Abstract
By means of fixed point index theory for multivalued maps, we provide an analogue of the classical Birkhoff–Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in detail this fact for a class of second-order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.
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1 Introduction
The celebrated invariant-direction theorem due to Birkhoff and Kellogg [2] is an abstract existence result that, roughly speaking, gives conditions for the existence of a “nonlinear” eigenvalue and eigenvector for compact maps in normed linear spaces. Among its various extensions, one is set in cones and is due to Krasnosel’skiĭ and Ladyženskiĭ [20]. These classical functional analytic tools find applications, e.g., to eigenvalue problems for ODEs and PDEs (see for example the book [1], the recent papers [18, 19], and references therein); typically, the methodology in this context is to reformulate the given boundary value problem as a fixed point problem in a suitable Banach space.
Recently, the first two authors developed a Birkhoff–Kellogg type theorem in the framework of affine cones (cf. [4], see also [3, 6, 9]). The motivation for this new type of results is that the setting of affine cones seems to be helpful when dealing with equations with delay effects. A key ingredient in [4] is the continuity of the involved operator. On the other hand, there has been recently a rising attention towards discontinuous differential equations, that occur when modelling real-world phenomena. Here, we mention the classical books by Filippov [13], Carl and Heikkilä [7], and Heikkilä and Lakshmikantham [17] and the more recent book by Figueroa et al. [12].
In the present paper, we provide a discontinuous version of the Birkhoff–Kellogg type result in the setting of affine wedges in Banach spaces; see Theorem 2.13. The proof of Theorem 2.13 is based on the fixed point index theory for discontinuous operators developed in [12]. We stress that a crucial point in the construction of the index for discontinuous operators is its equivalence with the corresponding one of a suitable multivalued map, for which it is already defined; see [14]. Note that this newly constructed topological tool for discontinuous operators inherits the key properties of the classical one. This construction is sketched in Sect. 2 for completeness.
In Sect. 3, we illustrate the applicability of our results to boundary value problems; see Theorem 3.4. In more details, we consider the following second-order parameter-dependent differential equation with deviated argument:
with initial condition
and the final homogeneous boundary condition
where \(\lambda \ge 0\) is a parameter, \(r\ge 0,\) \(\sigma :[0,1]\rightarrow [-r,1]\) and \(\omega :[-r,0]\rightarrow [0,\infty )\) are suitable continuous functions, while the nonlinearity \(f:[0,1]\times [0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) may be discontinuous with respect to the second argument in an appropriate sense. We employ a concept of admissible discontinuity curve as in [12]. We conclude the paper by illustrating the applicability of our theory by means of a toy model with delay; see Example 3.9.
As far as we know, our results extend and complement the previous literature. This is highlighted in more details in Remarks 2.12, 2.15 and 3.10.
2 Birkhoff–Kellogg Type Results via Fixed Point Index Theory
2.1 On Fixed Point Index Theory for Discontinuous Operators
Let K be a nonempty closed and convex subset of a real Banach space \((X,\Vert \cdot \Vert ),\) \(U\subset K\) a relatively open subset and \(T:\overline{U}\subset K \longrightarrow K\) a mapping, not necessarily continuous.
Definition 2.1
The closed-convex envelope of an operator \(T:\overline{U}\subset K \longrightarrow K\) is the multivalued mapping \({\mathbb {T}}: \overline{U} \longrightarrow 2^X\) given by
where \(\overline{B}_{\varepsilon }(x)\) denotes the closed ball centered at x and radius \(\varepsilon ,\) and \(\overline{\textrm{co}}\) means closed-convex hull.
Example 2.2
-
1.
Consider the real function \(T:{\mathbb {R}}\rightarrow {\mathbb {R}}\) defined as \(T(x)=x,\) if \(x\le 0,\) and \(T(x)=x+1,\) if \(x>0.\) Its closed-convex envelope is the multivalued map \({\mathbb {T}}\) given by \({\mathbb {T}}(x)=\{x\},\) if \(x<0;\) \({\mathbb {T}}(x)=\{x+1\},\) if \(x>0;\) and \({\mathbb {T}}(0)=[0,1].\)
-
2.
The closed-convex envelope of any continuous map T is equal to T.
Now, we recall some useful properties of closed-convex envelopes (cc-envelopes for short) and the definition of the fixed point index that we will employ throughout this paper. The reader is referred to [11, 12] for details.
Proposition 2.3
Let \({\mathbb {T}}\) be the cc-envelope of an operator \(T: \overline{U} \longrightarrow K.\) Then, the following properties hold :
-
1.
If \(\tilde{{\mathbb {T}}}:\overline{U} \longrightarrow 2^X\) is an upper semicontinuous (usc) operator which assumes closed and convex values and \(Tx \in \tilde{{\mathbb {T}}} x\) for all \(x \in \overline{U},\) then \({\mathbb {T}} x \subset \tilde{{\mathbb {T}}} x\) for all \(x \in \overline{U};\)
-
2.
If T maps bounded sets into relatively compact sets, then \({\mathbb {T}}\) assumes compact values, and it is usc;
-
3.
If \(T \, \overline{U}\) is relatively compact, then \({\mathbb {T}} \, \overline{U}\) is relatively compact.
The fixed point index for a not necessarily continuous operator T was introduced in [10] using the degree theory developed in [11] and a retraction trick, just as in the classical case. Both topological degree and fixed point index theories are based on the available results for the multivalued cc-envelope \({\mathbb {T}}.\)
Definition 2.4
Let \(T:\overline{U}\subset K \longrightarrow K\) be an operator, such that \(T\,\overline{U}\) is relatively compact, T has no fixed points on \(\partial \, U\) and
where \({\mathbb {T}}\) is the cc-envelope of T.
