Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator

TL;DR

This paper establishes the continuous differentiability of solutions to a nonlinear fractional elliptic boundary value problem with respect to functional parameters, using a global implicit function theorem.

Contribution

It introduces a novel sensitivity analysis framework for nonlinear fractional elliptic systems with Dirichlet boundary conditions.

Findings

Existence and uniqueness of solutions for all functional parameters.

Continuous differentiability of solutions with respect to parameters.

Application of a global implicit function theorem to fractional elliptic problems.

Abstract

In the paper, we derive a sensitivity result for a nonlinear fractional ordinary elliptic system on a bounded interval with Dirichlet boundary conditions. More precisely, using a global implicit function theorem, we show that, for any functional parameter, there exists a unique solution to such a problem and dependence of solutions on functional parameters is continuously differentiable.

Equations

$$(-\Delta {)}^{\beta }x(t)=f(t,x(t),u(t)),t\in (0,\pi )\text{ a.e.,}$$

$$u⟼x_u$$

$$\lambda (-\Delta {)}^{\frac{\sigma }{2}}x(t)+h(t,x(t))=(-\Delta {)}^{\frac{\sigma }{2}}z(t),t\in (0,1),$$

$$x(t)=0,t\in (-\infty ,-1]\cup [1,\infty ).$$

$${\sum }_{i,j=0}^k{\alpha }_i{\alpha }_j[(-\Delta {)}_{\omega }{]}^{{\beta }_i+{\beta }_j}u(x)=D_{u}F(x,u(x)),x\in \Omega \text{ a.e.},$$

$$\mathcal{B}\ni P⟼E(P)x\in H$$

$$({\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ))x={\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda )x$$

$${\int }_{-\infty }^{\infty }{\left|b(\lambda )\right|}^2{\Vert E(d\lambda )x\Vert }^2<\infty$$

$$\lim {\int }_{-\infty }^{\infty }{b}_n(\lambda )E(d\lambda )x$$

$$b_{n}:\mathbb{R\ni\lambda\longmapsto}\left\{\begin{array}[c]{ccc}b(\lambda)&\text{as}&\left|b(\lambda)\right|\leq n\\ 0&\text{as}&\left|b(\lambda)\right|>n\end{array}\right..$$

$${\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ):D\subset H\to H$$

$$({\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ))x=\lim {\int }_{-\infty }^{\infty }{b}_n(\lambda )E(d\lambda )x.$$

$$\lim {\int }_{-\infty }^{\infty }{b}_n(\lambda )E(d\lambda )x={\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda )x$$

$${\Vert ({\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ))x\Vert }^2={\int }_{-\infty }^{\infty }{\left|b(\lambda )\right|}^2{\Vert E(d\lambda )x\Vert }^2.$$

$$({\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ){)}^{\ast }={\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda ),$$

$${\int }_{\sigma }b(\lambda )E(d\lambda )$$

$${\int }_{-\infty }^{\infty }{\chi }_{\sigma }(\lambda )b(\lambda )E(d\lambda )$$

$$A={\int }_{-\infty }^{\infty }\lambda E(d\lambda )={\int }_{\sigma (A)}\lambda E(d\lambda ).$$

$$b(A)={\int }_{-\infty }^{\infty }b(\lambda )E(d\lambda )={\int }_{\sigma (A)}b(\lambda )E(d\lambda ).$$

$$\sigma (b(A))=\overline{b(\sigma (A))}$$

$$(b\cdot d)(A)\supset b(A)\circ d(A)$$

$$(b\cdot d)(A)=b(A)\circ d(A)$$

$$D((b\cdot d)(A))\subset D(d(A)).$$

$$(b(A){)}^n=b^n(A).$$

$$A^n={\int }_{-\infty }^{\infty }{\lambda }^{n}E(d\lambda ).$$

$$I={\int }_{-\infty }^{\infty }1E(d\lambda ).$$

$$b:\mathbb{R}\ni\lambda\rightarrow\left\{\begin{array}[c]{ccc}0&;&\lambda<0\\ \lambda^{\frac{\beta}{2}}&;&\lambda\geq 0\end{array}\right.$$

$$A^{\frac{\beta }{2}}\circ A^{\frac{\beta }{2}}=A^{\beta }.$$

$$-\Delta :H_0^1\cap H^2\subset L^2\to L^2$$

$$-\Delta x(t)=-x^{{}^{\prime \prime }}(t)$$

Full text

2010 Mathematics Subject Classification. 34B15, 34A08, 34B08, 34L05.

Key words. Fractional Dirichlet-Laplace operator, Stone-von Neumann operator calculus, global implicit function theorem, Palais-Smale condition

1 Introduction

In our paper, we study a nonlinear ordinary boundary value problem on the interval $(0,\pi)$, involving a Dirichlet-Laplace operator $(-\Delta)^{\beta}$ of order $\beta>\frac{1}{2}$,

[TABLE]

where $(-\Delta):H_{0}^{1}\cap H^{2}\rightarrow L^{2}$ is the Dirichlet-Laplace operator, $f:(0,\pi)\times\mathbb{R}^{m}\times\mathbb{R}^{r}\rightarrow\mathbb{R}^{m}$, $x$ is an unknown function and $u$ - a functional parameter.

