It was well known that Pang and Stewart introduced and studied differential variational inequality in a finite-dimensional Euclidean space (see [1]). Recently, the existence of solutions for different types of differential variational inequalities problems (see [2-7]) is considered by many authors.

In this paper, we introduce a class of differential mixed equilibrium problem. Under various conditions, we obtain the existence theorem concerned with the mild solutions for this class of problems.

Now we introduce some preliminaries which will be used in the paper. For any nonempty set $ Y $, $ P(Y) $ denotes the family of all nonempty subsets of $ Y $. We denote

$ K(Y): = \{D\in P(Y)|D\ {\rm is\ compact}\}, $

$ K_{v}(Y): = \{D \in P(Y)|D\ {\rm is\ compact\ and\ convex}\}. $

Lemma 1.1 [8]  (Fan-KKM) Let $ K $ be a nonempty subset of a Hausdorff topological vector space $ E_{1} $ and let $ G:K\rightarrow P(E_{1}) $ be a set-valued mapping with the properties:

(ⅰ) $ G $ is a KKM mapping;

(ⅱ) $ G(v) $ is closed in $ E_{1} $ for every $ v\in K $;

(ⅲ) $ G(v_{0}) $ is compact in $ E_{1} $ for some $ v_{0} \in K $.

Then one has $ \bigcap\limits_{v\in K}G(v)\neq \varnothing $.

Lemma 1.2 [9]  Let $ K $ be a nonempty compact and convex subset of a Banach space $ X $, let $ \varphi:K\times K\rightarrow \mathbb{R} $ be a mapping. Suppose the following conditions hold:

(ⅰ) $ x\mapsto \varphi(x, y) $ is lower semicontinuous for every $ y \in K $;

(ⅱ) $ y\mapsto -\varphi(x, y) $ is convex for every $ x \in K $;

(ⅲ) $ \varphi(y, y)\leq 0 $ for every $ y \in K $.

Then there exists $ \overline{x}\in K $ such that $ \varphi(\overline{x}, y)\leq 0 $ for every $ y\in K $.

Definition 1.1  Let $ E _{1} $ be a topological vector space and let $ P $ be a pointed convex cone in $ E _{1} $. $ \preceq $ is a partial order relation on $ E _{1} $: $ x\preceq y $ if and only if $ y-x \in P $. A mapping $ Q:E _{1}\rightarrow \mathbb{R} $ is said to be order weak monotone increasing if for each $ x_{1}, x_{2} \in E _{1} $ satisfying $ x_{1}\preceq x_{2} $, it holds $ Q(x_{2}-x_{1})>0. $

Lemma 1.3 [10]  Let $ F:X\rightarrow P(Y) $ be a set-valued mapping, with $ X $ and $ Y $ be topological spaces. The statements below are equivalent:

(ⅰ) $ F $ is upper semicontinuous;

(ⅱ) for every closed set $ C\subset Y $, the set $ F^{-1}(C) $ is closed in $ X $, where

$ \begin{equation*} F^{-1}(C): = \{x \in X:F(x)\cap C\neq \emptyset \}; \end{equation*} $

(ⅲ) for every open set $ O\subset Y $, the set $ F^{+1}(O) $ is open in $ X $, where

$ \begin{equation*} F^{+1}(O): = \{x \in X:F(x)\subset O\}. \end{equation*} $

(ⅰ') $ F $ is lower semicontinuous;

(ⅱ') for every closed set $ C\subset Y $, the set $ F^{+1}(C) $ is closed in $ X $, where

$ \begin{equation*} F^{+1}(C): = \{x \in X:F(x)\subset C \}; \end{equation*} $

(ⅲ') for every open set $ O\subset Y $, the set $ F^{-1}(O) $ is open in $ X $, where

$ \begin{equation*} F^{-1}(O): = \{x \in X:F(x)\cap O\neq \emptyset \}. \end{equation*} $

Definition 1.2 [11]  Let $ X $ be a Banach space. A set-valued mapping $ F:[0, T]\rightarrow P(X) $ is said to be measurable if for every closed set $ C \subset X $, the set $ F^{-1}(C): = \{x \in [0, T]:F(x)\cap C \neq \emptyset \} $ on $ \mathbb{R} $ is measurable.