We define the fixed point index of T in K over U as
where r is a continuous retraction of X onto K and \(\deg \) is the degree introduced in [11].
Remark 2.5
Note that condition (2.5) means that the set of fixed points of \({\mathbb {T}}\) (i.e., the set of points x, such that \(x\in {\mathbb {T}}x)\) is contained in the set of fixed points of T. This is a weaker condition than the continuity of T; indeed, if T is continuous, then \({\mathbb {T}}x=\{Tx\}\) for all \(x\in \overline{U}\), and thus, (2.5) is trivially satisfied.
We now recall a useful proposition from [10] that relates the fixed point index of the discontinuous operator T with that of its associated multivalued mapping \({\mathbb {T}}.\)
Proposition 2.6
[10, Proposition 2.12] Let T be a mapping that satisfies the conditions of Definition 2.4. Then, the fixed point index of T is such that
where the right-hand index is the fixed point index defined for multivalued mappings; see [14].
As a straightforward consequence of the fixed point index theory for usc multivalued mappings, the following properties can be derived (see [12]).
Theorem 2.7
Let T be a mapping that satisfies the conditions of Definition 2.4. Then, the following properties hold :
-
(i)
(Homotopy invariance) Let \(H:\overline{U}\times \left[ 0,1\right] \longrightarrow K\) be a mapping, such that
-
(a)
for each \((x,t)\in \overline{U}\times [0,1]\) and all \(\varepsilon >0\), there exists \(\delta =\delta (\varepsilon ,x,t)>0\), such that
$$\begin{aligned} s\in [0,1], \ \left| t-s\right|<\delta \ \Longrightarrow \ \left\| H(z,t)-H(z,s)\right\| <\varepsilon \quad \forall \,z\in \overline{B}_{\delta }(x)\cap \overline{U}; \end{aligned}$$ -
(b)
\(H\left( \overline{U}\times \left[ 0,1\right] \right) \) is relatively compact;
-
(c)
\(\left\{ x\right\} \cap {\mathbb {H}}_{t}(x)\subset \left\{ H_{t}(x)\right\} \) for all \(t\in \left[ 0,1\right] \) and all \(x\in \overline{U}\cap {\mathbb {H}}_t \overline{U},\) where \(H_{t}(\cdot ):= H(\cdot ,t)\) and \({\mathbb {H}}_{t}\) denotes the cc-envelope of \(H_{t}.\)
If \(x\ne H(x,t)\) for all \((x,t)\in \partial \,U\times \left[ 0,1\right] \), then the index \(i_{K}\left( H_{t},U\right) \) does not depend on \(t\in [0,1].\)
-
(a)
-
(ii)
(Additivity) Let U be the disjoint union of two open sets \(U_{1}\) and \(U_{2}.\) If \(0\not \in \left( I-T\right) \left( \overline{U}{\setminus }(U_{1}\cup U_{2})\right) ,\) then
$$\begin{aligned} i_{K}\left( T,U\right) =i_{K}\left( T,U_{1}\right) +i_{K}\left( T,U_{2}\right) . \end{aligned}$$ -
(iii)
(Excision) Let \(A\subset U\) be a closed set. If \(0\not \in \left( I-T\right) \left( \partial \,U\right) \cup \left( I-T\right) (A),\) then
$$\begin{aligned} i_{K}\left( T,U\right) =i_{K}\left( T,U{\setminus } A\right) . \end{aligned}$$ -
(iv)
(Existence) If \(i_{K}\left( T,U\right) \ne 0,\) then there exists \(x\in U\), such that \(Tx=x.\)
-
(v)
(Normalization) For every constant map T, such that \(T\,\overline{U}\subset U,\) \(i_{K}\left( T,U\right) =1.\)
2.2 Birkhoff–Kellogg Theorem and Discontinuous Operators
The following notions will be used along the text. A closed-convex subset K of a Banach space \((X,\left\| \cdot \right\| )\) is a wedge if \(\mu \,x\in K\) for every \(x\in K\) and for all \(\mu \ge 0.\) Furthermore, if a wedge K satisfies that \(K\cap (-K)=\{0\},\) then it is said to be a cone. A cone K induces the partial order in X given by \(u\preceq v\) if and only if \(v-u\in K.\) The cone K is called normal if there exists \(c>0\), such that \(\left\| u\right\| \le c\left\| v\right\| \) for all \(u,v\in X\) with \(0 \preceq u \preceq v.\)
Let K be a wedge of a Banach space \((X,\left\| \cdot \right\| ).\) For a given \(y\in X,\) the translate of the wedge K is defined as follows:
Given an open-bounded subset \(D\subset X\) with \(0\in D,\) we will denote \(D_{K_y}:=(y+D)\cap K_y,\) which is a relatively open subset of \(K_y.\) By \(\overline{D}_{K_y}\) and \(\partial \,D_{K_y}\), we will mean, respectively, the closure and the boundary of \(D_{K_y}\) relative to \(K_y.\)
For the convenience of the reader, we recall here the classical Birkhoff–Kellogg Theorem [2] and a variant of it set in cones. The latter result is due to Krasnosel’skiĭ and Ladyženskiĭ [20] (see also [16, Theorem 2.3.6]).
