Structure of solution sets to the nonlocal problems
 Yi Cheng^{1}Email author,
 Ben Niu^{1} and
 Cuiying Li^{2}
Received: 15 October 2015
Accepted: 12 January 2016
Published: 28 January 2016
Abstract
This paper deals with the structural properties of the solution set for a class of nonlinear evolution inclusions with nonlocal conditions. For the nonlocal problems with a convexvalued righthand side it is proved that the solution set is compact \(R_{\delta}\); it is the intersection of a decreasing sequence of nonempty compact absolute retracts. Then for the cases of a nonconvexvalued perturbation term it is proved that the solution set is path connected. Finally some examples of nonlinear parabolic problems are given.
Keywords
MSC
1 Introduction
In this paper we first of all prove that the solution set of nonlinear timedependent evolution inclusions with a convexvalued righthand side is compact \(R_{\delta}\); it is the intersection of a decreasing sequence of nonempty compact absolute retracts in \(C(I,H)\). Second, we go further and show that the solution set is pathconnected in \(C(I,H)\) for the case of a nonconvexvalued orientor field. Finally, some examples are also given to illustrate the effectiveness of our results. In particular, control systems given in this paper with a prior feedback, and systems with discontinuities, have a builtin multivalued character which is modeled appropriately by evolution inclusions.
2 Preliminaries
Definition 2.1
A multifunction H: \(\Omega\rightarrow P_{f}(\mathcal{X})\) is called ‘measurable’, if for all \(y\in\mathcal{X}\), \(\mathbb{R}_{+}\)valued function \(x\rightarrow d(y,H(x))=\inf\{\Vert yv\Vert ,v\in H(x)\}\) is measurable.
Definition 2.2
Let Z be a separable Banach space, a multifunction \(H: \mathcal{X}\rightarrow2^{Z}\setminus\emptyset\) is called ‘hupper semicontinuous’ (husc) if, for every \(x\in\mathcal{X}\), the function \(x'\rightarrow d_{H}(H(x'),H(x))\) is continuous.
Let Z be a complete metric space, also \(H: \mathcal{X}\rightarrow P_{f}(Z)\) is called ‘hcontinuous’ (resp. ‘hLipschitz’) if it is continuous (resp. Lipschitz) as a function from Z into \(P_{f}(Z,d_{H})\). For details refer to [18].
Let (\(V,H,V^{*}\)) be an evolution triple where the embedding \(V\rightarrow H\rightarrow V^{*}\) is compact. Let \(\langle\cdot,\cdot\rangle_{*}\) denote the pairing of an element \(x\in{V^{*}}\) and an element \(y\in{V}\). Let \(\langle\cdot \cdot\rangle \) be the inner product on H, then \(\langle\cdot,\cdot\rangle_{*}=\langle\cdot \cdot\rangle\), if \(x,y\in {H}\). The dual space of \(L^{p}(I,V)\) is \(L^{q}(I,V^{*})\) where \(1< q\leq p<\infty\), \(\frac{1}{p}+\frac{1}{q}=1\), and I is an interval in \(\mathbb{R}\). The norm in Banach space \(L^{p}(I,V)\) will be denoted by \(\Vert \cdot \Vert _{L^{p}(I,V)}\). Due to the reflexivity of V, both \(L^{p}(I,V)\) and \(L^{q}(I,V^{*})\) are reflexive Banach spaces (see Zeidler [17], p.411).
We define a Banach space \(\mathcal{W}_{pq}(I)=\{x:x\in L^{p}(I,V),\dot{x}\in L^{q}(I,V^{*})\}\) furnished with the norm \(\Vert x\Vert _{\mathcal{W}}=\Vert x\Vert _{L^{p}(I,V)}+\Vert \dot{x}\Vert _{L^{q}(I,V^{*})}\). The pairing between \(L^{p}(I,V)\) and \(L^{q}(I,V^{*})\) is denoted by \(\langle \cdot,\cdot\rangle_{**}\).
By \(S_{G}^{p}\) we denote the set of all \(L^{p}(I,H)\)selectors of a multifunction G, i.e. \(S^{p}_{G}=\{f\in L^{p}(I,H): f(x)\in G(x)\mbox{ a.e. for }x\in I\}\). We say that the set \(S_{G}^{p}\) is decomposable if \(\chi_{A}f_{1}+\chi _{A^{c}}f_{2}\in S_{G}^{p}\) where \((f_{1},f_{2},A)\in S_{G}^{p}\times S_{G}^{p}\times\Sigma\).
