Differentiability with respect to initial functions of solutions to nonlinear hyperbolic functional differential systems

Abstract

A generalized Cauchy problem for nonlinear hyperbolic functional differential systems is considered. A theorem on the existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions of this system is proved by using a method of successive approximations. It is shown a result on the differentiability of solutions with respect to initial functions. This is the main result of the paper.

1 Introduction

For any metric spaces $U$ and $V$ we denote by $C(U, V)$ the class of all continuous functions from $U$ into $V$. We will use vectorial inequalities if the same inequalities hold between corresponding components. Suppose that $M \in C( [0,a], \mathbb {R}_+^n)$, $a > 0$, $\mathbb {R}_+ = [ 0, + \infty )$, $M$ is nondecreasing and $M(0) = 0_{[n]}$ where $0_{[n]}=(0,\ldots ,0) \in \mathbb {R}^n.$ Let $E$ be the Haar pyramid

$$\begin{aligned} E = \{(t,x)\in \mathbb {R}^{1+n}:\; t \in [0,a],\; -b + M(t) \le x \le b - M(t)\}, \end{aligned}$$

where $b\in \mathbb {R}^n$ and $b>M(a).$ Suppose that $b_0\in R_+$ and $M^-, M^+ \in C([-b_0, 0],R^n)$, $M^-=(M^-_1,\ldots ,M^-_n)$, $M^+ = (M^+_1,\ldots ,M^+_n )$ and $M^-(0) = -b$, $M^+(0) = b$ and $M^-(t) < M^+(t)$ for $t \in [-b_0, 0]$. Set $E_0 = \{ (t,x) \in \mathbb {R}^{1+n} \, : \, t\in [-b_0, 0], M^-(t) \le x \le M^+(t) \}.$ For $(t,x) \in E$ we define

$$\begin{aligned} D[t,x] = \{(\tau ,y)\in \mathbb {R}^{1+n}:\; \tau \le 0,\; (t+\tau , x+y) \in E_0 \cup E\}. \end{aligned}$$

Then $D[t,x] = D_0[t,x] \cup D_{\star }[t,x)$ where

$$\begin{aligned}&D_0[t,x] = \{ (\tau , y) \in \mathbb {R}^{1+n} \, : \, -b_0 -t \le \tau \le -t, \; -x + M^-(t+\tau ) \le y \le -x\nonumber \\&\qquad + M^+ (t+\tau )\},\\&D_{\star } [t,x]=\{(\tau , y ):\; - t \le \tau \le 0,\; - b - x + M (\tau + t)\le y\le b-x-M(\tau +t)\}. \end{aligned}$$

Set $b^- = (b^-_1,\ldots ,b^-_n )$, $b^+ = (b^+_1,\ldots ,b^+_n )$ where

$$\begin{aligned} b^-_i = \min \{ M^-_i(t) : t \in [-b_0, 0]\},\quad b^+_i = \max \{M^+_i(t) : t \in [-b_0, 0]\},\qquad 1 \le i \le n, \end{aligned}$$

and $B = [ - b_0 -a, 0] \times [ -b + b^-, b + b^+].$ Then $D[t,x] \subset B$ for $(t,x) \in E.$ Denote by $\mathbb {N}$ the set of natural numbers. Let $\mathbb {S}$ be the class of all sequences $p = \{p_k\}_{k\in \mathbb {N}}$ where $p_k \in \mathbb {R}$ for $k \in \mathbb {N}$. Write

$$\begin{aligned} E_{0.k} = (E_0 \cup E) \cap ([-b_0, a_k] \times \mathbb {R}^n),\quad k \in \mathbb {N}, \end{aligned}$$

where $a_k \ge 0$ for $k \in \mathbb {N}$ and $\kappa = \sup \{a_k \, : \, k \in \mathbb {N}\} < a$. For a function $z :E_0 \cup E \rightarrow \mathbb {S}$, $z = \{z_k\}_{k\in \mathbb {N}}$, and for a point $(t, x) \in E$ we define $z_{(t,x)} :D[t, x] \rightarrow \mathbb {S}$, $z_{(t,x)} = \{(z_k)_{(t,x)}\}_{k\in \mathbb {N}}$, by

$$\begin{aligned} (z_k)_{(t,x)}(\tau , y) = z_k(t + \tau , x + y),\quad (\tau , y) \in D[t, x],\ k \in \mathbb {N}. \end{aligned}$$