Theorem 2.8
(Birkhoff–Kellogg) Let U be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space X, and \(T:\partial \,U\longrightarrow X\) a compact map satisfying \(\left\| Tx\right\| \ge \alpha >0\) for all \(x\in \partial \,U.\) Then, there exist \(x_0\in \partial \,U\) and \(\lambda _0>0\), such that \(x_0=\lambda _0\,T x_0.\)
Theorem 2.9
(Krasnosel’skiĭ–Ladyženskiĭ) Let X be a real Banach space, \(U\subset X\) be an open-bounded set with \(0\in U,\) \(K\subset X\) be a cone, \(T:K\cap \overline{U}\longrightarrow K\) be compact, and suppose that
Then, there exist \(x_0\in K\cap \partial \,U\) and \(\lambda _0>0\), such that \(x_0=\lambda _0\, Tx_0.\)
In the context of affine cones, a Birkhoff–Kellogg type result was recently proved in [4, Theorem 2]. It reads as follows.
Theorem 2.10
Let \((X,\left\| \cdot \right\| )\) be a real Banach space, \(K\subset X\) be a cone and \(D\subset X\) be an open-bounded set with \(y\in D_{K_y}.\) Assume that \(T:\overline{D}_{K_y}\longrightarrow K\) is a compact map and consider the operator
Assume that there exists \(\bar{\lambda }>0\), such that \(i_{K_y}(T_{(y,\bar{\lambda })},D_{K_y})=0.\) Then, there exist \(x^*\in \partial \,D_{K_y}\) and \(\lambda ^*\in (0,\bar{\lambda })\), such that \(x^*=y+\lambda ^*\, T(x^*).\)
Now, we present a discontinuous version of this Birkhoff–Kellogg type result in affine wedges.
Theorem 2.11
Let \(D\subset X\) be an open-bounded set with \(0\in D,\) \(y\in X\) be fixed and K be a wedge. Assume that \(T:\overline{D}_{K_y} \longrightarrow K\) is a mapping, such that \(T\,\overline{D}_{K_y}\) is relatively compact and consider the operator
Moreover, assume that there exists \(\bar{\lambda }>0\), such that \(i_{K_y}(T_{(y,\bar{\lambda })},D_{K_y})\ne 1\) and for each \(\lambda \in (0,\bar{\lambda }]\)
where \({\mathbb {T}}_{(y,\lambda )}\) denotes the cc-envelope of \(T_{(y,\lambda )}.\)
Then, there exist \(x^*\in \partial \,D_{K_y}\) and \(\lambda ^*\in (0,\bar{\lambda })\), such that \(x^*=y+\lambda ^*\, T(x^*).\)
Proof
If \(T_{(y,\lambda )}\) has a fixed point on \(\partial \,D_{K_y}\) for some \(\lambda \in (0,\bar{\lambda })\), we are done. Otherwise, suppose that \(T_{(y,\lambda )}\) is fixed point free on \(\partial \,D_{K_y}.\) Now, observe that for each \(\lambda \in (0,\bar{\lambda }]\), the operator \(T_{(y,\lambda )}:\overline{D}_{K_y} \longrightarrow K_y\) satisfies condition (2.6) and that \(T_{(y,\lambda )}\left( \overline{D}_{K_y}\right) \) is relatively compact, which implies that the fixed point index \(i_{K_y}(T_{(y,\lambda )},D_{K_y})\) is well defined according to Definition 2.4.
Consider the map \(H:\overline{D}_{K_y}\times [0,1]\rightarrow K_y\) defined as
Note that H satisfies conditions (a)–(c) in Theorem 2.7, i. Hence, if \(x\ne H(x,t)\) for all \((x,t)\in \partial \,D_{K_y}\times \left[ 0,1\right] ,\) then
By the normalization property, since \(y\in D_{K_y},\) we have
a contradiction. In conclusion, there exist \(t^*\in (0,1)\) and \(x^*\in \partial \,D_{K_y}\), such that \(x^*=y+t^*\bar{\lambda }\, T(x^*).\) \(\square \)
Remark 2.12
Note that Theorem 2.11 has a twofold interest: not only is a generalization of Theorem 2.10 to the context of discontinuous operators, but also an improvement in the continuous case, since the conditions on the index are weakened and the result is extended to the setting of wedges.
We now prove a result in the setting of normal cones which can be of a more direct applicability due to the use of the norm, as in the classical Birkhoff–Kellogg Theorem.
Theorem 2.13
Let \(K\subset X\) be a normal cone with normal constant \(c>0\) in a Banach space X, \(D\subset X\) be an open-bounded set with \(0\in D\), and \(y\in X\) be fixed. Assume that \(T:\overline{D}_{K_y} \longrightarrow K\) is a mapping, such that \(T\,\overline{D}_{K_y}\) is relatively compact and
If there exists a positive number
such that the operator \(T_{(y,\lambda )}\) satisfies condition (2.6) for each \(\lambda \in (0,\bar{\lambda }],\) then there exist \(x^*\in \partial \,D_{K_y}\) and \(\lambda ^*\in (0,\bar{\lambda })\), such that \(x^*=y+\lambda ^*\, T(x^*).\)
Proof
We shall show that \(i_{K_y}(T_{(y,\bar{\lambda })},D_{K_y})=0\), and so, the conclusion is obtained as a consequence of Theorem 2.11.
