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A topological degree for operators of generalized \((S_{+})\) type
Fixed Point Theory and Applications volume 2015, Article number: 194 (2015)
Abstract
As an extension of the LeraySchauder degree, we introduce a topological degree theory for a class of demicontinuous operators of generalized \((S_{+})\) type in real reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we show that the Borsuk theorem holds true for this class. Moreover, we study the Dirichlet boundary value problem involving the pLaplacian by way of an abstract Hammerstein equation.
Introduction
Topological degree theory may be one of the most effective tools in solving nonlinear equations. As a measure of the number of solutions of equation \(Fx = h\) for a fixed h, the degree has fundamental properties such as existence, normalization, additivity, and homotopy invariance. The most powerful one that the value of the degree is invariant under appropriate perturbations plays a crucial role in the study of nonlinear differential and integral equations.
Brouwer [1] initiated a topological degree for continuous maps in the Euclidean space. Leray and Schauder [2] developed the degree theory for compact operators in infinitedimensional Banach spaces. Since then numerous generalizations and applications have been investigated in various ways of approach; see, e.g., [3–6]. Browder [7, 8] introduced a topological degree for nonlinear operators of monotone type in reflexive Banach spaces, where the Galerkin method is used to apply the Brouwer degree. Berkovits [9, 10] gave a new construction of the Browder degree, based on the LeraySchauder degree. In this point of view, he studied in [11] an extension of the LeraySchauder degree for operators of generalized monotone type.
We consider an abstract Hammerstein equation of the form
where X is a reflexive Banach space with dual space \(X^{*}\) and \(T \colon X \to X^{*}\) and \(S \colon X^{*} \to X\) are operators of monotone type. In the case where S is linear, the solvability of abstract Hammerstein equation was systematically dealt with in [12], with application to Hammerstein integral equations. When S is quasimonotone and T satisfies condition \((S_{+})\), it was studied in [11].
In the present paper, our goal is to study the Berkovits degree theory for demicontinuous operators of generalized \((S_{+})\) type in real reflexive Banach spaces. To do this, we first observe a class of not necessarily bounded operators satisfying a generalized condition \((S_{+})\) with respect to T that contains abstract Hammerstein operators; see Lemma 2.3 below. As the next step for the construction of a new degree, we show that a demicontinuous operator can be reduced to some bounded operator on a suitable domain. Based on the Berkovits degree in [11], we introduce a topological degree for a wider class of demicontinuous operators satisfying a generalized condition \((S_{+})\) with respect to T. For the case of unbounded operators of class \((S_{+})\), we refer to [9]. Applying the degree theory, we prove that the Borsuk theorem holds true for this class. When a given boundary value problem is transformed to the corresponding integral equation, it may be often written in the form of abstract Hammerstein equation. As an example, we investigate the Dirichlet boundary value problem involving the pLaplacian by using our degree theory in terms of the Hammerstein equation.
This paper is organized as follows. In Section 2, we introduce some classes of operators of generalized \((S_{+})\) type and present some elementary facts which will be later needed. For the construction of a new degree, we show that demicontinuous homotopies can be reduced to bounded homotopies on a suitable domain. In Section 3, we prove that a topological degree for demicontinuous operators satisfying a generalized condition \((S_{+})\) with respect to T is well defined and satisfies some of fundamental properties. Using the degree theory, we show that the Borsuk theorem is still valid for our class. In Section 4, we study the solvability of elliptic boundary value problem by way of an abstract Hammerstein equation.
Some classes of operators
Let X and Y be two real Banach spaces. Given a nonempty subset Ω of X, let Ω̅ and ∂Ω denote the closure and the boundary of Ω in X, respectively. Let \(B_{r}(a)\) denote the open ball in X of radius \(r>0\) centered at a. The symbol \(\rightarrow(\rightharpoonup)\) stands for strong (weak) convergence.
An operator \(F \colon\Omega\subset X \to Y\) is said to be bounded if it takes any bounded set into a bounded set; F is said to be locally bounded if for each \(u\in\Omega\) there exists a neighborhood U of u such that the set \(F(U)\) is bounded. F is said to be demicontinuous if for each \(u\in\Omega\) and any sequence \((u_{k})\) in Ω, \(u_{k} \to u\) implies \(Fu_{k} \rightharpoonup Fu\); F is said to be compact if it is continuous and the image of any bounded set is relatively compact.
Let X be a real reflexive Banach space with dual space \(X^{*}\). The symbol \(\langle\cdot, \cdot\rangle_{X}\) denotes the usual dual paring between \(X^{*}\) and X in this order. In the reflexive case where the bidual space \(X^{**}\) is identified with X, we sometimes write \(\langle y, x \rangle\) for \(\langle x, y \rangle_{X^{*}}\) for \(x \in X\) and \(y \in X^{*}\).
We say that an operator \(F \colon\Omega\subset X \to X^{*}\) satisfies condition \((S_{+})\) if for any sequence \((u_{k})\) in Ω with \(u_{k} \rightharpoonup u\) and \(\limsup\langle Fu_{k}, u_{k}  u \rangle\leq0\), we have \(u_{k} \rightarrow u\); F is said to be quasimonotone if for any sequence \((u_{k})\) in Ω with \(u_{k} \rightharpoonup u\), we have \(\limsup\langle Fu_{k}, u_{k}  u \rangle\geq0\).
For any operator \(F \colon\Omega\subset X \to X\) and any bounded operator \(T \colon\Omega_{1} \subset X \to X^{*}\) such that \(\Omega \subset\Omega_{1}\), we say that F satisfies condition \((S_{+})_{T}\) if for any sequence \((u_{k})\) in Ω with \(u_{k} \rightharpoonup u\), \(y_{k} :=Tu_{k} \rightharpoonup y\) and \(\limsup\langle Fu_{k},y_{k}y\rangle\leq0\), we have \(u_{k} \rightarrow u\); we say that F has the property \((QM)_{T}\) if for any sequence \((u_{k})\) in Ω with \(u_{k} \rightharpoonup u\), \(y_{k} :=Tu_{k} \rightharpoonup y\), we have \(\limsup\langle Fu_{k},y_{k}y\rangle\geq0\).
We consider the following classes of operators:
for any \(\Omega\subset D_{F}\) and any \(T \in\mathcal{F}_{1}(\Omega)\), where \(D_{F}\) denotes the domain of F.
Let
where \(\mathcal{O}\) denotes the collection of all bounded open sets in X. Here, \(T \in\mathcal{F}_{1}(\overline{G})\) is called an essential inner map to F.
First, we establish the relationship between operators satisfying \((S_{+})_{T}\) and \((QM)_{T}\).
Lemma 2.1
Let T \(\colon\overline{G} \to X^{*}\) be a bounded operator, where G is a bounded open set in a real reflexive Banach space X. Then it has the following properties:

(a)
If \(F \colon\overline{G} \to X\) is locally bounded and satisfies condition \((S_{+})_{T}\) and T is continuous, then F has the property \((QM)_{T}\).

(b)
If \(F \colon\overline{G} \to X\) has the property \((QM)_{T}\), then for any sequence \((u_{k})\) in G̅ with \(u_{k} \rightharpoonup u\) and \(y_{k}:=Tu_{k} \rightharpoonup y\), we have
$$\liminf\langle Fu_{k}, y_{k}  y\rangle\geq0. $$ 
(c)
If \(F_{1}, F_{2} \colon\overline{G} \to X\) have the property \((QM)_{T}\), then so do \(F_{1}+F_{2}\) and \(\alpha F_{1}\) for any positive number α.

(d)
If \(F \colon\overline{G} \to X\) satisfies condition \((S_{+})_{T}\) and \(S \colon\overline{G} \to X\) has the property \((QM)_{T}\), then F+S satisfies condition \((S_{+})_{T}\).
Proof
(a) Assume to the contrary that there exists a sequence \((u_{k})\) in G̅ with \(u_{k} \rightharpoonup u\), \(y_{k} :=Tu_{k} \rightharpoonup y\) and
Then condition \((S_{+})_{T}\) on F implies that \(u_{k} \rightarrow u\) and hence, by the continuity of T, \(y_{k} = Tu_{k} \rightarrow y\). Since F is locally bounded, the set \(F(B_{r}(u))\) is bounded in X for some positive number r. By the boundedness of the sequence \((Fu_{k})\), we get
which is a contradiction to our assumption (2.1). We conclude that the operator F has the property \((QM)_{T}\).
(b) If \(q :=\liminf\langle Fu_{k}, y_{k}  y \rangle< 0\) for some sequence \((u_{k})\) in G̅ with \(u_{k} \rightharpoonup u\) and \(y_{k} :=Tu_{k} \rightharpoonup y\), then there is a subsequence \((u_{j})\) of \((u_{k})\) such that
and thus F does not have the property \((QM)_{T}\).
(c) Let \((u_{k})\) be any sequence in G̅ with \(u_{k} \rightharpoonup u\) and \(y_{k} :=Tu_{k} \rightharpoonup y\). Since \(F_{1}\) and \(F_{2}\) have the property \((QM)_{T}\), we have by (b)
Therefore, the sum \(F_{1} + F_{2}\) has the property \((QM)_{T}\). If \(\alpha> 0\), then it is clear that \(\alpha F_{1}\) has the property \((QM)_{T}\).
(d) Let \((u_{k})\) be any sequence in G̅ such that
Then we have by (b)
Since F satisfies condition \((S_{+})_{T}\), we obtain that \(u_{k} \rightarrow u\). Therefore, the operator \(F+S\) satisfies condition \((S_{+})_{T}\). This completes the proof. □
Remark 2.2
Note that each demicontinuous operator \(F: \overline{G}\to X\) is locally bounded. In particular, if \(F\in\mathcal{F}_{T}(\overline{G})\) and T is continuous, then F has the property \((QM)_{T}\).
The following result shows that the Hammerstein operator of the form \(I + S \circ T\) belongs to the class \(\mathcal{F}(X)\); see [11], Lemma 2.2, for the class \(\mathcal{F}_{B}(X)\).
Lemma 2.3
Suppose that \(T \in\mathcal{F}_{1}(\overline{G})\) is continuous and \(S \colon D_{S} \subset X^{*} \to X\) is demicontinuous such that \(T(\overline {G})\subset D_{S}\), where G is a bounded open set in a real reflexive Banach space X. Then the following statements are true:

(a)
If S is quasimonotone, then \(I + S \circ T\in\mathcal {F}_{T}(\overline{G})\), where I denotes the identity operator.

(b)
If S satisfies condition \((S_{+})\), then \(S \circ T\in \mathcal{F}_{T}(\overline{G})\).
Proof
(a) Set \(F :=I + S \circ T\). Let \((u_{k})\) be any sequence in G̅ such that
Since the sequence \((\langle Tu_{k}, u_{k}  u \rangle)\) is bounded in \(\Bbb{R}\), there is a subsequence \((u_{j})\) of \((u_{k})\) such that \(\lim\langle Tu_{j}, u_{j}  u \rangle\) exists. In view of \(X^{**}\cong X\), we know that
By the quasimonotonicity of S, (2.2), and (2.3), we get
Since T satisfies condition \((S_{+})\), we have \(u_{j}\to u\). By the convergence principle in [6], Proposition 10.13, the entire sequence \((u_{k})\) converges strongly to u. Thus, the operator F satisfies condition \((S_{+})_{T}\). Since F is demicontinuous on G̅, we conclude that \(F\in\mathcal {F}_{T}(\overline{G})\).
(b) Let \((u_{k})\) be any sequence in G̅ such that
Since S satisfies condition \((S_{+})\), it follows that \(y_{k}\to y\). Since \(\lim\langle Tu_{k}, u_{k}  u \rangle= 0\) and T satisfies condition \((S_{+})\), we have \(u_{k}\to u\). Consequently, we obtain that \(S \circ T\in\mathcal{F}_{T}(\overline{G})\). This completes the proof. □
We give a simple example of an operator which satisfies condition \((S_{+})_{T}\) but not condition \((S_{+})\); see also [11], Example 3.3.
Example 2.4
Let \((X,\langle\cdot, \cdot\rangle)\) be an infinitedimensional real Hilbert space with orthonormal basis \(\{e_{n}\}_{n\in\Bbb{N}}\). If we define a linear operator \(T:X\to X\) by setting
then the operator \(F:=T\circ T\) satisfies condition \((S_{+})_{T}\). However, F does not satisfy condition \((S_{+})\).
Proof
From \(\langle Tu, u\rangle= \Vert u\Vert ^{2}\) for all \(u\in X\) it follows that T is bounded, continuous and satisfies condition \((S_{+})\). In virtue of Lemma 2.3(b), the operator \(F=T\circ T\) satisfies condition \((S_{+})_{T}\). If we take a sequence \((u_{k})\), where \(u_{k}=e_{2k}\), then it is easy to see that \(u_{k} \rightharpoonup0\) and
but the sequence \((u_{k})\) does not converge strongly. Thus, F does not satisfy condition \((S_{+})\). This completes the proof. □
For a bounded operator \(T \colon\overline{G} \subset X \to X^{*}\), we say that a homotopy \(H \colon[0,1] \times\overline{G} \to X\) satisfies condition \((S_{+})_{T}\) if for any sequence \((t_{k},u_{k})\) in \([0,1]\times\overline{G}\) such that
we have \(u_{k} \rightarrow u\).
The following result tells us that each affine homotopy with a common essential inner map satisfies condition \((S_{+})_{T}\).
Lemma 2.5
Let G be a bounded open subset of a real reflexive Banach space X, and let \(T \in\mathcal{F}_{1}(\overline{G})\) be continuous. If \(F, S \in\mathcal{F}_{T}(\overline{G})\), then an affine homotopy \(H \colon[0,1] \times\overline{G} \to X\) defined by
satisfies condition \((S_{+})_{T}\).
In this case, the homotopy is called an admissible affine homotopy with the common essential inner map T.
Proof of Lemma 2.5
Let \((u_{k})\) be any sequence in G̅ and \((t_{k})\) any sequence in \([0,1]\) such that
Note that
If \(t=1\), then it follows from \(S \in\mathcal{F}_{T}(\overline{G})\) that
implies \(u_{k} \rightarrow u\). If \(t \in[0, 1)\), then the property \((QM)_{T}\) of S, in view of Lemma 2.1, implies that
Since F satisfies condition \((S_{+})_{T}\), we have \(u_{k} \rightarrow u\). In both cases, we have shown that \(u_{k} \rightarrow u\). This completes the proof. □
For our aim, we make the following observation to reduce suitably the domain of a demicontinuous homotopy.
Theorem 2.6
Let G be an open subset of a real reflexive Banach space X, and let Y be a real normed space. Suppose that \(H \colon[0,1] \times \overline{G} \to Y\) is a demicontinuous homotopy. If \(S \subset G\) is a nonempty compact set, then there exists an open set \(G_{0}\) and a positive constant R such that