Definition 1.3 [12]  Let $ E $ and $ E_{1} $ be Banach spaces, and let an interval $ I\subset \mathbb{R} $. We say that a mapping $ F:I\times E\rightarrow P(E_{1}) $ is super volitionally measurable if, for every measurable set-valued mapping $ Q:I\rightarrow K(E) $, the composition $ \phi:I\rightarrow P(E_{1}) $ given by $ \phi(t) = F(t, Q(t)) $ is measurable.

Lemma 1.4 [11]  Let $ E $ and $ E_{1} $ be Banach spaces, and let an interval $ I\subset \mathbb{R} $. Assume $ E $ is separable. If the mapping $ F:I\times E\rightarrow K(E_{1}) $ satisfies the Carathéodory condition or is upper or lower semicontinuous, then $ F $ is superpositionally measurable.

Lemma 1.5 [11]  Let $ E $ and $ E_{1} $ be Banach spaces. Suppose that the set-valued mapping $ G:[0, T]\times E\rightarrow K(E_{1}) $ satisfies:

(ⅰ) for every $ x\in E, \ G(\cdot, x):[0, T]\rightarrow K(E_{1}) $ has a strongly measurable selection;

(ⅱ) for a.e. $ t\in [0, T], \ G(t, \cdot):E\rightarrow K(E_{1}) $ is upper semicontinuous.

Then for every strongly measurable function $ q:[0, T]\rightarrow E $, there exists a strongly measurable selection $ g:[0, T]\rightarrow E_{1} $ of the composition $ M:[0, T]\rightarrow K(E_{1}) $ given by $ M(t) = G(t, q(t)) $ for a.e. $ t\in [0, T] $.

Now we recall the classical definition of measure of noncompactness.

Definition 1.4 [11]  Let $ X $ be a Banach space and $ (\mathfrak{A}, \preceq) $ be a partially ordered set. A mapping $ \beta:P(X)\rightarrow \mathfrak{A} $ is called a measure of noncompactness(MNC, for short) in $ X $ if $ \beta(\overline{co}\Omega) = \beta(\Omega) $ for every $ \Omega \in P(X) $. A measure of noncompactness $ \beta $ is called:

(ⅰ) monotone if $ \Omega_{0}, \Omega_{1}\in P(X), \Omega_{0}\subset \Omega_{1} $ implies $ \beta(\Omega_{0})\leq \beta(\Omega_{1}) $;

(ⅱ) nonsingular if $ \beta(\{a\}\cup \Omega) = \beta(\Omega) $ for every $ a\in X, \Omega\in P(X) $.

An example is the Hausdorff MNC$ _{\chi} $, which is defined by

$ \begin{equation*} \chi(\Omega): = {\rm inf}\{\varepsilon >0:\exists\Omega_{i}\in P(E), {\rm diam}(\Omega_{i})\leq \varepsilon, i = 1, \cdot, \cdot, n\ {\rm s.t.}\ \Omega\subseteq \bigcup\limits_{i = 1}^{n}\Omega_{i}\}, \end{equation*} $

Another example we mentioned here is the monotone nonsingular MNC in the space $ C([0, T];E) $. Namely, for every nonempty bounded set $ \Omega \subset C([0, T];E) $, it is equal to

$ \begin{equation} v(\Omega): = \mathop{\rm max}\limits_{\omega\in \Delta(\Omega)}(\gamma(\omega), {\rm mod}_{C}(\omega)), \end{equation} $

(1.1)

where $ \Delta(\Omega) $ denotes the collection of all countable subsets of $ \Omega $, and

$ \begin{equation*} \gamma(\omega): = \mathop{\rm sup}\limits_{t\in [0, T]}e^{-Lt}\chi(\omega(t)), \end{equation*} $

$ \begin{equation*} {\rm mod}_{C}(\omega): = \mathop{\rm lim}\limits_{\delta\rightarrow 0}\mathop{\rm sup}\limits_{x\in \omega}\mathop{\rm max}\limits_{\mid t_{1}-t_{2}\mid\leq\delta}\|x(t_{1})-x(t_{2})\|_{E}. \end{equation*} $

Definition 1.5 [11]  Let $ X $ be a closed subset of a Banach space $ E $ and let $ \beta $ be a MNC in $ E $. A set-valued mapping $ F:X\rightarrow K(E) $ is said to be $ \beta $-condensing if there exists some $ 0\leq k \leq1 $ such that $ \beta(F(\Omega))\leq k\beta(\Omega) $ for every $ \Omega \in P(X) $.