Take \(x_0\in K{\setminus }\{0\}\) and let us see that
Indeed, suppose that there exist \(x_1\in \partial \,D_{K_y},\) \(v\in {\mathbb {T}}x_1\) and \(\beta _0\ge 0\), such that
Then, \(\bar{\lambda }\, v\preceq \bar{\lambda }\, v+\beta _0\, x_0=x_1-y\) and since K is normal
Observe that \(x_1-y\in \partial \, D\) and so
a contradiction with the choice of \(\bar{\lambda }.\)
On the other hand, since \(\overline{D}_{K_y}\) and \({\mathbb {T}}\left( \overline{D}_{K_y} \right) \) are bounded, there exists \(\bar{\beta }>0\), such that
Consider the multivalued homotopy \({\mathcal {H}}:\overline{D}_{K_y}\times [0,1]\rightarrow 2^{K_y}\) defined as
By the homotopy invariance property of the index for usc multivalued maps [14]
Therefore, it follows from Proposition 2.6 that \(i_{K_y}(T_{(y,\bar{\lambda })},D_{K_y})=i_{K_y}({\mathbb {T}}_{(y,\bar{\lambda })},D_{K_y})=0.\) \(\square \)
The following corollary can be seen as an analogue of the classical result of Krasnosel’skiĭ and Ladyženskiĭ.
Corollary 2.14
Let \(K\subset X\) be a normal cone in a Banach space X and \(D\subset X\) be an open-bounded set with \(y\in D_{K_y}.\) Assume that \(T:\overline{D}_{K_y} \longrightarrow K\) is a mapping, such that \(T\,\overline{D}_{K_y}\) is relatively compact and, for each \(\lambda >0,\) the operator \(T_{(y,\lambda )}\) satisfies condition (2.6). If
then there exist \(x^*\in \partial \,D_{K_y}\) and \(\lambda ^*>0\), such that \(x^*=y+\lambda ^*\, T(x^*).\)
Remark 2.15
Note that, in the non-affine case, Corollary 2.14 extends Theorem 2.9 within the setting of discontinuous operators in normal cones. We stress that, in the non-affine case, Corollary 2.14 can also be deduced as a consequence of the multivalued generalization of the Birkhoff–Kellogg theorem given in [15].
3 Applications
Consider the second-order parameter-dependent differential equation
with initial conditions of the form
and the final homogeneous boundary condition
where \(\lambda \) is a positive parameter, \(r\ge 0,\) and \(\sigma :[0,1]\rightarrow [-r,1]\) and \(\omega :[-r,0]\rightarrow [0,\infty )\) are continuous functions. The nonlinearity \(f:[0,1]\times [0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) may be discontinuous with respect to the second argument in a sense which will be specified later.
To study the problem (3.7)–(3.9), we shall use a superposition principle as in [5]. To do so, first consider the Dirichlet BVP
whose unique solution is given by
where G is the corresponding Green’s function. It is well known that
and moreover (see [21])
with \(\Phi (s):=s(1-s).\) Associated to the Green’s function, we consider the kernel \(k:[-r,1]\times [0,1]\rightarrow {\mathbb {R}}\) defined as
On the other hand, note that the function \(\hat{y}(t)=1-t\) solves the Dirichlet BVP
so we define the function
which will be the vertex of our affine cone.
To apply the theory of the previous section, we will work in the Banach space of continuous functions \(X={\mathcal {C}}([-r,1]),\) endowed with the usual sup-norm, \(\left\| \cdot \right\| _{[-r,1]},\) and the cone
Observe that K is a normal cone with normal constant \(c=1\) and that \(\left\| u\right\| _{[-r,1]}=\left\| u\right\| _{[0,1]}\) for all \(u\in K.\) Now, for the vertex y defined in (3.11), we consider the translate of the cone K given by
and for each \(\rho >0,\) we denote by \(K_{y,\rho }\), the relatively open-bounded set
We will look for solutions of the following perturbed Hammerstein integral equation:
located in the affine cone \(K_y.\)
Definition 3.1
By a solution of the problem (3.7)–(3.9), we mean a solution \(u \in {\mathcal {C}}([-r, 1], {{\mathbb R}})\) of the integral equation (3.12).
Before doing so, we need to define the type of regions where f is allowed to be discontinuous. The concept of admissible discontinuity curve used here has been widely employed in [12].
Definition 3.2
A \(\lambda \)-admissible discontinuity curve for the second-order parameter-dependent differential equation \(u''+\lambda \,f(t,u,u(\sigma ))=0\) is a \(W^{2,1}\) function \(\gamma :[a,b]\subset [0,1]\rightarrow [0,\infty )\) satisfying that there exist \(\varepsilon >0\) and \(\psi \in L^1(a,b),\) \(\psi (t)>0\) for a.a. \(t\in [a,b]\), such that either
or
Remark 3.3
Since f is non-negative, to have that \(\gamma \) is a \(\lambda \)-admissible discontinuity curve for the differential equation (3.7) and any \(\lambda >0,\) it suffices that
Indeed, one may check that condition (3.13) holds with \(\psi (t)=\gamma ''(t)/2,\) \(t\in [a,b].\)
Let us now state and prove the main result of this section.