Problems involving fractional Laplacians are extensively investigated in resent years due to their numerous applications, among others in probability, fluid mechanics, hydrodynamics (see, for example, [3], [4], [8], [9], [15] and references therein).

Definition of the fractional Laplacian adopted in our paper comes from the Stone-von Neumann operator calculus and is based on the spectral integral representation theorem for a self-adjoint operator in Hilbert space. It reduces to a series form which is taken by other authors as a definition ([3], [6], [8]). Our more general approach allows us to obtain useful properties of this fractional operator in a smart way. This approach has also been used in [12].

In the first part of the paper, we recall some facts from the theory of spectral integral and Stone-von Neumann operator calculus. Next, we derive some properties of positive powers of the ordinary Dirichlet-Laplace operator and their domains (among other, some embedding theorems). In the second part, we use a global implicit function theorem ([10], [11]) to prove existence and uniqueness of a solution to problem (1) as well as its sensitivity. By sensitivity we mean continuous differentiability of the mapping

[TABLE]

where $x_{u}$ is a unique solution to the problem, corresponding to a parameter $u$. This property can be used in optimal control for system (1).

Similar method but based on a global diffeomorphism theorem ([13]) and applied to a nonlinear integral Hammerstein equation is presented in [5] with an application to the problem

[TABLE]

with the exterior Dirichlet boundary condition

[TABLE]

In [6], the problem of type (1) on a bounded Lipschitzian domain $\Omega\subset\mathbb{R}^{n}$ ($n\geq 2$) and with an exterior Dirichlet boundary condition, is studied. Continuous dependence of solutions on parameters (stability) is investigated therein.

In [12], using a variational method, we derive an existence result for the so-called bipolynomial fractional Dirichlet-Laplace problem

[TABLE]

where $\alpha_{i}>0$ for $i=0,...,k$ ($k\in\mathbb{N}\cup\{0\}$) and $0\leq\beta_{0}<\beta_{1}<...<\beta_{k}$, $(-\Delta)_{\omega}:D((-\Delta)_{\omega})\subset L^{2}\rightarrow L^{2}$ is a weak Dirichlet-Laplace operator, $\Omega\subset\mathbb{R}^{N}$ is a bounded open set, $F:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$, $D_{u}F$ is the partial derivative of $F$ with respect to $u$.

2 Integral representation of a self-adjoint operator

Results presented in this section can be found, in the case of complex Hilbert space, for example in [1], [14]. Their proofs can be moved without any or with small changes to the case of real Hilbert space. In the next, we shall deal only with real Hilbert spaces. Such a preliminary section has also been included in the paper [12].

Let $H$ be a real Hilbert space with a scalar product $\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{R}$. Let us denote by $\Pi(H)$ the set of all projections of $H$ on closed linear subspaces and by $\mathcal{B}$ - the $\sigma$-algebra of Borel subsets of $\mathbb{R}$. By the spectral measure in $\mathbb{R}$ we mean a set function $E:\mathcal{B}\rightarrow\Pi(H)$ that satisfies the following conditions:

• $\cdot$

for any $x\in H$, the function

[TABLE]

is a vector measure

• $\cdot$

$E(\mathbb{R})=I$

• $\cdot$

$E(P\cap Q)=E(P)\circ E(Q)$ for $P,Q\in\mathcal{B}$.

By a support of a spectral measure $E$ we mean the complement of the sum of all open subsets of $\mathbb{R}$ with zero spectral measure.

If $b:\mathbb{R}\rightarrow\mathbb{R}$ is a bounded Borel measurable function, defined $E$ - a.e., then the integral ${\displaystyle\int\nolimits_{-\infty}^{\infty}}b(\lambda)E(d\lambda)$ is defined by

[TABLE]

for any $x\in H$ where the integral ${\displaystyle\int\nolimits_{-\infty}^{\infty}}b(\lambda)E(d\lambda)x$ (with respect to the vector measure) is defined in a standard way, with the aid of the sequence of simple functions converging $E(d\lambda)x$ - a.e. to $b$ (cf. [1]).

If $b:\mathbb{R}\rightarrow\mathbb{R}$ is an unbounded Borel measurable function, defined $E$ - a.e., then, for any $x\in H$ such that

[TABLE]

(the above integral is taken with respect to the nonnegative measure $\mathcal{B\ni}P\longmapsto\left\|E(P)x\right\|^{2}\in\mathbb{R}_{0}^{+}$), there exists the limit

[TABLE]

of integrals (with respect to the vector measure (2)) where

[TABLE]

Let us denote the set of all points $x$ with property (3) by $D$. One proves that $D$ is dense linear subspace of $H$ and by ${\displaystyle\int\nolimits_{-\infty}^{\infty}}b(\lambda)E(d\lambda)$ one denotes the operator

[TABLE]

given by

[TABLE]

Of course, $D=H$ and

[TABLE]

when $b:\mathbb{R}\rightarrow\mathbb{R}$ is a bounded Borel measurable function, defined $E$ - a.e.

For $x\in D$, we have

[TABLE]

Moreover,

[TABLE]

i.e. the operator ${\displaystyle\int\nolimits_{-\infty}^{\infty}}b(\lambda)E(d\lambda)$ is self-adjoint.