Lemma 1.6 [11]  Let $ E $ be a Banach space and $ M\subset E $ be a nonempty closed and convex subset. If $ F:M\rightarrow K_{v}(M) $ is a closed $ \beta $-condensing set-valued mapping with $ \beta $ be a nonsingular MNC $ \beta $ in $ E $, then the set Fix$ F $ of fixed points of $ F $ is nonempty.

Lemma 1.7 [11]  Let $ E $ be a Banach space and let CCE be a nonempty closed subset. $ F:C\rightarrow K(E) $ is a closed set-valued mapping, which is $ \beta $-condensing on every bounded subset of $ E $ with a monotone MNC $ \beta $ in $ E $. If Fix$ F $ is bounded, then it is compact.

Let $ E $ and $ E_{1} $ be real Banach spaces and let $ K $ be a nonempty compact and convex subset of $ E_{1} $. Let $ A:D(A)\subset E \rightarrow E $ be the infinitesimal generator of a $ C_{0}- $semigroup $ e^{At} $ in $ E $ and let $ \phi:E_{1}\rightarrow \mathbb{R} $ be a convex, lower semicontinuous functional. Let $ f:[0, T]\times E \times E_{1}\rightarrow E $ and $ g:[0, T]\times E \times E_{1} \rightarrow \mathbb{R} $ be two fixed mappings with some constant $ T>0 $. In this paper, we investigate a new class of differential mixed equilibrium problems((DME), for short):

$ \begin{eqnarray} \left\{ \begin{array}{lll} \ \dot{x}(t) = Ax(t)+f(t, x(t), u(t)), \quad t\in [0, T] \\ \ u(t)\in {\rm SOL}(K, g(t, x(t), \cdot), \phi), \quad t\in [0, T]\\ \ x(0) = x_{0}, \\ \end{array} \right. \end{eqnarray} $

(2.1)

where SOL$ (K, g(t, x(t), \cdot), \phi) $ stands for the solution set of the mixed equilibrium problem ((MEP), for short): find $ u:[0, T]\rightarrow K $ such that

$ \begin{equation} g(t, x(t), v-u(t))+\phi(v)-\phi(u(t))\geq 0, \forall v\in K. \end{equation} $

(2.2)

Definition 2.1  A pair of functions $ (x, u) $, with $ x\in C([0, T];E) $ and $ u:[0, T]\rightarrow K $ measurable, is said to be a mild solution of problem (DME) if

$ \begin{equation} x(t) = e^{At}x_{0}+\int_{0}^{t}e^{A(t-s)}f(s, x(s), u(s))\, ds, \quad t \in [0, T], \end{equation} $

(2.3)

and $ u(s) \in {\rm SOL}(K, g(s, x(s), \cdot), \phi), s\in [0, T] $. If $ (x, u) $ is a mild solution of problem (DME), then $ x $ is called the mild trajectory and $ u $ is called the variational control trajectory.

Let $ E_{1} $ be a real Banach space and let $ K $ be a nonempty subset of $ E_{1} $. Let $ Q:E_{1}\rightarrow \mathbb{R} $ and $ \phi:E_{1}\rightarrow \mathbb{R} $ be two fixed mappings. Now we consider the following mixed equilibrium problem: find $ u\in K $ such that

$ \begin{equation} Q(v-u)+\phi(v) -\phi(u)\geq 0, \forall v\in K. \end{equation} $

(3.1)

In this section, we will use Fan-KKM theorem and Ky Fan's minmax inequality separately to prove the existence and properties of solutions for (3.1).

Theorem 3.1  Let ($ E_{1}, \preceq $) be a totally ordered real Banach space and let $ K $ be a nonempty compact and convex subset of $ E_{1} $. $ Q:E_{1}\rightarrow \mathbb{R} $ is an order weak monotone increasing mapping with $ Q(0) = 1 $. Assume that:

(ⅰ) $ Q $ is concave, continuous and $ Q(x)-Q(-x) = 0 $ on $ E_{1} $;

(ⅱ) $ \phi:E_{1}\rightarrow \mathbb{R} $ is convex and lower semicontinuous on $ E_{1} $.

Then the solution set of (3.1) is nonempty, convex and closed in $ K $.