Theorem 3.4
Let \(\rho >0\) and assume that the following conditions hold :
- \((H_1)\):
-
any composition \(t\in [0,1]\mapsto f(t,u(t),v(t))\) is measurable provided that \(u,v\in {\mathcal {C}}([0,1],[0,\infty ));\)
- \((H_2)\):
-
there exists \(M_{\rho }\in L^1([0,1])\), such that
$$\begin{aligned} f(t,u,v){} {} \le M_{\rho }(t) \quad \mathrm{for\, a.a. }\, t\in [0,1]{} & {} \mathrm{and\, all }\, (u,v)\, \textrm{with }\, \\ {}{} & {} 0\le u,v \le \rho +\left\| \omega \right\| _{[-r,0]}; \end{aligned}$$ - \((H_3)\):
-
there exists \(\delta _{\rho }\in L^1([1/4,3/4])\), such that
$$\begin{aligned} f(t,u,v){} {} \ge \delta _{\rho }(t) \quad \mathrm{for\, a.a.}\, t\in \left[ 1/4,3/4\right]{} & {} \mathrm{and\, all }\, (u,v) \,{ \mathrm with } \\{}{} & {} {} 0 \le u,v \le \rho +\left\| \omega \right\| _{[-r,0]} \end{aligned}$$and
$$\begin{aligned} \bar{\delta }:=\sup _{t\in [1/4,3/4]}\int _{1/4}^{3/4}k(t,s)\delta _{\rho }(s)\,{\textrm{d}}s>0; \end{aligned}$$ - \((H_4)\):
-
there exists a countable number of curves \(\gamma _n:I_n=[a_n,b_n]\rightarrow [0,\infty ),\) \(n\in {\mathbb {N}},\) such that for a.a. \(t\in [0,1]\), the function \(f(t,\cdot ,\cdot )\) is continuous on \(\left( [0,\infty ){\setminus } \bigcup _{n:t\in I_n}\{\gamma _n(t) \} \right) \times [0,\infty )\) and, moreover, each \(\gamma _n\) is a \(\lambda \)-admissible discontinuity curve for each \(\lambda \in (0,\bar{\lambda }]\) and some \(\bar{\lambda }>\rho /\bar{\delta }.\)
Then, there exist \(\lambda _{\rho }\in (0,\bar{\lambda })\) and \(u_{\rho }\in \partial \,K_{y,\rho }\) that satisfy the integral equation (3.12).
Proof
Let us divide the proof in several steps.
Step 1. The operator T, defined in (3.12), maps the set \(\overline{K}_{y,\rho }\) into the cone K and, moreover, \(T\,\overline{K}_{y,\rho }\) is relatively compact.
First, let \(u\in \overline{K}_{y,\rho }\) be arbitrarily fixed and let us show that \(T u\in K.\) By definition
The continuity of the kernel k, jointly with hypothesis \((H_1),\) \((H_2)\) and the constant sign of f and k, imply that \(T u\in {\mathcal {C}}([-r,1],[0,\infty )).\) Moreover, since \(k(t,s)=0\) for all \(t\le 0,\) we have that \(Tu(t)=0\) for all \(t\in [-r,0].\) Now, for \(t\in [1/4,3/4],\) we have
as a consequence of the properties of the Green’s function G stated above. In conclusion, \(Tu\in K.\)
On the other hand, the compactness of the set \(\overline{T\,\overline{K}_{y,\rho }}\) follows from assumption \((H_2)\) and the continuity of the kernel k, combined with a careful use of the Arzelà–Ascoli theorem (see [22]).
Step 2. For each \(\lambda \in (0,\bar{\lambda }],\) the operator \(y+\lambda \,T\) satisfies that
where \(\bar{\lambda }\) is fixed by hypothesis \((H_4).\)
Fix arbitrary \(\lambda \in (0,\bar{\lambda }]\) and \(u\in \overline{K}_{y,\rho }.\) Now, consider two different cases:
Case 1. \(m\left( \left\{ t\in I_n:u(t)=\gamma _n(t) \right\} \right) =0\) for all \(n\in {\mathbb {N}}\) (where m denotes Lebesgue measure).
Let us prove that T is continuous at u, which implies that \({\mathbb {T}}(u)=\left\{ T(u) \right\} \), and thus, condition (3.15) holds for such u. Indeed, in this case, we have that for a.a. \(t\in [0,1]\), the function \(f(t,\cdot ,\cdot )\) is continuous at \((u(t),u(\sigma (t))).\) Hence, if \(u_k\rightarrow u\) uniformly in \([-r,1],\) then
which implies, due to Lebesgue’s dominated convergence theorem, that \(T u_k\rightarrow Tu\) in \({\mathcal {C}}([-r,1]).\)
Case 2. \(m\left( \left\{ t\in I_n:u(t)=\gamma _n(t) \right\} \right) >0\) for some \(n\in {\mathbb {N}}.\)
In this case, one can show that \(u\notin y+\lambda \,{\mathbb {T}}(u),\) which implies that condition (3.15) holds for such u. The proof is based on condition \((H_4)\) and the fact that the function \(\gamma _n\) is a \(\lambda \)-admissible discontinuity curve for the problem. It can be replicated following the reasoning in the proof of Proposition 4.7, Case 2, in [10].
Step 3. It holds that
For \(u\in \partial \,K_{y,\rho }\) and \(\varepsilon >0,\) take \(u_i\in \overline{B}_{\varepsilon }(u)\cap \overline{K}_{y,\rho }\) and \(\lambda _i\ge 0\) with \(\sum \lambda _i=1,\) \(i=1,2,\ldots ,m.\) Then, by assumption \((H_3),\) we have for \(t\in [1/4,3/4]\)
Hence, for any \(v\in {\textrm{co}} \, T\left( \overline{B}_{\varepsilon }(u)\cap \overline{K}_{y,\rho }\right) ,\) we have
Since \({\mathbb {T}} u\subset \overline{\textrm{co}} \, T\left( \overline{B}_{\varepsilon }(u)\cap \overline{K}_{y,\rho }\right) ,\) it follows that \(\left\| v\right\| _{[-r,1]}\ge \bar{\delta }>0\) for any \(v\in {\mathbb {T}} u,\) as wished.