Then $z_{(t,x)}$ is the restriction of $z$ to the set $( E_0 \cup E)\cap ([-b_0, t] \times \mathbb {R}^n)$ and this restriction is shifted to the set $D[t,x]$.

Suppose the $\phi _0 : [0,a] \rightarrow \mathbb {R}$ and $\phi : E \rightarrow \mathbb {R}^n$, $\phi = ( \phi _1, \ldots ,\phi _n)$ are given functions. The requirements on $\phi _0$ and $\phi$ are that $0 \le \phi _0(t) \le t$ and $( \phi _0(t), \phi (t,x)) \in E$ for $(t,x) \in E.$ Write $\varphi (t,x) = ( \phi _0(t), \phi (t,x) )$ on $E$. Let $l^\infty$ be the class of all sequences $p = \{ p_k\}_{k \in \mathbb {N}}$ such that $\Vert p\Vert _\infty = \sup \{ |p_k| \, : \, k \in \mathbb {N}\} < \infty$. Set $\Omega = E \times C(B, l^\infty ) \times C(B, l^\infty ) \times \mathbb {R}^n$ and suppose that

$$\begin{aligned} F :\Omega \rightarrow \mathbb {S},\; F = \{F_k\}_{k\in \mathbb {N}},\text { and }\psi = \{\psi _k\}_{k\in \mathbb {N}}, \; \psi _k :E_{0.k} \rightarrow \mathbb {R}\quad \text {for } k \in \mathbb {N}, \end{aligned}$$

are given functions. We will say that $F$ satisfies condition $(V)$ if for each $(t,x, q) \in E \times \mathbb {R}^n$ and for $v, \tilde{v}, w, \tilde{w} \in C(B,l^\infty )$ such that $v(\tau ,y) = \tilde{v} ( \tau , y)$ for $(\tau , y) \in D[t,x]$ and $w(\tau ,y) = \bar{w} ( \tau , y)$ for $( \tau ,y) \in D[ \varphi (t,x)]$ we have $F(t,x,v,w, q ) = F(t,x,\tilde{v}, \tilde{w}, q )$. Note that the condition $(V)$ means that the values of $F$ at the point $(t,x, v,w, q) \in \Omega$ depends on $(t,x, q)$ and on the restriction of $v$ and $w$ to the sets $D[t,x]$ and $D[\varphi (t,x)]$ only.

Let us denote by $z = \{ z_k \}_{k \in \mathbb {N}}$ an unknown function of the variables $(t,x)$, $x = (x_1,\ldots ,x_n)$. We consider the system of functional differential equations

$$\begin{aligned} \partial _t z_k(t,x)=F_{k}(t,x,z_{(t,x)},z_{\varphi (t,x)}, \partial _x z_k(t,x)),\quad k\in \mathbb {N}, \end{aligned}$$

(1)

with the initial conditions

$$\begin{aligned} z_k(t,x) = \psi _k(t,x)\text { on }E_{0.k},\quad k \in \mathbb {N}, \end{aligned}$$

(2)

where $\partial _x z_k = ( \partial _{x_1}z_k, \ldots ,\partial _{x_n} z_k)$, $1 \le i \le k.$ We assume that $F$ satisfies the condition $(V)$.

