# Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

Authors: Teffera M. Asfaw
Journal: Abstract and Applied Analysis
 (2018)

---

## 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 a maximal monotone operator and C:X⊇D(C)→X⁎ be bounded and continuous with D(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the type T+C provided that C is compact or T is of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition on T+C. The operator C is neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.


---

## 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 RT=⋃x∈D(T)Tx. The symbol G(T) denotes the graph of T given by {(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. The mapping J:X→2X∗ defined by (1)Jx=x∗∈X∗:x∗,x=x2,x∗=xis called the “normalized duality mapping”. It is known due to Hahn-Banach theorem that J(x)≠∅. In addition, the local uniform convexity of X and X∗ implies that J is single valued, bounded, monotone, bicontinuous, and of type (S+).Definition 1.
An operatorT:X⊃D(T)→2X∗ is said to be (i)
“monotone” if〈u∗-v∗,x-y〉≥0 for all (x,u∗) and (y,v∗) in G(T);(ii)
“maximal monotone” ifT 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≥ψxx for all x∈D(T) and y∗∈Tx;(iv)
“expansive” if there existsα>0 such that u∗-v∗≥αx-y for all x∈D(T), y∈D(T), u∗∈Tx, and v∗∈Ty.It is well-known that a monotone operatorT is maximal monotone if and only if R(T+λJ)=X∗ for every λ>0 (cf. Theorem 2.2 [1]) and (T+λJ)-1:X∗→D(T) is single valued, monotone, and demicontinuous. For each λ>0, the operator Tλ:X→X∗, defined by Tλx=(T-1+λJ-1)-1x, is the “Yosida approximant” of T. It is bounded, continuous, and maximal monotone such that Tλx⇀T(0)x as λ→0+, for every x∈D(T), where T0x=inf{y∗:y∗∈Tx}. The operator Jλ:X→D(T) defined by Jλx=x-λJ-1(Tλx), is called the “Yosida resolvent” of T. It is continuous, Tλx∈T(Jλx) for every x∈X and limλ→0Jλx=x for all x∈coD(T)¯, where coD(T) is the convex hull of D(T). For each x∈D(T), Tλx≤|Tx| for all λ>0, where Tx denotes T0x. A maximal monotone operator T is called of compact resolvents if Jλ is compact for all λ>0. For further references on monotonicity theory, the reader is referred to Pascali and Sburlan [2], Barbu [1], Zeidler [3], Kenmochi [4], and the references therein. The following Lemma is due to Brèzis, Crandal, and Pazy [5].Lemma 2.
LetB be a maximal monotone set in X×X∗. If (un,un∗)∈B such that un⇀u in X, un∗⇀u∗ in X∗, and (2)limsupn→∞un∗-u∗,un-u≤0,then (u,u∗)∈B and 〈un∗,un〉→〈u∗,u〉 as n→∞.The main objective of this paper is to establish sufficient conditions which guarantee existence of solution for inclusions of the type(T+C)(D(T)∩BR(0))∋f∗ (where T:X⊇D(T)→2X∗ is maximal monotone of compact resolvents and C:X⊇D(C)→X∗ is bounded and continuous with D(T)⊆D(C)) provided that there exists R>0 such that (T+C+εJ)(D(T)∩∂BR(0))∌f∗ for all ε>0. Under this boundary condition, inclusion result of the type f∗∈(T+C)(D(T)∩BR(0))¯ is included if C is compact with D(T)⊆D(C) and T is arbitrary maximal monotone. The case where T is expansive maximal monotone and C is compact is also included. The operators of type C are available in the form of lower order term in many nonlinear differential equations in appropriate function spaces.The paper is organized as follows. In Section2, the main existence results (Theorems 3 and 6) are proved. The arguments of the proofs of Theorem 3 are based on Nagumo homotopy invariance result. Theorem 6 follows as a result of Theorem 3. Nagumo [6] developed a degree theory in a setting of linear convex topological space Y for operators of the type I-B, where B:G¯→Y is a compact operator, I is the identity mapping on Y, and G is a nonempty and open subset of X. The important contributions of Nagumo are (i) G is a nonempty and open subset of X (not necessarily bounded) and (ii) the degree is invariant under the homotopy H(x,t)=x-W(t,x), (t,x)∈[0,1]×G¯ provided that W:[0,1]×G¯→X is compact and 0∉H(t,∂G) for all t∈[0,1]. For further references on Nagumo degree and related results, the reader is referred to the paper due to Nagumo [6, Theorem 5, 6, 7]. Throughout the paper dNA stands for Nagumo degree. In Section 3, we demonstrated the applicability of the abstract results to prove existence of solution(s) for variational inequality problems.Existence results concerning pseudomonotone perturbations of maximal monotone operators under coercivity condition can be found in the papers due to Kenmochi [4, 7, 8], Asfaw and Kartsatos [9], Le [10], Asfaw [11], and the references therein. For related results concerning existence of solution for inclusion problems of the type Tu+Su∋f∗ in D(T), where S is an everywhere defined bounded pseudomonotone operator under Leray-Schauder type boundary condition on T+S, we cite the results due to Asfaw and Kartsatos [9, Theorem 11, Theorem 13, pp. 127-133]. Analogous result for possibly unbounded single multivalued pseudomonotone operator S is due to Figueiredo [12]. However, the cases where S is not necessarily pseudomonotone type is not studied earlier. It is the purpose of the present paper to address analogous result for T+C, where C is possibly not everywhere defined compact or C is bounded and continuous, and T is of compact resolvents.


