ScorzaDragoni approach to Dirichlet problem in Banach spaces
 Jan Andres^{1}Email author,
 Luisa Malaguti^{2} and
 Martina Pavlačková^{1}
https://doi.org/10.1186/16872770201423
© Andres et al.; licensee Springer. 2014
Received: 16 October 2013
Accepted: 9 January 2014
Published: 27 January 2014
Abstract
Hartmantype conditions are presented for the solvability of a multivalued Dirichlet problem in a Banach space by means of topological degree arguments, bounding functions, and a ScorzaDragoni approximation technique. The required transversality conditions are strictly localized on the boundaries of given bound sets. The main existence and localization result is applied to a partial integrodifferential equation involving possible discontinuities in state variables. Two illustrative examples are supplied. The comparison with classical singlevalued results in this field is also made.
MSC:34A60, 34B15, 47H04.
Keywords
Dirichlet problem ScorzaDragonitype technique strictly localized bounding functions solutions in a given set condensing multivalued operators1 Introduction
In this paper, we will establish sufficient conditions for the existence and localization of strong solutions to a multivalued Dirichlet problem in a Banach space via degree arguments combined with a bound sets technique. More precisely, Hartmantype conditions (cf. [1]), i.e. sign conditions w.r.t. the first state variable and growth conditions w.r.t. the second state variable, will be presented, provided the righthand side is a multivalued upperCarathéodory mapping which is γregular w.r.t. the Hausdorff measure of noncompactness γ.
The main aim will be twofold: (i) strict localization of sign conditions on the boundaries of bound sets by means of a technique originated by ScorzaDragoni [2], and (ii) the application of the obtained abstract result (see Theorem 3.1 below) to an integrodifferential equation involving possible discontinuities in a state variable. The first aim allows us, under some additional restrictions, to extend our earlier results obtained for globally upper semicontinuous righthand sides and partly improve those for upperCarathéodory righthand sides (see [3]). As we shall see, the latter aim justifies such an abstract setting, because the problem can be transformed into the form of a differential inclusion in a Hilbert ${L}^{2}$space. Roughly speaking, problems of this type naturally require such an abstract setting. In order to understand in a deeper way what we did and why, let us briefly recall classical results in this field and some of their extensions.
where $f:[0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ is, for the sake of simplicity allowing the comparison of the related results, a continuous function.
The first existence results, for a bounded f in (1), are due to ScorzaDragoni [4, 5]. Let us note that his name in the title is nevertheless related to the technique developed in [2] rather than to the existence results in [4, 5].
It is well known (see e.g. [3, 6–13]) that the problem (1) is solvable on various levels of generality provided:
(i_{sign}) $\mathrm{\exists}R>0$ such that $\u3008f(t,x,y),x\u3009>0$, for $(t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$ with $\parallel x\parallel =R$,
(ii_{growth}) $\mathrm{\exists}{C}_{1}\ge 0,{C}_{2}\ge 0$ such that ${C}_{1}R<1$ and $\parallel f(t,x,y)\parallel \le {C}_{1}{\parallel y\parallel}^{2}+{C}_{2}$, for $(t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$ with $\parallel x\parallel \le R$.
Let us note that the existence of the same constant $R>0$ in (i_{sign}) and (ii_{growth}) can be assumed either explicitly as in [6, 7, 9, 11, 13] or it follows from the assumptions as those in [8, 10, 12].
(1) Hartmann [9] (cf. also [1]) generalized both conditions as follows:
(i_{H}) $\mathrm{\exists}R>0$ such that $\u3008f(t,x,y),x\u3009+{\parallel y\parallel}^{2}>0$, for $t\in [0,1]$ and $(x,y)\in {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$ such that $\parallel x\parallel =R$ and $\u3008x,y\u3009=0$,
(i_{H}) the wellknown BernsteinNagumoHartman condition (for its definition and more details, see e.g. [1, 14]).
 (2)
Lasota and Yorke [10] improved condition (i_{sign}) with suitable constants ${K}_{1}\ge 0$ and ${K}_{2}>0$ in the following way:
(i_{LY}) $\u3008f(t,x,y),x\u3009+{\parallel y\parallel}^{2}\ge {K}_{1}(1+\parallel x\parallel +\u3008x,y\u3009)+{K}_{2}\parallel y\parallel $,
but for $t\in [0,1]$, $(x,y)\in {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$, and replaced (ii_{growth}) by the BernsteinNagumoHartman condition.
for $(t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$, the sign condition (i_{LY}) is obviously more liberal than (i_{sign}) as well as than (i_{H}), on the intersection of their domains.
If ${K}_{1}>0$ in (i_{LY}), then constant ${K}_{2}$ can be even equal to zero, i.e. ${K}_{2}=0$, in (i_{LY}) (see e.g. [[7], Corollary V.26 on p.74]). Moreover, the related BernsteinNagumoHartman condition can only hold for x in a suitable convex, closed, bounded subset of ${\mathbb{R}}^{n}$ (see again e.g. [7]).
(3) Following the ideas of Mawhin in [7, 11, 12], Amster and Haddad [6] demonstrated that an open, bounded subset of ${\mathbb{R}}^{n}$, say $D\subset {\mathbb{R}}^{n}$, need not be convex, provided it has a ${C}^{2}$boundary ∂D such that condition (i_{H}) can be generalized as follows:
(i_{AH}) $\u3008f(t,x,y),{n}_{x}\u3009\ge {I}_{x}(y)$, $(t,x,y)\in [0,1]\times T\partial D\times {\mathbb{R}}^{n}$, with $\u3008{n}_{x},y\u3009=0$,
where ${n}_{x}$ is the outerpointing normal unit vector field, $T\partial D$ denotes the tangent vector bundle and ${I}_{x}(y)$ stands for the second fundamental form of the hypersurface.
condition (i_{AH}) is obviously more general than the original Hartman condition (i_{H}).
Nevertheless, the growth condition takes there only the form (ii_{growth}), namely with $\parallel x\parallel \le R$ replaced by $x\in \overline{D}$, where R denotes, this time, the radius of D.
For a convex, open, bounded subset $D\subset {\mathbb{R}}^{n}$, the particular case of (i_{AH}) can read as follows:
(i_{conv}) $\u3008f(t,x,y),{n}_{x}\u3009>0$, for $(t,x,y)\in [0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$ with $x\in \partial D$ and $\u3008{n}_{x},y\u3009=0$,
which is another wellknown generalization of (i_{sign}).
(4) In a Hilbert space H, for a completely continuous mapping f, Mawhin [12] has shown that, for real constants a, b, c such that $a+b<1$, condition (i_{sign}) can be replaced in particular by
(i_{M}) $\u3008f(t,x,y),x\u3009\ge (a{\parallel x\parallel}^{2}+b\parallel x\parallel \parallel y\parallel +c\parallel x\parallel )$, $(t,x,y)\in [0,1]\times H\times H$,
and (ii_{growth}) by an appropriate version of the BernsteinNagumoHartman condition.
(5) In a Banach space E, Schmitt and Thompson [13] improved, for a completely continuous mapping f, condition (i_{conv}) in the sense that the strict inequality in (i_{conv}) can be replaced by a nonstrict one. More concretely, if there exists a convex, open, bounded subset $D\subset E$ of E with $0\in D$ such that
(i_{ST}) $\u3008f(t,x,y),{n}_{x}\u3009\ge 0$, for $(t,x,y)\in [0,1]\times E\times E$, with $x\in \partial D$ and $\u3008{n}_{x},y\u3009=0$,
where $\u3008\cdot ,\cdot \u3009$ denotes this time the pairing between E and its dual ${E}^{\prime}$, jointly with the appropriate BernsteinNagumoHartman condition, then the problem (1) admits a solution whose values are located in $\overline{D}$ (see [[13], Theorem 4.1]).
In the Carathéodory case of $f:[0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ in (1), for instance, the strict inequality in condition (i_{sign}) can be replaced, according to [[8], Theorem 6.1], by a nonstrict one and the constants ${C}_{1}$, ${C}_{2}$ can be replaced without the requirement ${C}_{1}R<1$, but globally in $[0,1]\times {\mathbb{R}}^{n}\times {\mathbb{R}}^{n}$, by functions ${c}_{1}(t,x)$, ${c}_{2}(t,x)$ which are bounded on bounded sets. Moreover, system (1) can be additively perturbed, for the same goal, by another Carathéodory function which is sublinear in both states variables x and y.
On the other hand, the Carathéodory case brings about some obstructions in a strict localization of sign conditions on the boundaries of bound sets (see e.g. [3, 15]). The same is also true for other boundary value problems (for Floquet problems, see e.g. [16–18]). Therefore, there naturally exist some extensions of classical results in this way. Further extensions concern problems in abstract spaces, functional problems, multivalued problems, etc. For the panorama of results in abstract spaces, see e.g. [19], where multivalued problems are also considered.
Nevertheless, let us note that in abstract spaces, it is extremely difficult (if not impossible) to avoid the convexity of given bound sets, provided the degree arguments are applied for noncompact maps (for more details, see [20]).
In this light, we would like to modify in the present paper the Hartmantype conditions (i_{sign}), (ii_{growth}) at least in the following way:

the given space E to be Banach (or, more practically, Hilbert),

the righthand side to be a multivalued upperCarathéodory mapping F which is γregular w.r.t. $(x,y)\in E\times E$ and either globally measurable or globally quasicompact,

the inequality in (i_{sign}) to hold w.r.t. x strictly on the boundary ∂D of a convex, bounded subset $D\subset E$ (or, more practically, of the ball $B(0,R)\subset E$),

condition (ii_{growth}) to be replaced by a suitable growth condition which would allow us reasonable applications (the usage of the BernsteinNagumoHartmantype condition will be employed in this context by ourselves elsewhere).
where $F:[0,T]\times E\times E\u22b8E$ is an upperCarathéodory multivalued mapping.
Let us note that in the entire paper all derivatives will be always understood in the sense of Fréchet and, by the measurability, we mean the one with respect to the Lebesgue σalgebra in $[0,T]$ and the Borel σalgebra in E.
The notion of a solution will be understood in a strong (i.e. Carathéodory) sense. Namely, by a solution of problem (2) we mean a function $x:[0,T]\to E$ whose first derivative $\dot{x}(\cdot )$ is absolutely continuous and satisfies (2), for almost all $t\in [0,T]$.
The solution of the b.v.p. (2) will be obtained as the limit of a sequence of solutions of approximating problems that we construct by means of a ScorzaDragonitype result developed in [22]. The approximating problems will be treated by means of the continuation principle developed in [19].
2 Preliminaries
Let E be as above and $[0,T]\subset \mathbb{R}$ be a closed interval. By the symbol ${L}^{1}([0,T],E)$, we shall mean the set of all Bochner integrable functions $x:[0,T]\to E$. For the definition and properties of Bochner integrals, see e.g. [[21], pp.693701]. The symbol $A{C}^{1}([0,T],E)$ will be reserved for the set of functions $x:[0,T]\to E$ whose first derivative $\dot{x}(\cdot )$ is absolutely continuous. Then $\ddot{x}\in {L}^{1}([0,T],E)$ and the fundamental theorem of calculus (the NewtonLeibniz formula) holds (see e.g. [[21], pp.695696], [[23], pp.243244]). In the sequel, we shall always consider $A{C}^{1}([0,T],E)$ as a subspace of the Banach space ${C}^{1}([0,T],E)$ and by the symbol $\mathcal{L}(E)$ we shall mean the Banach space of all linear, bounded transformations $L:E\to E$ endowed with the supnorm.
Given $C\subset E$ and $\epsilon >0$, the symbol $B(C,\epsilon )$ will denote, as usually, the set $C+\epsilon B$, where B is the open unit ball in E centered at 0, i.e. $B=\{x\in E\mid \parallel x\parallel <1\}$. In what follows, the symbol μ will denote the Lebesgue measure on ℝ.
Let ${E}^{\prime}$ be the Banach space dual to E and let us denote by $\u3008\cdot ,\cdot \u3009$ the pairing (the duality relation) between E and ${E}^{\prime}$, i.e., for all $\mathrm{\Phi}\in {E}^{\prime}$ and $x\in E$, we put $\mathrm{\Phi}(x)=:\u3008\mathrm{\Phi},x\u3009$.
We recall also the Pettis measurability theorem which will be used in Section 4 and which we state here in the form of proposition.
Proposition 2.1 [[24], p.278]
Let $(X,\mathrm{\Sigma})$ be a measure space, E be a separable Banach space. Then $f:X\to E$ is measurable if and only if for every $e\in {E}^{\prime}$ the function $e\circ f:X\to \mathbb{R}$ is measurable with respect to Σ and the Borel σalgebra in ℝ.
We shall also need the following definitions and notions from multivalued analysis. Let X, Y be two metric spaces. We say that F is a multivalued mapping from X to Y (written $F:X\u22b8Y$) if, for every $x\in X$, a nonempty subset $F(x)$ of Y is given. We associate with F its graph ${\mathrm{\Gamma}}_{F}$, the subset of $X\times Y$, defined by ${\mathrm{\Gamma}}_{F}:=\{(x,y)\in X\times Y\mid y\in F(x)\}$.
A multivalued mapping $F:X\u22b8Y$ is called upper semicontinuous (shortly, u.s.c.) if, for each open subset $U\subset Y$, the set $\{x\in X\mid F(x)\subset U\}$ is open in X.
Let $J\subset \mathbb{R}$ be a compact interval. A mapping $F:J\u22b8Y$, where Y is a separable metric space, is called measurable if, for each open subset $U\subset Y$, the set $\{t\in J\mid F(t)\subset U\}$ belongs to a σalgebra of subsets of J.
A multivalued mapping $F:X\u22b8Y$ is called compact if the set $F(X)={\bigcup}_{x\in X}F(x)$ is contained in a compact subset of Y and it is called quasicompact if it maps compact sets onto relatively compact sets.
Let $J\subset \mathbb{R}$ be a given compact interval. A multivalued mapping $F:J\times X\u22b8Y$, where Y is a separable Banach space, is called an upperCarathéodory mapping if the map $F(\cdot ,x):J\u22b8Y$ is measurable, for all $x\in X$, the map $F(t,\cdot ):X\u22b8Y$ is u.s.c., for almost all $t\in J$, and the set $F(t,x)$ is compact and convex, for all $(t,x)\in J\times X$.
The technique that will be used for proving the existence and localization result consists in constructing a sequence of approximating problems. This construction will be made on the basis of the ScorzaDragonitype result developed in [22] (cf. also [25]).
For more details concerning multivalued analysis, see e.g. [23, 26, 27].
 (i)
${F}_{0}(t,x,y)\subset F(t,x,y)$, for all $(t,x,y)\in [0,T]\times X\times X$,
 (ii)
if $u,v:[0,T]\to X$ are measurable functions with $v(t)\in F(t,u(t),\dot{u}(t))$, for a.a. $t\in [0,T]$, then also $v(t)\in {F}_{0}(t,u(t),\dot{u}(t))$, for a.a. $t\in [0,T]$,
 (iii)
for every $\epsilon >0$, there exists a closed ${I}_{\epsilon}\subset [0,T]$ such that $\mu ([0,T]\setminus {I}_{\epsilon})<\epsilon $, ${F}_{0}(t,x,y)\ne \mathrm{\varnothing}$, for all $(t,x,y)\in {I}_{\epsilon}\times X\times X$, and ${F}_{0}$ is u.s.c. on ${I}_{\epsilon}\times X\times X$.
The following two propositions are crucial in our investigation. The first one is almost a direct consequence of the main result in [22] (cf. [25] and [[16], Proposition 2]). The second one allows us to construct a sequence of approximating problems of (2).
Proposition 2.2 Let E be a separable Banach space and $F:[0,T]\times E\times E\u22b8E$ be an upperCarathéodory mapping. If F is globally measurable or quasicompact, then F has the ScorzaDragoni property.
Proposition 2.3 (cf. [[18], Theorem 2.2])
Let E be a Banach space and $K\subset E$ a nonempty, open, convex, bounded set such that $0\in K$. Moreover, let $\epsilon >0$ and $V:E\to \mathbb{R}$ be a Fréchet differentiable function with $\dot{V}$ Lipschitzian in $\overline{B(\partial K,\epsilon )}$ satisfying