Write

$$\begin{aligned}&E_t=(E_0 \cup E)\cap ([-b_0,t]\times \mathbb {R}^n),\quad S_t=[-b+ M(t), b - M(t)],\qquad t \in [0, a],\\&I_{c.k}[x]=\{t\in [a_k,c]:-b+M(t)\le x \le b-M(t)\},\quad x \in [-b,b],\ k \in \mathbb {N}, \end{aligned}$$

where $\kappa <c\le a.$ We consider weak solutions of initial problems. A function ${\tilde{z}} : E_c \rightarrow \mathbb {S}$, ${\tilde{z}}=\{{\tilde{z}}_k\}_{k\in \mathbb {N}}$, where $\kappa <c\le a$, is a solution to (1), (2) provides

• (i) $\tilde{z}_{(t,x)} \in C(B, l^\infty )$ for $(t,x) \in E_c$, $0 \le t \le c$ and $\partial _x {\tilde{z}}_k$ exist and they are continuous on $E \cap ( [a_k,c] \times \mathbb {R}^n)$ for $k \in \mathbb {N},$

• (ii) for each $k \in \mathbb {N}$ and $x \in [ - b,b]$ the function ${\tilde{z}}_k ( \, \cdot \, ,x) : I_{c.k}[x] \rightarrow \mathbb {R}$ is absolutely continuous,

• (iii) for each $x \in [ - b, b]$ and for $k \in \mathbb {N}$, the $k$-th equation in (1) is satisfied for almost all $t \in I_{c.k}[x]$ and conditions (2) hold.

System (1) with initial conditions (2) is called a generalized Cauchy problem. If $a_k = 0$ for $k \in \mathbb {N}$ then (1), (2) reduces to the classical Cauchy problem. The following question is considered in the paper. We prove that under natural assumptions on given functions there exists exactly one solution to (1), (2) defined on $E_c$ and we give an estimate of $c\in (\kappa , a].$ Let us denote by $\mathbb {X}$ the class of all $\psi = \{ \psi _k\}_{k \in \mathbb {N}}$, $\psi _k : E_{0.k} \rightarrow \mathbb {R}$ for $k \in \mathbb {N},$ such that there exists exactly one solution $\Xi [ \psi ] : E_c \rightarrow l^\infty$ to (1), (2). We give a construction of the space $\mathbb {X}.$ We prove that there is $\mathbb {Y}\subset \mathbb {X}$ such that for each $\psi \in \mathbb {Y}$ there exists the Fréchet derivative $\partial \Xi [\psi ]$ of $\Xi$ at $\psi$. Moreover, if $\psi \in \mathbb {Y}$ and $\pi \in \mathbb {X}$ and ${\bar{z}} = \partial \Xi [\psi ]\pi$ then ${\bar{z}}$ is a solution of an integral functional system generated by (1), (2).

Until now there have not been any results on the differentiability with respect to initial functions for solutions of nonlinear hyperbolic functional differential systems. Our theorems are new also in the case when (1), (2) reduces to a finite functional differential system.

In recent years, a number of papers concerning first order partial functional differential equations have been published. The following questions were considered: functional differential inequalities generated by initial or mixed problems and their applications [1, 5, 6, 12], existence theory of classical or weak solutions of equations or finite systems with initial or initial boundary conditions [2–4, 9, 14, 22] approximate solutions of functional differential problems [15–17, 25]. Essential extensions of some ideas concerning generalized solution of Hamilton–Jacobi equations are given in [20, 21] where viscosity solutions are considered.

Infinite systems of first order partial functional differential equations were first treated in [18, 19]. The existence result presented in [18] is based on a method of successive approximations which was introduced by Ważewski [23] for systems without the functional dependence. Existence results for initial problems [11] and for mixed problems [8] related to infinite systems of nonlinear equations are obtained by a quasilinearization procedure and by construction of functional integral systems for unknown functions and for their derivatives with respect to spatial variables. This method was initiated in [7] for nonlinear systems without functional variables. Differential inequalities and suitable comparison results for infinite systems of hyperbolic functional differential inequalities are given in [13, 19].

