We begin with the Bochner-Sobolev space of range-fixed type, that is

$$\begin{array}{}{\displaystyle {W}^{1,{\varrho}_{\alpha}}(I,{X}_{\alpha})=\{u\in {L}^{{\varrho}_{\alpha}}(I,{X}_{\alpha}):{u}^{\prime}\in {L}^{{\varrho}_{\alpha}}(I,{X}_{\alpha})\},}\end{array}$$

where *u*′ denotes the derivative of *u* in the sense of distribution, i.e. for all *ξ* ∈
$\begin{array}{}{\displaystyle {X}_{\alpha}^{\ast}}\end{array}$ and all *γ* ∈
$\begin{array}{}{\displaystyle {C}_{0}^{\mathrm{\infty}}}\end{array}$(*I*), equality

$$\begin{array}{}{\displaystyle \underset{I}{\int}\u27e8{u}^{\prime}(t),\gamma (t)\xi {\u27e9}_{\alpha}dt=-\underset{I}{\int}\u27e8u(t),{\gamma}^{\prime}(t)\xi {\u27e9}_{\alpha}dt}\end{array}$$

holds. It is easy to check that, endowed with the norm

$$\begin{array}{}{\displaystyle \parallel u{\parallel}_{{W}^{1,{\varrho}_{\alpha}}(I,{X}_{\alpha})}=\parallel u{\parallel}_{{L}^{{\varrho}_{\alpha}}(I,{X}_{\alpha})}+\parallel {u}^{\prime}{\parallel}_{{L}^{{\varrho}_{\alpha}}(I,{X}_{\alpha})},}\end{array}$$

which is equivalent to

$$\begin{array}{}{\displaystyle inf\{\lambda >0:{\mathit{\Phi}}_{{\varrho}_{\alpha}}(u/\lambda )+{\mathit{\Phi}}_{{\varrho}_{\alpha}}({u}^{\prime}/\lambda )\le 1\},}\end{array}$$

*W*^{1,ϱα}(*I*, *X*_{α}) turns to be a Banach space.

#### Theorem 5.1

*Function space* *W*^{1,ϱα}(*I*, *X*_{α}) *can be embedded into the space of continuous functions* *C*(*I*, *X*_{α}). *If in addition* *V*_{β} *is embedded into* *X*_{α} *compactly*, *then* *W*^{1,ϱα}(*I*, *X*_{α}) ∩ *L*^{φβ}(*I*, *V*_{β}) *is embedded compactly into* *L*^{ϱα}(*I*, *X*_{α}).

#### Proof

Firstly, from the inequality

$$\begin{array}{}{\displaystyle \parallel u{\parallel}_{{X}_{\alpha}}\le {\varrho}_{\alpha}(u)+1,\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}\phantom{\rule{thickmathspace}{0ex}}u\in {X}_{\alpha},}\end{array}$$

we can deduce that *L*^{ϱα}(*I*, *X*_{α}) ↪ *L*^{1}(*I*, *X*_{α}). Similarly, we have *W*^{1,ϱα}(*I*, *X*_{α}) ↪ *W*^{1,1}(*I*, *X*_{α}) and *L*^{φβ}(*I*, *V*_{β}) ↪ *L*^{1}(*I*, *V*_{β}). The first conclusion comes since the embedding *W*^{1,1}(*I*, *X*_{α})) ↪ *C*(*I*, *X*_{α}) holds.

Given a bounded subset *F* of *W*^{1,ϱα}(*I*, *X*_{α}) ∩ *L*^{φβ}(*I*, *V*_{β}), it is also bounded in *W*^{1,1}(*I*, *X*_{α})) ∩ *L*^{1}(*I*, *V*_{β})). Assume that

$$\begin{array}{}{\displaystyle {\mathit{\Phi}}_{{\varrho}_{\alpha}}({u}^{\prime})+{\mathit{\Phi}}_{{\varrho}_{\alpha}}(u)+{\mathit{\Phi}}_{{\phi}_{\beta}}(u)\le C}\end{array}$$

for some *C* > 0 independent of *u* ∈ *F*. Then for any *u* ∈ *F* and 0 < *h* < min{1, *T*/2} and *t*, *t* + *h* ∈ *I*, we have

$$\begin{array}{}{\displaystyle {\varrho}_{\alpha}(u(t+h)-u(t))\le \underset{t}{\overset{t+h}{\int}}{\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau ,}\end{array}$$

consequently

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\underset{0}{\overset{T-h}{\int}}{\varrho}_{\alpha}(u(t+h)-u(t))dt\le \underset{0}{\overset{T-h}{\int}}\underset{t}{\overset{t+h}{\int}}{\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau dt}\\ {\displaystyle =(\underset{0}{\overset{h}{\int}}\underset{0}{\overset{\tau}{\int}}+\underset{h}{\overset{T-h}{\int}}\underset{\tau -h}{\overset{\tau}{\int}}+\underset{T-h}{\overset{T}{\int}}\underset{\tau -h}{\overset{T-h}{\int}}){\varrho}_{\alpha}({u}^{\prime}(\tau ))dtd\tau}\\ {\displaystyle =\underset{0}{\overset{h}{\int}}\tau {\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau +\underset{h}{\overset{T-h}{\int}}h{\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau +\underset{T-h}{\overset{T}{\int}}(T-\tau ){\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau}\\ {\displaystyle \le h\underset{I}{\int}{\varrho}_{\alpha}({u}^{\prime}(\tau ))d\tau \le Ch.}\end{array}$$(21)