We recall some of the topological concepts which will be used to characterize the solution set of the evolution inclusion.
Definition 2.3
A subset \(\mathcal{A}\subset\mathcal{X}\) is called ‘path connected’, if for every \(x,y\subset \mathcal{A}\), there exists a path \(h: [0,1] \rightarrow \mathcal{A}\) which joins x to y.
For \(\mathcal{A}\subset\mathcal{X}\) nonempty, we claim that \(\mathcal{A}\) is a retract of \(\mathcal{X}\), if there exists a continuous map \(f : \mathcal{X} \rightarrow\mathcal{A}\) such that \(f\mid_{\mathcal{A}}= \mathrm{identity}\). It is clear to see that a retract \(\mathcal{A}\subset\mathcal{X}\) is closed.
Definition 2.4
A closed subset \(\mathcal{A}\) of \(\mathcal{X}\) is called an absolute retract, if for any closed subset C in every metric space Y, every continuous map \(f:C\rightarrow\mathcal{A}\) can be extended to be a continuous function, \(\widehat{f}: Y\rightarrow\mathcal{A}\).
Definition 2.5
A subset \(\mathcal{A}\) of a metric space \(\mathcal{X}\) is said to be contractible, if and only if there exist a continuous function \(g:[0,1]\times\mathcal{A}\rightarrow\mathcal{X}\) and a point \(a\in \mathcal{A}\) such that for all \(x\in\mathcal{A}\) we have \(g(0,x)=a\) and \(g(1,x)=x\).
A subset \(\mathcal{A}\) of a metric space \(\mathcal{X}\) is said to be simply connected if every closed path in it is contractible to a point. A contractible set is both simply connected and path connected.
Definition 2.6
Lemma 2.1
If \(F:I\times H\rightarrow P_{fc}(V^{*})\) is measurable in t, husc in x, and \(\vert F(t,x)\vert \leq\psi(t)\) for almost all \(t\in I\) with \(\psi(t)\in L^{q}(I)\), then there exists a sequence of multifunctions \(F_{n}: I\times H\rightarrow P_{fc}(V^{*})\), \(n\geq1\), such that for every \(t\in I\) there exist \(\mu(t)> 0\) and \(\epsilon> 0\) such that if \(x_{1}, x_{2}\in B_{\epsilon}(x)=\{y\in H:\Vert yx\Vert _{H}\leq\epsilon\}\), then \(d_{H}(F_{n}(t, x_{1}), F_{n}(t, x_{2}))\leq\mu(t)\psi(t)\Vert x_{1}x_{2}\Vert _{H}\) for almost all \(t\in I\) (i.e., \(F_{n}(t,x)\) is locally hLipschitz), \(F(t,x)\subseteq\cdots\subseteq F_{n}(t,x)\subseteq F_{n+1}(t,x) \subseteq \cdots, \vert F_{n}(t,x)\vert \leq\psi(t)\), \(n\in\mathbb{N}\), \(F_{n}(t, x)\longrightarrow F(t, x)\) as \(n\longrightarrow\infty\) for every \((t,x)\in I\times H\), and there exists \(u_{n}: I\times H\rightarrow H\), measurable in t, locally Lipschitz in x (as \(F_{n}(t,x)\)) and \(u_{n}(t,x)\in F_{n}(t, x)\) for every \((t,x)\in I\times H\). Moreover, if \(F(t,\cdot)\) is hcontinuous, then \(t\rightarrow F_{n}(t, x)\) is measurable (hence \((t,x)\rightarrow F_{n}(t,x)\) is measurable too; see [19]).
3 Main results
 (i)
\(\tau(x)=x(T)\);
 (ii)
\(\tau(x)=x(T)\);
 (iii)
\(\tau(x)=\frac{1}{T}\int_{0}^{T} x(t)\,dt\);
 (iv)
\(\tau(x)=\sum_{i=1}^{n}\theta_{i}x(t_{i})\), where \(0\leq t_{1}<\cdots<t_{n}\leq T\) are arbitrary, but fixed and \(\sum_{i=1}^{n}\vert \theta _{i}\vert \leq1\).
