ii)
“maximal monotone" ifT is monotone and R(T+λJ)=X∗ for every λ>0; that is, T is maximal monotone if and only if T is monotone and 〈u∗-u0∗,x-x0〉≥0 for every (x,u∗)∈G(T) implies x0∈D(T) and u0∗∈Tx0;(iii)
“coercive" if eitherD(T) is bounded or there exists a function ψ:[0,∞)→(-∞,∞) such that ψ(t)→∞ as t→∞ and 〈y∗,x〉≥ψ(x)x for all x∈D(T) and y∗∈Tx;(iv)
“weakly coercive” if eitherD(T) is bounded or |Tx|→∞ as x→∞, where for each x∈D(T), Tx=inf⁡{v∗:v∗∈Tx}.It is important to note here that the class of weakly coercive operators includes the classes of coercive operators. For a maximal monotone operatorT:X⊇D(T)→2X∗, we know that R(T+λJ)=X∗ for all λ>0 and (T+λJ)-1:X∗→D(T) is single valued and demicontinuous. In addition, the operator Tt:X→X∗, t∈(0,∞), defined by Ttx=(T-1+tJ-1)-1x, is the “Yosida approximant" of T. It is bounded, continuous, and maximal monotone with domain X such that Ttx⇀T(0)x as t→0+, for every x∈D(T), where T(0)x=inf⁡{y∗:y∗∈Tx}. Furthermore, the operator Jt:X→D(T), defined by Jtx=x-tJ-1(Ttx), is called the “Yosida resolvent" of T. It is continuous, Ttx∈T(Jtx) for every x∈X, and limt→0Jtx=x for all x∈coD(T)¯, where coD(T) is the convex hull of the set D(T). Furthermore, for each x∈D(T), Ttx≤Tx for all t>0. Browder and Hess [1] introduced the following definitions. The original definition of single valued pseudomonotone operator is due to Brèzis [2].Definition 2.
An operatorT:X⊃D(T)→2X∗ is said to be (a)
“pseudomonotone" if the following conditions are satisfied:
(i)
For everyx∈D(T), Tx is nonempty, closed, convex, and bounded subset of X∗;(ii)
T is finitely continuous; that is, for every x0∈D(T)∩F and every weak neighborhood V of Tx0 in X∗, there exists a neighborhood U of x0 in F such that TU⊂V;(iii)
for each sequence{xn}⊂D(T) with yn∗∈Txn such that xn⇀x0∈D(T) and (2)limsupn→∞⁡yn∗,xn-x0≤0,we have that, for everyx∈D(T), there exists y∗(x)∈Tx0 such that (3)y∗x,x0-x≤liminfn→∞yn∗,xn-x;in particular, lettingx0 in place of x in the above inequality, the pseudomonotonicity of T implies (4)liminfn→∞yn∗,xn-x0≥0;(b)
“of type(S+)" if (i) and (ii) of (a) hold and for each sequence {xn} in D(T) such that xn⇀x0 in X as n→∞ and every wn∗∈Sxn with (5)limsupn→∞wn∗,xn-x0≤0,we havexn→x0∈D(T) and there exists a subsequence of {wn∗}, denoted again by {wn∗}, such that wn∗⇀w0∗∈Tx0 as n→∞;(c)
“of type(S)” if (i) and (ii) of (a) hold and for any sequence xn∈D(T), vn∗∈Txn such that vn∗→v0∗ as n→∞, it follows that there exists a subsequence of {xn}, denoted again by {xn}, such that xn→x0 as n→∞.It is not difficult to see that the class of operators of type(S) includes the classes of operators of type (S+). Furthermore, it holds that T+C is of type (S) provided that T is of type (S) and C is compact. The main goals of this paper are(i)
to develop suitable degree theory for operators of the typeT+S+C, where T:X⊇D(T)→2X∗ is maximal monotone, S:X→2X∗ is bounded of type (S+), and C:D(C)→X∗ is compact with D(T)⊆D(C) and sublinear; that is, there exist τ≥0 and σ≥0 such that Cx≤τx+σ for all x∈D(C). The existing degree theories for operators of the type T+S cannot be used to treat inclusions involving operators of the type T+S+C because the compact operator is not everywhere defined. For recent degree theories for multivalued bounded (S+) or bounded pseudomonotone perturbations of arbitrary maximal monotone operators, the reader is referred to the papers by Asfaw and Kartsatos [3], Asfaw [4], Adhikari and Kartsatos [5], and the references therein. In these theories, the maximal monotone operator is arbitrary and (S+) and/or pseudomonotone operator is everywhere defined. The original degree mapping due to Browder [6] is for operators of the type T+f, where f is single valued bounded operator of type (S+) defined from the closure of a nonempty, bounded, and open subset G of X. Hu and Papageorgiou [7] generalized Browder’s theory for multivalued compact perturbation of T+f, where the compact operator is defined on G¯. All these theories do not include the case where C is not defined on G¯, in particular, when D(C) contains D(T). In view of these, our work in developing a degree theory for operators of the type T+S+C, where C is a compact operator with D(T)⊆D(C), is essential. It is worth mentioning that the theory associated with (i) is a generalization of the previous degree theories for bounded (S+) perturbations of maximal monotone operators due to Browder [6], Kobayashi and Otani [8], Hu and Papageorgiou [7], Asfaw and Kartsatos [3], and the references therein. The most general degree theory currently available which is due to Asfaw [9] is for pseudomonotone perturbations of the sum of two maximal monotone operators with one of the maximal monotone operators which is of type Γϕβ;(ii)
to derive existence theorem(s) in order to establish solvability of operator inclusion problems involving operators of the typeT+S+C. Consequently, the theory developed in (i) is applied to prove existence of solution for the inclusion problem f∗∈(T+S+C)(D(T)∩BR(0))¯ provided that there exists R=R(f∗)>0 such that (6)v∗+w∗+Cx-f∗,x>0for allx∈D(T)∩∂BR(0), v∗∈Tx, and w∗∈Sx; that is, R(T+S+C)¯=X∗ provided that T+S+C is coercive. The result is a generalization of the existence result due to Asfaw and Kartsatos [3, Theorem 17] for the operator T+S. This result yields the surjectivity of T+S+C provided that T+S+C is coercive and either S is bounded of type (S+) or T+S is operator of type (S).Throughout the paper, we shall use the following definition of a homotopy of class(S+).Definition 3.
Lett∈[0,1] and St:X⊃D(St)→2X∗. The family {St}t∈[0,1] is said to be a “homotopy of type (S+)” if the following are true: (i)
For eacht∈[0,1], x∈D(St), Stx is a nonempty, closed, convex, and bounded subset of X∗.(ii)
For eacht∈[0,1],St is finitely continuous.(iii)
Let{tn}⊂[0,1],xn∈D(Stn) be such that tn→t0 and xn⇀x0∈X. Let fn∈Stnxn be such that (7)limsupn→∞⁡fn,xn-x0≤0.Then xn→x0∈D(St0) and there exists a subsequence of {fn}, denoted again by {fn}, such that fn⇀f∈St0x0 as n→∞.The following lemma is due to Ibrahimou and Kartsatos [10].Lemma 4.
LetT:X⊃D(T)→2X∗ be maximal monotone and G⊂X be bounded. Let 0<s1≤s2,0<t1<t2. Let Ts≔sT. Then there exists a constant K1>0, independent of t and s, such that Ttsu≤K1 for all u∈G¯, s∈[s1,s2], and t∈[t1,t2].For basic definitions and further properties of mappings of monotone type, the reader is referred to Barbu [11], Pascali and Sburlan [12], Browder and Hess [1], and Zeidler [13].The content of the following important lemma is due to Brezis et al. [14].Lemma 5.
LetB be a maximal monotone set in X×X∗. If (un,un∗)∈B such that un⇀u in X, un∗⇀u∗ in X∗, and (8)limsupn→∞⁡un∗-u∗,un-u≤0,then (u,u∗)∈B and 〈un∗,un〉→〈u∗,u〉 as n→∞.Browder [6] introduced the concept of a pseudomonotone homotopy as given below.Definition 6.
Let{Tt}t∈[0,1] be a family of maximal monotone operators from X to 2X∗ such that 0∈Tt(0),  t∈[0,1]. Then {Tt}t∈[0,1] is called a “pseudomonotone homotopy" if it satisfies the following equivalent conditions: (i)
Suppose thattn→t0∈[0,1] and (xn,yn)∈G(Ttn) are such that xn⇀x0 in X, yn⇀y0 in X∗ and (9)limsupn→∞yn,xn≤y0,x0.Then (x0,y0)∈G(Tt0) and limn→∞〈yn,xn〉=〈y0,x0〉.(ii)
The mappingϕ:X∗×[0,1]→X defined by (10)ϕw,t≔Tt+J-1wis continuous.(iii)
For eachw∈X∗, the mapping ϕw:[0,1]→X defined by (11)ϕwt≔Tt+J-1wis continuous.(iv)
For any(x,y)∈G(Tt0) and any sequence tn→t0, there exists a sequence (xn,yn)∈G(Ttn) such that xn→x and yn→y as n→∞.For a maximal monotone operatorT:X⊇D(T)→2X∗, Kobayashi and Otani [8] proved that the family {tT}t∈[0,1] is a pseudomonotone homotopy of maximal monotone operators if and only if T is densely defined. It is worth mentioning that the proof of this fact does not require the hypothesis 0∈T(0). It is essential herein to mention that the original degree theory for single-value (S+) perturbations of maximal monotone operators is due to Browder [6]. For a generalization of Browder’s degree for multivalued compact perturbations of T+f, where T:X⊃D(T)→2X∗ is maximal monotone and f:G¯→X∗ is bounded demicontinuous of type (S+), the reader is referred to the paper due to Hu and Papageorgiou [7]. For existence results for compact perturbation of maximal monotone operators, the reader is referred to the paper due to Kartsatos [15]. For a relevant degree mapping for single multivalued operator of type (S+), we cite the paper of Zhang and Chen [16]. Recent developments on degree theories for perturbations of the sum of two maximal monotone operators can be found in the papers due to Adhikari and Kartsatos [5] and Asfaw [4].In Section2 we construct a degree mapping for operators of the type T+S+C, where T:X⊇D(T)→2X∗ is maximal monotone, S:X→2X∗ is bounded and of type (S+) or bounded pseudomonotone, and C:D(C)→X∗ is compact with D(T)⊆D(C) and satisfies a sublinearity condition. The existence of solutions for operator inclusion problems of the type Tu+Su+Cu∋f∗ is included in Section 3. In Section 4, the theory is applied to establish existence of weak solution(s) for a nonlinear parabolic problem in appropriate Sobolve spaces.