## 2. Main Results

In this section, we prove the main existence theorem.Theorem 3.
LetT:X⊇D(T)→2X∗ be maximal monotone and C:X⊇D(C)→X∗ with D(T)⊆D(C). Let f∗∈X∗. Suppose there exists R>0 such that D(T)∩BR(0)≠∅ and (3)T+C+εJDT∩∂BR0∌f∗for all ε>0. Then (i)
f∗∈(T+C)(D(T)∩B¯R(0))¯ if C is compact;(ii)
f∗∈(T+C)(D(T)∩B¯R(0)) if T is expansive and C is compact;(iii)
f∗∈(T+C)(D(T)∩B¯R(0)) if D(T)¯⊆D(C) and C:D(C)→X∗ is completely continuous;(iv)
f∗∈(T+C)(D(T)∩B¯R(0)) if T is of compact resolvent and, C is bounded and continuous.Proof.
Suppose the hypotheses hold. We divide the proof into two steps.
Step 1. Let ∂IR:B¯R(0)→2X∗ be the subdifferential of the indicator function IR on B¯R(0) and A=T+∂IR. The operator A is maximal monotone with bounded domain D(A)=D(T)∩B¯R(0) because D(T)∩D(∂∘IR)=D(T)∩BR(0)≠∅. For each λ>0, let Aλ:X→X∗ and Jλ:X→D(A) be the Yosida approximant and resolvent of A, respectively. It is well-known that Aλ is bounded, continuous, and maximal monotone, and Jλ is bounded and continuous such that Aλx=(1/λ)J(x-Jλx), Jλx∈D(A), and Aλx∈A(Jλx) for all x∈X. Let (4)Hελt,x=x-Wt,x,t,x∈0,1×B¯R0,where W:[0,1]×B¯R(0)→X is given by (5)Wt,x=εJ+tAλ-1-tCJλx-f∗.For each ε>0 and λ>0, we shall show that W is a compact operator. Let (tn,xn)→(t0,x0), i.e., tn→t0 and xn→x0 as n→∞. The continuity of C and Jλ implies -tn(CJλxn-f∗)→-t0(CJλx0-f∗) as n→∞. Let (6)zn=Wtn,xn=εJ+tnAλ-1-tnCJλxn-f∗for all n, i.e., εJzn+tnAλzn+tn(CJλxn-f∗)=0 for all n. Fix a0∈D(A). By applying the monotonicity of Aλ and boundedness of Jλ and C, we obtain that (7)εzn2=εJzn,zn-a0+εJzn,a0=εJzn,a0-tnAλzn,zn-a0-tnCJλxn-f∗,zn-a0=εJzn,a0-tnAλzn-Aλa0,zn-a0+tnAλa0,zn-a0-tnCJλxn-f∗,zn-a0≤tnAλa0,zn-a0-tnCJλxn-f∗,zn-a0+εJzn,a0≤Aλa0+Kzn-a0+εzna0≤Aa0+Kzn-a0+εzna0for all n; i.e., εzn2≤(|Aa0|+K)zn-a0+εzna0 for all n, where K is an upper bound for {tnCxn-f∗} and |Aa0|=inf⁡{x∗:x∗∈Aa0}. Thus, we get the boundedness of {zn}. Assume without loss of generality that zn⇀z0 as n→∞. By the monotonicity of Aλ, we arrive at (8)limsupn→∞Jzn,zn-z0≤limsupn→∞⁡-tnAλzn,zn-z0+limsupn→∞⁡-tnCJλxn-f∗,zn-z0≤-liminfn→∞⁡tnAλzn,zn-z0-liminfn→∞⁡tnCJλxn-f∗,zn-z0=-liminfn→∞⁡tnAλzn-Aλz0,zn-z0-liminfn→∞⁡tnCJλxn-f∗,zn-z0+limn→∞⁡tnAλz0,zn-z0≤-limn→∞⁡tnCJλxn-f∗,zn-z0+limn→∞⁡tnAλz0,zn-z0=0.Thus we conclude that zn→z0 as n→∞ because J is of type (S+). Since J and Aλ are continuous, it follows that εJz0+t0(Aλz0+(CJλx0-f∗))=0; i.e., we have (9)z0=εJ+t0Aλ-1-t0CJλx0-f∗=Wt0,x0.We notice here that the above argument holds for any subsequence of {zn} (i.e., every subsequence of {zn} admits a convergent subsequence). Consequently, we conclude that {zn} converges z0; i.e., the continuity of W is proved. Next we assume {(sn,yn)} is bounded in [0,1]×G¯. The sequence {Jλyn} is bounded because the sequences {sn} and {yn} are bounded. Since Jλyn∈D(A)⊆D(C) for all n and C is compact, we can extract a subsequence, denoted again by {-sn(CJλyn-f∗)}, such that -sn(CJλyn-f∗)→a0∗ as n→∞. We notice here that (10)wn=Wsn,yn=εJ+snAλ-1-snCJλyn-f∗for all n implies εJwn+sn(Aλwn+(CJλyn-f∗))=0 for all n. By applying the argument used in the proof of continuity of W, it follows that {wn} is bounded and admits a convergence subsequence. Thus the compactness of W is proved. The same argument shows that W is compact if C is completely continuous with D(T)¯⊆D(C).