Remark 2.1

To integrate a Borel measurable function $b:B\rightarrow\mathbb{R}$ where $B$ is a Borel set containing the support of the measure $E$, it is sufficient to extend $b$ on $\mathbb{R}$ to a whichever Borel measurable function (putting, for example, $b(\lambda)=0$ for $\lambda\notin B$).

If $b:\mathbb{R}\rightarrow\mathbb{R}$ is Borel measurable and $\sigma\in\mathcal{B}$, then by the integral

[TABLE]

we mean the integral

[TABLE]

where $\chi_{\sigma}$ is the characteristic function of the set $\sigma$ (111Integral ${\displaystyle\int\nolimits_{\sigma}}b(\lambda)E(d\lambda)$ can be also defined with the aid of the restriction of $E$ to the set $\sigma$).

Next theorem plays the fundamental role in the spectral theory of self-adjoint operators.

Theorem 2.1

If $A:D(A)\subset H\rightarrow H$ is self-adjoint and the resolvent set $\rho(A)$ is non-empty, then there exists a unique spectral measure $E$ with the closed support $\Lambda=\sigma(A)$, such that

[TABLE]

The basic notion in the Stone-von Neumann operator calculus is a function of a self-adjoint operator. Namely, if $A:D(A)\subset H\rightarrow H$ is self-adjoint and $E$ is the spectral measure determined according to the above theorem, then, for any Borel measurable function $b:\mathbb{R}\rightarrow\mathbb{R}$, one defines the operator $b(A)$ by

[TABLE]

It is known that the spectrum $\sigma(b(A))$ of $b(A)$ is given by

[TABLE]

provided that $b$ is continuous (it is sufficient to assume that $b$ is continuous on $\sigma(A)$).

We have the following general result.

Proposition 2.1

If $b,d:\mathbb{R}\rightarrow\mathbb{R}$ are Borel measurable functions and $E$ is the spectral measure for a self-adjoint operator $A:D(A)\subset H\rightarrow H$ with non-empty resolvent set, then

[TABLE]

and

[TABLE]

if and only if

[TABLE]

Using the above proposition one can deduce that, for any $n\in\mathbb{N}$, $n\geq 2$, and a Borel measurable function $b:\mathbb{R}\rightarrow\mathbb{R}$,

[TABLE]

When $b(\lambda)=\lambda$, (7) gives

[TABLE]

If $n=1$, then (8) follows from Theorem 2.1. Since $E(\mathbb{R})=I$, therefore the identity operator $I$ can be written as

[TABLE]

If $\beta>0$, then formula (7) with

[TABLE]

and $n=2$ implies the following proposition (cf. Remark 2.1).

Proposition 2.2

If $\sigma(A)\subset[0,\infty)$, then

[TABLE]

3 Fractional Dirichlet-Laplace operator

Consider the one-dimensional Dirichlet-Laplace operator on the interval $(0,\pi)$

[TABLE]

given by

[TABLE]

where $H_{0}^{1}=H_{0}^{1}((0,\pi),\mathbb{R}^{m})$, $H^{2}=H^{2}((0,\pi),\mathbb{R}^{m})$ are classical Sobolev spaces and $L^{2}=L^{2}((0,\pi),\mathbb{R}^{m})$. In an elementary way, one can check that this operator is self-adjoint,

[TABLE]

($\sigma_{p}(-\Delta)$ is the pointwise spectrum of $(-\Delta)$) and the eigenspace $N(j^{2})$ corresponding to the eigenvalue $\lambda_{j}=j^{2}$ is the set $\{c\sin jt;\ c\in\mathbb{R}^{m}\}$. The system of functions

[TABLE]

is the hilbertian basis (complete ortonormal system) in $L^{2}$.

Now, let us fix any $\beta>0$ and consider the operator

[TABLE]

where

[TABLE]

(here $E$ is the spectral measure given by Theorem 2.1 for the operator $(-\Delta)$, $a_{j}\sqrt{\frac{2}{\pi}}\sin jt$ is the projection of $x$ on the $m$-dimensional eigenspace $N(j^{2})$ of the operator $(-\Delta)$ and

[TABLE]

for $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt\in D((-\Delta)^{\beta})$ (222The series is meant in $L^{2}$ but from the Carleson theorem it follows that $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt$ a.e. on $(0,\pi)$ (cf. [7, Theorem 5.17]).).

Equality (5) and the fact that isolated points of the spectrum of a self-adjoint operator are the eigenvalues imply that

[TABLE]

The corresponding eigenspaces for $(-\Delta)$ and $(-\Delta)^{\beta}$ are the same (it follows from a general result concerning the power of any self-adjoint operator).

The operator $(-\Delta)^{\beta}$ will be called the Dirichlet-Laplace operator of order $\beta$, and the function $(-\Delta)^{\beta}x$ - the Dirichlet-Laplacian of order $\beta$ of $x$.

We also have

Lemma 3.1

$D((-\Delta)^{\beta})$* with the scalar product*

[TABLE]

is the Hilbert space.

Proof. The assertion follows from the fact that the operator $(-\Delta)^{\beta}$ being self-adjoint (cf. (4)) is closed.

The scalar product $\left\langle\cdot,\cdot\right\rangle_{\beta}$ and the scalar product

[TABLE]

generate equivalent norms in $D((-\Delta)^{\beta})$. Indeed, it is sufficient to observe that the following Poincare inequality holds true:

[TABLE]

for any $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt\in D((-\Delta)^{\beta})$. In the next, we shall consider $D((-\Delta)^{\beta})$ with the norm $\left\|\cdot\right\|_{\thicksim\beta}$.