## 2. Degree Theory forT+S+C with D(T)⊆D(C)

### 2.1. Degree Theory forT+S+C with S Bounded and of Type (S+)

The goal of this section is to develop a degree theory for operators of the typeT+S+C, where T:X⊃D(T)→2X∗ is maximal monotone, S:X→2X∗ is bounded and of type (S+), and C:D(C)→X∗ is compact with D(T)⊆D(C). Throughout the paper, we assume that C belongs to Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). To this end, we start by proving the following useful lemma.Lemma 7.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊇D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) such that C belongs to class Γστ. Assume, further, that f∗∉T+S+C(DT∩∂G). Then there exists ε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined and independent of ε∈(0,ε0].Proof.
In the first step, we claim that there existsε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined for all ε∈(0,ε0]. Suppose that this is false; that is, there exist εn↓0+, xn∈∂G, and wn∗∈Sxn such that(12)vn∗+wn∗+CJεnxn=f∗∀n,where vn∗=Tεnxn. By the definitions of Tεn and Jεn, we have (13)Jεnxn=xn-εnJ-1vn∗∈DT,vn∗∈TJεnxn∀n.Since {xn} and S are bounded, it follows that {wn∗} is bounded. Since C belongs to Γστ, we get that (14)vn∗≤f∗-wn∗+CJεnxn≤κ0+τJεnxn+σ=κ0+τxn-εnJ-1vn∗+σ≤κ0+τxn+τεnvn∗+σfor all n, where κ0 is an upper bound for {f∗-wn∗}. This yields the estimate (15)1-τεnvn∗≤κ0+τxn+σfor all n. Since εn↓0+ and {xn} is bounded, it follows that {vn∗} and {Jεnxn} are bounded. The compactness of C implies the boundedness of {CJεnxn}. Now, assume without loss of generality that xn⇀x0, vn∗⇀v0∗, and wn∗⇀w0∗ as n→∞. Since C is compact, we may assume, by passing into a subsequence if necessary, that CJεnxn→g0∗ as n→∞. The maximality of T along with Lemma 5 gives (16)liminfn→∞⁡vn∗,xn-x0≥0;
that is, we obtain from (12) that (17)limsupn→∞⁡wn∗,xn-x0=limsupn→∞⁡-vn∗+CJεnxn-f∗,xn-x0=-liminfn→∞⁡vn∗+CJεnxn-f∗,xn-x0≤-liminfn→∞vn∗,xn-x0-liminfn→∞⁡CJεnxn-f∗,xn-x0=-liminfn→∞⁡vn∗,xn-x0≤0.Since S is of type (S+), we conclude that xn→x0∈∂G as n→∞ and w0∗∈Sx0. Consequently, using (12) we arrive at (18)limsupn→∞⁡vn∗,xn-x0≤0.The maximality of T along with Lemma 5 yields x0∈D(T)∩∂G and v0∗∈Tx0 and 〈vn∗,xn〉→〈v0∗,x0〉 as n→∞. Since C is compact and Jεnxn=xn-εnJ-1(vn∗)→x0∈D(T)⊆D(C) as n→∞, it follows that CJεnxn→Cx0=g0∗ as n→∞. Letting n→∞ in (12), we get f∗∈(T+S+C)(D(T)∩∂G). However, this is impossible. Thus, there exists ε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined for all ε∈(0,ε0].
Next, we shall prove thatd(Tε+S+CJε,G,f∗) is independent of ε∈(0,ε0]. Let εi∈(0,ε0]  (i=1,2) be such that 0<ε1<ε2≤ε0, q(t)=tε1+(1-t)ε2, t∈[0,1]. We consider the homotopy operator (19)Ht,x=Tqtx+Sx+CJqtx,t,x∈0,1×G¯.We will show that the family {H(t,·)}t∈[0,1] is a homotopy of class (S+) such that 0∉H(t,∂G) for all t∈[0,1]. To this end, let xn∈G¯, wn∗∈Sxn,  tn∈[0,1], fn∗=Tq(tn)xn+wn∗+CJq(tn)xn, xn⇀x0, and tn→t0 as n→∞ be such that (20)limsupn→∞⁡fn∗,xn-x0≤0.Since {xn} and S are bounded, it follows that {wn∗} is also bounded. Since q(tn)∈[ε1,ε2] for all n, we apply Lemma 4 to conclude that {Tq(tn)xn} and {Jq(tn)xn} are bounded. On the other hand, we see that (21)limsupn→∞⁡wn∗,xn-x0≤limsupn→∞⁡fn∗-Tqtnxn-Tqtnx0+CJqtnxn,xn-x0-limn→∞⁡Tqtnx0,xn-x0.By the compactness of C, we may assume without loss of generality that CJq(tn)xn→h0∗ as n→∞. Since q(tn)→q=q(t0)>0 as n→∞, we use the continuity of (0,∞)×X∋(t,x)→Tt(x) ([3], Lemma 6) to conclude that Tq(tn)x0→Tqx0 as n→∞. Combining these along with the monotonicity of Tq(tn), we obtain (22)limsupn→∞⁡wn∗,xn-x0≤0.Since S is of type (S+), we see that xn→x0 as n→∞ and there exists a subsequence of {wn∗}, denoted again by {wn∗}, such that wn∗⇀w0∗∈Sx0 as n→∞. Since (0,∞)×X∋(t,x)↦Ttx is continuous, we conclude that (23)fn∗⇀Tqx0+w0∗+CJqx0∈Tqx0+Sx0+CJqx0asn⟶∞;that is, {H(t,·)}t∈[0,1] is a homotopy of class (S+) such that 0∉H(t,∂G) for all t∈[0,1]. Therefore, d(H(t,·),G,f∗) is independent of t∈[0,1]; that is, d(Tε1+S+CJε1,G,f∗)=d(Tε2+S+CJε2,G,f∗). Since ε1 and ε2 are arbitrary in (0,ε0], we conclude that d(Tε+S+CJε,G,f∗) is well-defined and independent of ε∈(0,ε0]. This completes the proof.Based on Lemma7, the associated degree mapping is defined as follows.Definition 8.
LetG be a nonempty, bounded, and open subset of X, T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Assume, further, that f∗∉(T+S+C)(D(T)∩∂G). Then the degree mapping d for T+S+C at f∗∈X∗ with respect to G is defined by (24)dT+S+C,G,f∗=limε↓0+⁡dS+Tε+S+CJε,G,f∗,where dS+ is the degree mapping for multivalued bounded operators of type (S+) from [16].