Information on applications of functional differential equations can be found in [12, 24]. The monograph [10] contains results on differentiability with respect to initial functions for solutions of ordinary functional differential equations.

The paper is organized as follows. In Sect. 2 we transform the generalized Cauchy problem into a system of integral functional equations. This system is solved in Sect. 3 by the method of successive approximations. As a consequence we obtain a theorem on the existence of solution to (1), (2) an on continuous dependence of solutions on initial functions.

A theorem on the differentiability of solutions with respect to initial functions is presented in Sect. 4. It is the main result of the paper.

Two types of assumptions are needed in theorems on the existence of solutions to initial or initial boundary value problems related to hyperbolic functional differential systems. The first type conditions concern the bicharacteristics. The second type assumptions concern the regularity of given functions. The authors of the papers [2–4, 8, 9, 11, 18, 22] have assumed that the partial derivatives of given functions satisfy the Lipschitz condition with respect to all variables except for $t$. These conditions are global. Our assumptions on the regularity of given functions are more general. We assume that the partial derivatives of $F$ satisfy the Lipschitz condition and suitable estimates are local with respect to all variables. It is clear that there are differential systems with deviated variables and differential integral systems such that local estimates hold and global inequalities are not satisfied.

Motivations for investigations of functional differential systems with two functional variables are given in Remark 2.4. We give examples of functional differential systems which can be derived from (1) be specializing the operator $F$.

Example 1.1

Suppose that $G : E \times l^\infty \times l^\infty \times \mathbb {R}^n \rightarrow \mathbb {S}$, $G = \{G_k\}_{k \in \mathbb {N}}$, is a given function and $F$ is defined by

$$\begin{aligned} F(t,x,v,w,q)=G(t,x,v(0,0_{[n]}),w(0,0_{[n]}),q)\quad \text {on } \Omega . \end{aligned}$$

(3)

Then (1) reduces to the system of differential equations with deviated variables

$$\begin{aligned} \partial _t z_k(t, x)=G_k(t, x, z(t, x), z(\varphi (t, x)), \partial _x z_k(t, x)),\quad k\in \mathbb {N}. \end{aligned}$$

(4)

Example 1.2

Suppose that $b_0 > 0$ and that there is ${\bar{M}} \in \mathbb {R}^n_+$ such that $M(t) ={\bar{M}} t$ for $t \in [0, a]$. Then $E$ is the classical Haar pyramid

$$\begin{aligned} E=\{(t,x)\in \mathbb {R}^{1+n} \, : \, t \in [0, a], \, -b + {\bar{M}} t \le x \le b-{\bar{M}}t\}. \end{aligned}$$

(5)

Set $M^-(t) = -b + {\bar{M}} t$ and $M^+(t) = b - {\bar{M}} t$ for $t \in [-b_0, 0]$. Then

$$\begin{aligned} E_0 = \{ (t, x) \in \mathbb {R}^{1+n} \, : \, t \in [-b_0, 0], \, -b + {\bar{M}} t \le x \le b - {\bar{M}} t\}. \end{aligned}$$

(6)

Suppose that $0 < \nu < \mu \le b_0$ and $0_{[n]} < h \le {\bar{M}} \nu$. For the above $G$ we put

$$\begin{aligned} F(t,x,v,w,q)=G\left( t,x,\int _{-h}^h v(-\mu ,y)~dy,w(0, 0_{[n]}), q\right) \quad \text {on }\ \Omega . \end{aligned}$$

(7)

Then (1) reduces to the differential integral system

$$\begin{aligned} \partial _t z_k (t,x) = G\left( t, x, \int _{x-h}^{x+h} z(t-\mu , y)~dy, z(\varphi (t,x)), \partial _x z_k(t,x)\right) ,\quad k \in \mathbb {N}. \end{aligned}$$

(8)

It is clear that more complicated examples of differential functional systems can be derived from (1).