Taking any *r* ∈ (0, *T*), consider the average operator *M*_{r} on *L*^{ϱα}(*I*, *X*_{α}) defined by

$$\begin{array}{}{\displaystyle {M}_{r}u(t)=\frac{1}{r}\underset{t}{\overset{t+r}{\int}}u(\tau )d\tau ,\phantom{\rule{thickmathspace}{0ex}}t\in [0,T-r].}\end{array}$$

Obviously, for all *u* ∈ *L*^{φβ}(*I*, *V*_{β}), *M*_{r}u ∈ *C*([0, *T* – *r*], *V*_{β}) with the estimate

$$\begin{array}{}{\displaystyle {\phi}_{\beta}({M}_{r}u(t))\le \frac{1}{r}\underset{t}{\overset{t+r}{\int}}{\phi}_{\beta}(u(\tau ))d\tau ,\phantom{\rule{thickmathspace}{0ex}}t\in [0,T-r].}\end{array}$$

Moreover, due to the boundedness of *F* in *W*^{1,ϱα}(*I*, *X*_{α}) and the estimate (21), precompactness of the set *F*_{r} = {*M*_{r}u : *u* ∈ *F*} in *C*([0, *T* – *r*], *X*_{α}) can be reached (refer to [16, 17]).

In addition, from (21), one can deduce that

$$\begin{array}{}{\displaystyle \underset{0}{\overset{T-h}{\int}}{\varrho}_{\alpha}({M}_{r}u(t)-u(t))dt=\underset{0}{\overset{T-h}{\int}}{\varrho}_{\alpha}(\frac{1}{r}\underset{0}{\overset{r}{\int}}(u(t+\tau )-u(t))d\tau )dt}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\le \frac{1}{r}\underset{0}{\overset{T-h}{\int}}\underset{0}{\overset{r}{\int}}{\varrho}_{\alpha}(u(t+\tau )-u(t))d\tau dt}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\le \frac{1}{r}\underset{0}{\overset{r}{\int}}\underset{0}{\overset{T-\tau}{\int}}{\varrho}_{\alpha}(u(t+\tau )-u(t))dtd\tau}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\le Ch}\end{array}$$

provided 0 < *r* ≤ *h*, which means that *M*_{r}u → *u* in *L*_{ϱα}([0, *T* – *h*], *X*_{α}) as *r* → 0 uniformly for *u* ∈ *F*. This fact, combined with the precompactness of *F*_{r} in *C*([0, *T* – *r*], *X*_{α}) for every fixed *r* ∈ (0, *h*], leads to the precompactness of *F* in *L*_{ϱα1}([0, *T* – *h*], *X*_{α}). The final conclusion comes if we make the same discussions on the set *F͠* = {*u͠*(*t*) = *u*(*T* – *t*) : *u* ∈ *F*} (see [16]).□

Using the facts *L*^{ϱα}(*I*, *X*_{α}) ↪ *L*^{1}(*I*, *X*_{α}) and *W*^{1,ϱα}(*I*, *X*_{α}) ↪ *C*(*I*, *X*_{α}), we can also deduce that

#### Corollary 5.2

*Under hypotheses of the above theorem*, *W*^{1,ϱα}(*I*, *X*_{α}) ∩ *L*^{φβ}(*I*, *V*_{β}) *can be embedded compactly into* *L*^{p}(*I*, *X*_{α}) *for any* 1 ≤ *p* < ∞, *hence* *L*^{p(⋅)}(*I*, *X*_{α}) *for all* *p* ∈ 𝓟_{b}(*I*, *X*_{α}).

Given two 𝓒𝓜𝓝s {*ϱ*_{α} :∈ 𝓐} and {*φ*_{β} : *β* ∈ 𝓑} satisfying *H*(*A*) + *H*(*B*) and *H*(*A*) + *H*(*B*)′ respectively, and the latter stronger than the former, introduce the Bochner-Sobolev space of range-varying type

$$\begin{array}{}{\displaystyle {W}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})=\{u\in {W}^{1,{\varrho}_{{\alpha}^{-}}}(I,{X}_{{\alpha}^{-}}):u,{u}^{\prime}\in {L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\}.}\end{array}$$

Similarly, equipped with the norm

$$\begin{array}{}{\displaystyle \parallel u{\parallel}_{{W}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}=\parallel u{\parallel}_{{L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}+\parallel {u}^{\prime}{\parallel}_{{L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})},}\end{array}$$

which is equivalent to

$$\begin{array}{}{\displaystyle inf\{\lambda >0:{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}(u/\lambda )+{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}^{\prime}/\lambda )\le 1\},}\end{array}$$

*W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) becomes a Banach space.