Proof  We consider the set-valued mapping $ G:K\rightarrow P(K) $ defined by

$ \begin{equation*} G(v): = \{u\in K :Q(v-u)+\phi(v)-\phi(u)\geq 0\}, \forall v\in K . \end{equation*} $

For each $ v \in K, G(v) $ is nonempty since $ v\in G(v) $.

First, we claim that $ G(v) $ is closed in $ K $ for all $ v\in K $. Indeed, we take a sequence $ \{u_{n}\}\subset G(v) $ such that $ u_{n}\rightarrow u_{0} $ as $ n\rightarrow \infty $. We get

$ \begin{equation*} Q(v-u_{n})+\phi(v)-\phi(u_{n})\geq 0, \forall n\in \mathbb{N}. \end{equation*} $

Applying the continuity of $ Q $ and the lower semicontinuity of $ \phi $, let $ n\rightarrow \infty $, then we have

$ \begin{equation*} Q(v-u_{0})+\phi(v)-\phi(u_{0})\geq 0, \end{equation*} $

which means $ u_{0}\in G(v) $, so $ G(v) $ is closed in $ K $.

Second, we claim that $ G $ is a KKM mapping. Arguing by contradiction, we assume that there exist a finite subset $ \{v_{1}, v_{2}, \cdot, \cdot, \cdot, v_{n}\}\subset K $ and $ u_{0} = \sum\limits_{i = 0}^{n}\lambda_{i}v_{i} $($ \sum\limits_{i = 1}^{n}\lambda_{i} = 1 $, $ 0\leq \lambda_{i}\leq 1 $) such that $ u_{0}\not\in \bigcup\limits_{i = 1}^{n}G(v_{i}) $, then we have

$ \begin{equation*} Q(v_{i}-u_{0})+\phi(v_{i})-\phi(u_{0})<0, \forall i\in \{1, 2, \cdot \cdot \cdot, n \}, \end{equation*} $

which implies

$ \begin{equation*} \sum\limits_{i = 1}^{n}\lambda_{i}Q(v_{i}-u_{0})+\sum\limits_{i = 1}^{n}\lambda_{i}\phi(v_{i})-\phi(u_{0})<0. \end{equation*} $

Since $ \phi $ is convex, we further obtain $ \sum\limits_{i = 1}^{n}\lambda_{i}Q(v_{i}-u_{0})<0, $ which means $ \exists j\in \{1, 2, \cdot \cdot \cdot , n\} $ such that $ Q(v_{j}-u_{0})<0. $ Since $ Q(x)-Q(-x) = 0 $ on $ E_{1} $, we have

$ \begin{equation} Q(u_{0}-v_{j}) = Q(v_{j}-u_{0})<0. \end{equation} $

(3.2)

Since $ E_{1} $ is totally ordered, we have either $ v_{j}\preceq u_{0} $ or $ u_{0}\preceq v_{j} $ holds. Since $ Q $ is order weak monotone increasing, we get

$ \begin{equation*} Q(u_{0}-v_{j})>0\ {\rm or}\ Q(v_{j}-u_{0})>0, \end{equation*} $

which contradicts (3.2). Therefore $ G $ is a KKM mapping.

Third, for any $ v_{0}\in K $, since $ G(v_{0}) $ is a closed subset of the compact set $ K $, we know $ G(v_{0}) $ is a compact set.

Using Lemma 1.1, we derive

$ \begin{equation*} \bigcap\limits_{v \in K}G(v)\neq \emptyset, \end{equation*} $

which ensures that the solution set of (3.1) is nonempty.

Finally, we verify that the solution set of problem (3.1) is closed and convex.

Let $ \{u_{n}\} $ be a sequence in the solution set satisfying $ u_{n}\rightarrow u_{0} $ as $ n\rightarrow \infty $, then we have

$ \begin{equation*} Q(v-u_{n})+\phi(v)-\phi(u_{n})\geq 0, \forall v\in K, \end{equation*} $

which yields in the limit

$ \begin{equation*} Q(v-u_{0})+\phi(v)-\phi(u_{0})\geq 0, \forall v\in K. \end{equation*} $

Therefore $ u_{0} $ solves problem (3.1), thus the solution set of (3.1) is closed.