3.1 Embeddings

From the description of the domain $D((-\Delta)^{\beta})$ it follows that

[TABLE]

for any $0<\beta_{1}<\beta_{2}$. Using this relation and equality (8) with $A=(-\Delta)$ we assert that

[TABLE]

for any $\beta>0$ ($C_{c}^{\infty}=C_{c}^{\infty}((0,\pi),\mathbb{R}^{m})$ is the set of smooth functions with the supports contained in $(0,\pi)$).

We also have the following three lemmas.

Lemma 3.2

If $\beta>\frac{1}{4}$, then

[TABLE]

and this embedding is continuous, more precisely,

[TABLE]

for $x\in D((-\Delta)^{\beta})$, where $\zeta(4\beta)$ is the value of the Riemann zeta function $\zeta(\gamma)={\displaystyle\sum\nolimits_{j=1}^{\infty}}\frac{1}{j^{\gamma}\text{ }}$ at $\gamma=4\beta$.

Proof. Let $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt\in D((-\Delta)^{\beta})$. Since

[TABLE]

and $\beta>\frac{1}{4}$, therefore, for $t\in(0,\pi)$ a.e., we have

[TABLE]

and the proof is completed.

Lemma 3.3

If $\beta\geq\frac{1}{2}$, then

[TABLE]

and, consequently,

[TABLE]

Proof. Of course (cf. (12)), it is sufficient to show that $D((-\Delta)^{\frac{1}{2}})\subset H_{0}^{1}$. Indeed, let $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt\in D((-\Delta)^{\frac{1}{2}})$ and consider this series on the interval $[0,\pi]$. The sequence $(S_{n})$ of partial sums converges in $L^{2}$ to $x$. From the convergence of the series ${\displaystyle\sum\nolimits_{j=1}^{\infty}}j^{2}a_{j}^{2}$ it follows that the sequence $(S_{n}^{\prime})$ of derivatives converges in $L^{2}$ to a function. So (cf. [7]), one can choose a subsequence $(S_{n_{k}}^{\prime})$ convergent a.e. on $[0,\pi]$ to this function and bounded pointwise a.e. on $[0,\pi]$ by a function $g\in L^{2}$. Consequently, the sequence $(S_{n_{k}}^{\prime})$ is equiabsolutely integrable on $[0,\pi]$. So, the sequence $(S_{n_{k}})$ is equiabsolutely continuous on $[0,\pi]$. Of course, $S_{n_{k}}(0)=0$, thus

[TABLE]

for $t\in[0,\pi]$. It means that elements of the sequence $(S_{n_{k}})$ satisfy the assumptions of the Ascoli-Arzela theorem for absolutely continuous functions and, in consequence, there exists a subsequence $(S_{n_{k_{i}}})$ converging uniformly on $[0,\pi]$ to an absolutely continuous function $\overline{x}$. Clearly, $(S_{n_{k_{i}}})$ converges to $\overline{x}$ in $L^{2}$. The uniqueness of the limit in $L^{2}$ means that $x=\overline{x}$ a.e. on $(0,\pi)$. So, $x$ has a representative which is absolutely continuous on $[0,\pi]$ and satisfies Dirichlet boundary conditions, i.e. $x\in W_{0}^{1,1}((0,\pi),\mathbb{R}^{m})$ (the classical Sobolev space). Consequently, there exists a function $g\in L^{1}$ such that

[TABLE]

for any $\varphi\in C_{c}^{\infty}$. But

[TABLE]

for $\varphi\in C_{c}^{\infty}$ (the last equality follows from the fact that ${\displaystyle\sum\nolimits_{j=1}^{\infty}}j^{2}a_{j}^{2}<\infty$ and, consequently, ${\displaystyle\sum\nolimits_{n=1}^{\infty}}ja_{j}\sqrt{\frac{2}{\pi}}\cos jt\in L^{2}$). Thus,

[TABLE]

and, finally, $x\in H_{0}^{1}$.

The second part of the theorem follows from known property of Sobolev space $W^{1,1}((0,\pi),\mathbb{R}^{m})$.

Lemma 3.4

If $\beta>\frac{3}{4}$, then any bounded the set $B\subset D((-\Delta)^{\beta})$ is equicontinuous on $[0,\pi]$.

Proof. Similarly as in the proof of Lemma 3.2 we obtain

[TABLE]

for $t_{1},t_{2}\in(0,\pi)$ a.e., where $x(t)={\displaystyle\sum\nolimits_{j=1}^{\infty}}a_{j}\sqrt{\frac{2}{\pi}}\sin jt\in D((-\Delta)^{\beta})$. Identifying $x$ with its absolutely continuous representative on $[0,\pi]$ we assert that the above estimation holds true for all $t_{1},t_{2}\in[0,\pi]$.

Using Lemmas 3.2, 3.3, 3.4 we obtain

Corollary 3.1

If $\beta>\frac{3}{4}$, then the embedding

[TABLE]

is compact.

3.2 Equivalence of equations

Fact that the operator $(-\Delta)^{\beta}$ ($\beta>0$) is self-adjoint means that its domain satisfies the equality

[TABLE]

and

[TABLE]

for $x\in D((-\Delta)^{\beta})$.