 (H1)\(B:I\times V\rightarrow{V^{*}}\) is an operator such that
 (i)
\(t\rightarrow B(t,u)\) is measurable;
 (ii)for almost all \(t\in I\), there exists a constant \(C_{1}>0\) such thatfor all \(u_{1},u_{2}\in V\), and the map \(s\mapsto\langle B(t,u+sz),y\rangle_{*}\) is continuous on \([0,1]\) for all \(u,y,z\in V\), and \(p>1\);$$\bigl\langle B(t,u_{1})B(t,u_{2}),u_{1}u_{2} \bigr\rangle _{*}\geq C_{1}\Vert u_{1}u_{2}\Vert _{H}^{p} $$
 (iii)
there exist a constant \(C_{2}>0\), a function \(a(\cdot)\in L^{q}_{+}(I)\) where q is the conjugate of \(p>1\), and a nondecreasing continuous function \(\alpha(\cdot)\) such that \(\Vert B(t,u)\Vert _{V^{*}}\leq a(t)+C_{2}\alpha(\Vert u\Vert _{V})\) for all \(u\in V\), a.e. for \(t\in I\);
 (iv)there exist \(C_{3}>0\), \(C_{4}>0\), \(b(\cdot)\in L^{1}(t)\) such thatand$$\begin{aligned} \bigl\langle B(t,u),u\bigr\rangle _{*}\geq{}& C_{3}\Vert u\Vert _{V}^{p}C_{4}\Vert u\Vert _{V}^{p1}\\ &{}+\frac{1}{2T}\bigl\Vert u(0)\bigr\Vert ^{2}b(t) \quad \mbox{a.e. }t\in I, \forall u\in V, 1< p\leq2, \end{aligned}$$$$\bigl\langle B(t,u),u\bigr\rangle _{*}\geq C_{3}\Vert u\Vert _{V}^{p}C_{4}\Vert u\Vert _{V}^{p1}b(t) \quad \mbox{a.e. }t\in I, \forall u\in V, p>2. $$
 (i)
 (H2)
 (i)
\(D: V\rightarrow V^{*}\) is a bounded linear selfadjoint operator such that \(\langle Du,u\rangle_{*}\geq0\), for all \(u\in V\), for almost all \(t\in I\);
 (ii)there exists a continuous function \(\tau:C(I,H)\rightarrow H\) such thatand \(\tau(0)=0\).$$\bigl\Vert \tau(u_{1})\tau(u_{2})\bigr\Vert _{H}\leq \Vert u_{1}u_{2}\Vert _{C(I,H)}, \quad \forall u_{1},u_{2}\in C(I,H), $$
 (i)
Let us give a proposition which is essential for our results.
Proposition 3.1
If hypotheses (H1) and (H2) hold, then \(P:L^{q}(I,H)_{w}\rightarrow C(I,H)\) is sequentially continuous.
Proof
 (H3)\(\mathcal{F}:I\times H\rightarrow2^{H}\) is a multifunction with closed and convex values such that
 (i)
\((t,u)\rightarrow\mathcal{F}(t,u)\) is graph measurable;
 (ii)
for almost all \(t\in I\), \(u\rightarrow\mathcal{F}(t,u)\) has a closed graph;
 (iii)there exist a function \(b_{1}(\cdot)\in L_{+}^{q}(I)\) and a constant \(C_{5}>0\) such thatwhere \(1\leq k< p\).$$\begin{aligned} \bigl\vert \mathcal{F}(t,u)\bigr\vert &=\sup\bigl\{ \Vert f\Vert _{H}:f\in\mathcal{F}(t,u)\bigr\} \\ &\leq b_{1}(t)+C_{5} \Vert u\Vert _{H}^{k1},\quad \forall x\in H,\mbox{ a.e. for } t \in I, \end{aligned}$$
 (i)
Theorem 3.1
Under assumptions (H1)(H3), S is an \(R_{\delta}\) set in \(C(I,H)\).
Proof
Step 1. This set \(S_{n}\) is contractible.
To prove that \(S_{n}\) is contractible, we first note that \(\mu :[0,1]\times S_{n}\rightarrow S_{n}\) and \(\mu(0,u)(t)=\widehat{u}(t)\) and \(\mu(1,u)(t)=u(t)\) for every \(u\in S_{n}\). Next, it remains to show that \(\mu(\delta,u)(t)\) is continuous in \([0,1]\times C(I,H)\). To this aim, let \((\delta_{m},u_{m})\longrightarrow(\delta,u)\) in \([0,1]\times S_{n}\). Next, we will distinguish two distinct cases to proceed.