(a)
\(S \subset G_{0} \subset G\) and

(b)
\(\Vert H(t,u) \Vert \leq R\) for all \(t \in[0,1]\) and all \(u \in \overline{G}_{0}\).
If, in addition, the sets G and S are symmetric with respect to the origin \(0\in S\), then \(G_{0}\) is also symmetric.
Proof
Let S be a nonempty compact set with \(S\subset G\), and let
The compactness of S enables us to write it in the form
Setting \(G_{n} :=D_{n} \cap G\), we see that \(G_{n}\) is open and \(S\subset G_{n}\subset G\), that is, (a) holds for each \(G_{n}\). We now prove that at least one of the sets \(G_{n}\) possesses property (b). If none of the sets \(G_{n}\) satisfies (b), we find sequences \((t_{n})\) in \([0,1] \) and \((u_{n})\) in \(\overline{G}_{n}\) such that
In view of \(u_{n} \in\overline{D}_{n}\), we can choose a sequence \((z_{n})\) in S such that \(\Vert u_{n}  z_{n} \Vert \leq2/n\). By the compactness of the set S, there exists a subsequence \((z_{k})\) of \((z_{n})\) which converges to some \(z \in S\). Hence it follows from the inequality
that \(u_{k} \to z\). We may suppose that \(t_{k} \to t \in[0,1]\). It follows from the demicontinuity of H that \(H(t_{k},u_{k}) \rightharpoonup H(t,z)\), which contradicts (2.4), by noting that every weakly convergent sequence in the normed space Y is bounded. Therefore, at least one of the sets \(G_{n}\) satisfies (a) and (b), say \(G_{n_{0}}\). Set \(G_{0}:= G_{n_{0}}\).
Next, to show that \(G_{0}\) is symmetric, let \(u\in G_{0}\), then there exists \(z \in S\) such that \(\Vert u  z \Vert < 1/{n_{0}}\). Since \(\Vert (u)  (z) \Vert < 1/{n_{0}}\), it follows from \(u\in G\) and \(z \in S\) that \(u \in G_{0}\). This completes the proof. □
We show that any demicontinuous operator satisfying condition \((S_{+})_{T}\) is proper on closed bounded sets; see [9], Lemma 2.5, for the case of class \((S_{+})\).
Lemma 2.7
Let G be a bounded open set in a real reflexive Banach space X, and let \(H: [0,1]\times\overline{G}\to X\) be a demicontinuous homotopy satisfying condition \((S_{+})_{T}\), where \(T: \overline{G} \to X^{*}\) is bounded. For any compact set \(A \subset X\),
is a compact subset of X.
Proof
Let \((u_{k})\) be any sequence in K. Then there exists a sequence \((t_{k})\) in \([0,1]\) such that \(H(t_{k}, u_{k}) \in A\) for all \(k\in\mathbb{N}\). Since A is compact, we can choose a subsequence \((u_{j})\) of \((u_{k})\) in G̅ and a subsequence \((t_{j})\) of \((t_{k})\) in \([0,1]\) such that \(H(t_{j}, u_{j}) \to w \in A\). By the boundedness of the set G and the map T, we may suppose, without loss of generality, that
Since we have \(\lim\langle H(t_{j}, u_{j}), y_{j}  y \rangle= 0\), the assumptions on the homotopy H imply that \(u_{j} \to u\) and \(H(t_{j}, u_{j}) \rightharpoonup H(t, u)\). Consequently, we have \(H(t, u) = w \in A\) and \(u \in\overline{G}\), which means that \(u \in K\). Thus, the set K is compact. This completes the proof. □
Corollary 2.8
Suppose that \(F \colon\overline{G} \to X\) is demicontinuous and satisfies condition \((S_{+})_{T}\), where G is a bounded open subset of X and T is bounded on G̅. For every \(h\notin F(\partial G)\), there exists an open set \(G_{0}\) such that \(F^{1}(h) \subset G_{0} \subset G\) and F is bounded on \({\overline{G}_{0}}\).
Proof
Let \(h\notin F(\partial G)\). By Lemma 2.7, \(F^{1}(h)\) is a compact subset of X and \(F^{1}(h)\subset G\). Applying Theorem 2.6 with the constant homotopy F and \(S=F^{1}(h)\), there exists an open set \(G_{0}\) such that \(F^{1}(h) \subset G_{0} \subset G\) and F is bounded on \({\overline{G}_{0}}\). □
Corollary 2.9
Let G be a bounded open symmetric subset of X with respect to the origin \(0\in G\). Suppose that \(F \colon\overline{G} \to X\) is odd, demicontinuous and satisfies condition \((S_{+})_{T}\) with \(0\notin F(\partial G)\), where T is bounded on G̅. Then there exists a symmetric open set \(G_{0}\) such that \(F^{1}(0) \subset G_{0} \subset G\) and F is bounded on \({\overline{G}_{0}}\).
Proof
Since F is odd on G̅ and \(0\notin F(\partial G)\), it is obvious from Lemma 2.7 that \(0\in F^{1}(0)\subset G\) and \(F^{1}(0)\) is symmetric and compact. By Theorem 2.6, there exists a symmetric open set \(G_{0}\) such that \(F^{1}(0) \subset G_{0} \subset G\) and F is bounded on \({\overline{G}_{0}}\). □
Degree theory
In this section, we extend the degree theory of Berkovits to all demicontinuous operators satisfying condition \((S_{+})_{T}\) in the class \(\mathcal{F}(X)\), and this is used to establish the Borsuk theorem.
In what follows, X will always be an infinitedimensional real reflexive separable Banach space which has been renormed so that both X and \(X^{*}\) are locally uniformly convex.
We first introduce the topological degree for the class \(\mathcal {F}_{B}(X)\) due to Berkovits [11]. For the details on the class \(\mathcal{F}_{S_{+}}(X)\), we refer to [9, 10].
Theorem 3.1
There exists a unique degree function
that satisfies the following properties:

(a)
(Existence) If \(d_{B}(F,G,h) \neq0\), then the equation \(Fu = h\) has a solution in G.

(b)
(Additivity) Let \(F\in\mathcal{F}_{T,B}(\overline {G})\). If \(G_{1}\) and \(G_{2}\) are two disjoint open subsets of G such that \(h \notin F(\overline{G}\setminus(G_{1} \cup G_{2}))\), then we have
$$d_{B}(F,G,h) = d_{B}(F,G_{1},h) + d_{B}(F,G_{2},h). $$ 
(c)
(Homotopy invariance) If \(H \colon[0,1] \times \overline{G} \to X\) is a bounded admissible affine homotopy with a common continuous essential inner map and \(h: [0,1] \to X\) is a continuous path in X such that \(h(t) \notin H(t,\partial G)\) for all \(t \in[0,1]\), then the value of \(d_{B}(H(t, \cdot), G, h(t))\) is constant for all \(t \in[0,1]\).

(d)
(Normalization) For any \(h \in G\), we have \(d_{B}(I,G,h) = +1\).
Lemma 3.2
Let \(F\in\mathcal{F}_{T}(\overline{G})\) be an operator, where G is a bounded open set in X and \(T \in\mathcal{F}_{1}(\overline{G})\). Suppose that for \(i=1,2\), \(G_{i}\) is an open subset of G such that
Then the degree \(d_{B}(F,G_{i},h)\) is well defined for \(i=1,2\) and
Proof
For \(i=1,2\), since \(F\in\mathcal{F}_{T,B}(\overline{G}_{i})\) and \(h\notin F(\partial G_{i})\), the degree \(d_{B}(F,G_{i},h)\) is well defined and \(F^{1}(h) \subset G_{1}\cap G_{2} \subset G_{i}\) implies \(h\notin F(\overline{G}_{i} \setminus(G_{1}\cap G_{2}))\). Applying Theorem 3.1(b) twice, we get
□
Definition 3.3
Let
Then we define a degree function \(d \colon M \to\mathbb{Z}\) as follows:
where \(G_{0}\) is any open subset of G with \(F^{1}(h) \subset G_{0}\) and F is bounded on \({\overline{G}_{0}}\), according to Corollary 2.8. Here, \(F_{\overline{G}_{0}}\) denotes the restriction of F to \(\overline{G}_{0}\).
In view of Lemma 3.2, the degree d does not depend on the choice of the set \(G_{0}\). Especially, if F is bounded on G̅, then we may take \(G_{0}=G\) and \(d(F, G, h) = d_{B}(F,G,h)\), which means that d and \(d_{B}\) coincide on \(\mathcal{F}_{T,B}(\overline{G})\).
Theorem 3.4
The above degree d for the class \(\mathcal{F}(X)\) has the following properties:

(a)
(Existence) If \(d(F,G,h) \neq0\), then the equation \(Fu = h\) has a solution in G.