From (9) it follows that $x\in D((-\Delta)^{\beta})$ if and only if $x\in D((-\Delta)^{\frac{\beta}{2}})$, $(-\Delta)^{\frac{\beta}{2}}x\in D((-\Delta)^{\frac{\beta}{2}})$ and, in such a case,

[TABLE]

Using this fact and (13), (14), we obtain

Lemma 3.5

If $\beta>0$ and $g\in L^{2}$, then $x\in D((-\Delta)^{\beta})$ and

[TABLE]

if and only if $x\in D((-\Delta)^{\frac{\beta}{2}})$ and

[TABLE]

for any $y\in D((-\Delta)^{\frac{\beta}{2}})$.

4 Global implicit function theorem

Let $X$ be a real Banach space and $I:X\rightarrow\mathbb{R}$ - a functional of class $C^{1}$. We say that $I$ satisfies Palais-Smale (PS) condition if any sequence $(x_{k})$ such that

• $\cdot$

$\left|I(x_{k})\right|\leq M$ for all $k\in\mathbb{N}$ and some $M>0,$

• $\cdot$

$I^{\prime}(x_{k})\longrightarrow 0,$

admits a convergent subsequence ($I^{\prime}(x_{k})$ denotes the Frechet differential of $I$ at $x_{k}$). A sequence $(x_{k})$ satisfying the above conditions is called the (PS) sequence for $I$.

We have (cf. [10], [11])

Theorem 4.1

Let $X$, $U$ be real Banach spaces, $H$ - a real Hilbert space. If $F:X\times U\rightarrow H$ is continuously differentiable with respect to $(x,u)\in X\times U$ and

• $\cdot$

for any $u\in U$, the functional

[TABLE]

satisfies (PS) condition

• $\cdot$

$F_{x}^{\prime}(x,u):X\rightarrow H$* is bijective for any $(x,u)\in X\times U$,*

then there exists a unique function $\lambda:U\rightarrow X$ such that $F(\lambda(u),u)=0$ for any $u\in U$ and this function is of class $C^{1}$ with differential $\lambda^{\prime}(u)$ at $u$ given by

[TABLE]

5 A boundary value problem

Let us consider boundary value problem (1). Using the global implicit function theorem, we shall show that (under suitable assumptions) this problem has a unique solution $x_{u}\in D((-\Delta)^{\beta})$ corresponding to any fixed $u\in L_{r}^{\infty}=L^{\infty}((0,\pi),\mathbb{R}^{r})$ and the mapping

[TABLE]

is continuously differentiable.

Consider the mapping

[TABLE]

We shall formulate conditions guaranteeing that

• $\cdot$

$F$ is of class $C^{1}$

• $\cdot$

differential $F_{x}(x,u):D((-\Delta)^{\beta})\rightarrow L^{2}$ is bijective for any $(x,u)\in D((-\Delta)^{\beta})\times L_{r}^{\infty}$

• $\cdot$

for any $u\in L_{r}^{\infty}$, functional

[TABLE]

satisfies (PS) condition.

5.1 Smoothness of $F$

Assume that function $f$ is measurable in $t\in(0,\pi)$, continuously differentiable in $(x,u)\in\mathbb{R}^{m}\times\mathbb{R}^{r}$ and

[TABLE]

for $(t,x,u)\in(0,\pi)\times\mathbb{R}^{m}\times\mathbb{R}^{r}$, where $a,b\in L^{2}$ and $\gamma,\delta:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}$ are continuous functions. We have

Proposition 5.1

If $\beta>\frac{1}{4}$, then $F$ is of class $C^{1}$ and the differential $F^{\prime}(x,u):D((-\Delta)^{\beta})\times L_{r}^{\infty}\rightarrow L^{2}$ of $F$ at $(x,u)$ is given by

[TABLE]

for $(h,v)\in D((-\Delta)^{\beta})\times L_{r}^{\infty}$.

Proof. Smoothness of the first term of $F$ is obvious. So, let us consider the mapping

[TABLE]

We shall show that the mappings

[TABLE]

[TABLE]

are partial Frechet differentials of $G$ at $(x,u)$ and mappings

[TABLE]

[TABLE]

are continuous. Of course, it is sufficient to check the differentiability in Gateaux sense because continuity of the above two mappings implies that the Gateaux differentials are Frechet ones.

So, let us consider differentiability of $G$ with respect to $x$. Linearity and continuity of the mapping $G_{x}(x,u)$ are obvious (in view of Lemma 3.2). To prove that $G_{x}(x,u)$ is Gateaux differential of $G$ with respect to $x$, we shall show that

[TABLE]

for any sequence $(\lambda_{k})\subset(-1,1)$ such that $\lambda_{k}\rightarrow 0$. Indeed, the sequence of functions

[TABLE]

converges pointwise a.e. on $(0,\pi)$ to the zero function (by differentiability of $f$ in $x$). Moreover, from the mean value theorem it follows that this sequence is bounded by a function from $L^{2}$:

[TABLE]

where $s_{t,k}\in(0,1)$ and $const_{x,u,h}$ is a constant depending on $x,u,h$. Thus, using the Lebesgue dominated convergence theorem we assert that $G_{x}(x,u)$ is Gateaux differential of $G$ with respect to $x$.