(H1) $V{}_{\partial K}=0$,

(H2) $V(x)\le 0$, for all $x\in \overline{K}$,

(H3) $\parallel \dot{V}(x)\parallel \ge \delta $, for all $x\in \partial K$, where $\delta >0$ is given.
Then there exist $k\in (0,\epsilon ]$ and a bounded Lipschitzian function $\varphi :\overline{B(\partial K,k)}\to E$ such that $\u3008{\dot{V}}_{x},\varphi (x)\u3009=1$, for every $x\in \overline{B(\partial K,k)}$.
Remark 2.1 Let us note that the function $x\to \varphi (x)\parallel {\dot{V}}_{x}\parallel $, where ϕ and ${\dot{V}}_{x}$ are the same as in Proposition 2.3, is Lipschitzian and bounded in $\overline{B(\partial K,k)}$. The symbol ${\dot{V}}_{x}$ denotes as usually the first Fréchet derivative of V at x.
which satisfies all the properties mentioned in Proposition 2.3.
Definition 2.2 Let N be a partially ordered set, E be a Banach space and let $P(E)$ denote the family of all nonempty bounded subsets of E. A function $\beta :P(E)\to N$ is called a measure of noncompactness (m.n.c.) in E if $\beta (\overline{co\mathrm{\Omega}})=\beta (\mathrm{\Omega})$, for all $\mathrm{\Omega}\in P(E)$, where $\overline{co\mathrm{\Omega}}$ denotes the closed convex hull of Ω.
 (i)
monotone if $\beta ({\mathrm{\Omega}}_{1})\le \beta ({\mathrm{\Omega}}_{2})$, for all ${\mathrm{\Omega}}_{1}\subset {\mathrm{\Omega}}_{2}\subset E$,
 (ii)
nonsingular if $\beta (\{x\}\cup \mathrm{\Omega})=\beta (\mathrm{\Omega})$, for all $x\in E$ and $\mathrm{\Omega}\subset E$.
 (iii)
semihomogeneous if $\beta (t\mathrm{\Omega})=t\beta (\mathrm{\Omega})$, for every $t\in \mathbb{R}$ and every $\mathrm{\Omega}\subset E$,
 (iv)
regular when $\beta (\mathrm{\Omega})=0$ if and only if Ω is relatively compact,
 (v)
algebraically subadditive if $\gamma ({\mathrm{\Omega}}_{1}+{\mathrm{\Omega}}_{2})\le \gamma ({\mathrm{\Omega}}_{1})+\gamma ({\mathrm{\Omega}}_{2})$, for all ${\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{2}\subset E$.
for all bounded $\mathrm{\Omega}\subset E$.
defined on the bounded $\mathrm{\Omega}\subset {C}^{1}([0,T],E)$, where the ordering is induced by the positive cone in ${\mathbb{R}}^{2}$ and where ${mod}_{C}(\mathrm{\Omega})$ denotes the modulus of continuity of a subset $\mathrm{\Omega}\subset C([0,T],E)$.^{a} It was proved in [19] that the function α given by (7) is an m.n.c. in ${C}^{1}([0,T],E)$ that is monotone, nonsingular and regular.
Definition 2.3 Let E be a Banach space and $X\subset E$. A multivalued mapping $F:X\u22b8E$ with compact values is called condensing with respect to an m.n.c. β (shortly, βcondensing) if, for every bounded $\mathrm{\Omega}\subset X$ such that $\beta (F(\mathrm{\Omega}))\ge \beta (\mathrm{\Omega})$, we see that Ω is relatively compact.
A family of mappings $G:X\times [0,1]\u22b8E$ with compact values is called βcondensing if, for every bounded $\mathrm{\Omega}\subset X$ such that $\beta (G(\mathrm{\Omega}\times [0,1]))\ge \beta (\mathrm{\Omega})$, we see that Ω is relatively compact.
The proof of the main result (cf. Theorem 3.1 below) will be based on the following slight modification of the continuation principle developed in [19]. Since the proof of this modified version differs from the one in [19] only slightly in technical details, we omit it here.
 (i)There exist a closed set ${S}_{1}\subset S$ and a closed, convex set $Q\subset {C}^{1}([0,T],E)$ with a nonempty interior IntQ such that each associated problem$P(q,\lambda )\phantom{\rule{1em}{0ex}}\begin{array}{l}\ddot{x}(t)\in H(t,x(t),\dot{x}(t),q(t),\dot{q}(t),\lambda ),\phantom{\rule{1em}{0ex}}\mathit{\text{for a.a.}}t\in [0,T],\\ x\in {S}_{1},\end{array}\}$
 (ii)For every nonempty, bounded set $\mathrm{\Omega}\subset E\times E\times E\times E$, there exists ${\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T],[0,\mathrm{\infty}))$ such that$\parallel H(t,x,y,u,v,\lambda )\parallel \le {\nu}_{\mathrm{\Omega}}(t),$
 (iii)
The solution mapping $\mathfrak{T}$ is quasicompact and μcondensing with respect to a monotone and nonsingular m.n.c. μ defined on ${C}^{1}([0,T],E)$.
 (iv)
For each $q\in Q$, the set of solutions of problem $P(q,0)$ is a subset of IntQ, i.e. $\mathfrak{T}(q,0)\subset IntQ$, for all $q\in Q$.
 (v)
For each $\lambda \in (0,1)$, the solution mapping $\mathfrak{T}(\cdot ,\lambda )$ has no fixed points on the boundary ∂Q of Q.
Then the b.v.p. (8) has a solution in Q.
3 Main result
Combining the foregoing continuation principle with the ScorzaDragonitype technique (cf. Proposition 2.2), we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (2).
Theorem 3.1 Consider the Dirichlet b.v.p. (2). Suppose that $F:[0,T]\times E\times E\u22b8E$ is an upperCarathéodory mapping which is either globally measurable or quasicompact. Furthermore, let $K\subset E$ be a nonempty, open, convex, bounded subset containing 0 of a separable Banach space E satisfying the RadonNikodym property. Let the following conditions (${2}_{\mathrm{i}}$)(${2}_{\mathrm{iii}}$) be satisfied:
(${2}_{\mathrm{i}}$) $\gamma (F(t,{\mathrm{\Omega}}_{1}\times {\mathrm{\Omega}}_{2}))\le g(t)(\gamma ({\mathrm{\Omega}}_{1})+\gamma ({\mathrm{\Omega}}_{2}))$, for a.a. $t\in [0,T]$ and each ${\mathrm{\Omega}}_{1}\subset \overline{K}$, and each bounded ${\mathrm{\Omega}}_{2}\subset E$, where $g\in {L}^{1}([0,T],[0,\mathrm{\infty}))$ and γ is the Hausdorff m.n.c. in E.
for a.a. $t\in [0,T]$ and all $(x,y)\in \mathrm{\Omega}\times E$.
for a.a. $t\in (0,T)$ and all $x\in \partial K$, $v\in E$, and $w\in F(t,x,v)$.
Then the Dirichlet b.v.p. (2) admits a solution whose values are located in $\overline{K}$. If, moreover, $0\notin F(t,0,0)$, for a.a. $t\in [0,T]$, then the obtained solution is nontrivial.
is well defined, continuous and bounded.
Since the mapping $(t,x,y)\u22b8F(t,x,y)$ has, according to Proposition 2.2, the ScorzaDragoni property, we are able to find a decreasing sequence ${\{{J}_{m}\}}_{m}$ of subsets of $[0,T]$ and a mapping ${F}_{0}:[0,T]\times E\times E\u22b8E\cup \{\mathrm{\varnothing}\}$ with compact, convex values such that, for all $m\in \mathbb{N}$,

$\mu ({J}_{m})<\frac{1}{m}$,

$[0,T]\setminus {J}_{m}$ is closed,

$(t,x,y)\u22b8{F}_{0}(t,x,y)$ is u.s.c. on $[0,T]\setminus {J}_{m}\times E\times E$,