Therefore, the conclusion follows from Theorem 2.13. \(\square \)
Remark 3.5
We emphasize that hypotheses \((H_1),\) \((H_2)\), and \((H_4)\) do not imply that f be a Carathéodory map, since, due to \((H_4)\), the function f can be discontinuous with respect to the last variables.
Furthermore, note that if, for each \((x,y)\in [0,\infty )\times [0,\infty ),\) the map \(t\in [0,1]\mapsto f(t,x,y)\) is measurable and, for a.a. \(t\in [0,1],\) the map \((x,y)\mapsto f(t,x,y)\) is continuous, then condition \((H_1)\) holds. However, the measurability of the map \(t\in [0,1]\mapsto f(t,x,y)\) together with \((H_4)\) does not imply necessarily that condition \((H_1)\) holds. More information about the measurability of compositions in this setting can be found in [8, Section 3.1].
Corollary 3.6
Let \(\rho >0\) and assume that conditions \((H_1)\)–\((H_3)\) hold and, moreover,
- \((H_4^*)\):
-
there exist a countable number of curves \(\gamma _n:I_n=[a_n,b_n]\rightarrow [0,\infty ),\) \(n\in {\mathbb {N}},\) such that \(\gamma _n''>0\) and for a.a. \(t\in [0,1]\), the function \(f(t,\cdot ,\cdot )\) is continuous on \(\left( [0,\infty ){\setminus } \bigcup _{n:t\in I_n}\{\gamma _n(t) \} \right) \times [0,\infty ).\)
Then, there exist \(\lambda _{\rho }>0\) and \(u_{\rho }\in \partial \,K_{y,\rho }\) that satisfy the integral equation (3.12).
Proof
It follows from Theorem 3.4 together with Remark 3.3. \(\square \)
Consider the special case of (3.7) where the nonlinearity can be seen as a discontinuous perturbation of a Carathéodory function, that is
where \(g:[0,1]\times [0,\infty )\rightarrow [0,\infty )\) is a Carathéodory function and \(h:[0,\infty )\rightarrow [0,\infty )\) is locally bounded and continuous except at most at a countable number of points.
Corollary 3.7
Assume that the following conditions hold :
- \((C_1)\):
-
g satisfies the Carathéodory conditions, namely,
- (a):
-
\(g(\cdot ,v)\) is measurable for each fixed \(v\in [0,\infty );\)
- (b):
-
\(g(t,\cdot )\) is continuous for a.a. \(t\in [0,1];\)
- (c):
-
for each \(R>0,\) there exists \(M_R\in L^1([0,1])\), such that
$$\begin{aligned} g(t,v)\le M_R(t) \quad \text {for a.a. } t\in [0,1] \text { and all } v\in [0,R]; \end{aligned}$$
- \((C_2)\):
-
h is locally bounded, \(t\mapsto h(u(t))\) is measurable for each non-negative continuous function u, and there exists a countable set A, such that h is continuous in \([0,\infty ){\setminus } A;\)
- \((C_3)\):
-
there exists \(\delta \in L^1([0,1]),\) \(\delta (t)>0\) for a.a. \(t\in [0,1]\), such that
$$\begin{aligned} g(t,v)\ge \delta (t) \quad \text {for a.a. } t\in [0,1] \text { and all } v\ge 0. \end{aligned}$$
Then, for each \(\rho >0\), there exist \(\lambda _{\rho }>0\) and \(u_{\rho }\in \partial \,K_{y,\rho }\) that satisfy the BVP (3.16)–(3.8)–(3.9).
Proof
Observe that, for each \(\rho >0,\) Theorem 3.4 can be applied to the function
Note that hypotheses \((H_1)\)–\((H_3)\) are satisfied.
Now, consider the countable set A where h may be discontinuous and denote \(A=\{a_k:k\in {\mathbb {N}} \}.\) Define the constant functions \(\gamma _n:[0,1]\rightarrow [0,\infty )\) given by \(\gamma _n(t)=a_n,\) \(t\in [0,1],\) \(n\in {\mathbb {N}}.\) For each \(\lambda >0\) fixed, choose the \(L^1\)-function \(\psi (t)=\lambda \,\delta (t),\) \(t\in [0,1].\) Then, each function \(\gamma _n\) satisfies condition (3.13), and so, it is a \(\lambda \)-admissible discontinuity curve. \(\square \)
Now, let us restrict our efforts to the particular case of problem (3.7)–(3.9) in which the deviated argument is given by a continuously differentiable function \(\sigma \) with constant derivative equal to 1 or \(-1.\) Notice that it covers the meaningful situations of equations with delay (where \(\sigma (t)=t-r)\) or with reflection of the argument (where, for instance, \(\sigma (t)=1-r-t).\)
In this case, we are able to prove another version of Theorem 3.4 where the nonlinearity f may be more discontinuous. More precisely, we weaken assumption \((H_4)\) allowing f to be discontinuous w.r.t. the second and third variables over the graphs of two countable families of functions.