Let $ u_{1} $ and $ u_{2} $ be arbitrary points in the solution set of (3.1), $ \lambda \in [0, 1] $. Since $ Q $ is concave and $ \phi $ is convex, for any $ v\in K $, we have

$ \begin{align*} &Q(v-(\lambda u_{1}+(1-\lambda)u_{2}))+\phi(v)-\phi(\lambda u_{1}+(1-\lambda)u_{2})\\ = &Q(\lambda(v-u_{1})+(1-\lambda)(v-u_{2}))+\lambda\phi(v)+(1-\lambda)\phi(v)-\phi(\lambda u_{1}+(1-\lambda)u_{2})\\ \geq &\lambda Q(v-u_{1})+(1-\lambda)Q(v-u_{0})+\lambda(\phi(v)-\phi(u_{1}))+(1-\lambda)(\phi(v)-\phi(u_{2}))\\ = &\lambda(Q(v-u_{1})+\phi(v)-\phi(u_{1}))+(1-\lambda)(Q(v-u_{2})+\phi(v)-\phi(u_{2}))\\ \geq&0. \end{align*} $

This implies that $ \lambda u_{1}+(1-\lambda)u_{2} $ solves problem (3.1), thus the solution set of (3.1) is convex. The proof is complete.

Example 3.1  Let $ E_{1} = \mathbb{R} $, $ K = [0, 1] $ and $ P = [0, +\infty) $ be a cone, we define the mappings $ Q:E_{1} \rightarrow \mathbb{R} $ and $ \phi:E_{1} \rightarrow \mathbb{R} $ by

$ \begin{equation*} Q(x) = \sqrt{|x|}+1 \quad and \quad \phi (x) = x. \end{equation*} $

We can check that the solution set of (3.1) is $ [0, 1] $.

Theorem 3.2  Let $ E_{1} $ be a real Banach space and let $ K $ be a nonempty compact and convex subset of $ E_{1} $. Let $ Q:E_{1}\rightarrow \mathbb{R} $ be a mapping satisfies $ Q(0) = 0 $. Assume that:

(ⅰ) $ Q $ is concave, continuous on $ E_{1} $;

(ⅱ) $ \phi:E_{1}\rightarrow \mathbb{R} $ is convex and lower semicontinuous on $ E_{1} $.

Then the solution set of (3.1) is nonempty, convex and closed in $ K $.

Proof  We consider the mapping $ \varphi:K\times K\rightarrow R $ defined by

$ \begin{equation*} \varphi(u, v): = -Q(v-u)+\phi(u)-\phi(v). \end{equation*} $

One can check that $ \varphi(\cdot, \cdot) $ satisfies conditions (ⅰ), (ⅱ) and (ⅲ) in Lemma 1.2.

Thus, by Lemma 1.2, we obtain that there exists $ \overline{u}\in K $, such that

$ \begin{equation*} \varphi(\overline{u}, v) = -Q(v-\overline{u})+\phi(\overline{u})-\phi(v)\leq 0, \forall v\in K, \end{equation*} $

or equivalently,

$ \begin{equation*} Q(v-\overline{u})+\phi(v)-\phi(\overline{u})\geq 0, \forall v\in K. \end{equation*} $

Hence, the solution set of (3.1) is nonempty.

In fact, the solution set of (3.1) is convex and closed. The proof can be done by arguing in the same way as Theorem 3.1, so we omit it.

Example 3.2  Let $ E_{1} = \mathbb{R} $, $ K = [0, 1] $, we define the mappings $ Q:E_{1} \rightarrow \mathbb{R} $ and $ \phi:E_{1} \rightarrow \mathbb{R} $ by

$ \begin{equation*} Q(x) = x \quad and \quad \phi (x) = x(x-2). \end{equation*} $

We can check that the solution set of (3.1) is $ \{0 \} $.

Remark 3.1  The hypotheses of Theorem 3.1 and Theorem 3.2 are different, while we obtain the same conclusion. The reason is that the method we use is different, we use Fan-KKM theorem to prove Theorem 3.1, while using Ky Fan's minmax inequality in the proof of Theorem 3.2.

Theorem 4.1  Let ($ E_{1}, \preceq $) be a totally ordered real Banach space and let $ K $ be a nonempty compact and convex subset of $ E_{1} $. $ E $ is a real separable Banach space. The mappings $ Q(\cdot): = g(t, x, \cdot) $ and $ \phi:E_{1}\rightarrow \mathbb{R} $ satisfy hypotheses in Theorem 3.1. Assume that:

(ⅰ) $ (t, x)\mapsto g(t, x, u) $ is continuous on $ [0, T]\times E $;

(ⅱ) $ u\mapsto g(t, x, u) $ is upper semicontinuous on $ K $.