Next we fixε>0 temporarily and show that 0∉Hελ(t,∂BR(0)) for all t∈[0,1] and sufficiently small λ>0. Suppose not, i.e., there exists λn↓0+, τn∈[0,1] and un∈∂G such that(11)εJun+τnAλnun+CJλnun-f∗=0for all n. If τn=0 for some n, then it follows that εJun=0; i.e., un=0. But this is impossible because un∈∂BR(0). Assume τn∈(0,1] for all n and τn→τ0∈[0,1] as n→∞. The boundedness of {CJλnun} follows because of the boundedness of C and D(A) is bounded and Jλnun∈D(A) for all n; i.e., {Aλnun} is bounded. Assume without loss of generality that un⇀u0, Aλnun⇀u0∗, and CJλnun→b0∗ as n→∞. By the monotonicity of J and (11), we claim that (12)α=liminfn→∞⁡Aλnun,un-u0≥0.Suppose not, i.e., α<0. Then there exists a subsequence {〈Aεnun,un-u0〉} that converges to α as n→∞. The fact that Jλnun∈D(A), Aλnun∈A(Jλnun), and Jλnun=un-λnJ-1(Aλnun) for all n implies (13)limsupn→∞⁡Aλnun,Jλnun-u0≤limsupn→∞⁡Aλnun,Jλnun-un+limsupn→∞⁡Aλnun,un-u0<-liminfn→∞⁡λnAλnun2+α≤α<0.The maximality of A along with Lemma 2 implies u0∈D(A), u0∗∈Au0, and 〈Aλnun,Jλnun-u0〉→0 as n→∞; i.e., we get 〈Aλnun,un-u0〉→0=α. However, this is impossible; i.e., the claim holds. The case τ0=0 implies un→0∈∂BR(0) as n→∞. However this is impossible. Assume that τ0∈(0,1]. By using α≥0 in (11), we obtain that (14)limsupn→∞⁡Jun,un-u0=limsupn→∞⁡-τnAλnun+CJλnun-f∗,un-u0=-τ0liminfn→∞⁡Aλnun+CJλnun-f∗,un-u0≤-τ0liminfn→∞⁡Aλnun,un-u0+liminfn→∞⁡CJλnun-f∗,un-u0≤0.The (S+) condition on J implies un→u0∈D(T)∩∂BR(0) and Jλnun→u0 as n→∞. The maximal monotonicity of A together with Lemma 2 implies that u0∈D(A) and u0∗∈Au0. The continuity of C and J implies (15)εJun+τnAλnun+CJλnun-f∗⟶εJu0+τ0u0∗+Cu0-f∗as n→∞. Consequently, letting n→∞ in (11) gives ετ0-1Ju0+u0∗+Cu0=f∗; i.e., we obtain that (16)u0∗+Cu0+ετ0-1Ju0∈Tu0+Cu0+∂IRu0.This implies u0∗=a0∗+b0∗ for some a0∗∈Tu0 and b0∗∈∂IR(u0). However, it is well-known that(17)∂IRx=0ifx∈BR0,λJx:λ≥0ifx∈∂BR0,∅ifx∈X∖BR0¯.As a result, b0∗=μJu0 for some μ≥0; i.e., we get (T+C+(τ0-1ε+μ)J)(D(T)∩∂BR(0))∋f∗. However, this is impossible because of the hypothesis of the theorem. Therefore, for fixed ε>0, there exists λ0>0 such that 0∉Hελ(t,∂BR(0)) for all t∈[0,1] and λ∈(0,λ0); i.e., {Hλε(t,.)}t∈[0,1] is an acceptable Nagumo homotopy.