### 2.2. Basic Properties of the Degree

Theorem 9.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:D(C)→X∗ be compact with D(T)⊆D(C) such that C belongs to Γστ. Then the following properties hold: (i)
(Normalization)d(J,G,0)=1 if 0∈G and d(J,G,0)=0 if 0∉G¯.(ii)
(Existence) iff∗∉(T+S+C)(D(T)∩∂G) and d(T+S+C,G,f∗)≠0, then f∗∈(T+S+C)(D(T)∩G).(iii)
(Decomposition) letG1 and G2 be nonempty, disjoint, and open subsets of G such that f∗∉(T+S+C)(D(T)∩(G¯∖(G1∪G2))). Then (25)dT+S+C,G,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗.(iv)
(Translation invariance) letf∗∉(T+S+C)(D(T)∩∂G). Then we have (26)dT+S+C-f∗,G,0=dT+S+C,G,f∗.(v)
LetM(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), where Si:X→2X∗(i=1,2) is bounded and of type (S+) and 0∉M(t,D(T)∩∂G) for all t∈[0,1]. Then d(M(t,·),G,0) is independent of t∈[0,1].(vi)
Let0∈G, N(t,x)=t(Tx+S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), where T:X⊃D(T)→X∗ is densely defined maximal monotone and positively homogeneous of order α>0, Si:X→2X∗(i=1,2) is bounded and of type (S+) such that 〈u∗,x〉≥x2 for all x∈X,u∗∈S2x, and 0∉N(t,D(T)∩∂G) for all t∈[0,1]. Assume, further, that 0∉M(t,D(T)∩∂G) for all t∈[0,1]. Then d(N(t,·),G,0) is independent of t∈[0,1].Proof.
The proof of (i) follows by settingT={0} and C={0}. To prove (ii), assume that f∗∉(T+S+C)(D(T)∩∂G) and d(T+S+C,G,f∗)≠0. By the definition of d, there exists ε0>0 such that d(Tε+S+CJε,G,f∗)≠0 for all ε∈(0,ε0]; that is, for each εn↓0+ there exist xn∈D(T)∩G and wn∗∈Sxn such that(27)vn∗+wn∗+CJεnxn=f∗,vn∗=Tεnxn∀n.Since S is bounded, it follows that {wn∗} is bounded. By using Γστ condition on C along with the arguments used in the proofs of Lemma 7, it is easy to see that {vn∗} and {Jεnxn} are bounded. Assume without loss of generality that xn⇀x0, wn∗⇀w0∗, vn∗⇀v0∗, and CJεnxn→g0∗ as n→∞. By the maximality of T, the (S+) condition on S, and the arguments used in the proof of Lemma 7, we conclude that x0∈D(T)∩G, v0∗∈Tx0, and w0∗∈Sx0 such that v0∗+w0∗+Cx0=f∗. This shows that f∗∈(T+S+C)(D(T)∩G).
Next we prove (iii). Suppose the hypotheses in (iii) hold. By the definition ofd, we see that d(T+S+C,G,f∗)=d(Tε+S+CJε,G,f∗) for all sufficiently small ε>0. Since Tε+S+CJε is bounded and of type (S+), the decomposition property of the degree mapping for multivalued (S+) operators implies (28)dT+S+C,G,f∗=limε↓0+⁡dTε+S+CJε,G,f∗=limε↓0+⁡dTε+S+CJε,G1,f∗+limε↓0+⁡dTε+S+CJε,G2,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗;
that is, (iii) holds.(iv) Suppose thatf∗∉(T+S+C)(D(T)∩∂G); that is, 0∉(T+S+C-f∗)(D(T)∩∂G). This implies that d(T+S+C-f∗,G,0) is well-defined. Since d(T+S+C-f∗,G,0)=d(Tε+S+CJε-f∗,G,0), by the translation property of the degree mapping for multivalued bounded operators of type (S+), we see that d(Tε+S+CJε-f∗,G,0)=d(Tε+S+CJε,G,f∗). Thus, (29)dT+S+C-f∗,G,0=limε↓0+dTε+S+CJε-f∗,G,0=limε↓0+dTε+S+CJε,G,f∗=dT+S+C,G,f∗.(v) Suppose that0∉M(t,D(T)∩∂G) for all t∈[0,1], where M(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯). For every ε>0, we consider (30)Mεt,x=Tεx+tS1x+CJεx+1-tS2x,t,x∈0,1×G¯.We shall show that there existsε0>0 such that d(Mε(t,·),G,0) is well-defined and independent of all (t,ε)∈[0,1]×(0,ε0]. To do this, we assume to the contrary that there exist εn↓0+, xn∈∂G, tn∈[0,1], wn∗∈S1xn, and un∗∈S2xn such that(31)vn∗+tnwn∗+CJεnxn+1-tnun∗=0,vn∗=Tεnxn∀n.Since {xn}, S1, and S2 are bounded, it follows that {wn∗} and {un∗} are bounded. By the Γστ condition on C, the boundedness of S1 and S2, and the arguments used in the proof of Lemma 7, we conclude that {Jεnxn} and {vn∗} are bounded. Assume without loss of generality that tn→t0, xn⇀x0, wn∗⇀w0∗, un∗⇀u0∗, vn∗⇀v0∗, Jεnxn=xn-εnJ-1(vn∗)⇀x0, and CJεnxn→g0∗ as n→∞. Suppose that t0=0. We have vn∗+un∗→0 as n→∞. Since S2 is of type (S+), it follows that xn→x0∈∂G, v0∗∈D(T), and 0∈Tx0+S2x0; that is, 0∈(T+S2)(D(T)∩∂G), that is, 0∈M(0,D(T)∩∂G). However, this is impossible. A similar proof covers the case t0≠1. Assume t0∈(0,1). Suppose there exists a subsequence of {tn}, denoted again by {tn}, such that tn≥τ0>0 for all n. Since T is maximal monotone, Lemma 5 implies liminfn→∞〈vn∗,xn-x0〉≥0. As a result, (31) implies (32)limsupn→∞wn∗,xn-x0=-liminfn→∞1tnvn∗+1-tnun∗,xn-x0-limsupn→∞CJεnxn,xn-x0=-1t0liminfn→∞vn∗+1-tnun∗,xn-x0-liminfn→∞CJεnxn,xn-x0≤0.Since S1 is of type (S+), it follows that xn→x0∈∂G as n→∞ and w0∗∈Sx0. Moreover, one can show that x0∈D(T), v0∗∈Tx0, and u0∗∈S2x0 so that 0=v0∗+t0(w0∗+Cx0)+(1-t0)u0∗; that is, 0∈(T+t0(S+C))(D(T)∩∂G). However, this is a contradiction.To show thatd(Mε(t,·),G,0) is constant for all t∈[0,1] and ε∈(0,ε0], with ε0 as in the proof of (ii), we let 0<ε1<ε2≤ε0 and consider the homotopy operator (33)M~t,x=Tqtx+tS1x+CJqtx+1-tS2x,qt=tε1+1-tε2,t,x∈0,1×G¯.Since for each t∈[0,1]Tq(t) is monotone, S1 and S2 are bounded and of type (S+) and CJq(t) is compact, it follows that M1(t,·) is bounded demicontinuous and of type (S+). It is not hard to verify that 0∉M~(t,∂G) for all t∈[0,1]. As in the arguments used in the proof of Lemma 7, we shall show that {M~(t,·)}t∈[0,1] is a homotopy of class (S+). To this end, let xn∈G¯ and tn∈[0,1] be such that xn⇀x0 and tn→t0∈[0,1] as n→∞, wn∗∈S1xn and un∗∈S2xn be such that (34)limsupn→∞Tqtnxn+tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0.SinceTq(tn) is monotone with domain X, it follows that (35)Tqtnxn-Tqtnx0,xn-x0≥0for all n. Since q(tn)→q(t0)>0 as n→∞ and (0,∞)×X∋(t,x)→Ttx is continuous, we get Tq(tn)x0→Tq(t0)x0 as n→∞. As a result of this, we get (36)limsupn→∞⁡Tqtnx0+tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0;that is,(37)limsupn→∞⁡tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0.Since q(t)∈[ε1,ε2] for all t∈[0,1], an application of Lemma 4 says that there exists γ0>0 independent of n such that Tq(tn)xn≤γ0 for all n. In addition, by the definition of Jq(tn), we see that (38)Jqtnxn=xn-qtnJ-1Tqtnxn≤xn+qtnJ-1Tqtnxn=xn+qtnTqtnxn≤xn+qtnγ0for all n. Since {xn} is bounded, the boundedness of {Jq(tn)xn} follows. By the compactness of C, we may assume without loss of generality that CJq(tn)xn→g0∗ as n→∞. As a result of this, we get (39)0≥limsupn→∞⁡tnwn∗+CJqtnxn+1-tnun∗,xn-x0=limsupn→∞⁡tnwn∗+1-tnun∗,xn-x0+limn→∞⁡tnCJqtnxn,xn-x0=limsupn→∞⁡tnwn∗+1-tnun∗,xn-x0.Lett0∈(0,1). The boundedness of {wn∗} and {un∗} imply (40)limsupn→∞t0wn∗+1-t0un∗,xn-x0≤0.SinceS1 and S2 are bounded and of type (S+), it follows that xn→x0 and there exist subsequences of {wn∗} and {un∗}, denoted again by {wn∗} and {un∗}, respectively, such that wn∗⇀w0∗∈S1x0 and un∗⇀u0∗∈S2x0 as n→∞. Moreover, by the continuity of (0,∞)×X∋(t,x)↦Ttx, it follows that Tq(tn)xn→Tq(t0)x0 as n→∞. From the continuity of J and C, we obtain that (41)Jqtnxn=xn-qtnJ-1Tqtnxn⟶x0-qt0J-1Tqt0x0=y0∈DT⊂DCas n→∞ and CJq(tn)xn→Cy0=g0∗ as n→∞. Thus, we arrive at (42)Tqtnxn=1qtnJxn-Jqtnxn⟶1qt0Jx0-y0=h0∗as n→∞. Since Tq(tn)xn∈T(Jq(tn)xn) for all n, by the maximality of T, we conclude that y0∈D(T)⊂D(C), h0∗∈Ty0, and 〈Tq(tn)xn,Jq(tn)xn〉→〈h0∗,y0〉 as n→∞. Therefore, we get (43)Tqtnxn+tnwn∗+CJqtnxn+1-tnun∗⇀Tqt0x0+t0w0∗+Cy0+1-t0u0∗as n→∞. The proofs of the cases t0=0 and t0=1 can be completed in an analogous manner. The details are omitted here. Thus the family {M~(t,·)}t∈[0,1] is a homotopy of class (S+); that is, d(M~(t,·),G,0) is independent of t∈[0,1]. This implies(44)dM~t,·.G,0=dM~1,·,G,0=dTε1+S1+CJε1,G,0=dM~0,·,G,0=dTε2+S2,G,0∀t∈0,1. Consequently, by the definition of d(M(t,·),G,0), there exists ε0>0 such that (45)dMt,·,G,0=dT+tS1+1-tS2+tC,G,0=dTε+tS1+1-tS2+tCJε,G,0=dTε2+S2,G,0∀t∈0,1,ε∈0,ε0.This proves thatd(M(t,·),G,0) is independent of t∈[0,1]; that is, the proof (iv) is complete.(vi) Suppose the hypotheses in (vi) hold. Since, for eacht∈[0,1], tT is maximal monotone, let Tεtx=(tT)εx,  x∈X, be Yosida approximant of tT and Jεt be the Yosida resolvent of tT. Since tS1+(1-t)S2 is bounded and of type (S+) and tC is compact with D(tC)=D(C) for all t∈[0,1], we shall show that there exists ε0>0 such that d(Tεt+tS1+(1-t)S2+tCJεt,G,0) is independent of t∈[0,1] and ε∈(0,ε0]. Assume that this does not hold; that is, there exist εn↓0+, xn∈D(T)∩∂G, wn∗∈S1xn, un∗∈S2xn, and tn∈[0,1] such that (46)vn∗+tnwn∗+1-tnun∗+tnCyn=0for all n, where vn∗=Tεntnxn∈tnT(yn) and yn=Jεntnxn∈D(T). By using the Γστ condition on C, we arrive at(47)vn∗≤tnwn∗+1-tnun∗+tnCyn≤κ1+τyn+σ≤κ1+τxn+εnvn∗+σ,for all n, where κ1 is an upper bound for {wn∗+un∗}. This gives the boundedness of {vn∗} and {yn}. Since xn⇀x0 as n→∞, yn=xn-εnJ-1(vn∗) and {vn∗} is bounded, it follows that yn⇀x0 as n→∞. Assume without loss of generality that Cyn→g0∗ as n→∞. Since yn-xn→0 as n→∞, the quasimonotonicity of S1 and S2 implies (48)limsupn→∞vn∗,yn-x0=-liminfn→∞tnwn∗+1-tnun∗+tnCyn,yn-x0≤-liminfn→∞tnwn∗+1-tnun∗,yn-xn+xn-x0-limn→∞tnCyn,yn-x0≤-liminfn→∞tnwn∗+1-tnun∗,yn-xn-liminfn→∞tnwn∗+1-tnun∗,xn-x0=-liminfn→∞tnwn∗+1-tnun∗,xn-x0≤-liminfn→∞tnwn∗,xn-x0-liminfn→∞1-tnun∗,xn-x0≤0.Therefore, we get(49)limsupn→∞vn∗,yn≤v0∗,x0.Since D(T)¯=X, the result of Kobayashi and Otani [8] says that the family {tT}t∈[0,1] is a pseudomonotone homotopy of maximal monotone operators. By (i) of Definition 6, we conclude that x0∈D(Tt0=t0T), v0∗∈t0Tx0, and 〈vn∗,yn〉→〈v0∗,x0〉 as n→∞. Applying analogous arguments to those of the proof of (iv) along with the (S+) condition on S1 and S2, one can easily verify that xn→x0∈D(T)∩∂G, Cyn→Cx0, wn∗→w0∗∈S1x0, and un∗⇀u0∗∈S2x0 so that v0∗+t0w0∗+(1-t0)u0∗+t0Cx0=0; that is, 0∈(t0(T+S1+C)+(1-t0)S2)(D(T)∩∂G), which is impossible by the hypotheses. In conclusion, we have proved that d(Tεt+tS1+(1-t)S2+tCJε,G,0) is well-defined for all t∈[0,1] and sufficiently small ε>0.Finally, we shall show thatd(Tεt+tS1+(1-t)S2+tCJε,G,0) is independent of t∈[0,1] and ε∈(0,ε0]. To this end, let 0<ε1<ε2≤ε0, q(t)=tε1+(1-t)ε2, 0<t1<t2≤1, and γt=tt1+(1-t)t2, t∈[0,1]. To complete the proof, we consider the homotopy operator (50)N1t,x=Tqtγtx+tS1x+1-tS2x+tCJqttx,t,x∈0,1×G¯.It is sufficient to show that {N1(t,·)}t∈[0,1] is a homotopy of class (S+). For each t∈[0,1], it is easy to see that N1(t,·):X→2X∗ is bounded and of type (S+). Let {xn}⊂X be such that xn⇀x0, tn→t0 as n→∞, wn∗∈S1xn, and un∗∈S2xn so that limsupn→∞〈Tq(tn)γtnxn+gn∗,xn-x0〉≤0, where gn∗=tnwn∗+(1-tn)un∗+tnCJq(tn)tnxn. Let (51)zn∗=Tqtnγtnxn,zn=Jqtnγtnxn∀n.We show that {zn∗} and {zn} are bounded. Since 0<ε1≤q(tn)≤ε2 and 0<t1≤γt≤t2 for all n, we conclude from Lemma 4 that {zn∗} is bounded. Since zn=xn-q(tn)J-1(zn∗) for all n and {zn∗}, {q(tn)} and {xn} are bounded, we get the boundedness of {zn}. Since C is compact, we assume without loss of generality that Czn→h0∗ as n→∞. By the boundedness of {xn}, S1, and S2, we assume, by passing into subsequences if necessary, that xn⇀x0, wn∗⇀w0∗, and un∗⇀u0∗ as n→∞. On the other hand, the pseudomonotonicity of S1 and S2 gives (52)liminfn→∞⁡gn∗,xn-x0≥0.Consequently, we get(53)limsupn→∞⁡zn∗,xn-x0≤limsupn→∞⁡zn∗+gn∗,xn-x0-liminfn→∞⁡gn∗,xn-x0≤-liminfn→∞⁡gn∗,xn-x0≤0.Since T is positively homogeneous of order α>0, it is not difficult to see that T-1:R(T)→D(T) is maximal monotone and positively homogeneous of order α-1>0. It also holds that (λT)-1(x)=T-1(1/λ)x=λ-1/αT-1(x) for all x∈R(T). In addition, we see that(54)zn∗=Tqtnγtnxn=γtnTqtnxn=γtnT-1+qtnJ-1-1xn=1γtn1/αT-1+qtnJ-1-1xn=1γtn1/αT-1+qtnγtn1/αJ-1-1xn=γtn1/αT-1+qtnγtn1/αJ-1-1xn=γtn1/αTqtnγtn1/αxn∀n.In fact, it is true that, Tεa(x)=a1/αTεa1/α(x) for all x∈X, ε>0 and a>0. For each n, letting λn=γtn1/α, we get(55)zn∗,xn-x0=λnTqtnλnxn,xn-x0=λnTqtnλnxn,xn-x0=λnTqtnλnxn-Tqtnλnx0,xn-x0+λnTqtnλnx0,xn-x0∀n.Since (0,∞)×X∋(t,x)→Ttx is continuous, it follows that Tq(tn)λnx0→Tq(t0)λ0x0 as n→∞, where λn→λ0>0 as n→∞. By the monotonicity of Tq(tn)λn for all n, we have (56)liminfn→∞⁡zn∗,xn-x0≥0;that is,(57)0≤liminfn→∞⁡zn∗,xn-x0≤limsupn→∞⁡zn∗,xn-x0≤0,which implies 〈zn∗,xn〉→〈z0∗,x0〉 as n→∞. Consequently, we arrive at (58)limsupn→∞wn∗,xn-x0≤0orlimsupn→∞un∗,xn-x0≤0.Since both S1 and S2 are bounded and of type (S+), it follows that xn→x0 as n→∞. As a result of this, we get (59)zn∗=λnTqtnλnxn⟶λ0Tqt0λ0x0=z0∗,zn=xn-qtnJ-1zn∗⟶x0-qt0λ0J-1Tqt0λ0x0=z0=Jqt0λ0x0as n→∞; that is, we have limsupn→∞〈zn∗,zn〉≤〈z0∗,z0〉. Since {tT}t∈[0,1] is a pseudomonotone homotopy of maximal monotone operators, it follows that z0∈D(t0T) and z0∗∈t0T(z0). In conclusion, we obtain that x0∈∂G, w0∗∈S1x0, u0∗∈S2x0, z0∈D(T), and (60)z0∗+t0w0∗+1-t0u0∗+t0Cz0∈Tqt0γt0x0+t0S1x0+1-t0S2x0+Cz0.Therefore, for any c0∈(0,1], the family {N1(t,·)}t∈[c0,1] is a homotopy of class (S+). Thus, d(N1(t,·),G,0) is independent of t∈(0,1] and ε∈(0,ε0]; that is, (61)dN1t,·,G,0=dN11,·,G,0=dTε1+S1+CJε1,G,0∀t∈0,1.On the other hand, by the definition ofd, we have that (62)dNt,·,G,0=dTεt+tS1+1-tS2+CJεt,G,0is independent of t∈(0,1] and ε∈(0,ε0]. In particular, for t=1, we have d(N(t,·),G,0)=d(Tε+S1+CJε,G,0) for all t∈(0,1]. But, for t=0, we see that N(0,x)=S2x for all x∈X. To complete the proof, it is sufficient to show that (63)dTε+S1+CJε,G,0=dS2,G,0.For each ε>0, we consider the homotopy (64)Nεt,x=Tεx+tS1x+CJεx+1-tS2x,t,x∈0,1×G¯.Suppose that there exist εn↓0+, tn∈[0,1], xn∈∂G, wn∗∈S1xn, and un∗∈S2xn such that(65)Tεnxn+tnwn∗+CJεnxn+1-tnun∗=0∀n.We assume without loss of generality that tn→t0∈[0,1], xn⇀x0, wn∗⇀w0∗, and un∗⇀u0∗ as n→∞. By the Γστ condition on C, we get (66)Tεnxn≤tnτJεnxn+κ2≤τxn-εnJ-1Tεnxn+κ2≤τxn+τεnTεnxn+κ2∀n,where κ2 is an upper bound for the sequence {σ+wn∗+un∗}. This shows the boundedness of {Tεnxn} and {Jεnxn}. By the maximality of T along with Lemma 5, the compactness of C, and the (S+) condition on S1 and S2 and analogous arguments to those in the proof of Theorem 9, we conclude that x0∈D(T)∩∂G, v0∗∈Tx0, w0∗∈S1x0, and u0∗∈S2x0 so that v0∗+t0(w0∗+Cx0)+(1-t0)u0∗=0; that is, 0∈(T+t0(S1+C)+(1-t0)S2)(D(T)∩∂G). However, this is impossible. In addition, the boundary condition on M in (v) implies that {Nε(t,·)}t∈[0,1] is an admissible homotopy; that is, d(Tε+S1+CJε,G,0)=d(Tε+S2,G,0) for all ε∈(0,ε0]. Since 0∈G and S2 satisfies the condition 〈u∗,x〉≥x2 for all x∈X and u∗∈S2x and 0∈T(0), it follows that 0≠t(Tε+S2x)+(1-t)S2x for all x∈∂G and t∈[0,1] and d(t(Tε+S2)+(1-t)S2,G,0) is independent of t∈[0,1]. In particular, d(Tε+S2,G,0)=d(S2,G,0) for all ε∈(0,ε0]. Therefore, we conclude that d(N(t,·),G,0) is independent of all t∈[0,1]. This completes the proof.