In the same way, we check that $G_{u}(x,u)$ is Gateaux differential of $G$ with respect to $u$.

To finish the proof we shall show that the mappings (18), (19) are continuous. Really, let $(x_{k},u_{k})\rightarrow(x_{0},u_{0})$ in $D((-\Delta)^{\beta})\times L_{r}^{\infty}$. Then

[TABLE]

Consequently,

[TABLE]

Using Lemma 3.2, assumption (17) and the Lebesgue dominated convergence theorem we assert that $G_{x}(x_{k},u_{k})\rightarrow G_{x}(x_{0},u_{0})$ in $\mathcal{L}(D((-\Delta)^{\beta}),L^{2})$.

In a similar way, we check the continuity of the mapping

[TABLE]

The proof is completed.

5.2 Bijectivity of $F_{x}(x,u)$

In view of the previous theorem and its proof, it is clear that if $\beta>\frac{1}{4}$ and functions $f$, $f_{x}$ satisfy growth condition (17), then the partial differential of $F$ with respect to $x$ is of the form

[TABLE]

for any $(x,u)\in D((-\Delta)^{\beta})\times L_{r}^{\infty}$. We also have

Proposition 5.2

Assume that functions $f$, $f_{x}$ satisfy growth condition (17). If $\beta>\frac{1}{2}$ and one of the following conditions is satisfied

• a)

$\left\|\Lambda\right\|_{L_{m\times m}^{1}}<\frac{\pi}{2\zeta(2\beta)}$**

• b)

$\Lambda(t)\leq 0$, i.e. matrix $\Lambda(t)$ is nonpositive, for $t\in(0,\pi)$ a.e.

• c)

$\Lambda\in L_{m\times m}^{\infty}$* and $\left\|\Lambda\right\|_{\infty}<1$,*

where $\Lambda(t):=f_{x}(t,x(t),u(t))$, $L_{m\times m}^{p}=L^{p}((0,\pi),\mathbb{R}^{m\times m})$ for $p=1,\infty$, then differential $F_{x}(x,u):D((-\Delta)^{\beta})\rightarrow L^{2}$ is bijective ( 333By the norm of a matrix $C=[c_{i,j}]\in\mathbb{R}^{m\times m}$ we mean the value $({\displaystyle\sum\nolimits_{i,j=1}^{m}}\left|c_{i,j}\right|^{2})^{\frac{1}{2}}$.).

Remark 5.1

In Part c) one can assume that $\beta>\frac{1}{4}$. In such a case the proof of coercivity of $a$ remains unchanged and to show its continuity one estimates

[TABLE]

Proof of the Proposition. We shall show that, for any function $g\in L^{2}$, equation

[TABLE]

has a unique solution in $D((-\Delta)^{\beta})$. Using Lemma 3.5, we see that it is equivalent to show that there exists a unique function $h\in D((-\Delta)^{\frac{\beta}{2}})$ such that

[TABLE]

for any $y\in D((-\Delta)^{\frac{\beta}{2}})$. So, let us define a bilinear form $a:D((-\Delta)^{\frac{\beta}{2}})\times D((-\Delta)^{\frac{\beta}{2}})\rightarrow\mathbb{R}$ by

[TABLE]

This form is continuous. Indeed (cf. Lemma 3.2),

[TABLE]

for $h,y\in D((-\Delta)^{\frac{\beta}{2}})$. Moreover,

Part a.

[TABLE]

Part b.

[TABLE]

Part c.

[TABLE]

So, $a$ is coercive. From Lax-Milgram theorem it follows that for any linear continuous functional $l:D((-\Delta)^{\frac{\beta}{2}})\rightarrow\mathbb{R}$ there exists a unique $h\in D((-\Delta)^{\frac{\beta}{2}})$ such that

[TABLE]

for any $y\in D((-\Delta)^{\frac{\beta}{2}})$. Since the functional

[TABLE]

is linear and continuous, therefore there exists a unique $h\in D((-\Delta)^{\frac{\beta}{2}})$ such that

[TABLE]

for any $y\in D((-\Delta)^{\frac{\beta}{2}})$. The proof is completed.

5.3 (PS) condition

In the same way as in the proof of Proposition 5.2 one can show that, for any $\beta>0$ and any function $g\in L^{2}$, there exists a unique function $x_{g}\in D((-\Delta)^{\frac{\beta}{2}})$ such that

[TABLE]

for any $y\in D((-\Delta)^{\frac{\beta}{2}})$. It means, in view of Lemma 3.5, that the following lemma holds true.

Lemma 5.1

For any $\beta>0$ and any $g\in L^{2}$, there exists a unique solution $x_{g}\in D((-\Delta)^{\beta})$ of the equation

[TABLE]

Moreover, we have

Lemma 5.2

If $\beta>\frac{1}{2}$, then the operator

[TABLE]

is compact, i.e. the image of any bounded set in $L^{2}$ is relatively compact in $L^{2}$.

Proof. Since $x_{(g_{1},...,g_{m})}=(x_{g_{1}},...,x_{g_{m}})$ for any $(g_{1},...,g_{m})\in L^{2}$, therefore one can assume that $m=1$.