Step 2.Fix ε>0. By using the admissible Nagumo homotopy {Hελ(t,.)}t∈[0,1] obtained in Step 1 and applying the homotopy invariance properties of dNA, we see that dNA(Hλε(t,.),G,0) is independent of t∈[0,1]; i.e., (18)dNAHελ1,BR0,0=dNAHελ0,BR0,0=dNAI,BR0,0=1for all sufficiently small λ>0. Thus for each λn↓0+, there exists vn∈BR(0) such that(19)εJvn+Aλnvn+CJλnvn=f∗ for all n. The boundedness of {CJλnvn} and {Aλnvn} follows because of the boundedness of {vn}, D(A), and C. As a result we can easily follow the arguments used in the last part of Step 1 to conclude the existence of zε∈D(T)∩BR(0) and zε∗∈Azε such that εJzε+zε∗+Czε=f∗; i.e., for each εn↓0+, there exist zn∈D(T)∩BR(0) and zn∗∈Azn such that (20)εnJzn+zn∗+Czn=f∗for all n; i.e., we get (21)εnJzn+an∗+bn∗+Czn=f∗for all n and for some an∗∈∂IR(zn) and bn∗∈Tzn. However, applying the boundary hypothesis on T+C, we conclude that zn∈BR(0) for all n and an∗=0 for all n. Consequently, we arrive at(22)εnJzn+bn∗+Czn=f∗for all n. The boundedness of {Jzn} implies that bn∗+Czn→f∗; i.e., f∗∈(T+C)(D(T)∩BR(0))¯. This completes the proof of (i). Next we prove (ii). Suppose T is expansive and C is compact. By the compactness of C we assume without loss of generality that Czn→g0∗; i.e., bn∗→f∗-g0∗. The expansiveness of T implies zn→z0∈B¯R(0) as n→∞. The maximality of T along with Lemma 2 yields z0∈D(T)⊆D(C) and f∗-g0∗∈Tz0. As a result we conclude that f∗-Cz0∈Tz0 ( i.e., f∗∈(T+C)(D(T)∩B¯R(0))) because of the continuity of C; i.e., (ii) holds.
(iii) SupposeC is completely continuous and D(T)¯⊆D(C). Assume by passing into a subsequence that zn⇀d0 as n→∞. The maximality of T implies that D(T)¯∩B¯R(0) is closed and convex; i.e., it is weakly closed and d0∈D(T)¯∩B¯R(0). The complete continuity of C implies Czn→Cd0 and bn∗→f∗-Cd0; i.e., d0∈D(T)∩B¯R(0) and f∗∈(T+C)(D(T)∩B¯R(0)); i.e., (iii) is proved.
(iv) SupposeT is of resolvent compact and C is bounded and continuous such that the boundary condition on T+C holds. It is known due to Kartsatos [13, Lemma 3, pp. 1684] that Jλ is compact if and only if (λT+J)-1 is compact, and Jλ is compact for all λ>0 if Jμ is compact for some μ>0. As a result, the compactness of  (T+J)-1 and J1 is used equivalently. Since C is bounded and Jλ is compact, it follows that CJλ is a compact operator; i.e., we can follow the arguments used in the proof of the first part of Theorem 3 to conclude that W:[0,1]×B¯R(0)→X given by (23)Wt,x=εJ+tTλ-1-tCJλx-f∗is a compact operator. By following exactly analogous arguments used in the proof of (i) through (iii) of this theorem, for each εn↓0+ there exist τn∈D(T)∩BR(0) and νn∗∈Tτn such that (24)νn∗+Cτn+εnJτn=f∗for all n; i.e., we get νn∗+Jτn=-Cτn-εnJτn+Jτn for all n; i.e., (25)τn=T+J-1bnfor all n, where bn=-Cτn-εnJτn+Jτn for all n. By the compactness of (J+T)-1, there exists a subsequence of {bn}, denoted again by {bn}, such that τn=(T+J)-1(bn)→τ0 as n→∞. Assume without loss of generality (by passing into a subsequence) that Cτn⇀g∗ and vn∗⇀f∗-g∗ as n→∞. The maximal monotonicity of T along with Lemma 2 implies that τ0∈D(T) and f∗-g∗∈Tτ0. On the other hand, the continuity of C implies Cτn→Cτ0=g∗; i.e., vn∗→f∗-Cτ0; i.e., f∗∈(T+C)(D(T)∩B¯R(0)). The proof is complete.The following corollary holds.Corollary 4.