2 Integral functional equations

Let $\mathbb {L}( [\tau , t], \mathbb {R}^n_+)$, $[\tau , t] \subset \mathbb {R}$, be the class of all $\omega : [\tau , t] \rightarrow \mathbb {R}^n_+$ which are integrable on $[\tau , t]$. For $x \in \mathbb {R}^n$, $x = (x_1, \ldots ,x_n)$, we put $\Vert x \Vert = |x_1|+\ldots +|x_n|$. We use the symbol “$\circ$” to denote the scalar product in $\mathbb {R}^n$. We denote by $M_{n\times n}$ be the class of all $n \times n$ matrices with real elements. For $A \in M_{n \times n}$ where $A = [ a_{ij}]_{i,j=1,\ldots ,n}$ we put

$$\begin{aligned} \Vert A\Vert _{n\times n}=\max \left\{ \sum _{j=1}^n | a_{i j}|\, :\; 1 \le i \le n\right\} . \end{aligned}$$

Let $M_{\infty \times n}$ be the class of all real matrices $B = [b_{ij}]_{i \in \mathbb {N},\, 1\le j \le n}$ with the finite norm

$$\begin{aligned} \Vert B\Vert _{\infty \times n}=\sup \left\{ \sum _{j=1}^n | b_{i j}|\, :\; i \in \mathbb {N}\right\} . \end{aligned}$$

We will use the symbol $M_{\infty \times \infty }$to denote the set of real matrices $C = [c_{ij}]_{i, j \in \mathbb {N}}$ with the finite norm:

$$\begin{aligned} \Vert C\Vert _{\infty \times \infty }=\sup \left\{ \sum _{j=1}^\infty |c_{i j}|\, :\; i \in \mathbb {N}\right\} . \end{aligned}$$

For the above $B \in M_{\infty \times n}$ and $C \in M_{\infty \times \infty }$ we write $b_{[i]} = (b_{i1}, \ldots ,b_{in})$, and $c_{[i]} = (c_{i1}, c_{i2}, \ldots )$, $i \in \mathbb {N}$.

We will denote by $CL(B,\mathbb {R})$ the class of all linear and continuous operators defined on $C(B,\mathbb {R})$ and taking values in $\mathbb {R}$. The norm in the space $CL(B,\mathbb {R})$ generated by the maximum norm in $C(B,\mathbb {R})$ will be denoted by $\Vert \, \cdot \, \Vert _\star$. Let $CL(B,M_{\infty \times \infty })$ be the class of all $\Xi = [ \Xi _{ij}]_{i, j \in \mathbb {N}}$ such that

$$\begin{aligned} \Vert \Xi \Vert _{\infty \times \infty ; \star } = \sup \{ \Vert \Xi _{[i]} \Vert _{\infty ; \star } \, : \, i \in \mathbb {N}\} < +\infty , \end{aligned}$$

where

$$\begin{aligned} \Vert \Xi _{[i]} \Vert _{\infty ; \star } = \sum _{j = 1}^\infty \Vert \Xi _{ij} \Vert _\star ,\quad \Xi _{[i]} = (\Xi _{i1}, \Xi _{i2}, \ldots )\qquad \text {for }\quad i\in \mathbb {N}. \end{aligned}$$

Now we define some function spaces. Given ${\bar{c}} = (c_0, c_1, c_2) \in \mathbb {R}^3_+$, we denote by $\mathbb {X}$ the set of all $\psi = \{\psi _k\}_{k\in \mathbb {N}}$ such that for each $k \in \mathbb {N}$ we have

• (i) $\psi _k \in C(E_{0.k}, \mathbb {R})$, the derivatives $\partial _x \psi _k = (\partial _{x_1} \psi _k,\ldots ,\partial _{x_n} \psi _k)$ exist on $E_{0.k}$ and $\partial _x \psi _k \in C(E_{0.k}, \mathbb {R}^n)$,

• (ii) the estimates

  $$\begin{aligned}&|\psi _k(t,x)|\le c_0,\quad \Vert \partial _x\psi _k(t,x)\Vert \le c_1,\\&\Vert \partial _x\psi _k(t,x)-\partial _x \psi _k (t,{{\bar{x}}})\Vert \le c_2 \Vert x-{\bar{x}}\Vert \end{aligned}$$

  are satisfied on $E_{0.k}$.