(b)
(Additivity) Let \(F\in\mathcal{F}_{T}(\overline {G})\). If \(G_{1}\) and \(G_{2}\) are two disjoint open subsets of G such that \(h \notin F(\overline{G}\setminus(G_{1} \cup G_{2}))\), then we have
$$d(F,G,h) = d(F,G_{1},h) + d(F,G_{2},h). $$ 
(c)
(Homotopy invariance) Suppose that \(H \colon[0,1] \times\overline{G} \to X\) is an admissible affine homotopy with a common continuous essential inner map and \(h: [0,1] \to X\) is a continuous path in X such that \(h(t) \notin H(t,\partial G)\) for all \(t \in[0,1]\). Then the value of \(d(H(t, \cdot), G, h(t))\) is constant for all \(t \in[0,1]\).

(d)
(Normalization) For any \(h \in G\), we have \(d(I,G,h) = +1\).

(e)
(Boundary dependence) If \(F, S\in\mathcal {F}_{T}(\overline{G})\) coincide on ∂G and \(h \notin F(\partial G)\), then
$$d(F,G,h) = d(S,G,h). $$
Proof
(a) If \(d(F,G,h) \neq0\), then we have by Definition 3.3
for a suitable open set \(G_{0} \subset G\). By Theorem 3.1(a), the equation \(Fu = h\) has a solution in \(G_{0}\) which also belongs to G.
(b) Let \(K = \{ u \in G \colon Fu = h\}\) and \(K_{i} = \{ u \in G_{i} \colon Fu = h\}\) for \(i = 1, 2\). Note by hypotheses that K is the disjoint union of the sets \(K_{1}\) and \(K_{2}\). By Corollary 2.8, there exist open sets \(G_{0i}\) such that \(K_{i} \subset G_{0i} \subset G_{i}\), and F is bounded on \(\overline{G}_{0i}\) for \(i = 1, 2\). Set \(G_{0} :=G_{01} \cup G_{02}\). Obviously, F is bounded on \(\overline{G}_{0}\) and \(K \subset G_{0} \subset G\), and so \(h \notin F(\overline{G}_{0}\setminus(G_{01} \cup G_{02}))\). Hence it follows from Definition 3.3 and Theorem 3.1(b) that
(c) Note that \(A=\{h(t)\in X  t\in[0,1]\}\) is a compact subset of X. By Lemma 2.7,
is a compact subset of X. In particular, we have \(S\subset G\) in view of \(h(t)\notin H(t,\partial G)\) for all \(t\in[0,1]\). According to Theorem 2.6, there exists an open set \(G_{0}\) such that \(S\subset G_{0}\subset G\) and H is bounded on \([0,1]\times\overline{G}_{0}\). This implies that \(h(t)\notin H(t,\partial G_{0})\) and
By Theorem 3.1(c), we conclude that the value of \(d(H(t, \cdot), G, h(t))\) is constant for all \(t \in [0,1]\).
(d) Since the identity operator I is bounded, it is an immediate consequence of Theorem 3.1(d). Actually, the identity operator \(I=J^{1}\circ J\) belongs to \(\mathcal {F}_{J}(\overline{G})\), in view of Lemma 2.3, where J denotes the duality operator. It is known in, e.g., [12] that \(J:X\to X^{*}\) is bounded, continuous and satisfies condition \((S_{+})\) and \(J^{1}:X^{*}\to X\) is continuous and satisfies condition \((S_{+})\).
(e) Consider an affine homotopy \(H: [0,1]\times\overline{G} \to X\) given by
As \(h\notin F(\partial G)=H(t,\partial G)\) for all \(t \in[0,1]\), the homotopy invariance property (c) of the degree d implies that
This completes the proof. □
For the next aim, we need the Borsuk theorem for operators in \(\mathcal {F}_{B}(X)\) taken from [11], Theorem 8.1.
Lemma 3.5
Let G be a bounded open subset of X which is symmetric with respect to the origin \(0 \in G\), and let \(T \in\mathcal{F}_{1}(\overline{G})\) be continuous. If \(F \in\mathcal{F}_{T,B}(\overline{G})\) is odd on ∂G such that \(0 \notin F(\partial G)\), then \(d_{B}(F,G,0)\) is an odd number.
Now we give a new version of the Borsuk theorem for operators in \(\mathcal{F}(X)\).
Theorem 3.6
Let G be a bounded open set in X which is symmetric with respect to \(0 \in G\). Suppose that \(T \in\mathcal{F}_{1}(\overline{G})\) is continuous and odd on G̅ and \(F\in\mathcal{F}_{T}(\overline{G})\) is odd on ∂G with \(0 \notin F(\partial G)\). Then \(d(F,G,0)\) is an odd number, and the equation \(Fu = 0\) has at least one solution in G.
Proof
Let \(P \colon\overline{G} \to X\) be an operator defined by
Then P is odd and demicontinuous on G̅. Since F is odd on ∂G and \(0\notin F(\partial G)\), it is clear that \(0\notin P(\partial G)\). To show that P satisfies condition \((S_{+})_{T}\), let \((u_{k})\) be any sequence in G̅ such that
Since T is odd and continuous on G̅ and \(F\in\mathcal {F}_{T}(\overline{G})\), we have by Lemma 2.1(a) and (b)
Since F satisfies condition \((S_{+})_{T}\), this implies that \(u_{k}\to u\) and thus P satisfies condition \((S_{+})_{T}\). In view of Corollary 2.9, we can choose a symmetric open subset \(G_{0}\) of G such that
Since F and P coincide on ∂G, we have by Theorem 3.4(e)
It follows from Theorem 3.4(b) that
Since the restriction \(P_{\overline{G}_{0}}\in\mathcal {F}_{T,B}(\overline{G}_{0})\) is odd on \(\partial{G_{0}}\), Lemma 3.5 says that
Combining this with (3.1) and (3.2), we conclude that \(d(F, G, 0)\) is an odd number. By Theorem 3.4(a), the equation \(Fu = 0\) has a solution in G. This completes the proof. □
Application
In this section, we study the Dirichlet boundary value problem based on the degree theory in Section 3.
Let Ω be a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary. Let \(2< p<N\) and set \(p' = p/(p1)\). We consider a nonlinear equation of the form
where \(\Delta_{p}\) is the pLaplacian given by
Assume that \(f \colon\Omega\times\mathbb{R} \times\mathbb{R}^{N} \to\mathbb{R}\) is a realvalued function such that

