Scorza-Dragoni approach to Dirichlet problem in Banach spaces

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, Hartman-type 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 right-hand side is a multivalued upper-Carathéodory mapping which is γ-regular w.r.t. the Hausdorff measure of non-compactness γ.

The main aim will be two-fold: (i) strict localization of sign conditions on the boundaries of bound sets by means of a technique originated by Scorza-Dragoni [2], and (ii) the application of the obtained abstract result (see Theorem 3.1 below) to an integro-differential 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 right-hand sides and partly improve those for upper-Carathéodory right-hand 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 Scorza-Dragoni [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 $〈f(t,x,y),x〉>0$, 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 $〈f(t,x,y),x〉+{\parallel y\parallel }^2>0$, for $t\in [0,1]$ and $(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ such that $\parallel x\parallel =R$ and $〈x,y〉=0$,

(i_H) the well-known Bernstein-Nagumo-Hartman condition (for its definition and more details, see e.g. [1, 14]).

(i_LY) $〈f(t,x,y),x〉+{\parallel y\parallel }^2\ge -K_1(1+\parallel x\parallel +〈x,y〉)+K_2\parallel y\parallel$,

but for $t\in [0,1]$, $(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$, and replaced (ii_growth) by the Bernstein-Nagumo-Hartman 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 Bernstein-Nagumo-Hartman 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) $〈f(t,x,y),n_x〉\ge I_x(y)$, $(t,x,y)\in [0,1]\times T\partial D\times {\mathbb{R}}^n$, with $〈n_x,y〉=0$,

where $n_x$ is the outer-pointing 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) $〈f(t,x,y),n_x〉>0$, for $(t,x,y)\in [0,1]\times {\mathbb{R}}^n\times {\mathbb{R}}^n$ with $x\in \partial D$ and $〈n_x,y〉=0$,

which is another well-known 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) $〈f(t,x,y),x〉\ge -(a{\parallel x\parallel }^2+b\parallel x\parallel \parallel y\parallel +c\parallel x\parallel )$, $(t,x,y)\in [0,1]\times H\times H$,

and (ii_growth) by an appropriate version of the Bernstein-Nagumo-Hartman 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 non-strict one. More concretely, if there exists a convex, open, bounded subset $D\subset E$ of E with $0\in D$ such that

(i_ST) $〈f(t,x,y),n_x〉\ge 0$, for $(t,x,y)\in [0,1]\times E\times E$, with $x\in \partial D$ and $〈n_x,y〉=0$,

where $〈\cdot ,\cdot 〉$ denotes this time the pairing between E and its dual $E^{\prime }$, jointly with the appropriate Bernstein-Nagumo-Hartman 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 non-strict 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 non-compact maps (for more details, see [20]).

In this light, we would like to modify in the present paper the Hartman-type conditions (i_sign), (ii_growth) at least in the following way:

where $F:[0,T]\times E\times E⊸E$ is an upper-Carathéodory 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 σ-algebra in $[0,T]$ and the Borel σ-algebra 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 Scorza-Dragoni-type result developed in [22]. The approximating problems will be treated by means of the continuation principle developed in [19].

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.693-701]. 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 Newton-Leibniz formula) holds (see e.g. [[21], pp.695-696], [[23], pp.243-244]). 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 sup-norm.

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 $〈\cdot ,\cdot 〉$ 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 $\mathrm{\Phi }(x)=:〈\mathrm{\Phi },x〉$.

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 σ-algebra 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 $F:X⊸Y$) if, for every $x\in X$, a non-empty 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 $F:X⊸Y$ 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 $F:J⊸Y$, 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 σ-algebra of subsets of J.

A multivalued mapping $F:X⊸Y$ 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 quasi-compact if it maps compact sets onto relatively compact sets.

Let $J\subset \mathbb{R}$ be a given compact interval. A multivalued mapping $F:J\times X⊸Y$, where Y is a separable Banach space, is called an upper-Carathéodory mapping if the map $F(\cdot ,x):J⊸Y$ is measurable, for all $x\in X$, the map $F(t,\cdot ):X⊸Y$ 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 Scorza-Dragoni-type result developed in [22] (cf. also [25]).

For more details concerning multivalued analysis, see e.g. [23, 26, 27].

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 $F:[0,T]\times E\times E⊸E$ be an upper-Carathéodory mapping. If F is globally measurable or quasi-compact, then F has the Scorza-Dragoni property.

Proposition 2.3 (cf. [[18], Theorem 2.2])

Let E be a Banach space and $K\subset E$ a non-empty, 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

Then there exist $k\in (0,\epsilon ]$ and a bounded Lipschitzian function $\varphi :\overline{B(\partial K,k)}\to E$ such that $〈{\dot{V}}_x,\varphi (x)〉=1$, 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 non-empty bounded subsets of E. A function $\beta :P(E)\to N$ is called a measure of non-compactness (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.

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, non-singular and regular.

Definition 2.3 Let E be a Banach space and $X\subset E$. A multivalued mapping $F:X⊸E$ with compact values is called condensing with respect to an m.n.c. β (shortly, β-condensing) 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 $G:X\times [0,1]⊸E$ with compact values is called β-condensing 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.

Then the b.v.p. (8) has a solution in Q.

Combining the foregoing continuation principle with the Scorza-Dragoni-type 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 $F:[0,T]\times E\times E⊸E$ is an upper-Carathéodory mapping which is either globally measurable or quasi-compact. Furthermore, let $K\subset E$ be a non-empty, open, convex, bounded subset containing 0 of a separable Banach space E satisfying the Radon-Nikodym 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 non-trivial.

is well defined, continuous and bounded.

Since the mapping $(t,x,y)⊸F(t,x,y)$ has, according to Proposition 2.2, the Scorza-Dragoni property, we are able to find a decreasing sequence ${\{J_m\}}_m$ of subsets of $[0,T]$ and a mapping $F_0:[0,T]\times E\times E⊸E\cup \{\mathrm{\varnothing }\}$ with compact, convex values such that, for all $m\in \mathbb{N}$,

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 $(t,x,y)⊸F_0(t,x,y)$ 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 non-empty. 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: (iii₁) the quasi-compactness of the solution operator ${\mathfrak{T}}_m$, (iii₂) the condensity of ${\mathfrak{T}}_m$ w.r.t. the monotone and non-singular 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 $\varpi (t):={\nu }_{\mathrm{\Omega }}(t)+2{\nu }_{\overline{K}}(t)\cdot {max}_{x\in \overline{B(\partial K,\epsilon )}}\parallel \hat{\varphi }(x)\parallel$.

Hence, the sequences $\{x_n\}$ and $\{{\dot{x}}_n\}$ are bounded and $\{{\ddot{x}}_n\}$ is uniformly integrable.