Then the set-valued mapping $ U:[0, T]\times E \rightarrow K_{v}(K) $ defined by

$ \begin{equation} U(t, x): = \{u \in K:g(t, x, v-u)+\phi(v)-\phi(u)\geq 0, \forall v\in K\} \end{equation} $

(4.1)

is upper semicontinuous and superpositionally measurable.

Proof  Theorem 3.1 guarantees that for every $ (t, x) \in [0, T]\times E $, the set $ U(t, x) $ is nonempty, convex and compact in $ K $. Thus the mapping $ U(\cdot, \cdot) $ is well defined.

Now we claim that $ U(\cdot, \cdot) $ is upper semicontinuous. By Lemma 1.3, it's sufficient to prove that $ U^{-1}(C): = \{(t, x)\in [0, T]\times E:U(t, x)\cap C\neq \emptyset \} $ is closed for each closed subset $ C $ of $ K $. In fact, if $ (t_{n}, x_{n})\in U^{-1}(C) $ and $ (t_{n}, x_{n})\rightarrow (t_{0}, x_{0}) $. Since $ (t_{n}, x_{n})\in U^{-1}(C) $, there exists $ u_{n}\in U(t_{n}, x_{n})\cap C, \forall n\in \mathbb{N} $. Therefore,

$ \begin{equation} g(t_{n}, x_{n}, v-u_{n})+\phi(v)-\phi(u_{n})\geq 0, \forall v\in K. \end{equation} $

(4.2)

Since $ C $ is a closed subset of the compact set $ K $, we obtain $ C $ is also a compact set. Hence there is a subsequence $ \{u_{n_{k}}\} $ such that $ u_{n_{k}}\rightarrow u_{0} \in K $. Let $ k\rightarrow \infty $ in (4.2), by the conditions (ⅰ) and (ⅱ), we have

$ \begin{equation*} g(t_{0}, x_{0}, v-u_{0})+\phi(v)-\phi(u_{0})\geq 0 \quad , \forall v\in K. \end{equation*} $

In other words, we proved $ u_{0}\in U(t_{0}, x_{0})\cap C $, so $ (t_{0}, x_{0})\in U^{-1}(C) $. Therefore, $ U(\cdot, \cdot) $ is upper semicontinuous.

Finally, we conclude that $ U(t, x) $ is superpositionally measurable by applying Lemma 1.4. This completes the proof.

In this section, we will show the existence of solutions for (DME). First, we assume the following hypotheses on the mapping $ f:[0, T]\times E \times E_{1} \rightarrow E $ in (2.1):

($ f $1) for every $ (t, x)\in [0, T]\times E $, the set $ f(t, x, D) $ is convex in $ E $ for every convex set $ D \subset K $;

($ f $2) there exists $ \psi \in L^{1}([0, T]) $ such that

$ \begin{equation*} \| f(t, x, u)\|_{E}\leq \psi(t)(1+\|x\|_{E}), \forall (t, x, u)\in [0, T]\times E\times K; \end{equation*} $

($ f $3) $ f(\cdot, x, u):[0, T]\rightarrow E $ is measurable for every $ (x, u)\in E\times E_{1} $;

($ f $4) $ f(t, \cdot, \cdot):E\times E_{1}\rightarrow E $ is continuous for a.e. $ t\in [0, T] $;

($ f $5) there exists $ k \in L^{1}([0, T]) $ such that

$ \begin{equation*} \|f(t, x_{0}, u)-f(t, x_{1}, u)\|_{E}\leq k(t)\|x_{0}-x_{1}\|_{E} \end{equation*} $

for a.e. $ t \in [0, T], \forall x_{0}, x_{1} \in E, \forall u\in K $.

We next study the properties of the set-valued mapping $ F:[0, T]\times E \rightarrow P(E) $ given by

$ \begin{equation*} F(t, x) = f(t, x, U(t, x)), \end{equation*} $

with $ U $ introduced in (4.1).