#### Theorem 5.3

*Besides the assumptions upon* {*ϱ*_{α}} *and* {*φ*_{β}} *as above*, *assume that there are scalar functions* *p̃*_{i}, *q̃*_{i}, *δ* ∈ *C*(*I*) (*i* = 1, 2) *such that* 1 ≤ *p̃*_{1}(*t*) ≤ *p̃*_{2}(*t*) < ∞, 1 ≤ *q̃*_{1}(*t*) ≤ *q̃*_{2}(*t*) < ∞, *δ*(*t*) ∈ (0, 1), *and* *p̃*_{2}(*t*)*δ*(*t*) < *q̃*_{2}(*t*) *for all* *t* ∈ *I*. *Suppose also*

–

*V*_{ϑ–} ↪↪ *X*_{α+} *and there is a constant* *C* > 0 *such that*

–

(*X*_{α–}, *V*_{ϑ(t)})_{δ(t)} ↪ *X*_{θ(t)} *uniformly for* *t* ∈ *I*, *in other words*,

$$\begin{array}{}{\displaystyle \parallel u{\parallel}_{\theta (t)}\le C\parallel u{\parallel}_{{\alpha}^{-}}^{1-\delta (t)}\parallel u{\parallel}_{\vartheta (t)}^{\delta (t)}\phantom{\rule{thickmathspace}{0ex}}for\text{\hspace{0.17em}}all\phantom{\rule{thickmathspace}{0ex}}u\in {V}_{\vartheta (t)},}\end{array}$$(22)

*where notation* (*X*_{α–}, *V*_{ϑ(t)})_{δ(t)} *represents the real or complex interpolation space between* *X*_{α–} *and* *V*_{ϑ(t)} *with the index* *δ*(*t*);

–

*for all* *t* ∈ *I*,

$$\begin{array}{}{\displaystyle {\varrho}_{\theta (t)}(u)\le Cmax\{\parallel u{\parallel}_{\theta (t)}^{{\stackrel{~}{p}}_{1}(t)},\parallel u{\parallel}_{\theta (t)}^{{\stackrel{~}{p}}_{2}(t)}\}\phantom{\rule{thickmathspace}{0ex}}u\in {X}_{\theta (t)},}\end{array}$$(23)

and

$$\begin{array}{}{\displaystyle min\{\parallel u{\parallel}_{\vartheta (t)}^{{\stackrel{~}{q}}_{1}(t)},\parallel u{\parallel}_{\vartheta (t)}^{{\stackrel{~}{q}}_{2}(t)}\}\le C{\phi}_{\vartheta (t)}(u),\phantom{\rule{thickmathspace}{0ex}}u\in {V}_{\vartheta (t)}.}\end{array}$$

*Then space* *W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) ∩ *L*^{φϑ(⋅)}(*I*, *V*_{ϑ(⋅)}) *is embedded into* *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) *compactly*.

#### Proof

Firstly, by (23) and Remark 4.8 in [11], we have that

$$\begin{array}{}{\displaystyle {L}^{{\stackrel{~}{p}}_{2}(\cdot )}(I,{X}_{\theta (\cdot )})\hookrightarrow {L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )}).}\end{array}$$

Thus from Theorem 5.1 and its corollary, it suffices to show that a sequence {*u*_{k}} bounded in *L*^{φϑ(⋅)}(*I*, *V*_{ϑ(⋅)}) and convergent in *L*^{p(⋅)}(*I*, *X*_{α–}) for all *p* ∈ 𝓟_{b}(*I*, *X*_{α}) is convergent in *L*^{p̃2(⋅)}(*I*, *X*_{θ(⋅)}) definitely. Without loss of generality, assume that the limit of {*u*_{k}} is 0. Take *K* > 1 so close to 1 that *p̃*_{2}(*t*)*δ*(*t*)*K* ≤ *q̃*_{2}(*t*) and *p̃*_{2}(*t*)(1 – *δ*(*t*))*K*′ ≥ 1 (1/*K* + 1/*K*′ = 1) for all *t* ∈ *I*, then by (22), we have

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{\theta (t)}^{{\stackrel{~}{p}}_{2}(t)}dt}\\ {\displaystyle \le C\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{{\alpha}^{-}}^{{\stackrel{~}{p}}_{2}(t)(1-\delta (t))}\parallel {u}_{k}(t){\parallel}_{\vartheta (t)}^{{\stackrel{~}{p}}_{2}(t)\delta (t)}dt}\\ {\displaystyle \le C(\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{{\alpha}^{-}}^{{\stackrel{~}{p}}_{2}(t)(1-\delta (t)){K}^{\prime}}dt{)}^{1/{K}^{\prime}}(\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{\vartheta (t)}^{{\stackrel{~}{p}}_{2}(t)\delta (t)K}dt{)}^{1/K}}\\ {\displaystyle \le C(\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{{\alpha}^{-}}^{{\stackrel{~}{p}}_{2}(t)(1-\delta (t)){K}^{\prime}}dt{)}^{1/{K}^{\prime}}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\cdot [{T}^{1/K}+(\underset{\{\parallel {u}_{k}(t){\parallel}_{\vartheta (t)}>1\}}{\int}\parallel {u}_{k}(t){\parallel}_{\vartheta (t)}^{{\stackrel{~}{p}}_{2}(t)\delta (t)K}dt{)}^{1/K}]}\\ {\displaystyle \le C(\underset{I}{\int}\parallel {u}_{k}(t){\parallel}_{{\alpha}^{-}}^{{\stackrel{~}{p}}_{2}(t)(1-\delta (t)){K}^{\prime}}dt{)}^{1/{K}^{\prime}}[{T}^{1/K}+(\underset{I}{\int}{\phi}_{\vartheta (t)}({u}_{k}(t))dt{)}^{1/K}],}\end{array}$$(24)

which leads to the desired conclusion.□

In spite of the estimate (26), we do not expect the control of the modular *Φ*_{ϱθ(⋅)}(*u*) by the sum *Φ*_{ϱθ(⋅)}(*u*′) + *Φ͠*_{φϑ(⋅)}(*u*). Conversely, we have

#### Proposition 5.5

*For every r* ≥ 0, *there is an* *ε*_{r} > 0 *such that for all* 0 ≤ *ε* ≤ *ε*_{r}, *the a priori estimate*

$$\begin{array}{}{\displaystyle {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}(u)\le {\epsilon}_{r}{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}(u)+r}\end{array}$$

*defines a bounded subset of* *W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}).