${\nu}_{\overline{K}}$ is continuous in $[0,T]\setminus {J}_{m}$ (cf. e.g.[2]).
If we put $J={\bigcap}_{m=1}^{\mathrm{\infty}}{J}_{m}$, then $\mu (J)=0$, ${F}_{0}(t,x,y)\ne \mathrm{\varnothing}$, for all $t\in [0,T]\setminus J$, the mapping $(t,x,y)\u22b8{F}_{0}(t,x,y)$ is u.s.c. on $[0,T]\setminus J\times E\times E$ and ${\nu}_{\overline{K}}$ is continuous in $[0,T]\setminus J$.
Let us show that, when $m\in \mathbb{N}$ is sufficiently large, all assumptions of Proposition 2.4 (for $\phi (t,x,\dot{x}):={F}_{m}(t,x,\dot{x})$) are satisfied.
and let the set Q of candidate solutions be defined as $Q:={C}^{1}([0,T],\overline{K})$. Because of the convexity of K, the set Q is closed and convex.
and denote by ${\mathfrak{T}}_{m}$ the solution mapping which assigns to each $(q,\lambda )\in Q\times [0,1]$ the set of solutions of ${P}_{m}(q,\lambda )$.
Thus, the set of solutions of ${P}_{m}(q,\lambda )$ is nonempty. The convexity of the solution sets follows immediately from the definition of ${H}_{m}$ and the fact that problems ${P}_{m}(q,\lambda )$ are fully linearized.
Therefore, the mapping ${H}_{m}(t,q(t),\dot{q}(t),\lambda )$ satisfies condition (ii) from Proposition 2.4.
ad (iii) Since the verification of condition (iii) in Proposition 2.4 is technically the most complicated, it will be split into two parts: (iii_{1}) the quasicompactness of the solution operator ${\mathfrak{T}}_{m}$, (iii_{2}) the condensity of ${\mathfrak{T}}_{m}$ w.r.t. the monotone and nonsingular m.n.c. α defined by (7).
Since ${q}_{n}\to q$ and ${\dot{q}}_{n}\to \dot{q}$ in $C([0,T],E)$, there exists a bounded $\mathrm{\Omega}\times \mathrm{\Omega}\subset E\times E$ such that $({q}_{n}(t),{\dot{q}}_{n}(t))\in \mathrm{\Omega}\times \mathrm{\Omega}$, for all $t\in [0,T]$ and $n\in \mathbb{N}$. Therefore, there exists, according to condition (${2}_{\mathrm{ii}}$), ${\nu}_{\mathrm{\Omega}}\in {L}^{1}([0,T],[0,\mathrm{\infty}))$ such that $\parallel {f}_{n}(t)\parallel \le \varpi (t)$, for every $n\in \mathbb{N}$ and a.a. $t\in [0,T]$, where $\varpi (t):={\nu}_{\mathrm{\Omega}}(t)+2{\nu}_{\overline{K}}(t)\cdot {max}_{x\in \overline{B(\partial K,\epsilon )}}\parallel \stackrel{\u02c6}{\varphi}(x)\parallel $.
Hence, the sequences $\{{x}_{n}\}$ and $\{{\dot{x}}_{n}\}$ are bounded and $\{{\ddot{x}}_{n}\}$ is uniformly integrable.
Since ${q}_{n}\to q$ and ${\dot{q}}_{n}\to \dot{q}$ in $C([0,T],E)$, we get, for all $t\in [0,T]$, $\gamma ({\{{q}_{n}(t)\}}_{n})=\gamma ({\{{\dot{q}}_{n}(t)\}}_{n})=0$, which implies that $\gamma ({\{{f}_{n}(t)\}}_{n})=0$, for all $t\in [0,T]$.
by which $\{{x}_{n}(t)\}$, $\{{\dot{x}}_{n}(t)\}$ are relatively compact, for all $t\in [0,T]$.
Moreover, since ${x}_{n}$ satisfies for all $n\in \mathbb{N}$ (13), $\{{\ddot{x}}_{n}(t)\}$ is relatively compact, for a.a. $t\in [0,T]$. Thus, according to [[23], Lemma III.1.30], there exist a subsequence of $\{{\dot{x}}_{n}\}$, for the sake of simplicity denoted in the same way as the sequence, and $x\in {C}^{1}([0,T],E)$ such that $\{{\dot{x}}_{n}\}$ converges to $\dot{x}$ in $C([0,T],E)$ and $\{{\ddot{x}}_{n}\}$ converges weakly to $\ddot{x}$ in ${L}^{1}([0,T],E)$. According to the classical closure results (cf. e.g. [[27], Lemma 5.1.1]), $x\in {\mathfrak{T}}_{m}(q,\lambda )$, which implies the quasicompactness of ${\mathfrak{T}}_{m}$.
with $f(t)=\lambda k(t)+{\nu}_{\overline{K}}(t)({\chi}_{{J}_{m}}(t)+\frac{1}{m})\stackrel{\u02c6}{\varphi}(q(t))$, for a.a. $t\in [0,T]$.
Thus, the set ${\mathfrak{T}}_{m}(\mathrm{\Theta}\times [0,1])$ is bounded.
Moreover, we can find $\{{q}_{n}\}\subset \mathrm{\Theta}$, $\{{\lambda}_{n}\}\subset [0,1]$ and $\{{k}_{n}\}$ satisfying, for a.a. $t\in [0,T]$, ${k}_{n}(t)\in {F}_{0}(t,{q}_{n}(t),{\dot{q}}_{n}(t))$, such that, for all $t\in [0,T]$, ${x}_{n}(t)$ and ${\dot{x}}_{n}(t)$ are defined by (15) and (16), respectively, where ${f}_{n}(t)$ is defined by (14).
This implies $\parallel G(t,s){f}_{n}(t)\parallel \le G(t,s)({\nu}_{\mathrm{\Omega}}(t)+2\cdot {\nu}_{\overline{K}}(t)\cdot {max}_{x\in \overline{B(\partial K,\epsilon )}}\parallel \stackrel{\u02c6}{\varphi}(x)\parallel )$, for a.a. $t,s\in [0,T]$ and all $n\in \mathbb{N}$.
Therefore, we get, for sufficiently large $m\in \mathbb{N}$, the contradiction $\mathcal{S}<\mathcal{S}$ which ensures the validity of condition (iii) in Proposition 2.4.
where ${\phi}_{m}(t):={\nu}_{\overline{K}}(t)({\chi}_{{J}_{m}}(t)+\frac{1}{m})\stackrel{\u02c6}{\varphi}(q(t))$.
Let us now consider $r>0$ such that $rB\subset K$. Then it follows from (21) that we are able to find ${m}_{0}\in \mathbb{N}$ such that, for all $m\in \mathbb{N}$, $m\ge {m}_{0}$, and $t\in [0,T]$, $\parallel {x}_{m}\parallel \le r$. Therefore, for all $m\in \mathbb{N}$, $m\ge {m}_{0}$, ${\mathfrak{T}}_{m}(q,0)\subset IntQ$, for all $q\in Q$, which ensures the validity of condition (iv) in Proposition 2.4.