Let $\psi \in \mathbb {X}$, $\psi = \{ \psi _k\}_{k \in \mathbb {N}}$, be given and $\kappa < c \le a$. We denote by $C_{\psi .c}$ the class of all $z \in C(E_c, l^\infty )$, $z = \{ z_k\}_{k \in \mathbb {N}}$, such that $z_k(t,x) = \psi _k(t,x)$ on $E_{0.k}$ for $k \in \mathbb {N}$. For the above $\psi$ and $\kappa < c \le a$ we denote by $C_{\partial \psi _k.c}$, $k \in \mathbb {N}$, the class of all $\vartheta \in C(E_c, \mathbb {R}^n)$ such that $\vartheta (t,x) = \partial _x \psi _k(t,x)$ on $E_{0.k}$.

Write $\Omega _I = [-b, b] \times C(B, l^\infty ) \times C(B, l^\infty ) \times \mathbb {R}^n$ and $\Omega _t = S_t \times C(B, l^\infty ) \times C(B, l^\infty ) \times \mathbb {R}^n$, $t \in [0,a]$.

Assumption $H_0[F]$. The function $F :\Omega \rightarrow \mathbb {S}$ satisfies the condition (V) and

• 1. for each $(x, v, w, q) \in \Omega _I$ the function $F : ( \, \cdot , x, v, w, q) :I[x] \rightarrow \mathbb {S}$ is measurable and there is $\alpha \in \mathbb {L}([0,a], \mathbb {R}_+)$ such that

  $$\begin{aligned} \Vert F(t, x,\theta ,\theta , 0_{[n]})\Vert _\infty \le \alpha (t)\quad \text {on }E \end{aligned}$$

  where $\theta \in C(B, l^\infty )$ is given by $\theta (\tau , s) = 0_{l^\infty }$ and $0_{l^\infty }$ is the zero in the space $l^\infty$,

• 2. for each $P=(t,x,v,w,q)\in \Omega$ there exist the derivatives

  $$\begin{aligned} \partial _x F(P)=[\partial _{x_j}F_i(P)]_{i \in \mathbb {N}, 1 \le j \le n},\quad \partial _q F(P)=[\partial _{q_j} F_i(P)]_{i\in \mathbb {N}, 1 \le j \le n}, \end{aligned}$$

  and the functions $\partial _x F(\, \cdot , x, v, w, q), \partial _q F (\, \cdot , x, v, w, q) :I[x] \rightarrow M_{\infty \times n}$ are measurable and the functions $\partial _x F(t, \, \cdot ), \partial _qF(t, \cdot ) :\Omega _t \rightarrow M_{\infty \times n}$ are continuous,

• 3. for each $P=(t,x,v,w,q)\in \Omega$ there exist the Frechét derivatives

  $$\begin{aligned} \partial _v F(P) = [\partial _{v_j} F_i(P)]_{i, j \in \mathbb {N}},\quad \partial _w F(P) = [\partial _{w_j} F_i(P)]_{i, j \in \mathbb {N}}, \end{aligned}$$

  and for each ${\tilde{w}} \in C(B, \mathbb {R})$ we have

  $$\begin{aligned} \partial _v F(\, \cdot , x, v, w, q) {\tilde{w}}, \partial _q F (\, \cdot , x, v, w, q) {\tilde{w}} :I[x] \rightarrow M_{\infty \times \infty }\text { are measurable} \end{aligned}$$