#### Proof

We proceed by contradiction. Assume that there is a sequence {*u*_{k}} in *W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) verifying

$$\begin{array}{}{\displaystyle {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}_{k}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}({u}_{k})\le \frac{1}{k}{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}_{k})+r,}\end{array}$$

but *Φ*_{ϱθ(⋅)}(*u*_{k}) ≥ *k*. Let *v*_{k} = *u*_{k}/*Φ*_{ϱθ(⋅)}(*u*_{k}). Then we have

$$\begin{array}{}{\displaystyle {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({v}_{k}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}({v}_{k})\le \frac{{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}_{k}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}({u}_{k})}{{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({u}_{k})}\le \frac{1+r}{k}.}\end{array}$$

Therefore *Φ*_{ϱθ(⋅)}(
$\begin{array}{}{\displaystyle {v}_{k}^{\prime}}\end{array}$) + *Φ͠*_{φϑ(⋅)}(*v*_{k}) → 0, and consequently *v*_{k} → 0 in *W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) as *k* → ∞. Thus lim_{k→∞} *Φ*_{ϱθ(⋅)}(*v*_{k}) = 0, which contradicts to fact that *Φ*_{ϱθ(⋅)}(*v*_{k}) ≥ 1/2 for all *k* ∈ ℕ.□

Define two function spaces with periodic boundary condition, one is

$$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\alpha}}(I,{X}_{\alpha})=\{u\in {W}^{1,{\varrho}_{\alpha}}(I,{X}_{\alpha}):u(0)=u(T)\},}\end{array}$$

the other is

$$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})={W}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\cap {W}_{\text{per}}^{1,{\varrho}_{{\alpha}^{-}}}(I,{X}_{{\alpha}^{-}})}\end{array}$$

under the condition *θ*(0) = *θ*(*t*). Evidently, the two spaces are closed subspaces of *W*^{1,ϱα}(*I*, *X*_{α}) and *W*^{1,ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) respectively.

Let us make some investigations on the operator
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ :
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)}) →
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)})^{*} defined through

$$\begin{array}{}{\displaystyle \u3008\u3008{D}_{\theta (\cdot )}^{2}(u),v\u3009\u3009=\underset{I}{\int}\u3008\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t)),{v}^{\prime}(t){\u3009}_{\theta (t)}dt,\phantom{\rule{thickmathspace}{0ex}}u,v\in {W}^{1,{\varrho}_{\alpha}}(I,{X}_{\alpha}).}\end{array}$$(27)

By the definition, it is easy to check that
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ is a single-valued, monotone and demicontinuous operator, hence it is maximal monotone. In addition, taking *v*(*t*) = *γ*(*t*)*w* in (27) with *γ* ∈
$\begin{array}{}{\displaystyle {C}_{0}^{\mathrm{\infty}}}\end{array}$(*I*) and *w* ∈ *X*_{α+}, we have

$$\begin{array}{}{\displaystyle \u3008\u3008{D}_{\theta (\cdot )}^{2}(u),v\u3009\u3009=\underset{I}{\int}\u3008\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t)),{\gamma}^{\prime}(t)w{\u3009}_{{\alpha}^{+}}dt}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}=-\underset{I}{\int}\u3008\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t)),\gamma (t)w{\u3009}_{{\alpha}^{+}}dt}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}=-\underset{I}{\int}\u3008\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t)),\gamma (t)w{\u3009}_{\theta (t)}dt,}\end{array}$$

which means that
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ is a second order nonlinear differential operator, i.e.

$$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}(u)=-\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (\cdot )}({u}^{\prime}(\cdot ))}\end{array}$$

in the sense of distribution.

Let *Ψ͠*(*u*) = *Φ*_{ϱθ(⋅)}(*u*′) at *u* ∈
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)}), we obtain a continuous and convex functional. A direct calculation shows that *Ψ͠* is Gâteaux differentiable everywhere with the Gâteaux differential *Ψ͠*′(*u*) =
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$(*u*). Thus the subdifferential operator *∂Ψ͠* is single-valued, and *∂Ψ͠*(*u*) =
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$(*u*) for all *u* ∈
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)}). Consider the extension of *Ψ͠* onto *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)})

$$\begin{array}{}{\displaystyle \mathit{\Psi}(u)=\left\{\begin{array}{ll}\mathrm{\infty},& \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\in {L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\setminus {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\\ \stackrel{~}{\mathit{\Psi}}(u),& \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\in {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )}).\end{array}\right.}\end{array}$$

It is also easy to verify that *Ψ* is a semicontinuous and convex proper functional, whose subdifferential operator has the domain

$$\begin{array}{}{\displaystyle \mathcal{D}(\mathrm{\partial}\mathit{\Psi})=\{u\in {W}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )}):\text{there is an}\phantom{\rule{thickmathspace}{0ex}}\xi \in {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})\phantom{\rule{thickmathspace}{0ex}}\text{such that}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\underset{I}{\int}\u3008\xi (t),v(t){\u3009}_{\theta (t)}dt=\underset{I}{\int}\u3008{Z}_{\theta (\cdot )}({u}^{\prime})(t),{v}^{\prime}(t){\u3009}_{\theta (t)}dt}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\text{for all}\phantom{\rule{thickmathspace}{0ex}}v\in {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\},}\end{array}$$(28)

together with *∂Ψ*(*u*) =
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$(*u*) = *ξ* in
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)})^{*} for the function *ξ* ∈
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$ satisfying (28) and all *u* ∈ 𝓓(*∂Ψ*). In this sense, we can say that *∂Ψ* ⊆
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$.