LetT and C be as given in Theorem 3. The conclusions in Theorem 3 hold if (a)
the boundary condition(26)T+C+εJDT∩∂BR0∌f∗for allε>0 is replaced by (27)T+C+εJ~DT∩∂BR0∌f∗for allε>0 and some u0∈D(T)∩BR(0), where J~(x)=J(x-u0) for x∈X;(b)
there existR>0 and v0∈B¯R(0) such that (28)u∗+Cx-f∗,x-v0>0for allx∈D(T)∩∂BR(0) and u∗∈Tx;(c)
0∈T(0) and B¯R(0) is replaced by a nonempty, closed, bounded, and convex subset K of X such that 0∈K∘ and εJ is replaced by ε∂IK, and (29)T+C+ε∂IKDT∩∂K∌f∗for allε>0, where ∂IK is the subdifferential of the indicator function on K.Proof.
(a) The proof for the analogous result under the boundary condition involvingJ~ follows because the operator J~:X→X∗ inherits all the properties of J; i.e., J~ is bounded, continuous, monotone, and of type (S+).
(b) The side condition in (b) is equivalent to(T+C+ε∂IR)(D(T)∩∂BR(0))∌f∗ for all ε>0 because of the definition of f∗-u∗-Cx∉∂IR(x) for all x∈D(T)∩∂BR(0), u∗∈Tx, and the fact that ∂IR(x)={εJx:ε≥0} for all x∈∂BR(0). This implies that the condition in (b) is equivalent to the boundary condition in Theorem 3.
(c) Suppose(T+C+ε∂IK)(D(T)∩∂K)∌f∗ for all ε>0. The maximal monotonicity of A=T+∂IK follows because D(T)∩K∘≠∅. It is not difficult to see that boundary condition on T+C+ε∂K implies that there exists y0∈K such that (30)u∗+Cx-f∗,x-y0>0for all x∈D(T)∩∂K∘ and u∗∈Tx. The proof of (c) follows based on the arguments of the proofs of (i) through (iii) of Theorem 3 by using G=K∘ instead of BR(0). The details are omitted here. To prove (iv) of Theorem 3 under (c), we shall show that A is resolvent compact. Let {fn∗} be a sequence in X∗ such that fn∗→f0∗ as n→∞ and xn=(J+A)-1(fn∗) for all n; i.e., xn∈D(A)∩K for all n. Let un∗∈Txn and vn∗∈∂IK(xn) such that (31)un∗+vn∗+Jxn=fn∗for all n. The boundedness of {xn} follows because K is bounded. Assume without loss of generality that xn⇀x0. Since K is closed and convex (i.e., it is weakly closed), we have x0∈K. In addition, the condition 0∈T(0) yields (32)vn∗,xn=-un∗,xn-xn2+fn∗xn≤fn∗xn≤Mfor all n, where M is an upper bound for {fn∗xn}. Since D(∂∘IK)=K°≠∅, it follows that ∂IK is strongly quasibounded, which yields the boundedness of {vn∗}, i.e., {un∗} is bounded. Assume without loss of generality that vn∗⇀v0∗ and un∗⇀u0∗ as n→∞. By following the arguments used in the proof of Theorem 3 and using the fact that J is bounded of type (S+), we get xn→x0 as n→∞ and u0∗+v0∗+Jx0=f0∗; i.e., x0=(J+A)-1(f0∗). Since this convergence holds for each subsequence of {xn}, we conclude that (J+A)-1 is continuous. Next we assume {fn∗} is bounded. Then the compactness of (J+T)-1 implies the existence of a subsequence, denoted by {fn∗-vn∗}, such that (33)xn=J+T-1fn∗-vn∗=J+A-1fn∗⟶y0asn⟶∞. Thus the compactness of (J+A)-1 follows; i.e., Jλ is compact for all λ>0. The remaining proof follows as in the arguments of the proof of Theorem 3. The detail is omitted here.Next we give the following surjectivity result.Corollary 5.
LetT and C be as given in Theorem 3. Assume that T+C is coercive; i.e., there exists v0∈X such that (34)infx∈DT,u∗∈Tx⁡u∗+Cx,x-v0x⟶∞asx⟶∞.Then the following conclusions hold. (i)
R(T+C)¯=X∗ if C is compact.(ii)
R(T+C)=X∗ if T is expansive and C is compact.(iii)
R(T+C)=X∗ if C is completely continuous.(iv)
R(T+C)=X∗ if T is of resolvent compact and C is bounded and continuous.Proof.