(f1)
f satisfies the Carathéodory condition, that is, \(f(\cdot,\eta,\zeta)\) is measurable on Ω for all \((\eta,\zeta)\in\mathbb{R} \times\mathbb{R}^{N}\) and \(f(x,\cdot,\cdot)\) is continuous on \(\mathbb{R} \times\mathbb{R}^{N}\) for almost all \(x\in\Omega\).

(f2)
f has the growth condition
$$\bigl\vert f(x, \eta, \zeta)\bigr\vert \leq c\bigl(k(x) + \vert \eta \vert ^{q1} +\vert \zeta \vert ^{q1}\bigr) $$for almost all \(x\in\Omega\) and all \((\eta,\zeta)\in\mathbb{R} \times\mathbb{R}^{N}\), where c is a positive constant, \(1< q< p\), and \(k \in L^{p'}(\Omega)\).
Let \(W_{0}^{1,p}(\Omega)\) be the closure of \(C_{0}^{\infty}(\Omega)\) in the Sobolev space
equipped with the norm
where \(\Vert \cdot \Vert _{p}\) stands for the norm on \(L^{p}(\Omega)\). Due to the Poincaré inequality, the norm \(\Vert \cdot \Vert _{1,p}\) on \(W_{0}^{1,p}(\Omega)\) is equivalent to the norm \(\Vert \cdot \Vert \) given by
Note that the Sobolev space \(W_{0}^{1,p}(\Omega)\) is a uniformly convex Banach space and the embedding \(I: W_{0}^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)\) is compact; see, e.g., [12].
A point \(u\in W_{0}^{1,p}(\Omega)\) is said to be a weak solution of (4.1) if
Lemma 4.1
Under assumptions (f1) and (f2), the operator \(S \colon W_{0}^{1,p}(\Omega) \to(W_{0}^{1,p}(\Omega))^{*}\) setting by
is compact.
Proof
Let \(X =W_{0}^{1,p}(\Omega)\) be the Sobolev space with the norm
Let \(\Phi: X \rightarrow L^{p'}(\Omega)\) be an operator defined by
We first show that the operator Φ is bounded and continuous. Note that the embedding \(L^{p}(\Omega)\hookrightarrow L^{(q1)p'}(\Omega)\) is continuous, that is,
where \(c_{1}\) is a positive constant. For each \(u\in X\), we have by the growth condition (f2) and (4.3)
This implies that Φ is bounded on X. To show that Φ is continuous, let \(u_{k}\rightarrow u\) in X. Then \(u_{k}\to u\) and \(D_{i}u_{k}\to D_{i}u\) in \(L^{p}(\Omega)\) for \(i=1,\dots,N\). Hence there exist a subsequence \((u_{j})\) of \((u_{k})\) and measurable functions \(v, w_{i}\) in \(L^{p}(\Omega)\) for \(i=1,\dots,N\) such that
for almost all \(x \in\Omega\) and for every \(i\in\{1,\ldots, N\}\) and all \(j\in\Bbb{N}\). Since f satisfies the Carathéodory condition, we obtain that
It follows from (f2) and \(v, w_{i}\in L^{(q1)p'}(\Omega)\) that
for almost all \(x \in\Omega\) and for all \(j\in\Bbb{N}\) and
Taking into account the identity
the Lebesgue dominated convergence theorem implies that
The convergence principle tells us that the entire sequence \((\Phi u_{k})\) converges to Φu in \(L^{p'}(\Omega)\). We have just proved that Φ is continuous on X. Since the embedding \(I:X\hookrightarrow L^{p}(\Omega)\) is compact, it is known that the adjoint operator \(I^{*}: L^{p'}(\Omega)\rightarrow X^{*}\) is also compact. Therefore, the composition \(I^{*}\circ\Phi: X\to X^{*}\) is compact. Moreover, considering the operator \(J: X\to X^{*}\) given by
it can be seen that J is compact, by noting that the embedding \(i: L^{p}(\Omega)\hookrightarrow L^{p'}(\Omega)\) is continuous and \(J=I^{*}\circ i\circ I\). We conclude that \(S= J+ I^{*}\circ\Phi\) is compact. This completes the proof. □