Taking the intersection

$$\begin{array}{}{\displaystyle \mathcal{W}={W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\cap {L}^{{\phi}_{\vartheta (\cdot )}}(I,{V}_{\vartheta (\cdot )})}\end{array}$$

as the work space, where the norm ∥⋅∥_{𝓦} takes the value
$\begin{array}{}{\displaystyle \parallel u{\parallel}_{{W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}+\parallel u{\parallel}_{{L}^{{\phi}_{\vartheta (\cdot )}}(I,{V}_{\vartheta (\cdot )})},}\end{array}$ and consider the sum

$$\begin{array}{}{\displaystyle \stackrel{~}{\mathit{\Phi}}(u)=\stackrel{~}{\mathit{\Psi}}({u}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}(u)}\end{array}$$

as the potential functional. Evidently, *Φ͠*(*u*) is a continuous modular on 𝓦 with the effective domain 𝓓(*Φ͠*) = 𝓦, its subdifferential operator

$$\begin{array}{}{\displaystyle \mathrm{\partial}\stackrel{~}{\mathit{\Phi}}={D}_{\theta (\cdot )}^{2}+\mathrm{\partial}{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}}\end{array}$$

is a maximal monotone subset of
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)}) ×
$\begin{array}{}{\displaystyle {W}_{\text{per}}^{1,{\varrho}_{\theta (\cdot )}}}\end{array}$(*I*, *X*_{θ(⋅)})^{*}. Moreover, by (26) and the coercivity of *Φ*_{ϱθ(⋅)}, *Φ*_{φϑ(⋅)}, we have

$$\begin{array}{}{\displaystyle \frac{\u3008\u3008\mathrm{\partial}\stackrel{~}{\mathit{\Phi}}(u),u\u3009\u3009}{\parallel u{\parallel}_{\mathcal{W}}}\ge C\frac{\stackrel{~}{\mathit{\Psi}}({u}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}(u)}{\parallel {u}^{\prime}{\parallel}_{{L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}+\parallel u{\parallel}_{{L}^{{\phi}_{\vartheta (\cdot )}}(I,{V}_{\vartheta (\cdot )})}}\to \mathrm{\infty}\phantom{\rule{thickmathspace}{0ex}}\text{as}\phantom{\rule{thickmathspace}{0ex}}\parallel u{\parallel}_{\mathcal{W}}\to \mathrm{\infty},}\end{array}$$

which yields the coercivity, hence surjectivity of *∂Φ͠*. Furthermore, suppose that for every *β* ∈ 𝓑, *φ*_{β} is strictly convex, then *Φ͠*_{φϑ(⋅)} hence *Φ͠* is also strictly convex, which in turn leads to the injectivity of *∂Φ͠*. Summing up, we conclude that

#### Theorem 5.7

*Suppose all the hypotheses mentioned above are satisfied*, *then for every*
$\begin{array}{}{\displaystyle f\in {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$ ⊆ 𝓦^{*}, *there is a unique* *u* ∈ 𝓦 *and a corresponding selection* *ξ* ∈ *∂Φ͠*_{φϑ(⋅)}(*u*) *such that*

$$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}(u)+\xi =f\phantom{\rule{thickmathspace}{0ex}}in\phantom{\rule{thickmathspace}{0ex}}{\mathcal{W}}^{\ast}.}\end{array}$$(29)

*In other words*, *for every*
$\begin{array}{}{\displaystyle f\in {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$, *second order differential inclusion*

$$\begin{array}{}{\displaystyle -\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t))+\mathrm{\partial}{\phi}_{\vartheta (t)}(u(t))\ni f(t)\phantom{\rule{thickmathspace}{0ex}}a.e.\text{\hspace{0.17em}}on\phantom{\rule{thickmathspace}{0ex}}I}\end{array}$$(30)

*with periodic boundary values admits a unique weak solution* *u* *in the sense*

$$\begin{array}{}{\displaystyle \underset{I}{\int}\u3008\mathrm{\partial}{\varrho}_{\theta (\cdot )}({u}^{\prime}(t)),{v}^{\prime}(t){\u3009}_{\theta (t)}dt+\underset{I}{\int}\u3008\xi (t),v(t){\u3009}_{\vartheta (t)}dt=\underset{I}{\int}\u3008f(t),v(t){\u3009}_{\theta (t)}dt}\end{array}$$

*for all* *v* ∈ 𝓦.