due to the Dirichlet boundary conditions, ${t}_{0}$ belongs to the open interval $(0,T)$,

since $A(t)=B(t)=0$, we have $p(t)={\nu}_{\overline{K}}(t)$.
In this way, we can prove that there exists ${m}_{0}\in \mathbb{N}$ such that every problem $({P}_{m})$, where $m\ge {m}_{0}$, satisfies all the assumptions of Proposition 2.4. This implies that every such $({P}_{m})$ admits a solution, denoted by ${x}_{m}$, with ${x}_{m}(t)\in \overline{K}$, for all $t\in [0,T]$. By similar arguments as in [16], but with the expression $Z(4Zk+1)$ replaced by $\frac{T}{4}$, according to condition (${2}_{\mathrm{ii}}$), we can obtain the result that there exists a subsequence, denoted as the sequence, and a function $x\in A{C}^{1}([0,T],E)$ such that ${x}_{m}\to x$ and ${\dot{x}}_{m}\to \dot{x}$ in $C([0,T],E)$ and also ${\ddot{x}}_{m}\rightharpoonup x$ in ${L}^{1}([0,T],E)$, when $m\to \mathrm{\infty}$. Thus, a classical closure result (see e.g. [[27], Lemma 5.1.1]) guarantees that x is a solution of (2) satisfying $x(t)\in \overline{K}$, for all $t\in [0,T]$, and the sketch of proof is so complete. □
The case when $F={F}_{1}+{F}_{2}$, with ${F}_{1}(t,\cdot ,\cdot )$ to be completely continuous and ${F}_{2}(t,\cdot ,\cdot )$ to be Lipschitzian, for a.a. $t\in [0,T]$, represents the most classical example of a map which is γregular w.r.t. the Hausdorff measure of noncompactness γ. The following corollary of Theorem 3.1 can be proved quite analogously as in [[3], Example 6.1 and Remark 6.1].
 (i)${F}_{1}:[0,T]\times H\times H\u22b8H$ is an upperCarathéodory, globally measurable, multivalued mapping and ${F}_{1}(t,\cdot ,\cdot ):H\times H\u22b8H$ is completely continuous, for a.a. $t\in [0,T]$, such that$\parallel {F}_{1}(t,x,y)\parallel \le {\nu}_{1}(t,{D}_{0}),$
 (ii)${F}_{2}:[0,T]\times H\times H\u22b8H$ is a Carathéodory multivalued mapping such that$\parallel {F}_{2}(t,0,0)\parallel \le {\nu}_{2}(t),\phantom{\rule{1em}{0ex}}\mathit{\text{for a.a.}}t\in [0,T],$
 (iii)there exists $R>0$ such that, for all $x\in H$ with $\parallel x\parallel =R$, $t\in (0,T)$, $y\in H$ and $w\in {F}_{1}(t,x,y)+{F}_{2}(t,x,y)$, we have$\u3008x,w\u3009>0.$
Then the Dirichlet problem (22) admits, according to Theorem 3.1, a solution $x(\cdot )$ such that $\parallel x(t)\parallel \le R$, for all $t\in [0,T]$.
Remark 3.1 For ${F}_{2}(t,x,y)\equiv 0$, the completely continuous mapping ${F}_{1}(t,x,y)$ allows us to make a comparison with classical singlevalued results recalled in the Introduction. Unfortunately, our ${F}_{1}$ in (i) (see also (${2}_{\mathrm{ii}}$) in Theorem 3.1) is the only mapping which is (unlike in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions quite liberal growth restrictions were permitted) globally bounded w.r.t. $y\in H$. Furthermore, our sign condition in (iii) is also (unlike again in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions the Hartmantype condition like (i_{H}) in the Introduction was employed) the most restrictive among their analogies in [6–13]. On the other hand, because of multivalued upperCarathéodory maps ${F}_{1}+{F}_{2}$ in a Hilbert space which are γregular, our result has still, as far as we know, no analogy at all.
4 Illustrative examples
involving discontinuities in a state variable. In this equation, the nonlocal diffusion term ${\int}_{\mathbb{R}}k(x,y)u(t,y)\phantom{\rule{0.2em}{0ex}}dy$ replaces the classical diffusion behavior given by ${u}_{xx}(t,x)$. In dispersal models such an integral term takes into account the longdistance interactions between individuals (see e.g. [29]). Moreover, when φ is linear in ${u}_{t}$, (23) can be considered as an alternative version of the classical telegraph equation (see e.g. [30] and the references therein), where the classical diffusivity is replaced by the present nonlocal diffusivity.
Telegraph equations appear in many fields such as modeling of an anomalous diffusion, a wave propagation phenomenon, subdiffusive systems or modeling of a pulsate blood flow in arteries (see e.g. [31, 32]).