In fact, we can always get a subsequence of \(\{\delta_{m}\}_{m\geq1}\) conforming to I or II. Thus we have established the continuity of \(\mu(\delta,u)\). Therefore, for every \(n\geq1\), \(S_{n}\subseteq C(I,H) \) is compact and contractible.
Step 2. \(S=\bigcap_{n\geq1}S_{n}\).
Evidently, \(S\subseteq\bigcap_{n\geq1}S_{n}\). Let \(u\in\bigcap_{n\geq1}S_{n}\). Then through definition \(u=P(f_{n})\), where \(f_{n}\in S^{q}_{\mathcal{F}_{n}(\cdot,u_{n}(\cdot))}\) (the set of all \(L^{q}(I,H)\)selectors of \(\mathcal{F}_{n}\)), \(n\geq1\). By passing to a subsequence if necessary, we may assume that \(f_{n}\longrightarrow f\) weakly in \(L^{q}(I,H)\). Then \(f\in S^{q}_{\mathcal{F}(\cdot,u(\cdot))}\) (see Theorem 3.2 of [1]). So \(u=P(f)\) with \(f\in S^{q}_{\mathcal{F}(\cdot,u(\cdot))}\) from which we conclude that \(S=\bigcap_{n\geq1}S_{n}\). Finally, by Steps 1 and 2, Hyman’s result [22] implies that S is an \(R_{\delta}\) set in \(C(I,H)\). □
An immediate conclusion of Theorem 3.1 is the following result for the multivalued problem (3.1).
Remark 3.1
Assume (H1)(H3), then for every \(t\in I\), \(S(t)=\{x(t)x\in S\}\) (the reachable set at time \(t\in I\)) is compact and connected in H.
We can establish an analogous result for the topological structure of a nonconvex evolution inclusion (i.e., \(\mathcal{F}(t,u)\) has nonconvex values) provided we strengthen our assumption on the continuity of \(\mathcal{F}(t,u)\). In fact, in this case we can see that the solution set is path connected.
 (H4):

\(\mathcal{F}:I\times H\rightarrow P_{f}(H)\) is a multifunction such that
 (i)
\(t\rightarrow\mathcal{F}(t,u)\) is measurable;
 (ii)
for every \(u_{1},u_{2}\in H\), \(h(\mathcal{F}(t,u_{1}),\mathcal{F}(t,u_{2}))\leq z(t)\Vert u_{1}u_{2}\Vert _{H}\) a.e. with \(z(t)\in L_{+}^{q}(I)\);
 (iii)there exist a function \(b_{2}(\cdot)\in L_{+}^{q}(I)\) and a constant \(C_{5}>0\) such thatwith \(1\leq k< p\).$$\bigl\vert \mathcal{F}(t,u)\bigr\vert =\sup\bigl\{ \Vert f\Vert _{H}:f\in\mathcal{F}(t,u)\bigr\} \leq b_{2}(t)+C_{5} \Vert u\Vert _{H}^{k1} \quad \forall u\in V\mbox{ a.e. for } t\in I $$
 (i)
 (H2)_{1} :

 (i)
\(D: V\rightarrow V^{*}\) is a bounded linear selfadjoint operator such that \(\langle Du,u\rangle_{*}\geq0\), for all \(u\in V\), a.e. for \(t\in I\);
 (ii)there exists a continuous function \(\tau:C(I,H)\rightarrow H\) such thatwhere \(\mathcal{L}\in[0,1)\), \(\tau(0)=0\).$$\bigl\Vert \tau(u_{1})\tau(u_{2})\bigr\Vert _{H}\leq\mathcal{L}\Vert u_{1}u_{2}\Vert _{C(I,H)}, \quad \forall u_{1},u_{2}\in C(I,H), $$
 (i)
Theorem 3.2
If hypotheses (H1), (H2)_{1}, and (H4) are satisfied, then \(S\subseteq C(I, H)\) is nonempty and path connected.
Proof
Remark 3.2
In [25], it is proved that the set of extremal solutions of a differential inclusion in \({\mathbb{R}}^{N}\) is path connected based on the Baire category method.