Let us recall the Kolmogorov-Frechet-Riesz theorem (cf. [7]): if $\mathcal{F}$ is a bounded set in $L^{p}(\mathbb{R}^{n})$ ($1\leq p<\infty$) and

[TABLE]

(here, $\tau_{h}f(x)=f(x+h)$), then $\mathcal{F\mid}_{\Omega}$ is relatively compact in $L^{p}(\Omega)$ for any measurable set $\Omega\subset\mathbb{R}^{n}$ with finite Lebesgue measure.

Let $G\subset L^{2}((0,\pi),\mathbb{R})$ be a set bounded by a constant $C$ in $L^{2}((0,\pi),\mathbb{R})$. Consider a function

[TABLE]

and the function

[TABLE]

(both equalities and convergences are meant in $L^{2}$ and, in view of the Carleson theorem, a.e. on $(0,\pi)$). Since

[TABLE]

i.e.

[TABLE]

therefore

[TABLE]

for $j\in\mathbb{N}$. Now, we shall show that the set of functions $\{\widetilde{x}_{g};\ g\in G\}$ where

[TABLE]

satisfies condition (21) (of course, it is bounded in $L^{2}(\mathbb{R},\mathbb{R})$). Let us fix $0<h<\pi$ and consider the integral

[TABLE]

The first term of the above sum can be estimated in the following way (to obtain third inequality we use Hölder inequality for series)

[TABLE]

In the same way one can estimate third term of (22).

For the second term, we have

[TABLE]

If $-\pi<h<0$, we proceed in the same way.

Finally,

[TABLE]

for $\left|h\right|<\pi$. So, the set $\{\left.\widetilde{x}_{g}\right|_{(0,\pi)};\ g\in G\}=\{x_{g};\ g\in G\}$ is relatively compact in $L^{2}$. The proof is completed.

Using the above lemma we obtain

Lemma 5.3

If $\beta>\frac{1}{2}$ and $x_{k}\rightharpoonup x_{0}$ weakly in $D((-\Delta)^{\beta})$, then $x_{k}\rightarrow x_{0}$ strongly in $L^{2}$ and $(-\Delta)^{\beta}x_{k}\rightharpoonup(-\Delta)^{\beta}x_{0}$ weakly in $L^{2}$.

Proof. From the continuity of the linear operators

[TABLE]

[TABLE]

it follows that $x_{k}\rightharpoonup x_{0}$ weakly in $L^{2}$ and $(-\Delta)^{\beta}x_{k}\rightharpoonup(-\Delta)^{\beta}x_{0}$ weakly in $L^{2}$. Lemma 5.2 implies that the sequence $(x_{k})$ contains a subsequence $(x_{k_{i}})$ converging strongly in $L^{2}$ to a limit. Of course, this limit is the function $x_{0}$, i.e. $x_{k_{i}}\rightarrow x_{0}$ strongly in $L^{2}$. Supposing contrary and repeating the above argumentation we assert that $x_{k}\rightarrow x_{0}$ strongly in $L^{2}$.

Remark 5.2

Lemmas 5.2 and Lemma 5.3 are valid for any $\beta>0$. The proofs of such stronger results, in the case of bounded open set $\Omega\subset\mathbb{R}^{n}$ ($n\geq 1$), can be found in [12]. We give here weaker theorems for two reasons. First, to prove more general results (in fact, a counterpart of Lemma 5.2 because the proof of Lemma 5.3 remains unchanged) some additional considerations, concerning the spectral representation of the inverse operator, are needed. Second, due to the other assumptions (cf. Proposition 5.2) assumption $\beta>\frac{1}{2}$ in Theorem 6.1 can not be omitted.

The main tool that we shall use to prove that $\varphi$ satisfies (PS) condition is the following lemma.

Lemma 5.4

If $\beta>\frac{1}{4}$, $f$ satisfies the growth condition

[TABLE]

for $(t,x,u)\in(0,\pi)\times\mathbb{R}^{m}\times\mathbb{R}^{r}$, where $a,b\in L^{2}$, $\delta:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}$ is a continuous function and

[TABLE]

then, for any $u\in L_{r}^{\infty}$, the functional

[TABLE]

is coercive, i.e.

[TABLE]

Proof. We have

[TABLE]

But

[TABLE]

where $D=({\displaystyle\int\nolimits_{0}^{\pi}}\left|b(t)\right|^{2}(\delta(\left|u(t)\right|))^{2}dt)^{\frac{1}{2}}$. Thus,

[TABLE]

It means that $\varphi$ is coercive.

Now, we are in a position to prove that the functional $\varphi$ satisfies (PS) condition. Namely, we have

Proposition 5.3

If $\beta>\frac{1}{2}$, $f$ and $f_{x}$ satisfy the growth conditions

[TABLE]

[TABLE]

for $(t,x,u)\in(0,\pi)\times\mathbb{R}^{m}\times\mathbb{R}^{r}$, where $a,b\in L^{2}$ and $\gamma,\delta:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}$ are continuous functions, and (23) holds true, then $\varphi$ (with any fixed $u\in L_{r}^{\infty}$) satisfies (PS) condition.