For the sake of simplicity, we will discuss here only the case when φ is globally bounded w.r.t. ${u}_{t}$. On the other hand, for nonstrictly localized transversality conditions as in [3], for instance, a suitable linear growth estimate w.r.t. ${u}_{t}$ can be permitted.
 (a)
φ is Carathéodory, i.e. $\phi (\cdot ,x,y)$ is measurable, for all $x,y\in \mathbb{R}$, and $\phi (t,\cdot ,\cdot )$ is continuous, for a.a. $t\in [0,T]$; $\phi (t,x,\cdot )$ is $L(t)$Lipschitzian with $L\in {L}^{1}([0,T])$; $\phi (t,x,y)\le {\phi}_{1}(t){\phi}_{2}(x)$, for a.a. $t\in [0,T]$ and all $x,y\in \mathbb{R}$, where ${\phi}_{1}\in {L}^{1}([0,T])$ and ${\phi}_{2}\in {L}^{2}(\mathbb{R})$; $\phi (t,x,0)\ne 0$, for all a.a. $t\in [0,T]$ and all $x\in \mathbb{R}$,
 (b)
$b\in {L}^{1}([0,T])$ and satisfies $b(t)\ge {b}_{0}>1$, for a.a. $t\in [0,T]$,
 (c)
$k\in {L}^{2}(\mathbb{R}\times \mathbb{R})$ with ${\parallel k\parallel}_{{L}^{2}(\mathbb{R}\times \mathbb{R})}=1$,
 (d)
$p(r)\ge 0$, for all $r\in \mathbb{R}$; and there can exist ${r}_{1}<{r}_{2}<\cdots <{r}_{k}$ such that $p(\cdot )$ is continuous, for $r\ne {r}_{i}$, and $p(\cdot )$ has discontinuities at ${r}_{i}$, for $i=1,\dots ,k$, with $p({r}_{i}^{\mp}):={lim}_{r\to {r}_{i}^{\mp}}p(r)\in \mathbb{R}$,
 (e)
f is LLipschitzian; $L>0$; $f(0)=0$; and $xf(x)>0$, for all $x\ne 0$,
 (f)
$\psi \in {L}^{2}(\mathbb{R})$ with ${\parallel \psi \parallel}_{{L}^{2}(\mathbb{R})}=1$.
and if it satisfies (24).
then the problem (23), (24) has a solution, in the sense of Filippov, satisfying ${\parallel u(t,\cdot )\parallel}_{{L}^{2}(\mathbb{R})}\le R$, for a.a. $t\in [0,T]$.
where $\stackrel{\u02c6}{\phi}:[0,T]\times {L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R})$, $(t,y)\mapsto (x\mapsto \phi (t,x,y(x)))$, $K:{L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R})$, $w\mapsto (x\mapsto {\int}_{\mathbb{R}}k(x,y)w(y)\phantom{\rule{0.2em}{0ex}}dy)$, $\stackrel{\u02c6}{f}:{L}^{2}(\mathbb{R})\to {L}^{2}(\mathbb{R})$, $y\mapsto (x\mapsto f(y(x)))$ and $\stackrel{\u02c6}{F}:{L}^{2}(\mathbb{R})\u22b8{L}^{2}(\mathbb{R})$, $y\u22b8\{p\stackrel{\u02c6}{f}(y):p\in P({\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx)\}$.
Let us now examine the properties of F. According to (a), $\stackrel{\u02c6}{\phi}$ is well defined. Given $y\in {L}^{2}(\mathbb{R})$, let us show that $\stackrel{\u02c6}{\phi}(\cdot ,y)$ is measurable. For this purpose, let Ψ be an arbitrary element in the dual space ${({L}^{2}(\mathbb{R}))}^{\prime}$ of ${L}^{2}(\mathbb{R})$. Hence, there exists $\psi \in {L}^{2}(\mathbb{R})$ such that $\mathrm{\Psi}(z)={\int}_{\mathbb{R}}\psi (x)z(x)\phantom{\rule{0.2em}{0ex}}dx$, for all $z\in {L}^{2}(\mathbb{R})$, and consequently the composition $\mathrm{\Psi}\circ \stackrel{\u02c6}{\phi}(\cdot ,y):[0,T]\to \mathbb{R}$ is such that $t\to {\int}_{\mathbb{R}}\psi (x)\phi (t,x,y(x))\phantom{\rule{0.2em}{0ex}}dx$. Since φ is Carathéodory, it is globally measurable, and so the mapping $(t,x)\to \psi (x)\phi (t,x,y(x))$ is globally measurable as well. This implies that, according to the Fubini Theorem, the mapping $\mathrm{\Psi}\circ \stackrel{\u02c6}{\phi}(\cdot ,y)$ is measurable, too. Finally, since Ψ was arbitrary, according to the Pettis Theorem (see Proposition 2.1), $\stackrel{\u02c6}{\phi}(\cdot ,y)$ is measurable.
 (i)
If ${r}_{0}:={\int}_{\mathbb{R}}\psi (x){y}_{0}(x)\phantom{\rule{0.2em}{0ex}}dx\ne {r}_{i}$, $i=1,2,\dots ,k$, then it is possible to find $\delta >0$ such that $\stackrel{\u02c6}{F}:B({y}_{0},\delta )\to {L}^{2}(\mathbb{R})$ is singlevalued, i.e. $\stackrel{\u02c6}{F}(y)=p(r)\stackrel{\u02c6}{f}(y)$, $r:={\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx\in [{r}_{0}\delta ,{r}_{0}+\delta ]$, for all $y\in B({y}_{0},\delta )$ and ${r}_{i}\notin [{r}_{0}\delta ,{r}_{0}+\delta ]$, for $i=1,2,\dots ,k$. Since p is continuous in $[{r}_{0}\delta ,{r}_{0}+\delta ]$ and $\stackrel{\u02c6}{f}$ is Lipschitzian, $\stackrel{\u02c6}{F}$ is continuous in $B({y}_{0},\delta )$.