The above theorem shows that the sum operator
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ + *∂Φ*_{φϑ(⋅)} is both injective and surjective, its inverse (
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ + *∂Φ*_{φϑ(⋅)})^{–1} is existing and single-valued. Moreover, in light of Proposition 5.5, together with the compact imbedding 𝓦 ↪↪ *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}), we can prove that

#### Theorem 5.8

*If all the hypotheses of Theorem 5.3 together with (25) are satisfied*, *then the inverse operator*

$$\begin{array}{}{\displaystyle ({D}_{\theta (\cdot )}^{2}+\mathrm{\partial}{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}{)}^{-1}:{L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})\to {L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}\end{array}$$

*is both bounded and strongly continuous in the sense that for any sequence* {*f*_{k}} *convergent to* *f* *in*
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$ *weakly*, *the corresponding sequence* {(
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ + *∂Φ͠*_{φϑ(⋅)})^{–1} *f*_{k}} *converges to* (
$\begin{array}{}{\displaystyle {D}_{\theta (\cdot )}^{2}}\end{array}$ + *∂Φ͠*_{φϑ(⋅)})^{–1} *f* *in* *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) *strongly*.

Let us consider the operator *f* : *I* × *X* → *X*. Assume that

–

for any *J* ∈ *Π*(*I*) and *u* ∈
$\begin{array}{}{\displaystyle {X}_{{\theta}_{J}^{+}}}\end{array}$, *t* ↦ *f*(*t*, *u*) lies in *L*^{0}(*I*,
$\begin{array}{}{\displaystyle {X}_{{\theta}_{J}^{+}}^{\ast}}\end{array}$),

–

for every *t* ∈ *I*, *f*(*t*, ⋅) : *X*_{θ(t)} →
$\begin{array}{}{\displaystyle {X}_{\theta (t)}^{\ast}}\end{array}$ is demicontinuous, and

–

there is an *μ* > 0 and a nonnegative integrable function *h* for which inequality

$$\begin{array}{}{\displaystyle {\varrho}_{\theta (t)}^{\ast}(f(t,u))\le \mu {\varrho}_{\theta (t)}(u)+h(t)}\end{array}$$(31)

holds for all *u* ∈ *X*_{θ(t)}.

Taken any *u* ∈ *L*^{0}(*I*, *X*_{θ(⋅)}), denote by *F*(*u*) = *f*(*t*, *u*). It is easy to see that under the first assumption upon *f*, if *u* ∈ 𝓢(*I*, *X*_{α+}), then *F*(*u*) ∈
$\begin{array}{}{\displaystyle {L}_{-}^{0}(I,{X}_{\theta (\cdot )}^{\ast}),}\end{array}$ and by (31), it comes *F*(*u*) ∈
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$. Consequently, from the density of 𝓢(*I*, *X*_{α+}) in *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) and the demicontinuity assumption upon *f*, we can derive that *F*(*u*) ∈
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$ provided *u* ∈ *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}). Moreover, we have

#### Proposition 5.9

*F* *is a bounded and demicontinuous operator from* *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) *to*
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$.

#### Proof

Boundedness of *F* comes from (31) immediately. For the weak continuity, suppose that {*u*_{n}} is a sequence of *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) converging to *u* strongly, that is

$$\begin{array}{}{\displaystyle \underset{I}{\int}{\varrho}_{\theta (t)}({u}_{n}(t)-u(t))dt\to 0\phantom{\rule{thickmathspace}{0ex}}\text{as}\phantom{\rule{thickmathspace}{0ex}}n\to \mathrm{\infty}.}\end{array}$$

Thus there is a subsequence, say {*u*_{n}} itself satisfying

$$\begin{array}{}{\displaystyle \underset{n\to \mathrm{\infty}}{lim}{\varrho}_{\theta (t)}({u}_{n}(t)-u(t))=0\phantom{\rule{thickmathspace}{0ex}}\text{a.e. on}\phantom{\rule{thickmathspace}{0ex}}I.}\end{array}$$

From the boundedness of {*u*_{n}} in *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) and (31), we get the boundedness of {*F*(*u*_{n})} in
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$. Consequently, there is a subsequence, without loss of generality, assuming also {*u*_{n}} itself, and a function *ξ* ∈
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$, such that *F*(*u*_{n}) → *ξ* weakly, or equivalently

$$\begin{array}{}{\displaystyle \underset{n\to \mathrm{\infty}}{lim}\underset{I}{\int}\u3008f(t,{u}_{n}(t))-\xi (t),v(t){\u3009}_{\theta (t)}dt=0}\end{array}$$

for all *v* ∈ *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}), which in turn yields

$$\begin{array}{}{\displaystyle \underset{n\to \mathrm{\infty}}{lim}\underset{E}{\int}\u3008f(t,{u}_{n}(t))-\xi (t),v{\u3009}_{\theta (t)}dt=0}\end{array}$$(32)

for all measurable subsets *E* of *I* and all elements *v* of *X*_{α+} since 𝓢(*I*, *X*_{α+}) is dense in *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}) by *H*(*A*).