### 2.3. Degree Theory forT+C+S with S Pseudomonotone

In this section we present a generalization of the theory developed in the previous section for operators of typeT+C+S, where S:X→2X∗ is bounded pseudomonotone and T,C satisfy the conditions of Section 2.1. For each ε>0, it is well-known that S+εJ is bounded and of type (S+). As a result of this, we may apply the arguments used in the proof of Lemma 7 to show that d(T+S+C+εJ,G,f∗) is well-defined and constant for all sufficiently small ε>0 provided that f∗∉(T+S+C)(D(T)∩∂G)¯, where d is given in Definition 6. We thus give the following definition.Definition 10.
LetG be a nonempty, bounded, and open subset of X, T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded pseudomonotone, and C:D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Assume, further, that f∗∉(T+S+C)(D(T)∩∂G). Then the degree mapping d for T+S+C at f∗∈X∗ with respect to G is defined by (67)dT+S+C,G,f∗=limε↓0+⁡dT+C+S+εJ,G,f∗,where d(T+C+S+εJ,G,f∗) denotes the degree mapping constructed in Section 2.1.The following theorem gives some basic properties and homotopy invariance results analogous to those of Theorem9.Theorem 11.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded pseudomonotone, and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Then the following properties hold: (i)
(Normalization)d(J,G,0)=1 if 0∈G and d(J,G,0)=0 if 0∉G¯.(ii)
(Existence) iff∗∉(T+S+C)(D(T)∩∂G)¯ and d(T+S+C,G,f∗)≠0, then f∗∈(T+S+C)(D(T)∩G)¯. If T+S is of type (S), then f∗∈(T+C+S)(D(T)∩G).(iii)
(Decomposition) letG1 and G2 be nonempty and disjoint open subsets of G such thatf∗∉(T+S+C)(D(T)∩(G¯∖(G1∪G2)))¯. Then (68)dT+S+C,G,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗.(iv)
(Translation invariance) letf∗∉(T+S+C)(D(T)∩∂G)¯. Then we have (69)dT+S+C-f∗,G,0=dT+S+C,G,f∗.(v)
LetM(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), and Si:X→2X∗(i=1,2) be bounded pseudomonotone such that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. Then d(M(t,·),G,0) is independent of t∈[0,1].(vi)
LetN(t,x)=t(Tx+S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), and T:X⊃D(T)→X∗ be densely defined maximal monotone and positively homogeneous of order α>0, S1:X→2X∗ be bounded pseudomonotone, S2:X→2X∗ be bounded and of type (S+), and 0∉N(t,D(T)∩∂G)¯ for all t∈[0,1]. Assume, further, that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. Then d(N(t,·),G,0) is independent of t∈[0,1].Proof.
The proofs for (i) through (iv) follow as in the analogous items in the proof of Theorem9. We shall give sketches of the proofs of (v) and (vi). To prove (v), for each ε>0, we consider the homotopy inclusion (70)Mεt,x=Tx+tS1x+Cx+1-tS2x+εJx,t,x∈0,1×DT∩G¯.Following the arguments used in the proof of (v) of Theorem 9, it can be shown that there exists ε0>0 such that 0∉Mε(t,D(T)∩∂G) for all t∈[0,1] and ε∈(0,ε0]. Otherwise, we would get 0∈M(t0,D(T)∩∂G)¯ for some t0∈[0,1], which is impossible. On the other hand, for each ε∈(0,ε0], we see that (71)Mεt,x=Tx+tS1x+Cx+εJx+1-tS2x+εJx,t,x∈0,1×DT∩G¯.Since S1+εJ and S2+εJ are bounded operators of type (S+), the proof of (v) of Theorem 9 implies that d(Mε(t,·),G,0) is independent of t∈[0,1]; that is, (72)dMεt,·,G,0=dT+S1+C+εJ,G,0=dT+S1+C,G,0for all t∈[0,1] and ε∈(0,ε0]. As a result of this, we get (73)dMt,·,G,0=limε↓0+⁡dMεt,·,G,0=dT+S1+C,G,0
for allt∈[0,1]. This proves that d(M(t,·),G,0) is independent of t∈[0,1] provided that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. The proof of (vi) can be completed in analogous manner. The details are omitted here.