 (ii)Let ${r}_{0}={r}_{j}$, for some $j\in \{i=1,2,\dots ,k\}$ and let $U\subset {L}^{2}(\mathbb{R})$ be open and such that $\stackrel{\u02c6}{F}({y}_{0})\subset U$. Moreover, let $\sigma >0$ be such that $r:={\int}_{\mathbb{R}}\psi (x)y(x)\phantom{\rule{0.2em}{0ex}}dx\ne {r}_{i}$, $i\ne j$, for any $y\in B({y}_{0},\sigma )$. This implies that $\stackrel{\u02c6}{F}(y)$ is equal either to $p(r)\stackrel{\u02c6}{f}(y)$ or to $P({r}_{j})\stackrel{\u02c6}{f}(y)$, for all $y\in B({y}_{0},\sigma )$. If $r<{r}_{j}$ is such that $\stackrel{\u02c6}{F}(y)=p(r)\stackrel{\u02c6}{f}(y)$, then$\begin{array}{rcl}{\parallel \stackrel{\u02c6}{F}(y)p\left({r}_{j}^{}\right)\stackrel{\u02c6}{f}({y}_{0})\parallel}_{{L}^{2}(\mathbb{R})}& =& {\parallel p(r)\stackrel{\u02c6}{f}(y)p\left({r}_{j}^{}\right)\stackrel{\u02c6}{f}({y}_{0})\parallel}_{{L}^{2}(\mathbb{R})}\\ \le & p(r)p\left({r}_{j}^{}\right)\cdot \parallel \stackrel{\u02c6}{f}(y)\parallel +p\left({r}_{j}^{}\right)\cdot \parallel \stackrel{\u02c6}{f}(y)\stackrel{\u02c6}{f}({y}_{0})\parallel ,\end{array}$
which implies that it is possible to find ${\sigma}_{1}>0$ such that $F(y)\subset U$, for all $y\in B({y}_{0},{\sigma}_{1})$. Similarly, we would obtain the same when assuming $r>{r}_{j}$.
which implies that also in this case it is possible to find ${\sigma}_{2}>0$ such that $F(y)\subset U$, for all $y\in B({y}_{0},{\sigma}_{2})$.
Moreover, according to (a) and (c), $\stackrel{\u02c6}{\phi}$ is a Carathéodory mapping such that $\stackrel{\u02c6}{\phi}(t,\cdot )$ is $L(t)$Lipschitzian, for all $t\in [0,T]$, and K is well defined and 1Lipschitzian. It can also be shown that, according to (d) and (e), $\stackrel{\u02c6}{F}$ has compact and convex values. Therefore, the mapping F is globally measurable, and so has the ScorzaDragoni property (cf. Proposition 2.2).
Let us now verify particular assumptions of Theorem 3.1.
where m is defined by (28).
and so condition (${2}_{\mathrm{i}}$) is satisfied with $g(t)=L(t)+b(t)+1+m\cdot L$. The obtained form of $g(t)$ together with assumption (27) directly guarantee the condition (${2}_{\mathrm{iii}}$). It can also be easily shown that properties of F ensure the validity of condition (${2}_{\mathrm{ii}}$).
in view of condition (26), (30), and (31).
Hence, the Dirichlet problem (29) admits, according to Theorem 3.1, a solution y satisfying ${\parallel y(t)\parallel}_{{L}^{2}(\mathbb{R})}\le R$, for a.a. $t\in (0,T)$. If $u(t,x):=y(t)(x)$, then u is a solution of (24), (25) which is the Filippov solution of the original problem (23), (24).
Finally, we can sum up the above result in the form of the following theorem.
Theorem 4.1 Let the assumptions (a)(f) be satisfied. If still conditions (26), (27) hold, then the problem (23), (24) admits a nontrivial solution u in the sense of Fillippov such that ${\parallel u(t,\cdot )\parallel}_{{L}^{2}(\mathbb{R})}\le R$.
was solved provided $\tilde{k}:[0,1]\times [0,1]\to (0,\mathrm{\infty})$ is a positive kernel of the HilbertSchmidttype and the norms ${\parallel {u}_{0}\parallel}_{{L}^{2}([0,1])}$ and ${\parallel {u}_{1}\parallel}_{{L}^{2}([0,1])}$ are finite.
where $\tilde{\phi}(t,x):={\int}_{0}^{1}\tilde{k}(x,y)\{[{u}_{1}(y){u}_{0}(y)]t+{u}_{0}(y)\}\phantom{\rule{0.2em}{0ex}}dy$.
and $b(t)\equiv 0$, $p(r)\equiv 0$ or $f(s)\equiv 0$.
The result in [[13], Example 5.2] cannot be, however, deduced from Theorem 3.1, because condition (b) in Example 4.1 cannot be satisfied in this way.
On the other hand, the linear term with coefficient b could not be implemented in their equation, because it is not completely continuous in (36) below, as required in [13].
In view of the arguments in Remark 4.1, we can conclude by the second illustrative example.
and $b\in {L}^{1}((0,1))$ is such that $b(t)\ge {b}_{0}>0$, for a.a. $t\in (0,1)$.
We will show that, under (33) and (34), problem (32) is solvable, in the abstract setting, by means of Corollary 3.1.
with $w(t,x):={\int}_{0}^{1}\tilde{k}(x,y)\{[{u}_{1}(y){u}_{0}(y)]t+{u}_{0}(y)\}\phantom{\rule{0.2em}{0ex}}dy$.
as claimed.
After all, we can sum up the sufficient conditions for the existence of a solution $\stackrel{\u02c6}{u}$ of (32) satisfying (38) as follows:

$\tilde{k}$ is a positive kernel of the HilbertSchmidt operator with the finite norm${k}_{0}:={\parallel \tilde{k}\parallel}_{{L}^{2}([0,1]\times [0,1])}<\mathrm{\infty},$

there exists ${b}_{0}>0$, $L<\frac{4}{5}$: ${b}_{0}\le b(t)\le L$, for a.a. $t\in (0,1)$,

condition (37) holds.
Endnotes
^{a}The m.n.c. ${mod}_{C}(\mathrm{\Omega})$ is a monotone, nonsingular and algebraically subadditive on $C([0,T],E)$ (cf. e.g. [27]) and it is equal to zero if and only if all the elements $x\in \mathrm{\Omega}$ are equicontinuous.
^{b}Since a ${C}^{2}$function V has only a locally Lipschitzian Fréchet derivative $\dot{V}$ (cf. e.g. [21]), we had to assume explicitly the global Lipschitzianity of $\dot{V}$ in a noncompact set $\overline{B(\partial K,\epsilon )}$.
Declarations
Acknowledgements
The first and third authors were supported by grant ‘Singularities and impulses in boundary value problems for nonlinear ordinary differential equations’. The second author was supported by the national research project PRIN ‘Ordinary Differential Equations and Applications’.
Authors’ Affiliations
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