Lemma 5.1 [2]  Let $ E $ and $ E_{1} $ be real Banach space, with $ E $ separable, and let $ K \subseteq E_{1} $ be a nonempty compact and convex subset. We assume that the hypotheses of Theorem 4.1 and conditions $ (f1)-(f5) $ are fulfilled. Then we have:

(ⅰ) $ F(t, x)\in K_{v}(E) $ for all $ (t, x)\in[0, T]\times E $;

(ⅱ) $ F(\cdot, x) $ has a strongly measurable selection for every $ x \in E $;

(ⅲ) $ F(t, \cdot) $ is upper semicontinuous for a.e. $ t\in [0, T] $;

(ⅳ) for every bounded subset $ D \subset E $, there exists $ l\in L^{1}([0, T]) $ such that

$ \begin{equation*} \chi(F(t, D))\leq l(t)\chi(D), \ {\rm for \ a.e.} \ t\in [0, T], \end{equation*} $

where $ \chi $ is the Hausdorff measure of noncompactness in $ E $.

By Lemma 5.1 we can define the set-valued mapping $ P_{F}:C([0, T];E)\rightarrow P(L^{1}([0, T];E)) $ by

$ \begin{equation*} P_{F}(q): = \{g|g \; {\rm is \; strongly \; measurable} \; {\rm and} \; g(t)\in F(t, q(t)) \; {\rm for \; a.e.} \; t \in [0, T]\}. \end{equation*} $

Furthermore, we can introduce $ \Gamma:C([0, T];E)\rightarrow K_{v}(C([0, T];E)) $ by

$ \begin{equation} \Gamma x: = \{ y \in C([0, T];E):y(t) = e^{At}x_{0}+ \int_{0}^{t}e^{A(t-s)}h(s)\, ds, h\in P_{F}(x)\}, \end{equation} $

(5.1)

where $ A:D(A)\subset E \rightarrow E $ is the infinitesimal generator of a $ C_{0}- $semigroup $ e^{At} $ in $ E $ as given in (2.1).

Lemma 5.2 [13]  Under the hypotheses of Lemma 5.1, the set-valued mapping $ \Gamma $ in (5.1) is upper semicontinuous and $ v $-condensing in the sense of Definition 1.5 on every closed bounded subset of $ C([0, T];E) $, with $ v $ constructed in (1.1).

Theorem 5.1  Under the hypotheses of Lemma 5.2, the solution set of problem (DME) in the sense of Definition 2.1 is nonempty and the set of all mild trajectories $ x $ of (DME) is compact in $ C([0, T];E) $.

Proof  We introduce the following evolutionary differential inclusion((EDI), for short):

$ \begin{eqnarray} \left\{ \begin{array}{lll} \ \dot{x}(t) = Ax(t)+F(t, x(t)), \quad t\in [0, T] \\ \ x(0) = x_{0}, \\ \end{array} \right. \end{eqnarray} $

(5.2)

Here $ F(t, x) = f(t, x, U(t, x)) $ with $ U(t, x) $ is defined in (4.1). The proof of the theorem is divided into three parts.

Step 1  We note that the solution set of (EDI) is nonempty if and only if the set of fixed points Fix$ \Gamma $ of $ \Gamma $ is nonempty. By Lemma 5.2, the set-valued mapping $ \Gamma:C([0, T];E)\rightarrow K_{v}(C[0, T];E) $ in (5.1) is upper semicontinuous and $ v $-condensing on every bounded subset of $ C([0, T];E) $.

Let $ L $ be a positive constant such that

$ \begin{equation} \int_{0}^{t}e^{-L(t-s)}\psi(s)\, ds<\cfrac{1}{M} \quad , \ \forall t\in[0, T], \end{equation} $

(5.3)

where $ \psi \in L^{1}([0, T]) $ and $ M: = \mathop{\rm max}\limits_{t\in [0, T]}\| e^{At}\| $.