## 2.1. Degree Theory forT+S+C with S Bounded and of Type (S+)

The goal of this section is to develop a degree theory for operators of the typeT+S+C, where T:X⊃D(T)→2X∗ is maximal monotone, S:X→2X∗ is bounded and of type (S+), and C:D(C)→X∗ is compact with D(T)⊆D(C). Throughout the paper, we assume that C belongs to Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). To this end, we start by proving the following useful lemma.Lemma 7.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊇D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) such that C belongs to class Γστ. Assume, further, that f∗∉T+S+C(DT∩∂G). Then there exists ε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined and independent of ε∈(0,ε0].Proof.
In the first step, we claim that there existsε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined for all ε∈(0,ε0]. Suppose that this is false; that is, there exist εn↓0+, xn∈∂G, and wn∗∈Sxn such that(12)vn∗+wn∗+CJεnxn=f∗∀n,where vn∗=Tεnxn. By the definitions of Tεn and Jεn, we have (13)Jεnxn=xn-εnJ-1vn∗∈DT,vn∗∈TJεnxn∀n.Since {xn} and S are bounded, it follows that {wn∗} is bounded. Since C belongs to Γστ, we get that (14)vn∗≤f∗-wn∗+CJεnxn≤κ0+τJεnxn+σ=κ0+τxn-εnJ-1vn∗+σ≤κ0+τxn+τεnvn∗+σfor all n, where κ0 is an upper bound for {f∗-wn∗}. This yields the estimate (15)1-τεnvn∗≤κ0+τxn+σfor all n. Since εn↓0+ and {xn} is bounded, it follows that {vn∗} and {Jεnxn} are bounded. The compactness of C implies the boundedness of {CJεnxn}. Now, assume without loss of generality that xn⇀x0, vn∗⇀v0∗, and wn∗⇀w0∗ as n→∞. Since C is compact, we may assume, by passing into a subsequence if necessary, that CJεnxn→g0∗ as n→∞. The maximality of T along with Lemma 5 gives (16)liminfn→∞⁡vn∗,xn-x0≥0;
that is, we obtain from (12) that (17)limsupn→∞⁡wn∗,xn-x0=limsupn→∞⁡-vn∗+CJεnxn-f∗,xn-x0=-liminfn→∞⁡vn∗+CJεnxn-f∗,xn-x0≤-liminfn→∞vn∗,xn-x0-liminfn→∞⁡CJεnxn-f∗,xn-x0=-liminfn→∞⁡vn∗,xn-x0≤0.Since S is of type (S+), we conclude that xn→x0∈∂G as n→∞ and w0∗∈Sx0. Consequently, using (12) we arrive at (18)limsupn→∞⁡vn∗,xn-x0≤0.The maximality of T along with Lemma 5 yields x0∈D(T)∩∂G and v0∗∈Tx0 and 〈vn∗,xn〉→〈v0∗,x0〉 as n→∞. Since C is compact and Jεnxn=xn-εnJ-1(vn∗)→x0∈D(T)⊆D(C) as n→∞, it follows that CJεnxn→Cx0=g0∗ as n→∞. Letting n→∞ in (12), we get f∗∈(T+S+C)(D(T)∩∂G). However, this is impossible. Thus, there exists ε0>0 such that d(Tε+S+CJε,G,f∗) is well-defined for all ε∈(0,ε0].
Next, we shall prove thatd(Tε+S+CJε,G,f∗) is independent of ε∈(0,ε0]. Let εi∈(0,ε0]  (i=1,2) be such that 0<ε1<ε2≤ε0, q(t)=tε1+(1-t)ε2, t∈[0,1]. We consider the homotopy operator (19)Ht,x=Tqtx+Sx+CJqtx,t,x∈0,1×G¯.We will show that the family {H(t,·)}t∈[0,1] is a homotopy of class (S+) such that 0∉H(t,∂G) for all t∈[0,1]. To this end, let xn∈G¯, wn∗∈Sxn,  tn∈[0,1], fn∗=Tq(tn)xn+wn∗+CJq(tn)xn, xn⇀x0, and tn→t0 as n→∞ be such that (20)limsupn→∞⁡fn∗,xn-x0≤0.Since {xn} and S are bounded, it follows that {wn∗} is also bounded. Since q(tn)∈[ε1,ε2] for all n, we apply Lemma 4 to conclude that {Tq(tn)xn} and {Jq(tn)xn} are bounded. On the other hand, we see that (21)limsupn→∞⁡wn∗,xn-x0≤limsupn→∞⁡fn∗-Tqtnxn-Tqtnx0+CJqtnxn,xn-x0-limn→∞⁡Tqtnx0,xn-x0.By the compactness of C, we may assume without loss of generality that CJq(tn)xn→h0∗ as n→∞. Since q(tn)→q=q(t0)>0 as n→∞, we use the continuity of (0,∞)×X∋(t,x)→Tt(x) ([3], Lemma 6) to conclude that Tq(tn)x0→Tqx0 as n→∞. Combining these along with the monotonicity of Tq(tn), we obtain (22)limsupn→∞⁡wn∗,xn-x0≤0.Since S is of type (S+), we see that xn→x0 as n→∞ and there exists a subsequence of {wn∗}, denoted again by {wn∗}, such that wn∗⇀w0∗∈Sx0 as n→∞. Since (0,∞)×X∋(t,x)↦Ttx is continuous, we conclude that (23)fn∗⇀Tqx0+w0∗+CJqx0∈Tqx0+Sx0+CJqx0asn⟶∞;that is, {H(t,·)}t∈[0,1] is a homotopy of class (S+) such that 0∉H(t,∂G) for all t∈[0,1]. Therefore, d(H(t,·),G,f∗) is independent of t∈[0,1]; that is, d(Tε1+S+CJε1,G,f∗)=d(Tε2+S+CJε2,G,f∗). Since ε1 and ε2 are arbitrary in (0,ε0], we conclude that d(Tε+S+CJε,G,f∗) is well-defined and independent of ε∈(0,ε0]. This completes the proof.Based on Lemma7, the associated degree mapping is defined as follows.Definition 8.
LetG be a nonempty, bounded, and open subset of X, T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Assume, further, that f∗∉(T+S+C)(D(T)∩∂G). Then the degree mapping d for T+S+C at f∗∈X∗ with respect to G is defined by (24)dT+S+C,G,f∗=limε↓0+⁡dS+Tε+S+CJε,G,f∗,where dS+ is the degree mapping for multivalued bounded operators of type (S+) from [16].