LetJ~x=J(x-v0) for x∈X and some u0∈D(T). It is known that J~ is bounded, monotone, continuous, and of type (S+). Fix f∗∈X∗. It is enough to show that the boundary condition in Theorem 3 holds. The coercivity of T+C implies(35)u∗+Cx-f∗,x-v0x≥u∗+Cx,x-v0x-f∗x-v0x≥u∗+Cx,x-v0x-f∗-f∗v0xfor all x∈D(T)∖{0} and u∗∈Tx. The coercivity of T+C implies that the right side in (35) approaches to ∞ as x→∞; i.e., there exists R=R(f∗)>0 such that (36)u∗+Cx-f∗,x-v0>0for all x∈D(T)∩∂BR(0) and u∗∈Tx. Thus we have (37)u∗+Cx+εJ~x-f∗,x-v0=u∗+Cx-f∗,x-v0+εJx-v0,x-v0>εx-v02for all x∈D(T)∩∂BR(0) and u∗∈Tx; i.e., we get (38)T+C+εJ~DT∩∂BR0∌f∗for all ε>0. Consequently, the conclusions (i) through (iv) follows based on the conclusion of Theorem 3.It is worth noticing that Theorem3 is new in the sense that the conclusion required only the boundary condition (T+C+εJ)(D(T)∩∂BR(0))∌f∗ for all ε>0. For analogous results under such boundary condition, the reader is referred to Figueiredo [12] (for single multivalued pseudomonotone operator S) and Asfaw and Kartsatos [9, Theorem 11 and Theorem 13] (for pseudomonotone perturbations of maximal monotone operator).Next we prove the following result.Theorem 6.
LetT:X⊇D(T)→2X∗ be maximal monotone and C:D(C)→X∗ be such that D(T)⊆D(C). Assume that d≥0, μ≥0, and Q>0 such that(39)u∗+Cx,x≥-dx2-μfor all x∈D(T) with x≥Q, and(40)limx∈DT,x→∞⁡Tx+Cxx2>0.Then the following hold. (i)
R(T+C)¯=X∗ if C is compact.(ii)
T+C is surjective if C is completely continuous and D(T)¯⊆D(C)(iii)
T+C is surjective if T is expansive and C is compact;(iv)
T+C is surjective if T is of compact resolvent and C is bounded and continuous.Proof.
Fixε>0 and f∗∈X∗. Let J~:X→X∗ be defined by J~x=xJx, x∈X. It is well-known that J~ is bounded, continuous, maximal monotone, and of type (S+). Then applying (39) gives(41)u∗+Cx+εJ~x,x≥x3ψεxfor all x∈D(T)∖{0} and u∗∈Tx, where (42)ψεt=ε-dt-1-μt-3-f∗t-2,t>0.As a result, we get(43)u∗+Cx+εJ~x,xx⟶∞as x→∞. The operator Aε=T+εJ~ is maximal monotone. By using the operators Aε and C in Corollary 5, we conclude that (T+C+εJ~)(D(T)∩B¯rε(0))¯∋f∗. Then there exist xn∈D(T)∩B¯rε(0) and un∗∈Txn such that (44)un∗+Cxn+εJ~xn⟶f∗as n→∞. The compactness of C implies that Cxn→a0∗ (for some subsequence {xn}). By applying the maximality of T together with Lemma 2 and following the arguments used in the proof of Theorem 3 we arrive at f∗∈(T+C+εJ~)(D(T)∩B¯rε(0)). The surjectivity of T+C+εJ~ follows because f∗∈X∗ is arbitrary. Next we give the proof of (i). Since R(T+C+εJ~)=X∗ for all ε>0, for each εn↓0+, there exist xn∈D(T) and un∗∈Txn such that(45)un∗+Cxn+εnxnJxn=f∗for all n; i.e., (39) gives(46)εnxn3=εnxnJxn,xn=-un∗+Cxn-f∗,xn≤dxn2+μ+f∗xnfor all n. We shall show that {xn} is bounded. Suppose not, i.e., there exists a subsequence, denoted again by {xn}, such that xn→∞ as n→∞. Dividing (46) by xn for all large n implies (47)εnxn2≤dxn+μxn+f∗for all large n; i.e., (45) implies(48)un∗+Cxn≤εnxn2+f∗≤dxn+μxn+2f∗for all large n; i.e., we get (49)un∗+Cxnxn2⟶0asn⟶∞.However, this is a contradiction to (40). As a result the sequence {xn} is bounded. Letting n→∞ in (45) gives f∗∈R(T+C)¯. The proofs of (ii), (iii), and (iv) follow based on the arguments used in the proof of (i) and Theorem 3. The details are omitted here.In [13], Kartsatos proved that R(T+C)¯=X∗ if C is compact and T is maximal monotone with D(C)=D(T) provided that there exists β:[0,∞)→[0,∞) such that β(t)→0 as t→∞ and Q>0 such that(50)u∗+Cx,x≥-βxx2for all x∈D(T), u∗∈Tx, and x≥Q and(51)liminfx∈DT,x→∞⁡Tx+Cxx>0.It is worth mentioning here that (50) implies that for each d>0 there exists Q>0 such that β(x)<d for all x≥Q; i.e., (50) implies (39) and (40) implies (51). On the other hand, Theorem 6 due to Kartsatos [13] holds if D(T)⊆D(C) is used instead of D(C)=D(T). The proof follows based on the proof of (i) of Theorem 6 by using J instead of J~. However, it is worth noticing that condition (39) is natural in applications than that of (50).