Proof. From Proposition 5.1 it follows that $\varphi$ is of class $C^{1}$ and its differential $\varphi^{\prime}(x):D((-\Delta)^{\beta})\rightarrow\mathbb{R}$ is given by

[TABLE]

for $h\in D((-\Delta)^{\beta})$. Consequently, for $x_{k},x_{0}\in D((-\Delta)^{\beta})$, we have

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now, let $(x_{k})$ be a (PS) sequence for $\varphi$. Since $\varphi$ is coercive, therefore $(x_{k})$ is bounded in $D((-\Delta)^{\beta})$. So, one can choose a subsequence $(x_{k_{i}})$ weakly converging in $D((-\Delta)^{\beta})$ to some $x_{0}$. From Lemma 5.3 it follows that $x_{k_{i}}\rightarrow x_{0}$ strongly in $L^{2}$ and $(-\Delta)^{\beta}x_{k_{i}}(t)\rightharpoonup(-\Delta)^{\beta}x_{0}(t)$ weakly in $L^{2}$. Since the sequence $(x_{k_{i}})$ is bounded in $D((-\Delta)^{\beta})$, therefore it is bounded in $L_{m}^{\infty}$ and, consequently ($\beta>\frac{1}{2}$), in $C$. Moreover, there exists a subsequence of the sequence $(x_{k_{i}})$ (let us denote it by $(x_{k_{i}})$) converging to $x_{0}$ pointwise a.e. on $(0,\pi)$.

Term $\psi_{1}(x_{k_{i}})$ tends to zero. Indeed, functions $f_{x}(t,x_{k_{i}}(t),u(t))$, $k\in\mathbb{N}$, are equibounded on $(0,\pi)$ by a squared integrable function. Functions $f_{x}(t,x_{k_{i}}(t),u(t))(x_{0}(t)-x_{m_{k}}(t))$ belong to $L^{2}$ and converge pointwise (a.e. on $(0,\pi)$) to zero function. Moreover, they are equibounded on $(0,\pi)$ by a squared integrable function. So, from the Lebesgue dominated convergence theorem it follows that the sequence

[TABLE]

converges in $L^{2}$ to the zero function. Thus, in view of the weak convergence of the sequence $((-\Delta)^{\beta}x_{k})$ to $(-\Delta)^{\beta}x_{0}$ in $L^{2}$, $\psi_{1}(x_{k_{i}})\rightarrow 0$.

Similarly, $\psi_{l}(x_{k_{i}})\rightarrow 0$ for remaining $l$.

Finally, since

[TABLE]

[TABLE]

therefore

[TABLE]

i.e. $\varphi$ satisfies (PS) condition.

6 Final result

Thus, we have proved

Theorem 6.1

Assume that $\beta>\frac{1}{2}$, function $f$ is measurable in $t\in(0,\pi)$, continuously differentiable in $(x,u)\in\mathbb{R}^{m}\times\mathbb{R}^{r}$ and

[TABLE]

[TABLE]

for $(t,x,u)\in(0,\pi)\times\mathbb{R}^{m}\times\mathbb{R}^{r}$, where $a,b\in L^{2}$, $\gamma,\delta:\mathbb{R}_{0}^{+}\rightarrow\mathbb{R}_{0}^{+}$ are continuous functions and

[TABLE]

If, for any pair $(x,u)\in D((-\Delta)^{\beta})\times L_{r}^{\infty}$ one of the following assumptions is satisfied

• a)

$\left\|f_{x}(t,x(t),u(t))\right\|_{L_{m\times m}^{1}}<\frac{\pi}{2\zeta(2\beta)}$**

• b)

$f_{x}(t,x(t),u(t))\leq 0$* for $t\in(0,\pi)$ a.e.*

• c)

$f_{x}(t,x(t),u(t))\in L_{m\times m}^{\infty}$* and $\left\|f_{x}(t,x(t),u(t))\right\|_{\infty}<1$,*

then, for any $u\in L_{r}^{\infty}$, there exists a unique solution $x_{u}\in D((-\Delta)^{\beta})$ of problem (1) and the mapping

[TABLE]

is continuously differentiable with the differential $\lambda^{\prime}(u)$ at $u\in L_{r}^{\infty}$ such that, for any $v\in L_{r}^{\infty}$,

[TABLE]

for $t\in(0,\pi)$ a.e.

Remark 6.1

Thus, for any $u\in L_{r}^{\infty}$, $v\in D((-\Delta)^{\beta})$ the function $\lambda^{\prime}(u)v\in D((-\Delta)^{\beta})$ is a solution to the equation

[TABLE]

Example 6.1

Let $\beta>\frac{1}{2},\ m=2,\ r=2$. It is easy to see that the function

[TABLE]

satisfies assumptions of Theorem 6.1 (including a)) with

[TABLE]

where $a,b\in\mathbb{R}$ are such that

[TABLE]

Consequently, for any $u=(u_{1},u_{2})\in L_{2}^{\infty}$, there exists a unique solution $x_{u}\in D((-\Delta)^{\beta})$ of the problem

[TABLE]

for $t\in(0,\pi)$ a.e., and the mapping $\lambda(u)=(x_{u}^{1},x_{u}^{2})$ is continuously differentiable with the differential $\lambda^{\prime}(u):L_{2}^{\infty}\rightarrow D((-\Delta)^{\beta})$ such that

[TABLE]

for any $v\in L_{2}^{\infty}$, i.e.

[TABLE]

for$\ t\in(0,\pi)$ a.e., and any $v=(v_{1},v_{2})\in L_{2}^{\infty}.$