## 2.2. Basic Properties of the Degree

Theorem 9.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded and of type (S+), and C:D(C)→X∗ be compact with D(T)⊆D(C) such that C belongs to Γστ. Then the following properties hold: (i)
(Normalization)d(J,G,0)=1 if 0∈G and d(J,G,0)=0 if 0∉G¯.(ii)
(Existence) iff∗∉(T+S+C)(D(T)∩∂G) and d(T+S+C,G,f∗)≠0, then f∗∈(T+S+C)(D(T)∩G).(iii)
(Decomposition) letG1 and G2 be nonempty, disjoint, and open subsets of G such that f∗∉(T+S+C)(D(T)∩(G¯∖(G1∪G2))). Then (25)dT+S+C,G,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗.(iv)
(Translation invariance) letf∗∉(T+S+C)(D(T)∩∂G). Then we have (26)dT+S+C-f∗,G,0=dT+S+C,G,f∗.(v)
LetM(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), where Si:X→2X∗(i=1,2) is bounded and of type (S+) and 0∉M(t,D(T)∩∂G) for all t∈[0,1]. Then d(M(t,·),G,0) is independent of t∈[0,1].(vi)
Let0∈G, N(t,x)=t(Tx+S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), where T:X⊃D(T)→X∗ is densely defined maximal monotone and positively homogeneous of order α>0, Si:X→2X∗(i=1,2) is bounded and of type (S+) such that 〈u∗,x〉≥x2 for all x∈X,u∗∈S2x, and 0∉N(t,D(T)∩∂G) for all t∈[0,1]. Assume, further, that 0∉M(t,D(T)∩∂G) for all t∈[0,1]. Then d(N(t,·),G,0) is independent of t∈[0,1].Proof.
The proof of (i) follows by settingT={0} and C={0}. To prove (ii), assume that f∗∉(T+S+C)(D(T)∩∂G) and d(T+S+C,G,f∗)≠0. By the definition of d, there exists ε0>0 such that d(Tε+S+CJε,G,f∗)≠0 for all ε∈(0,ε0]; that is, for each εn↓0+ there exist xn∈D(T)∩G and wn∗∈Sxn such that(27)vn∗+wn∗+CJεnxn=f∗,vn∗=Tεnxn∀n.Since S is bounded, it follows that {wn∗} is bounded. By using Γστ condition on C along with the arguments used in the proofs of Lemma 7, it is easy to see that {vn∗} and {Jεnxn} are bounded. Assume without loss of generality that xn⇀x0, wn∗⇀w0∗, vn∗⇀v0∗, and CJεnxn→g0∗ as n→∞. By the maximality of T, the (S+) condition on S, and the arguments used in the proof of Lemma 7, we conclude that x0∈D(T)∩G, v0∗∈Tx0, and w0∗∈Sx0 such that v0∗+w0∗+Cx0=f∗. This shows that f∗∈(T+S+C)(D(T)∩G).
Next we prove (iii). Suppose the hypotheses in (iii) hold. By the definition ofd, we see that d(T+S+C,G,f∗)=d(Tε+S+CJε,G,f∗) for all sufficiently small ε>0. Since Tε+S+CJε is bounded and of type (S+), the decomposition property of the degree mapping for multivalued (S+) operators implies (28)dT+S+C,G,f∗=limε↓0+⁡dTε+S+CJε,G,f∗=limε↓0+⁡dTε+S+CJε,G1,f∗+limε↓0+⁡dTε+S+CJε,G2,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗;
that is, (iii) holds.(iv) Suppose thatf∗∉(T+S+C)(D(T)∩∂G); that is, 0∉(T+S+C-f∗)(D(T)∩∂G). This implies that d(T+S+C-f∗,G,0) is well-defined. Since d(T+S+C-f∗,G,0)=d(Tε+S+CJε-f∗,G,0), by the translation property of the degree mapping for multivalued bounded operators of type (S+), we see that d(Tε+S+CJε-f∗,G,0)=d(Tε+S+CJε,G,f∗). Thus, (29)dT+S+C-f∗,G,0=limε↓0+dTε+S+CJε-f∗,G,0=limε↓0+dTε+S+CJε,G,f∗=dT+S+C,G,f∗.(v) Suppose that0∉M(t,D(T)∩∂G) for all t∈[0,1], where M(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯). For every ε>0, we consider (30)Mεt,x=Tεx+tS1x+CJεx+1-tS2x,t,x∈0,1×G¯.We shall show that there existsε0>0 such that d(Mε(t,·),G,0) is well-defined and independent of all (t,ε)∈[0,1]×(0,ε0]. To do this, we assume to the contrary that there exist εn↓0+, xn∈∂G, tn∈[0,1], wn∗∈S1xn, and un∗∈S2xn such that(31)vn∗+tnwn∗+CJεnxn+1-tnun∗=0,vn∗=Tεnxn∀n.Since {xn}, S1, and S2 are bounded, it follows that {wn∗} and {un∗} are bounded. By the Γστ condition on C, the boundedness of S1 and S2, and the arguments used in the proof of Lemma 7, we conclude that {Jεnxn} and {vn∗} are bounded. Assume without loss of generality that tn→t0, xn⇀x0, wn∗⇀w0∗, un∗⇀u0∗, vn∗⇀v0∗, Jεnxn=xn-εnJ-1(vn∗)⇀x0, and CJεnxn→g0∗ as n→∞. Suppose that t0=0. We have vn∗+un∗→0 as n→∞. Since S2 is of type (S+), it follows that xn→x0∈∂G, v0∗∈D(T), and 0∈Tx0+S2x0; that is, 0∈(T+S2)(D(T)∩∂G), that is, 0∈M(0,D(T)∩∂G). However, this is impossible. A similar proof covers the case t0≠1. Assume t0∈(0,1). Suppose there exists a subsequence of {tn}, denoted again by {tn}, such that tn≥τ0>0 for all n. Since T is maximal monotone, Lemma 5 implies liminfn→∞〈vn∗,xn-x0〉≥0. As a result, (31) implies (32)limsupn→∞wn∗,xn-x0=-liminfn→∞1tnvn∗+1-tnun∗,xn-x0-limsupn→∞CJεnxn,xn-x0=-1t0liminfn→∞vn∗+1-tnun∗,xn-x0-liminfn→∞CJεnxn,xn-x0≤0.Since S1 is of type (S+), it follows that xn→x0∈∂G as n→∞ and w0∗∈Sx0. Moreover, one can show that x0∈D(T), v0∗∈Tx0, and u0∗∈S2x0 so that 0=v0∗+t0(w0∗+Cx0)+(1-t0)u0∗; that is, 0∈(T+t0(S+C))(D(T)∩∂G). However, this is a contradiction.To show thatd(Mε(t,·),G,0) is constant for all t∈[0,1] and ε∈(0,ε0], with ε0 as in the proof of (ii), we let 0<ε1<ε2≤ε0 and consider the homotopy operator (33)M~t,x=Tqtx+tS1x+CJqtx+1-tS2x,qt=tε1+1-tε2,t,x∈0,1×G¯.Since for each t∈[0,1]Tq(t) is monotone, S1 and S2 are bounded and of type (S+) and CJq(t) is compact, it follows that M1(t,·) is bounded demicontinuous and of type (S+). It is not hard to verify that 0∉M~(t,∂G) for all t∈[0,1]. As in the arguments used in the proof of Lemma 7, we shall show that {M~(t,·)}t∈[0,1] is a homotopy of class (S+). To this end, let xn∈G¯ and tn∈[0,1] be such that xn⇀x0 and tn→t0∈[0,1] as n→∞, wn∗∈S1xn and un∗∈S2xn be such that (34)limsupn→∞Tqtnxn+tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0.SinceTq(tn) is monotone with domain X, it follows that (35)Tqtnxn-Tqtnx0,xn-x0≥0for all n. Since q(tn)→q(t0)>0 as n→∞ and (0,∞)×X∋(t,x)→Ttx is continuous, we get Tq(tn)x0→Tq(t0)x0 as n→∞. As a result of this, we get (36)limsupn→∞⁡Tqtnx0+tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0;that is,(37)limsupn→∞⁡tnwn∗+CJqtnxn+1-tnun∗,xn-x0≤0.Since q(t)∈[ε1,ε2] for all t∈[0,1], an application of Lemma 4 says that there exists γ0>0 independent of n such that Tq(tn)xn≤γ0 for all n. In addition, by the definition of Jq(tn), we see that (38)Jqtnxn=xn-qtnJ-1Tqtnxn≤xn+qtnJ-1Tqtnxn=xn+qtnTqtnxn≤xn+qtnγ0for all n. Since {xn} is bounded, the boundedness of {Jq(tn)xn} follows. By the compactness of C, we may assume without loss of generality that CJq(tn)xn→g0∗ as n→∞. As a result of this, we get (39)0≥limsupn→∞⁡tnwn∗+CJqtnxn+1-tnun∗,xn-x0=limsupn→∞⁡tnwn∗+1-tnun∗,xn-x0+limn→∞⁡tnCJqtnxn,xn-x0=limsupn→∞⁡tnwn∗+1-tnun∗,xn-x0.Lett0∈(0,1). The boundedness of {wn∗} and {un∗} imply (40)limsupn→∞t0wn∗+1-t0un∗,xn-x0≤0.SinceS1 and S2 are bounded and of type (S+), it follows that xn→x0 and there exist subsequences of {wn∗} and {un∗}, denoted again by {wn∗} and {un∗}, respectively, such that wn∗⇀w0∗∈S1x0 and un∗⇀u0∗∈S2x0 as n→∞. Moreover, by the continuity of (0,∞)×X∋(t,x)↦Ttx, it follows that Tq(tn)xn→Tq(t0)x0 as n→∞. From the continuity of J and C, we obtain that (41)Jqtnxn=xn-qtnJ-1Tqtnxn⟶x0-qt0J-1Tqt0x0=y0∈DT⊂DCas n→∞ and CJq(tn)xn→Cy0=g0∗ as n→∞. Thus, we arrive at (42)Tqtnxn=1qtnJxn-Jqtnxn⟶1qt0Jx0-y0=h0∗as n→∞. Since Tq(tn)xn∈T(Jq(tn)xn) for all n, by the maximality of T, we conclude that y0∈D(T)⊂D(C), h0∗∈Ty0, and 〈Tq(tn)xn,Jq(tn)xn〉→〈h0∗,y0〉 as n→∞. Therefore, we get (43)Tqtnxn+tnwn∗+CJqtnxn+1-tnun∗⇀Tqt0x0+t0w0∗+Cy0+1-t0u0∗as n→∞. The proofs of the cases t0=0 and t0=1 can be completed in an analogous manner. The details are omitted here. Thus the family {M~(t,·)}t∈[0,1] is a homotopy of class (S+); that is, d(M~(t,·),G,0) is independent of t∈[0,1]. This implies(44)dM~t,·.G,0=dM~1,·,G,0=dTε1+S1+CJε1,G,0=dM~0,·,G,0=dTε2+S2,G,0∀t∈0,1. Consequently, by the definition of d(M(t,·),G,0), there exists ε0>0 such that (45)dMt,·,G,0=dT+tS1+1-tS2+tC,G,0=dTε+tS1+1-tS2+tCJε,G,0=dTε2+S2,G,0∀t∈0,1,ε∈0,ε0.This proves thatd(M(t,·),G,0) is independent of t∈[0,1]; that is, the proof (iv) is complete.(vi) Suppose the hypotheses in (vi) hold. Since, for eacht∈[0,1], tT is maximal monotone, let Tεtx=(tT)εx,  x∈X, be Yosida approximant of tT and Jεt be the Yosida resolvent of tT. Since tS1+(1-t)S2 is bounded and of type (S+) and tC is compact with D(tC)=D(C) for all t∈[0,1], we shall show that there exists ε0>0 such that d(Tεt+tS1+(1-t)S2+tCJεt,G,0) is independent of t∈[0,1] and ε∈(0,ε0]. Assume that this does not hold; that is, there exist εn↓0+, xn∈D(T)∩∂G, wn∗∈S1xn, un∗∈S2xn, and tn∈[0,1] such that (46)vn∗+tnwn∗+1-tnun∗+tnCyn=0for all n, where vn∗=Tεntnxn∈tnT(yn) and yn=Jεntnxn∈D(T). By using the Γστ condition on C, we arrive at(47)vn∗≤tnwn∗+1-tnun∗+tnCyn≤κ1+τyn+σ≤κ1+τxn+εnvn∗+σ,for all n, where κ1 is an upper bound for {wn∗+un∗}. This gives the boundedness of {vn∗} and {yn}. Since xn⇀x0 as n→∞, yn=xn-εnJ-1(vn∗) and {vn∗} is bounded, it follows that yn⇀x0 as n→∞. Assume without loss of generality that Cyn→g0∗ as n→∞. Since yn-xn→0 as n→∞, the quasimonotonicity of S1 and S2 implies (48)limsupn→∞vn∗,yn-x0=-liminfn→∞tnwn∗+1-tnun∗+tnCyn,yn-x0≤-liminfn→∞tnwn∗+1-tnun∗,yn-xn+xn-x0-limn→∞tnCyn,yn-x0≤-liminfn→∞tnwn∗+1-tnun∗,yn-xn-liminfn→∞tnwn∗+1-tnun∗,xn-x0=-liminfn→∞tnwn∗+1-tnun∗,xn-x0≤-liminfn→∞tnwn∗,xn-x0-liminfn→∞1-tnun∗,xn-x0≤0.Therefore, we get(49)limsupn→∞vn∗,yn≤v0∗,x0.Since D(T)¯=X, the result of Kobayashi and Otani [8] says that the family {tT}t∈[0,1] is a pseudomonotone homotopy of maximal monotone operators. By (i) of Definition 6, we conclude that x0∈D(Tt0=t0T), v0∗∈t0Tx0, and 〈vn∗,yn〉→〈v0∗,x0〉 as n→∞. Applying analogous arguments to those of the proof of (iv) along with the (S+) condition on S1 and S2, one can easily verify that xn→x0∈D(T)∩∂G, Cyn→Cx0, wn∗→w0∗∈S1x0, and un∗⇀u0∗∈S2x0 so that v0∗+t0w0∗+(1-t0)u0∗+t0Cx0=0; that is, 0∈(t0(T+S1+C)+(1-t0)S2)(D(T)∩∂G), which is impossible by the hypotheses. In conclusion, we have proved that d(Tεt+tS1+(1-t)S2+tCJε,G,0) is well-defined for all t∈[0,1] and sufficiently small ε>0.Finally, we shall show thatd(Tεt+tS1+(1-t)S2+tCJε,G,0) is independent of t∈[0,1] and ε∈(0,ε0]. To this end, let 0<ε1<ε2≤ε0, q(t)=tε1+(1-t)ε2, 0<t1<t2≤1, and γt=tt1+(1-t)t2, t∈[0,1]. To complete the proof, we consider the homotopy operator (50)N1t,x=Tqtγtx+tS1x+1-tS2x+tCJqttx,t,x∈0,1×G¯.It is sufficient to show that {N1(t,·)}t∈[0,1] is a homotopy of class (S+). For each t∈[0,1], it is easy to see that N1(t,·):X→2X∗ is bounded and of type (S+). Let {xn}⊂X be such that xn⇀x0, tn→t0 as n→∞, wn∗∈S1xn, and un∗∈S2xn so that limsupn→∞〈Tq(tn)γtnxn+gn∗,xn-x0〉≤0, where gn∗=tnwn∗+(1-tn)un∗+tnCJq(tn)tnxn. Let (51)zn∗=Tqtnγtnxn,zn=Jqtnγtnxn∀n.We show that {zn∗} and {zn} are bounded. Since 0<ε1≤q(tn)≤ε2 and 0<t1≤γt≤t2 for all n, we conclude from Lemma 4 that {zn∗} is bounded. Since zn=xn-q(tn)J-1(zn∗) for all n and {zn∗}, {q(tn)} and {xn} are bounded, we get the boundedness of {zn}. Since C is compact, we assume without loss of generality that Czn→h0∗ as n→∞. By the boundedness of {xn}, S1, and S2, we assume, by passing into subsequences if necessary, that xn⇀x0, wn∗⇀w0∗, and un∗⇀u0∗ as n→∞. On the other hand, the pseudomonotonicity of S1 and S2 gives (52)liminfn→∞⁡gn∗,xn-x0≥0.Consequently, we get(53)limsupn→∞⁡zn∗,xn-x0≤limsupn→∞⁡zn∗+gn∗,xn-x0-liminfn→∞⁡gn∗,xn-x0≤-liminfn→∞⁡gn∗,xn-x0≤0.Since T is positively homogeneous of order α>0, it is not difficult to see that T-1:R(T)→D(T) is maximal monotone and positively homogeneous of order α-1>0. It also holds that (λT)-1(x)=T-1(1/λ)x=λ-1/αT-1(x) for all x∈R(T). In addition, we see that(54)zn∗=Tqtnγtnxn=γtnTqtnxn=γtnT-1+qtnJ-1-1xn=1γtn1/αT-1+qtnJ-1-1xn=1γtn1/αT-1+qtnγtn1/αJ-1-1xn=γtn1/αT-1+qtnγtn1/αJ-1-1xn=γtn1/αTqtnγtn1/αxn∀n.In fact, it is true that, Tεa(x)=a1/αTεa1/α(x) for all x∈X, ε>0 and a>0. For each n, letting λn=γtn1/α, we get(55)zn∗,xn-x0=λnTqtnλnxn,xn-x0=λnTqtnλnxn,xn-x0=λnTqtnλnxn-Tqtnλnx0,xn-x0+λnTqtnλnx0,xn-x0∀n.Since (0,∞)×X∋(t,x)→Ttx is continuous, it follows that Tq(tn)λnx0→Tq(t0)λ0x0 as n→∞, where λn→λ0>0 as n→∞. By the monotonicity of Tq(tn)λn for all n, we have (56)liminfn→∞⁡zn∗,xn-x0≥0;that is,(57)0≤liminfn→∞⁡zn∗,xn-x0≤limsupn→∞⁡zn∗,xn-x0≤0,which implies 〈zn∗,xn〉→〈z0∗,x0〉 as n→∞. Consequently, we arrive at (58)limsupn→∞wn∗,xn-x0≤0orlimsupn→∞un∗,xn-x0≤0.Since both S1 and S2 are bounded and of type (S+), it follows that xn→x0 as n→∞. As a result of this, we get (59)zn∗=λnTqtnλnxn⟶λ0Tqt0λ0x0=z0∗,zn=xn-qtnJ-1zn∗⟶x0-qt0λ0J-1Tqt0λ0x0=z0=Jqt0λ0x0as n→∞; that is, we have limsupn→∞〈zn∗,zn〉≤〈z0∗,z0〉. Since {tT}t∈[0,1] is a pseudomonotone homotopy of maximal monotone operators, it follows that z0∈D(t0T) and z0∗∈t0T(z0). In conclusion, we obtain that x0∈∂G, w0∗∈S1x0, u0∗∈S2x0, z0∈D(T), and (60)z0∗+t0w0∗+1-t0u0∗+t0Cz0∈Tqt0γt0x0+t0S1x0+1-t0S2x0+Cz0.Therefore, for any c0∈(0,1], the family {N1(t,·)}t∈[c0,1] is a homotopy of class (S+). Thus, d(N1(t,·),G,0) is independent of t∈(0,1] and ε∈(0,ε0]; that is, (61)dN1t,·,G,0=dN11,·,G,0=dTε1+S1+CJε1,G,0∀t∈0,1.On the other hand, by the definition ofd, we have that (62)dNt,·,G,0=dTεt+tS1+1-tS2+CJεt,G,0is independent of t∈(0,1] and ε∈(0,ε0]. In particular, for t=1, we have d(N(t,·),G,0)=d(Tε+S1+CJε,G,0) for all t∈(0,1]. But, for t=0, we see that N(0,x)=S2x for all x∈X. To complete the proof, it is sufficient to show that (63)dTε+S1+CJε,G,0=dS2,G,0.For each ε>0, we consider the homotopy (64)Nεt,x=Tεx+tS1x+CJεx+1-tS2x,t,x∈0,1×G¯.Suppose that there exist εn↓0+, tn∈[0,1], xn∈∂G, wn∗∈S1xn, and un∗∈S2xn such that(65)Tεnxn+tnwn∗+CJεnxn+1-tnun∗=0∀n.We assume without loss of generality that tn→t0∈[0,1], xn⇀x0, wn∗⇀w0∗, and un∗⇀u0∗ as n→∞. By the Γστ condition on C, we get (66)Tεnxn≤tnτJεnxn+κ2≤τxn-εnJ-1Tεnxn+κ2≤τxn+τεnTεnxn+κ2∀n,where κ2 is an upper bound for the sequence {σ+wn∗+un∗}. This shows the boundedness of {Tεnxn} and {Jεnxn}. By the maximality of T along with Lemma 5, the compactness of C, and the (S+) condition on S1 and S2 and analogous arguments to those in the proof of Theorem 9, we conclude that x0∈D(T)∩∂G, v0∗∈Tx0, w0∗∈S1x0, and u0∗∈S2x0 so that v0∗+t0(w0∗+Cx0)+(1-t0)u0∗=0; that is, 0∈(T+t0(S1+C)+(1-t0)S2)(D(T)∩∂G). However, this is impossible. In addition, the boundary condition on M in (v) implies that {Nε(t,·)}t∈[0,1] is an admissible homotopy; that is, d(Tε+S1+CJε,G,0)=d(Tε+S2,G,0) for all ε∈(0,ε0]. Since 0∈G and S2 satisfies the condition 〈u∗,x〉≥x2 for all x∈X and u∗∈S2x and 0∈T(0), it follows that 0≠t(Tε+S2x)+(1-t)S2x for all x∈∂G and t∈[0,1] and d(t(Tε+S2)+(1-t)S2,G,0) is independent of t∈[0,1]. In particular, d(Tε+S2,G,0)=d(S2,G,0) for all ε∈(0,ε0]. Therefore, we conclude that d(N(t,·),G,0) is independent of all t∈[0,1]. This completes the proof.