## 3. Applications

In this section, we demonstrate the applicability of the result to prove existence of solution for variational inequality problems.Example 7.
LetΩ be a nonempty, bounded, and open subset of Rn with n≥1 and Γ=∂Ω is of smooth boundary. Let X=L2(Ω) and K be a nonempty, closed, and convex subset of X with nonempty interior. Let j:R→(-∞,∞] be proper, convex, and lower semicontinuous function, p∈[2,∞) and λ>0. Let Φpλ:X→(-∞,∞] be defined by(52)Φpλu=1p∑i=1n∫Ω∂u∂xipdx+λp∫Ωupdx+∫Γjudσif u∈W1,p(Ω) and j(u)∈L1(Γ), and Φpλ(u)=∞ otherwise. It is known that ∂Φpλ:X⊇D(∂Φpλ)→2X is given by (53)∂Φpλu=-Δpλufor each u∈D(∂Φpλ), where (54)Δpλu=∑i=1n∂∂xi∂u∂xip-2∂u∂xi-λuup-2,domain of ∂Φpλ given by (55)D∂Φpλ=u∈W1,pΩ:Δλpu∈L2Ω,-∂u∂vpx∈∂juxa.e.x∈Γand (56)∂u∂vp=∑i=1n∂u∂xip-2∂u∂xicos⁡n,ei,where n is the outward normal on Γ and {e1,e2,…,en} is the canonical base in Rn. Assume, further, that (C1)
g:Ω×R↦R be Carathéodory function; i.e., x↦g(x,s) is measurable for almost all s∈R and s↦g(x,s) is continuous for almost all x∈Ω;(C2)
there existd≥0 and h∈L2(Q) such that (57)gx,s≤hx+dsfor all(x,s)∈Ω×R.
LetC:X→X be defined by Cu(.)=f(.,u(.)) for u∈X. In the following theorem, we prove the solvability of variational inequality problem VIP(C,Φpλ,K,f∗); i.e., we find u∈K∩D(Φpλ) such that(58)Cu-f∗,v-u≥Φpλu-Φpλvfor all x∈K.Theorem 8.
Letf∈X and Φpλ be as in (52). Suppose (C1) and (C2) are satisfied. If d<2-1λ for p=2 or λ>0 for p∈(2,∞), then the problem VIP(C,Φpλ,K,f∗) is solvable.Proof.
LetA=∂Φpλ be the subdifferential of Φpλ in the sense of convex analysis. It is well-known that A is an m-accretive operator (equivalently, A is maximal monotone) with compact resolvents (cf. Proposition 2.2.2 and Proposition 2.2.3 [14]). By applying conditions (C1) and (C2), it follows that C is bounded and continuous. Let ∂IK:K→2X be the subdifferential of the indicator function on K. The m-accretivity (maximal monotonicity) of T=A+∂IK follows because of the condition D(A)∩K∘≠∅. In addition, it is not difficult to see that T is of resolvent compact m-accretive operator. Next we show that T+C satisfies the boundary condition in Theorem 3. Let f∗∈X be the functional generated by f. Choose μ0∈D(Φpλ). By applying the definition of A and conditions (C1) and (C2), we arrive at (59)g∗+Cu,u-μ0≥Φpλu-Φpλμ0+Cu,u-μ0≥λpup-Φpλμ0-Cuu-Cuμ0≥λpup-Φpλμ0-h+duu-h+duμ0for all u∈D(A) and g∗∈Au. As a result, we obtain the estimate(60)g∗+Cu,u-μ0u≥λpup-1-u-1Φpλμ0-h+du-h+duμ0ufor u∈D(A)∖{0} and g∗∈Au. Since λ<2-1d (if p=2) or λ>0 (if p∈(2,∞)), the right side of (60) approaches to ∞ as u→∞ and (61)infu∈DA,g∗∈Au⁡g∗+Cu,u-μ0u⟶∞as u→∞; i.e., the coercivity condition on A+C is satisfied. Thus for each f∗∈X we conclude that the inclusion problem Au+Cu∋f∗ is solvable; i.e., the problem VIP(C,Φpλ,K,f∗) is solvable. This completes the proof.Next we present the following example.Example 9.