On the other hand, since

$$\begin{array}{}{\displaystyle \underset{n\to \mathrm{\infty}}{lim}\u3008f(t,{u}_{n}(t)),v{\u3009}_{\theta (t)}=\u3008f(t,u(t)),v{\u3009}_{\theta (t)}}\end{array}$$

a.e. on *I*, for each *ε* > 0, by Egorov’s theorem, there is a measurable set *E*_{ε} ⊆ *I* with |*I* ∖ *E*_{ε}| < *ε* verifying

$$\begin{array}{}{\displaystyle \u3008f(t,{u}_{n}(t)),v{\u3009}_{\theta (t)}\to \u3008f(t,u(t)),v{\u3009}_{\theta (t)}\phantom{\rule{thickmathspace}{0ex}}\text{uniformly on}\phantom{\rule{thickmathspace}{0ex}}{E}_{\epsilon}}\end{array}$$

as *n* → ∞, consequently for any subset *E* of *E*_{ε}, we have

$$\begin{array}{}{\displaystyle \underset{n\to \mathrm{\infty}}{lim}\underset{E}{\int}\u3008f(t,{u}_{n}(t))-f(t,u(t)),v{\u3009}_{\theta (t)}dt=0.}\end{array}$$(33)

Putting (32) and (33) together with the same subset *E*, we obtain

$$\begin{array}{}{\displaystyle \underset{E}{\int}\u3008f(t,u(t))-\xi (t),v{\u3009}_{\theta (t)}dt=0,}\end{array}$$

which implies

$$\begin{array}{}{\displaystyle \u3008f(t,u(t))-\xi (t),v{\u3009}_{\theta (t)}=0}\end{array}$$(34)

a.e. on *E*_{ε} and eventually a.e. on *I* by the arbitrariness of *E* and *ε* respectively.

Suppose that {*v*_{k}} is a dense and countable subset of *X*_{α+}, then there is a subset *E*_{0} of *I* with zero complement on which (34) holds with *v* replaced by *v*_{k} for all *k* ∈ ℕ. Finally, using the density of *X*_{α+} in *X*_{θ(t)}, we deduce that *f*(*t*, *u*(*t*)) = *ξ*(*t*) in
$\begin{array}{}{\displaystyle {X}_{\theta (t)}^{\ast}}\end{array}$ on *E*_{0}. Therefore *F*(*u*) = *ξ* and *F*(*u*_{n}) ⇀ *F*(*u*) in
$\begin{array}{}{\displaystyle {L}^{{\varrho}_{\theta (\cdot )}^{\ast}}(I,{X}_{\theta (\cdot )}^{\ast})}\end{array}$ as *n* → ∞. Thus the proof has been completed.□

Putting Theorem 5.8 and Proposition 5.9 together, we can easily see that the composite operator

$$\begin{array}{}{\displaystyle \mathcal{F}:=({D}_{\theta (\cdot )}^{2}+\mathrm{\partial}{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}{)}^{-1}\circ F:{L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})\to {L}^{{\varrho}_{\theta (\cdot )}}(I,{X}_{\theta (\cdot )})}\end{array}$$

is both continuous and compact. Moreover, we have

#### Theorem 5.10

*Under all the assumptions mentioned above*, *there is an* *μ*_{0} > 0 *such that as long as* 0 ≤ *μ* ≤ *μ*_{0}, 𝓕 *has a fixed point in* *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}), *or in other words*, *second order differential inclusion*

$$\begin{array}{}{\displaystyle -\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (t)}({u}^{\prime}(t))+\mathrm{\partial}{\phi}_{\vartheta (t)}(u(t))\ni f(t,u(t))\phantom{\rule{thickmathspace}{0ex}}a.e.\text{\hspace{0.17em}}on\phantom{\rule{thickmathspace}{0ex}}I}\end{array}$$(35)

*has a weak solution in* 𝓦.

#### Proof

Consider the set

$$\begin{array}{}{\displaystyle S=\{u\in \mathcal{W}:u=\lambda \mathcal{F}(u)\phantom{\rule{thickmathspace}{0ex}}\text{for some}\phantom{\rule{thickmathspace}{0ex}}0<\lambda <1\}.}\end{array}$$

Evidently, every member of *S* is a weak solution of the inclusion

$$\begin{array}{}{\displaystyle -\frac{d}{dt}\mathrm{\partial}{\varrho}_{\theta (t)}({\lambda}^{-1}{u}^{\prime}(t))+\mathrm{\partial}{\phi}_{\vartheta (t)}({\lambda}^{-1}u(t))\ni f(t,u(t))\phantom{\rule{thickmathspace}{0ex}}\text{a.e. on}\phantom{\rule{thickmathspace}{0ex}}I.}\end{array}$$

By multiplying both sides of the above inclusion by λ^{–1} *u*′(*t*), and integrate on *I*, then using the assumption (31), we can deduce that

$$\begin{array}{}{\displaystyle {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({\lambda}^{-1}{u}^{\prime})+{\stackrel{~}{\mathit{\Phi}}}_{{\phi}_{\vartheta (\cdot )}}({\lambda}^{-1}u)\le \u3008\u3008F(u),{\lambda}^{-1}u\u3009{\u3009}_{\theta (\cdot )}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\le {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}^{\ast}}({\delta}^{-1}F(u))+{\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}(\delta {\lambda}^{-1}u)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}\le \sigma ({\delta}^{-1})[\mu {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({\lambda}^{-1}u)+\parallel h{\parallel}_{{L}^{1}(I)}]+\delta {\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({\lambda}^{-1}u)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}=(\delta +\mu \sigma ({\delta}^{-1})){\mathit{\Phi}}_{{\varrho}_{\theta (\cdot )}}({\lambda}^{-1}u)+\parallel h{\parallel}_{{L}^{1}(I)}.}\end{array}$$