Now we can show the solvability of the given boundary value problem involving the pLaplacian by using the degree theory.
Theorem 4.2
Under assumptions (f1) and (f2), problem (4.1) has a weak solution u in \(W_{0}^{1,p}(\Omega)\).
Proof
Let Y = \(W_{0}^{1,p}(\Omega)\) be the Sobolev space, and let \(S: Y\to Y^{*}\) be as in Lemma 4.1. Define an operator \(F \colon Y \to Y^{*}\) by the relation
Then \(u \in Y\) is a weak solution of (4.1) if and only if
It is known in [12], Proposition 26.10, that the operator \(F: Y\to Y^{*}\) is bounded, continuous, and uniformly monotone. In particular, it is coercive and satisfies condition \((S_{+})\). Now let \(X = Y^{*}\) and identify \(X^{*}\) with Y. By the main theorem on monotone operators due to Browder and Minty in [12], Theorem 26.A, the inverse operator \(T :=F^{1} \colon X \to X^{*}\) is bounded, continuous and satisfies condition \((S_{+})\), where the last follows from the fact that F is continuous and satisfies condition \((S_{+})\) and T is bounded. Moreover, note by Lemma 4.1 that the operator \(S:X^{*}\to X\) is bounded, continuous, and quasimonotone. Consequently, equation (4.4) is equivalent to
To solve equation (4.5), we will apply the degree theory for \(\mathcal{F}(X)\). To do this, we first claim that the set
is bounded. Indeed, let \(v \in B\), that is, \(v + tS \circ Tv = 0\) for some \(t \in[0,1]\). Set \(u :=Tv\). Noting that the embeddings \(L^{p}(\Omega) \hookrightarrow L^{2}(\Omega)\), \(L^{p}(\Omega) \hookrightarrow L^{q}(\Omega)\), \(L^{p}(\Omega)\hookrightarrow L^{(q1)p'}(\Omega)\), and \(Y \hookrightarrow L^{p}(\Omega)\) are continuous, we get by the growth condition (f2) the estimate
where \(\Vert \cdot \Vert \) denotes the equivalent norm on Y given by (4.2). From \(p>2\) and \(p>q\) it follows that
Since the operator S is bounded, it is obvious from (4.5) that the set B is bounded in X. We can now choose a positive constant R such that
This says that
From Lemma 2.3 it follows that
Consider a homotopy \(H \colon[0,1] \times\overline{B_{R}(0)} \to X\) given by
Applying the homotopy invariance and normalization property of the degree d stated in Theorem 3.4, we get
and hence there exists a point \(v \in B_{R}(0)\) such that
We conclude that \(u = Tv\) is a weak solution of (4.1). This completes the proof. □
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Acknowledgements
This work was supported by Sungkyun Research Fund, Sungkyunkwan University, 2014.
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KI conceived of the study and drafted the manuscript. HS participated in coordination. All authors approved the final manuscript.
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Kim, I., Hong, S. A topological degree for operators of generalized \((S_{+})\) type. Fixed Point Theory Appl 2015, 194 (2015). https://doi.org/10.1186/s1366301504458
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MSC
 47J05
 47H05
 47H11
 47H30
Keywords
 operator of \((S_{+})\) type
 quasimonotone operator
 degree theory
 pLaplacian