LetΩ be a nonempty, bounded, and open subset of RN with smooth boundary and N≥1, T>0, Q=(0,T)×Ω, H=L2(0,T;H01(Ω)), X=L2(Q), and V=H01(Ω). Let λ>0 and A:X⊇D(A)→X be defined by(62)Au=∂u∂t-λΔufor u∈D(A), where (63)DA=u∈L20,T;H2Ω∩H01Ω:∂u∂t∈L2Q,ux,0=ux,T.Let , denote the inner product in X; i.e., for g∗∈X and u∈H; we have 〈g∗,u〉=〈g∗,u〉(X,X) with X∗ identified with X. The norm of u∈H is the normed induced from X. It is well-known that A is maximal monotone, surjective, N(A)={0}, and (64)Au,u≥c0uH2for all u∈D(A). Since H01(Ω) is compactly embedded in L2(Q), it follows that A-1:H→H is a compact operator; i.e., for each ε>0 the resolvent (A+εI)-1:H→H is compact. Assume, further, that (C3)
g:Q×R→R is Carathéodory function; i.e., (x,t)↦g(x,t,s) is measurable for almost all s∈R, and s↦g(x,t,s) is continuous for almost all (x,t)∈Q;(C4)
there existc≥0 and k∈L2(Q) such that (65)gx,t,s≤cs+kx,tfor all(x,t,s)∈Q×R;(C5)
Φ:X→(-∞,∞] is a proper, convex, and lower semicontinuous function;(C6)
there exists a nonnegative constantα such that D(Φ)⊆M, where (66)M=u∈X:∫Qgx,t,uudμ≥-αu.We notice here that the set M defined in (C6) is nonempty if s→g(x,t,s) is nondecreasing for almost all (x,t)∈Q such that g(x,t,0)=0 for all (x,t)∈Q. Actually, condition (C4) implies that the unit ball in X is contained in M with α=c+kL2(Q). In Theorem 10 below, we prove existence of solution(s) for variational inequality problem denoted by VIP(A+C,Φ,K,f∗), i.e., finding u∈D(A)∩D(ϕ)∩K such that(67)Au-f∗,v-u+∫Qgx,t,uv-udμ≥Φu-Φvfor all v∈K, where K is a nonempty, closed, and convex subset of X with K∘≠∅.Theorem 10.
LetA:X⊇D(A)→X be as given in (62) and f∈L2(Q). Assume, further, that K∘∩D(Φ)∘≠∅ and conditions (C3) through (C6) are satisfied. Then the problem VIP(A+C,Φ,K,f∗) is solvable.Proof.
Letf∈L2(Q), A be as given in (62), and C:X→X be defined by (68)Cu,v=∫Qgx,t,uudμ,u∈X.The operator A is linear maximal monotone, surjective, one to one, and densely defined. The density of D(A) in X implies D(A)∩D(∂∘IK)∩D(Φ)∘≠∅. As a result, we conclude that T=∂IK+∂ϕ+A is maximal monotone (cf. Rockafellar [15]). The arguments used in the proof of (c) of Corollary 4 imply that T is resolvent compact. In addition, the continuity of C follows based on the conditions (C3) and (C4). The proof follows if we prove the coercivity of T+C. Let us denote again the restriction of C on M by C. The definition on M=D(C) as given in (C6) implies that 〈Cu,u〉≥-αu for all u∈M. The conditions on Φ imply that there exist h∗∈X∗ and β∈R such that Φu≥h∗,u-β≥-h∗u-β for all u∈X. Choose w0∈D(A)∩D(Φ). The monotonicity of A and definition of ∂Φ(w0) imply (69)Tu+Cu,u-w0=Au-Aw0,u-w0+Aw0,u-w0+Φu-Φw0+Cu,u-w0≥c0u-w0H2-Aw0u-w0-h∗u-β-Φw0+Cu,u-Cu,w0≥c0u-w0H2-Aw0u-w0-h∗u-β-Φw0-αu-Cuw0≥c0u-w02-Aw0u-w0-h∗u-β-Φw0-αu-du+kw0for all u∈D(T)∩D(Φ). We notice that D(T)⊆M because D(Φ)⊆M. In addition, the right side of the above inequality approaches ∞ as u→∞; i.e., we get (70)Tu+Cu,u-w0u⟶∞asu⟶∞;that is, T+C is coercive. Consequently, we conclude that T+C is surjective. Thus for each f∈L2(Q), the problem in (67) admits at least one weak solution. The proof is completed.The reader can find plenty of resolvent compact maximal monotone operators in the paper due to Brèzis and Nirenberg [16] and in the books due to Vrabie [14], Barbu [1, 17], Showalter [18], and the references therein.



---

*Source: 6953649-2018-07-12.xml*