By (5.3), there exists $ r>0 $ such that

$ \begin{equation} M\| x_{0}\|_{E}+Mr \int_{0}^{t}e^{-L(t-s)}\psi(s)\, ds\leq r, \ \forall t\in [0, T]. \end{equation} $

(5.4)

Next, we introduce the equivalent norm on the space $ C([0, T];E) $ by

$ \begin{equation*} \|x\|_{\ast}: = \mathop{\rm max}\limits_{t\in [0, T]}e^{-Lt}\| x(t)\|_{E}, \end{equation*} $

we denote the closed ball centered at $ 0 $ with radius $ r $ in $ C([0, T];E) $ by

$ \begin{equation*} \overline{B_{r}}(0): = \{ x\in C([0, T];E): \parallel x\parallel_{\ast}\leq r\}. \end{equation*} $

Now we claim that $ \Gamma(\overline{B_{r}}(0))\subset \overline{B_{r}}(0) $. Let $ x\in \overline{B_{r}}(0) $ and $ y\in \Gamma x $. From (5.1), there exists $ h\in P_{F}(x) $ such that

$ \begin{equation*} y(t) = e^{At}x_{0}+\int_{0}^{t}e^{A(t-s)}\psi(s)h(s)\, ds, \ \forall t\in [0, T]. \end{equation*} $

Using ($ f $2), we obtain

$ \begin{align*} e^{-Lt}\|y(t)\|_{E} = &e^{-Lt}\| e^{At}+\int_{0}^{t}e^{A(t-s)}h(s)\, ds\|_{E}\\ \leq& e^{-Lt}\| e^{At}x_{0}\|+e^{-Lt}\int_{0}^{t}\| e ^{A(t-s)}\| \| h(s)\|_{E}\, ds \\ \leq& M(\| x_{0}\|_{E}+\| \psi \|_{L^{1}([0, T])})+M \| x\|_{\ast}\int_{0}^{t}e^{-L(t-s)}\psi(s)\, ds. \end{align*} $

Since $ x\in \overline{B_{r}}(0) $ and using (5.4), it follows that for every $ t\in [0, T] $,

$ \begin{equation*} e^{-Lt}\| y(t)\|_{E}\leq M(\| x_{0}\|_{E}+\| \psi \|_{L^{1}([0, T])})+Mr\int_{0}^{t}e^{-L(t-s)}\psi(s)\, ds\leq r, \end{equation*} $

which implies $ \| y\|_{\ast} \leq r $. Therefore, $ \Gamma (\overline{B_{r}}(0))\subset \overline{B_{r}}(0) $.

Applying Lemma 5.2 and Lemma 1.6 with $ M = \overline{B_{r}}(0) $ and $ F = \Gamma $, it follows that Fix$ \Gamma \neq \emptyset $. Hence the solution set of (EDI) is nonempty.

Step 2  We claim that the solution set of (EDI) is compact in $ C([0, T];E) $. Let $ x\in C([0, T];E) $ be a solution of (EDI), then we have

$ \begin{equation*} \| x(t)\|_{E}\leq \| e^{At}\| \| x_{0}\|_{E}+ \int_{0}^{t}\| e^{A(t-s)}\| \| h(s)\|_{E} \, ds, \ \forall t\in [0, T], \end{equation*} $

where $ h \in P_{F}(x) $. From the condition ($ f $2) we obtain

$ \begin{align*} \| x(t)\|_{E}&\leq M \| x_{0}\|_{E} +M \int_{0}^{t}\psi (s) (1+\| x(s)\|_{E}) \, ds\\ &\leq M( \| x_{0}\|_{E}+\| \psi \|_{L^{1}([0, T])}+\int_{0}^{t}\psi (s)\| x(s)\|_{E} \, ds). \end{align*} $

Using Gronwall inequality, we have the following estimate

$ \begin{equation*} \| x(t)\|_{E}\leq M( \| x_{0}\|_{E}+\| \psi \|_{L^{1}([0, T])})e^{M\| \psi \| _{L^{1}([0, T])}}. \end{equation*} $

Therefore, Fix$ \Gamma $ is bounded in $ C([0, T];E) $.

Applying Lemma 5.2 and Lemma 1.7 with $ F = \Gamma $, we know the solution set of problem (EDI), which equals to Fix$ \Gamma $, is compact in $ C([0, T];E) $.

Step 3  Note that the set-valued mapping $ U $ is superpositionally measurable from Theorem 4.1. Therefore, by Filippov implicit function lemma (see [11]), we deduce that for every solution $ x $ of (EDI), there exists a measurable selection $ u(t)\in U(t, x(t)) $ such that $ \dot{x}(t) = Ax(t)+f(t, x(t), u(t)), t\in [0, T]. $ Hence, $ (x, u) $ is a mild solution of problem (DME) in the sense of Definition 2.1, which implies the set of all mild trajectories of problem (DME) is consistent with the solution set of problem (EDI). This completes the proof.