By taking *δ*_{0} > 0 and *μ*_{0} > 0 as in Remark 5.6 and letting 0 < *δ* ≤ *δ*_{0}, 0 ≤ *μ* ≤ *μ*_{0}, we can claim that {λ^{–1}*u* : *u* ∈ *S*} is bounded in *L*^{ϱθ(⋅)}(*I*, *X*_{θ(⋅)}). Therefore, there is a constant *C* > 0 independent of λ such that ∥*u*∥_{Lϱθ(⋅)(I, Xθ(⋅))} ≤ *C* for all *u* ∈ *S*. Finally by invoking Leray-Schauder’s alternative theorem for the compact and strongly continuous operators (refer to [18, Ch. 13] or [19]), we can assert the existence of the fixed point of 𝓕.□

At the end of this paper, let us choose an anisotropic elliptic partial differential equation of second order to illustrate our results,

$$\begin{array}{}\left\{\begin{array}{l}-{D}_{t}(|{D}_{t}u{|}^{p(x,t)-2}{D}_{t}u)-\sum _{i=1}^{N}{D}_{i}(|{D}_{i}u{|}^{{q}_{i}(x,t)-2}{D}_{i}u)\\ \phantom{\rule{1em}{0ex}}=\mu g(t,x,u),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}(t,x)\in I\times \mathit{\Omega},\\ u(0,x)=u(T,x),\phantom{\rule{1em}{0ex}}x\in \mathit{\Omega},\\ u(t,x)=0,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}(t,x)\in I\times \mathrm{\partial}\mathit{\Omega}.\end{array}\right.\end{array}$$(36)

*Here D*_{t} denotes the partial differential derivative with respect to *t*, *p* and *q*_{i} are doubly variable exponents introduced in Example 3.22 and Remark 4.13 respectively.

Suppose that *g* : *I* × *Ω* × ℝ → ℝ is a Caratheodory function with a nonstandard growth, i.e. for a.e. (*t*, *x*) ∈ *I* × *Ω*, *u* ↦ *g*(*t*, *x*, *u*) is continuous, and for all *u* ∈ ℝ, (*t*, *x*) ↦ *g*(*t*, *x*, *u*) is measurable, together with

$$\begin{array}{}{\displaystyle |g(t,x,u)|\le C(1+|u{|}^{p(t,x)-1})}\end{array}$$(37)

holding for a.e. (*t*, *x*) ∈ *I* × *Ω* and all *u* ∈ ℝ.

Let *f*(*t*, *u*)(*x*) = *g*(*t*, *x*, *u*(*t*, *x*)). Since *L*^{0}(*I*, *L*^{0}(*Ω*)) is equivalent to *L*^{0}(*Q*), one can easily check that *f*(*t*, *u*) lies in *L*^{0}(*I*, *L*^{0}(*Ω*)) provided *u* does. Moreover, combining (37) with the continuity of *g*(*t*, *x*, *u*) with respect to *u*, we can find that *f*(*t*, *u*) ∈ *L*^{0}(*I*, *L*^{p̄*}(*Ω*)) for all *u* ∈ *L*^{0}(*I*, *L*^{p̄}(*Ω*)), and *f*(*t*, *u*) is weakly continuous from *L*^{p(t, x)}(*Ω*) to *L*^{p′(t, x)}(*Ω*) for a.e. *t* ∈ *I*, together with (31) verified.

Suppose that *p*(*t*, *x*) and *q*_{i}(*t*, *x*) (*i* = 1, 2, ⋯, *N*) are variable exponents as in Example 3.22 and Remark 4.13 fulfilling

$$\begin{array}{}{\displaystyle \underset{x\in \mathit{\Omega}}{inf}({q}_{i}^{\ast}(t,x)-p(t,x))>0,\phantom{\rule{thickmathspace}{0ex}}i=1,2,\cdots ,N.}\end{array}$$(38)

Using the notations and definitions in Examples 3.22, 4.11 and Remark 4.13, from equation (36), we then derive an abstract evolution equation (35) on the space

$$\begin{array}{}{\displaystyle \mathcal{W}={W}_{\text{per}}^{1,{\varrho}_{p(\cdot )}}(I,{L}^{p(\cdot ,x)}(\mathit{\Omega}))\cap {L}^{{\varrho}_{\mathbf{q}(\cdot )}}(I,{W}_{0}^{1,\mathbf{q}(\cdot ,x)}(\mathit{\Omega})).}\end{array}$$

From all the assumptions of Example 3.22 and Remark 4.13 together with (38), one can easily verify all the conditions listed in Theorem 5.3 as well as (25). Thus in terms of Theorem 5.10, equation (35) has a solution *u* in 𝓦, whose scalar version, still denoted by *u*, solves (36) in the sense of distribution, i.e.

$$\begin{array}{}{\displaystyle \underset{Q}{\int}(|{D}_{t}u{|}^{p(x,t)-2}{D}_{t}u){D}_{t}vdxdt+\sum _{i=1}^{N}\underset{Q}{\int}(|{D}_{i}u{|}^{{q}_{i}(x,t)-2}{D}_{i}u){D}_{i}vdxdt=\underset{Q}{\int}f(t,u)v(t,x)dxdt}\end{array}$$

for all *v* ∈
$\begin{array}{}{\displaystyle {C}_{\text{per}}^{1}}\end{array}$(*I*, *C*^{∞}(*Ω*)). In conclusion, under all the conditions upon *h*, *p* and *q*_{i} (*i* = 1, 2, ⋯, *N*) including (38), there is a *μ*_{0} > 0 such that for all 0 ≤ *μ* ≤ *μ*_{0}, Equation (36) has a weak solution.

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