Theorem 3.8
Let \(\rho >0\) and assume conditions \((H_1)\)–\((H_3)\) in Theorem 3.4 hold. Moreover, suppose that \(\sigma :[0,1]\rightarrow [-r,1]\) is a continuously differentiable function with constant derivative \(\sigma '=\pm 1\) and the following assumption holds :
- (D):
-
there exist two countable families of curves \(\gamma _n:I_n=[a_n,b_n]\rightarrow [0,\infty ),\) \(n\in {\mathbb {N}},\) and \(\Gamma _j:{\mathcal {I}}_j=[c_j,d_j]\rightarrow [0,\infty ),\) \(j\in {\mathbb {N}},\) such that for a.a. \(t\in [0,1]\), the function
$$\begin{aligned} f(t,\cdot ,\cdot ) \text { is continuous on } \left( [0,\infty ){\setminus } \bigcup _{n:t\in I_n}\{\gamma _n(t) \} \right) \times \left( [0,\infty ){\setminus } \bigcup _{j:t\in {\mathcal {I}}_j}\{\Gamma _j(t) \} \right) . \end{aligned}$$
For each \(\lambda \in (0,\bar{\lambda }]\) with \(\bar{\lambda }>\rho /\bar{\delta },\) each function \(\gamma _n\) is a \(\lambda \)-admissible discontinuity curve and each function \(\Gamma _j\) satisfies that
-
(a)
\(\Gamma _j(t)\ne \omega (\sigma (t))\) for a.a. \(t\in {\mathcal {I}}_j\cap \sigma ^{-1}([-r,0]);\)
-
(b)
the restriction of \(\Gamma _j\) to \({\mathcal {I}}_j\cap \sigma ^{-1}([0,1])\) satisfies either of the following conditions : there exists \(\varepsilon _j>0\) and \(\psi _j\in L^1({\mathcal {I}}_j),\) \(\psi _j(t)>0\) for a.a. \(t\in {\mathcal {I}}_j\cap \sigma ^{-1}([0,1])\), such that
-
(a)
\(-\Gamma _j''(t)+\psi _j(t)<\lambda \,f(\sigma (t),y,z)\) for a.a. \(t\in {\mathcal {I}}_j\cap \sigma ^{-1}([0,1]),\) all \(y\in [\Gamma _j(t)-\varepsilon _j,\Gamma _j(t)+\varepsilon _j]\) and all \(z\in [0,\infty );\) or
-
(b)
\(-\Gamma _j''(t)-\psi _j(t)>\lambda \,f(\sigma (t),y,z)\) for a.a. \(t\in {\mathcal {I}}_j\cap \sigma ^{-1}([0,1]),\) all \(y\in [\Gamma _j(t)-\varepsilon _j,\Gamma _j(t)+\varepsilon _j]\) and all \(z\in [0,\infty ).\)
-
(a)
Then, there exist \(\lambda _{\rho }\in (0,\bar{\lambda })\) and \(u_{\rho }\in \partial \,K_{y,\rho }\) that satisfy the integral equation (3.12), that is, they solve the problem (3.7)–(3.9).
Proof
It follows in line of the proof of Theorem 3.4 as a consequence of Theorem 2.13. Observe that it suffices to rewrite Step 2. Let us prove that for each \(\lambda \in (0,\bar{\lambda }],\) the operator \(y+\lambda \,T\) satisfies that
where \(\bar{\lambda }\) is fixed by assumption (D).
Fix arbitrary \(\lambda \in (0,\bar{\lambda }]\) and \(u\in \overline{K}_{y,\rho }.\) Now, consider three different cases:
Case 1. \(m\left( \left\{ t\in I_n:u(t)=\gamma _n(t) \right\} \cup \left\{ t\in {\mathcal {I}}_j:u(\sigma (t))=\Gamma _j(t) \right\} \right) =0\) for all \(j,n\in {\mathbb {N}}.\) Then, for a.a. \(t\in [0,1]\), the function \(f(t,\cdot ,\cdot )\) is continuous at \((u(t),u(\sigma (t)))\), and so, T is continuous at u.
Case 2. \(m\left( \left\{ t\in {\mathcal {I}}_j:u(\sigma (t))=\Gamma _j(t) \right\} \right) >0\) for some \(j\in {\mathbb {N}}.\) Let us prove that \(u\notin y +\lambda \,{\mathbb {T}} u,\) which can be justified as in the proof of [10, Proposition 4.7], but we include the reasoning here again for completeness.