4 Examples
 (H5)\(g:I\times\Omega\times{\mathbb{R}}\rightarrow{\mathbb {R}}\) is a function such that
 (i)
if \(u:I\times\Omega\rightarrow{\mathbb{R}}\) is a measurable function, then so are \((t,x)\rightarrow g_{1}(t,x,u(t,x)), g_{2}(t,x,u(t,x))\);
 (ii)for almost all \((t,x)\in I\times\Omega\) and all \(u\in{\mathbb {R}}\), there exist \(a_{2}(t,x)\in L_{+}^{q}(I,L^{2}(\Omega))\) and \(\widehat {c}(x)\in L^{\infty}(\Omega)\) such thatwith \(1\leq k< p\).$$\bigl\vert g(t,x,u)\bigr\vert \leq a_{2}(t,x)+\widehat{c}(x) \vert u\vert ^{k1}, $$
 (i)
Theorem 4.1
If hypothesis (H5) is satisfied, then the solution set S of (4.1) is nonempty and compact \(R_{\delta}\) in \(C(I,L^{2}(\Omega))\). Moreover, it is compact and connected in \(C(I,L^{2}(\Omega))\).
 (H6)\(B_{k}(k=1,2,\ldots,N):I\times{\Omega}\times {\mathbb{R}}\times{\mathbb{R}}^{N}\rightarrow{\mathbb{R}}\) are functions such that
 (i)
\((t,x)\rightarrow B_{k}(t,x,u,\xi)\) is measurable on \(I\times{\Omega }\) for every \((u,\xi)\in{\mathbb{R}}\times{\mathbb{R}}^{N}\), \((u,\xi )\rightarrow B_{k}(t,x,u,\xi)\) is continuous on \({\mathbb{R}}\times{\mathbb{R}}^{N}\) for almost all \((t,x)\in I\times{\Omega}\);
 (ii)
\(\vert B_{k}(t,x,u,\xi)\vert \leq \hat{\alpha}_{1}(t,x)+\hat{c}_{1}(x)(\vert u\vert +\vert \xi \vert )\) with a function \(\hat {\alpha}_{1}\in L^{2}_{+}(I,L^{2}(\Omega))\) and \(\hat{c}_{1}(z)\in L^{\infty }(\Omega)\) for almost all \(t\in I\);
 (iii)
\(\sum_{k=1}^{N}(B_{k}(t,x,u,\xi)B_{k}(t,x,u,\xi'))(\xi_{k}\xi _{k}')\geq \vert \xi\xi'\vert ^{2} \) for almost all \(t\in I\);
 (iv)
\(B_{k}(t,x,0,0)=0\) for all \((t,x)\in I\times{\Omega}\).
 (i)
 (H7)The functions \(f:I\times\Omega\times{\mathbb {R}}\rightarrow{\mathbb{R}}\) satisfies
 (i)
for all \(u\in{\mathbb{R}}\), \((t,x)\rightarrow f(t,x,u)\) is measurable;
 (ii)
for all \((t,x)\in I\times\Omega\), \(u\rightarrow f(t,x,u)\) is continuous;
 (iii)for almost all \((t,x)\in I\times\Omega\) and all \(u\in{\mathbb {R}}\), we havewhere \(\eta_{1}\in L^{2}(I,L^{2}(\Omega))\), \(\eta_{2}\in L_{+}^{\infty}(\Omega)\).$$\bigl\vert f(t,x,u)\bigr\vert \leq\eta_{1}(t,x)+ \eta_{2}(x)\vert u\vert , $$
 (i)
 (H8)\(U:I\times\Omega\times{{\mathbb{R}}}\rightarrow P_{f}({\mathbb{R}})\) is a multifunction such that
 (i)
for all \(u\in{\mathbb{R}}\), \((t,x)\rightarrow U(t,x,u)\) is measurable;
 (ii)
for almost all \((t,x)\in I\times\Omega\) and all \(u_{1},u_{2}\in L^{2}(\Omega)\), \(d_{H}(U(t,x,u_{1}),U(t,x,u_{2}))\leq z(t)\Vert u_{1}u_{2}\Vert _{L^{2}}\) with \(z(t)\in L_{+}^{1}(I)\);
 (iii)
for almost all \((t,x)\in I\times\Omega\) and all \(u\in{\mathbb{R}}\), \(\vert U(t,x,u)\vert \leq M\), where \(M>0\).
 (i)
By applying Theorem 3.2 on problem (4.4), we obtain the following theorem.
Theorem 4.2
If hypotheses (H6)(H8) are satisfied, then the solution set S of (4.3) is nonempty and path connected in \(C(I,L^{2}(\Omega))\).
Declarations
Acknowledgements
This work is partially supported by National Natural Science Foundation of China (No.11401042, 10902125, 61304054) and the Program for Liaoning Provincial Excellent Talents in University, China (LJQ2014122). The authors would like to express their sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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