## 2.3. Degree Theory forT+C+S with S Pseudomonotone

In this section we present a generalization of the theory developed in the previous section for operators of typeT+C+S, where S:X→2X∗ is bounded pseudomonotone and T,C satisfy the conditions of Section 2.1. For each ε>0, it is well-known that S+εJ is bounded and of type (S+). As a result of this, we may apply the arguments used in the proof of Lemma 7 to show that d(T+S+C+εJ,G,f∗) is well-defined and constant for all sufficiently small ε>0 provided that f∗∉(T+S+C)(D(T)∩∂G)¯, where d is given in Definition 6. We thus give the following definition.Definition 10.
LetG be a nonempty, bounded, and open subset of X, T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded pseudomonotone, and C:D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Assume, further, that f∗∉(T+S+C)(D(T)∩∂G). Then the degree mapping d for T+S+C at f∗∈X∗ with respect to G is defined by (67)dT+S+C,G,f∗=limε↓0+⁡dT+C+S+εJ,G,f∗,where d(T+C+S+εJ,G,f∗) denotes the degree mapping constructed in Section 2.1.The following theorem gives some basic properties and homotopy invariance results analogous to those of Theorem9.Theorem 11.
LetG be a nonempty, bounded, and open subset of X. Let T:X⊃D(T)→2X∗ be maximal monotone, S:X→2X∗ be bounded pseudomonotone, and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Then the following properties hold: (i)
(Normalization)d(J,G,0)=1 if 0∈G and d(J,G,0)=0 if 0∉G¯.(ii)
(Existence) iff∗∉(T+S+C)(D(T)∩∂G)¯ and d(T+S+C,G,f∗)≠0, then f∗∈(T+S+C)(D(T)∩G)¯. If T+S is of type (S), then f∗∈(T+C+S)(D(T)∩G).(iii)
(Decomposition) letG1 and G2 be nonempty and disjoint open subsets of G such thatf∗∉(T+S+C)(D(T)∩(G¯∖(G1∪G2)))¯. Then (68)dT+S+C,G,f∗=dT+S+C,G1,f∗+dT+S+C,G2,f∗.(iv)
(Translation invariance) letf∗∉(T+S+C)(D(T)∩∂G)¯. Then we have (69)dT+S+C-f∗,G,0=dT+S+C,G,f∗.(v)
LetM(t,x)=Tx+t(S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), and Si:X→2X∗(i=1,2) be bounded pseudomonotone such that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. Then d(M(t,·),G,0) is independent of t∈[0,1].(vi)
LetN(t,x)=t(Tx+S1x+Cx)+(1-t)S2x, (t,x)∈[0,1]×(D(T)∩G¯), and T:X⊃D(T)→X∗ be densely defined maximal monotone and positively homogeneous of order α>0, S1:X→2X∗ be bounded pseudomonotone, S2:X→2X∗ be bounded and of type (S+), and 0∉N(t,D(T)∩∂G)¯ for all t∈[0,1]. Assume, further, that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. Then d(N(t,·),G,0) is independent of t∈[0,1].Proof.
The proofs for (i) through (iv) follow as in the analogous items in the proof of Theorem9. We shall give sketches of the proofs of (v) and (vi). To prove (v), for each ε>0, we consider the homotopy inclusion (70)Mεt,x=Tx+tS1x+Cx+1-tS2x+εJx,t,x∈0,1×DT∩G¯.Following the arguments used in the proof of (v) of Theorem 9, it can be shown that there exists ε0>0 such that 0∉Mε(t,D(T)∩∂G) for all t∈[0,1] and ε∈(0,ε0]. Otherwise, we would get 0∈M(t0,D(T)∩∂G)¯ for some t0∈[0,1], which is impossible. On the other hand, for each ε∈(0,ε0], we see that (71)Mεt,x=Tx+tS1x+Cx+εJx+1-tS2x+εJx,t,x∈0,1×DT∩G¯.Since S1+εJ and S2+εJ are bounded operators of type (S+), the proof of (v) of Theorem 9 implies that d(Mε(t,·),G,0) is independent of t∈[0,1]; that is, (72)dMεt,·,G,0=dT+S1+C+εJ,G,0=dT+S1+C,G,0for all t∈[0,1] and ε∈(0,ε0]. As a result of this, we get (73)dMt,·,G,0=limε↓0+⁡dMεt,·,G,0=dT+S1+C,G,0
for allt∈[0,1]. This proves that d(M(t,·),G,0) is independent of t∈[0,1] provided that 0∉M(t,D(T)∩∂G)¯ for all t∈[0,1]. The proof of (vi) can be completed in analogous manner. The details are omitted here.