Since \(u\in K_y,\) we have \(u(s)=y(s)=\omega (s)\) for all \(s\in [-r,0]\), and thus, \(u(\sigma (t))=\omega (\sigma (t))\) for all \(t\in \sigma ^{-1}([-r,0]).\) Now, condition (D), (a), implies that
Hence, we can fix some \(j\in {\mathbb {N}}\), such that
By condition (D), (b), we can assume that there exist \(\varepsilon _j>0\) and \(\psi _j\in L^1({\mathcal {I}}_j),\) \(\psi _j(t)>0\) on \({\mathcal {I}}_j,\) such that
In what follows, let us denote \(J:=\left\{ t\in {\mathcal {I}}_j\cap \sigma ^{-1}([0,1]):u(\sigma (t))=\Gamma _j(t) \right\} \) and \(M:=\lambda \left( M_{\rho }\circ \sigma \right) .\) By technical results of Lebesgue measure (see [10, Lemma 4.2 and Corollary 4.3]), we know that there exists a measurable set \(J_0\subset J\) with \(m(J)=m(J_0)\), such that for all \(\tau _0\in J_0\)
and, moreover, there is \(J_1\subset J_0\) with \(m(J_0{\setminus } J_1)=0\), such that for all \(\tau _0\in J_1\)
Now, fix \(\tau _0\in J_1.\) By (3.19) and (3.20), there exists \(\bar{t}>0\) sufficiently close to 0, such that for all \(t\in [\bar{t},2\,\bar{t}]\), the following inequalities hold:
and
Define the positive number
Let \(\varepsilon _j>0\) be given above. Let us show that for every finite family \(u_i\in \overline{B}_{\varepsilon _j}(u)\cap \overline{K}_{y,\rho }\) and \(\mu _i\in [0,1]\) \((i=1,2,\ldots ,m),\) with \(\sum \mu _i=1,\) we have
which implies \(u\notin y +\lambda \,{\mathbb {T}} u.\) Notice that we can suppose without loss of generality that the restriction of u to [0, 1], denoted also as u, satisfies that \(u\in y+\lambda \,Q\) where Q is the subset of \({\mathcal {C}}([0,1])\) defined as
Indeed, due to assumption \((H_2),\) then
Therefore, \(T\left( \overline{K}_{y,\rho }\right) \subset Q\) and Q is a closed-convex subset of \({\mathcal {C}}([0,1])\) (see [10, Lemma 4.5]), which implies that \({\mathbb {T}}\left( \overline{K}_{y,\rho }\right) \subset Q.\)
To prove (3.21), for simplicity, let us denote \(z=\lambda \sum _{i=1}^{m}\mu _i Tu_i\) and \(v=u-y.\) For a.a. \(t\in J,\) we have by the chain rule that
and, since \(\sigma '=\pm 1,\)
On the other hand, for every \(i\in \{1,\ldots ,m \}\) and \(t\in J,\) we deduce from \(u_i\in \overline{B}_{\varepsilon _j}(u)\) that
and then, condition (3.18) ensures that for a.a. \(t\in J\)
Note that \(y''(s)=0\) for all \(s>0,\) so we obtain that for a.a. \(t\in J\)
By integration, for \(t\in [\bar{t},2\,\bar{t}]\)
Hence, we have for all \(t\in [\bar{t},2\,\bar{t}]\) that
In case \(z'(\sigma (\tau _0))\ge v'(\sigma (\tau _0)),\) then
so by integration
Then, either \(z(\sigma (\tau _0-2\,\bar{t}))-v(\sigma (\tau _0-2\,\bar{t}))<-r\) or \(z(\sigma (\tau _0-\bar{t}))-v(\sigma (\tau _0-\bar{t}))>r,\) and thus, \(\left\| v-z\right\| >r,\) as wished.
It can be seen in a similar way that
which ensures that \(\left\| v-z\right\| >r\) if \(z'(\sigma (\tau _0))< v'(\sigma (\tau _0)).\)
Case 3. \(m\left( \left\{ t\in I_n:u(t)=\gamma _n(t) \right\} \right) >0\) for some \(n\in {\mathbb {N}}.\) It follows as in Case 2.
Finally, Theorem 2.13 gives the conclusion. \(\square \)
Example 3.9
Consider the function \(\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) given by
where \(\{q_n \}_{n\in {\mathbb {N}}}\) is an enumeration of the rational numbers. Observe that \(\phi \) is discontinuous at the rational numbers and continuous at the irrational ones.
We study the existence of solutions for the following BVP with delay:
In this case
For a fixed \(\rho >0,\) we can choose
to check hypotheses \((H_2)\) and \((H_3).\) On the other hand, for each rational \(q_n\), we define the function \(\gamma _n:[0,1]\rightarrow {\mathbb {R}}\) as
and so, for a.a. \(t\in [0,1],\) \(f(t,\cdot ,\cdot )\) is continuous on \(\left( [0,\infty ){\setminus }\bigcup _n \{\gamma _n(t) \} \right) \times \left( [0,\infty ){\setminus }\bigcup _n \{\gamma _n(t) \} \right) .\) Note that for each \(n\in {\mathbb {N}}\) and each \(\lambda >0\)
and thus, condition (D) in Theorem 3.8 holds. Therefore, this result ensures that the BVP (3.22) has uncountable many pairs of solutions and parameters \((u_{\rho },\lambda _{\rho }).\)
Remark 3.10
We stress that the theory presented so far is applicable and represents a novelty even in the special case of eigenvalue problems for ODEs, in presence of discontinuities. To illustrate this fact, one may consider the eigenvalue problem
a classical problem studied in the book of Guo and Lakshmikantham [16, Example 2.3.2], where in our case, the nonlinearity can be allowed to be discontinuous.
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Acknowledgements
The authors would like to thank the anonymous referee for the careful reading of the manuscript and the constructive comments. A. Calamai and G. Infante are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). G. Infante is a member of the UMI Group TAA “Approximation Theory and Applications”. J. Rodríguez-López has been partially supported by the VIS Program of the University of Calabria, by Ministerio de Ciencia y Tecnología (Spain), AEI and Feder, Grant PID2020-113275GB-I00, and by Xunta de Galicia, Grant ED431C 2023/12. This study was partly funded by: Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2022 “Nonlinear differential problems with applications to real phenomena” (Grant No. 2022ZXZTN2).
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Calamai, A., Infante, G. & Rodríguez-López, J. A Birkhoff–Kellogg Type Theorem for Discontinuous Operators with Applications. Mediterr. J. Math. 21, 149 (2024). https://doi.org/10.1007/s00009-024-02692-3
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DOI: https://doi.org/10.1007/s00009-024-02692-3
Keywords
- Nontrivial solutions
- wedge
- Birkhoff–Kellogg type result
- multivalued map
- discontinuous differential equation
- deviated argument