## 3. An Existence Theorem

As a consequence of the degree theory developed in Section2, the following theorem gives a new existence result on the solvability of operator inclusions of the type Tu+Su+Cu∋f∗ in D(T) provided that T+S is of type (S) or S is bounded of type (S+).Theorem 12.
LetT:X⊃D(T)→2X∗ be maximal monotone with 0∈T(0), S:X→2X∗ be bounded pseudomonotone, and C:X⊇D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Let f∗∈X∗. Assume, further, that there exists R>0 such that (74)v∗+w∗+Cx-f∗,x>0for all x∈D(T)∩∂BR(0), v∗∈Tx, and w∗∈Sx. Then f∗∈(T+S+C)(D(T)∩BR(0))¯. Furthermore, R(T+S+C)¯=X∗ provided that T+S+C is coercive.Proof.
Letε>0. We shall show that 0∉Kε(t,D(T)∩∂BR(0)) for all t∈[0,1], where (75)Kεt,x=Tx+tSx+Cx+εJx-f∗+1-tεJx,t,x∈0,1×DT∩B¯R0.Since 0∈T(0), by using the boundary condition on T+S+C, we see that (76)tv∗+w∗+Cx-f∗,x+1-tv∗+εJx,x≥1-tv∗+εJx,x=1-tv∗,x+εJx,x≥εx2=εR2>0for all t∈[0,1], x∈D(T)∩∂BR(0), v∗∈Tx, and w∗∈Sx; that is, for each ε>0, it follows that 0∉Kε(t,D(T)∩∂BR(0)) for all t∈[0,1]. Since εJ and S+εJ are continuous, bounded, and of type (S+), (v) of Theorem 9 implies that {Kε(t,·)}t∈[0,1] is an admissible homotopy. Therefore, for each ε>0, we obtain (77)dKεt,·,BR0,0=dεJ,BR0,0=1∀t∈0,1;that is, d(T+S+C+εJ,BR(0),f∗)=1. By (ii) of Theorem 9, we conclude that f∗∈(T+S+C+εJ)(D(T)∩BR(0)); that is, for each εn↓0+, there exist xn∈D(T)∩BR(0), vn∗∈Txn, and wn∗∈Sxn such that (78)vn∗+wn∗+Cxn+εnJxn=f∗∀n.Since {xn} is bounded, we have εnJxn→0 as n→∞, which implies that f∗∈(T+S+C)(D(T)∩BR(0))¯. If T+S+C is coercive, then for each f∗∈X∗ there exists R=R(f∗)>0 such that the boundary condition holds. This implies that f∗∈(T+S+C)(D(T)∩BR(0))¯. Since f∗∈X∗ is arbitrary, we conclude that R(T+S+C)¯=X∗. The proof is complete.The arguments used in the proof of Theorem12 gives the following existence result on the surjectivity of T+S+C provided that either S is bounded and of type (S+) or T+S is of type (S).Corollary 13.
LetT:X⊃D(T)→2X∗ be maximal monotone with 0∈T(0), S:X→2X∗, and C:D(C)→X∗ be compact with D(T)⊆D(C) and belonging to the class Γστ. Let f∗∈X∗. Assume, further, that T+S+C is coercive. Then T+S+C is surjective provided that S is bounded of type (S+) or T+S is of type (S).Proof.
Letf∗∈X∗. Suppose T+S+C is coercive; that is, there exists ϕ:[0,∞)→(-∞,∞) and ϕ(t)→∞ as t→∞ such that (79)v∗+w∗+Cx,x≥ϕxx∀x∈DT,v∗∈Tx,w∗∈Sx.Then, there exists R=R(f∗)>0 such that (80)v∗+w∗+Cx-f∗,x>0for all x∈D(T)∩∂BR(0), v∗∈Tx, and w∗∈Sx. Assume that T+S is of type (S). By Theorem 12, we conclude that f∗∈(T+S+C)(D(T)∩BR(0))¯; that is, there exists xn∈D(T)∩BR(0), vn∗∈Txn, and wn∗∈Sxn such that vn∗+wn∗+Cxn→f∗ as n→∞. Since C is compact, we assume without loss of generality that Cxn→g0∗ as n→∞; that is, vn∗+wn→f∗-g0∗ as n→∞. Since T+S is of type (S), it follows that xn→x0 as n→∞. By the maximality of T along with Lemma 5, the continuity of C and generalized pseudomonotonicity of S, and the arguments used in the proof of Lemma 7, we conclude that x0∈D(T)∩BR(0) and f∗∈Tx0+Sx0+Cx0; that is, f∗∈(T+S+C)(D(T)∩BR(0)). Since f∗∈X∗ is arbitrary, we conclude that T+S+C is surjective. The case when S is bounded and of type (S+) can be reached by following analogous arguments. The details are omitted here.Theorem12 is a new result and Corollary 13 gives a surjectivity result for operators of the type T+S+C. For further existence results involving operators of the type T+S, the reader is referred to Kenmochi [17], Le [18], and Asfaw [19]. For various examples on pseudomonotone and quasimonotone operators, we cite the paper due to Mustonen [20].


## 4. An Example

LetH=L2(0,T;V) and V=W01,2(Ω). It is well-known that H and V are real Hilbert spaces with duality pairing between u∈H and v∈H denoted by 〈u,v〉 which is given by (81)u,v=∫0Tut,vtVdt,u∈H,v∈H,where 〈u(t),v(t)〉V denotes the duality pairing between u(t)∈V and v(t)∈V,  t∈[0,T]; that is, the norm of u∈H is given by u2=∫0Tu(t)V2dt, where u(t)V denotes the norm of u(t) in V. We shall apply the existence theorem(s) derived with the aid of the degree theory developed in this paper to establish existence of weak solution(s) in H for nonlinear problem given by(82)∂u∂t-∑i=1N∂∂xiaix,t,u,∇u+gx,t,u,∇u=fx,tx,t∈Qux,t=0,x,t∈∂Ω×0,Tux,0=ux,T,ut∈H,x∈Ω,where Q=Ω×(0,T) and f∈L2(Q) and the functions ai (i=1,2,…,N) and g satisfy the following measurability and sublinearity conditions:(C1)
ai(x,t,η,ζ) (i=1,2,…,N) is Carathèodory function; that is, (x,t)↦ai(x,t,η,ζ) is measurable for almost all (η,ζ)∈RN and (η,ζ)↦ai(x,t,η,ζ) is continuous for almost all (x,t)∈Ω×[0,T]. Assume, further, that there exist c1>0 and k1∈L2(Q) such that (83)aix,η,ζ≤c1η+ζ+k1x,tfor all(x,t)∈Ω×[0,T] and ξ∈RN.(C2)
There existsc2>0 such that ∑i=1Nai(x,η,ζ)ζi≥c2|ζ|2 for all (x,t)∈Q, (η,ζ)∈R×RN.(C3)
g:Q×R→R is Carathèodory function and there exist c3≥0 and k2∈L2(Q) such that (84)gx,t,η,ζ≤c3η+k2x,t,gx,η,ζη≥η2for all(x,t)∈Q, η∈R and ζ∈RN.A weak solutionu∈H is understood as follows.Definition 14.
A functionu=u(x,t) is a weak solution of (82) if u∈H and ut∈H such that the following are satisfied:(85)iux,t=0∀x∈Ω;(ii)ux,0=ux,T∀x∈∂Ω;iiiu′,ϕ+∑i=1N∫Qaix,t,u,∇u∂ϕ∂xidxdt+∫Qgx,t,u,∇u-fx,tϕx,tdxdt=0∀ϕ∈Hwhere u′ is understood in the sense of distributions; that is, (86)∫0Tu′tψtdt=-∫0Tutψ′tdt∀ψ∈C0∞0,T.Next we give the following theorem.Theorem 15.
Letf∈L2(Q). Assume that conditions (C1) through (C3) are satisfied. Then (82) admits at least one-weak solution.Proof.
LetS:H→H be given by (87)Su,v=∑i=1N∫Qaix,t,u,∇u∂v∂xidxdt,v∈H,u∈H.By using (C1) and (C2), it is well-known that S is bounded continuous of type (S+). For the proof of these facts and other relevant properties of pseudomonotone and (S+) type differential operators, the reader is referred to the papers by Browder [21], Berkovits and Mustonen [22], Hu and Papageorgiou [7], Landes and Mustonen [23], and the references therein. Let C:H⊇D(C)→H be defined by (88)Cu,v=∫Qgx,t,u,∇uvx,tdxdt,v∈H,where u∈D(C)={y∈H:y′∈H} and L:H⊇D(L)→H such that 〈Lu,v〉=∫0T〈u′(t),v(t)〉dt,v∈H, where u∈D(L)={y∈H:y′∈H,y(0)=y(T)}, that is, D(L)⊆D(C). It is well-known that L is a densely defined maximal monotone operator. The proof of this result is due to Brèzis which can be found in the book by Zeidler [13, Theorem 32. L, pp. 897–899]. Since D(C) is compactly embedded in L2(Q), it is known that C is a completely continuous operator; that is, C is a compact operator. Further reference on operators of the type C and existence results for parabolic problems, the reader is referred to the recent book due to Carl et al. [24]. Next we shall use Theorem 12 using the compact operator C, the maximal monotone operator L, and the (S+) operator S. It remains to show that C lies in Γστ and for each f∗∈H, there exists R=R(f∗)>0 such that 〈Lu+Su+Cu-f∗,u〉>0 for all u∈D(L)∩∂BR(0). To this end, by applying condition (C3), Hòlder’s inequality and observing that y(t)L2(Ω)≤y(t)V for all y∈H and t∈[0,1], we see that (89)Cu,v=∫0T∫Ωgx,t,u,∇uvx,tdxdt≤∫0T∫Ωgx,t,u,∇uvx,tdxdt≤c3∫0T∫Ωut+∇utvx,tdxdt+∫Qk2x,tvx,tdxdt≤c3∫0TutL2ΩvtL2Ωdt+c3∫0T∇utL2ΩvtL2Ωdt+∫0Tk2tL2ΩvtL2Ωdt≤c3∫0TutVvtVdt+c3∫0T∇utVvtVdt+∫0Tk2tL2ΩvtVdt≤2c3uv+k2L2Qv∀u∈H,v∈H;that is, we get that (90)Cu,v≤2c3uv+k2L2Qvfor all u∈H and v∈H. Consequently, taking supremum overall v∈H with v≤1, we conclude that Cu≤τu+σ for all u∈H, where τ=2c3 and σ=k2L2(Q); that is, C belongs to Γστ. Next we show the boundary condition in Theorem 12. To this end, by using conditions (C1) through (C3) and monotonicity of L (〈Lu,u〉≥0 for all u∈D(L)), we get (91)Lu+Su+Cu,u≥∑i=1N∫Qaix,t,u,∇ux,t∂u∂xidxdt+∫Qgx,u,∇uux,tdxdt≥∫Q∇ux,t2dxdt+∫Qux,t2dxdt=∫0T∫Ω∇ux,t2+ux,t2dxdt=∫0TutV2dt=u2for all u∈H. Since the right side of the above inequality approaches ∞ as u→∞, for each f∈L2(Q) there exists R=R(f)>0 such that (92)Lu+Su+Cu-f,u>0for all u∈D(L)∩∂BR(0). By applying Theorem 12, we conclude that the equation Lu+Su+Cu=f is solvable in D(L); that is, (82) admits at least one-weak solution.In conclusion, we like to notice that the functiong depends on both u and ∇u, sublinear, and possibly nonmonotone with respect to u. Consequently, Theorem 15 improves those analogous results under monotonicity condition on g with respect to u. Existence results in elliptic as well as parabolic problems under monotone nonlinearities independent of ∇u; the reader is referred to [7, 9, 17, 20, 21, 25–28] and the references therein.



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*Source: 7236103-2017-09-12.xml*