  and

  $$\begin{aligned} \partial _v F(t, \, \cdot ) {\tilde{w}}, \partial _w F(t, \cdot ) {\tilde{w}} :\Omega _I \rightarrow M_{\infty \times \infty } \text { are continuous} \end{aligned}$$

  where

  $$\begin{aligned} \partial _v F(P) {\tilde{w}} = [\partial _{v_j} F_i(P) {\tilde{w}}]_{i, j \in \mathbb {N}},\quad \partial _w F(P) {\tilde{w}} = [\partial _{w_j} F_i(P) {\tilde{w}}]_{i, j \in \mathbb {N}}, \end{aligned}$$

• 4. there are $\beta \in \mathbb {L}([0,a], \mathbb {R}_+)$ and $L \in \mathbb {L}([0,a], \mathbb {R}^n_+)$, $L = (L_1, \ldots , L_n)$, such that for $P = (t, x, v, w, q) \in \Omega$ we have

  $$\begin{aligned} \Vert \partial _x F(P)\Vert _{\infty \times n}\le \beta (t),\quad \Vert \partial _v F(P)\Vert _{\infty \times \infty ;\star }\le \beta (t),\quad \Vert \partial _w F(P)\Vert _{\infty \times \infty ;\star }\le \beta (t), \end{aligned}$$

  and

  $$\begin{aligned} (|\partial _{q_1} F_k(P)|, \ldots , |\partial _{q_n} F_k(P)|) \le L(t),\quad k \in \mathbb {N}, \end{aligned}$$

  and for $t \in [0,a]$ we have

  $$\begin{aligned} M(t) = \int _0^t L(\xi )~d\xi . \end{aligned}$$

Assumption $H[\varphi ]$. The functions $\phi _0 :[0,a] \rightarrow \mathbb {R}$, $\phi :E \rightarrow \mathbb {R}^n$, $\phi = (\phi _1, \ldots , \phi _n)$, are continuous and

• 1. $0 \le \phi _0 \le t$ for $t \in [0,a]$ and $\varphi (t,x) = (\phi _0(t), \phi (t,x)) \in E$,

• 2. there exist the derivatives

  $$\begin{aligned} \partial _x\phi (t,x)=[\partial _{x_j} \phi _i(t,x)]_{i, j = 1, \ldots , n} \end{aligned}$$

  and $\partial _x \phi \in C(E, M_{n \times n})$,

• 3. the constant $Q_0 \in \mathbb {R}_+$ is defined by the relation $\Vert \partial _x\phi (t,x) \Vert _{n \times n} \le Q_0$ for $(t,x) \in E$ and there is $Q \in \mathbb {R}_+$ such that

  $$\begin{aligned} \Vert \partial _x \phi (t,x) - \partial _x \phi (t, {\bar{x}})\Vert _{n \times n} \le Q\quad \text {on }E. \end{aligned}$$

Suppose that Assumptions $H_0[F]$, $H[\varphi ]$ are satisfied and $\psi \in \mathbb {X}$, $z \in C_{\psi .c}$, $u \in C(E_c, M_{\infty \times \infty })$ where $\kappa < c \le a$ and

$$\begin{aligned}&z=\{z_k\}_{k\in \mathbb {N}},\quad u=[u_{ij}]_{i \in \mathbb {N}, 1 \le j \le n},\\&u_{[i]} = (u_{i1}, \ldots , u_{in}),\quad u_{[i]} \in C_{\partial \psi _i.c}\qquad \text {for }i \in \mathbb {N}. \end{aligned}$$

Write $S[z, u_{[k]}](\tau , x) = (\tau , x, z_{(\tau , x)}, z_{\varphi (t, x)}, u_{[k]}(\tau , x))$, $k \in \mathbb {N}$. We consider the Cauchy problem

$$\begin{aligned} \omega '(\tau ) = - \partial _q F_k (S[z, u_{[k]}](\tau , \omega (\tau ))),\quad \omega (t) = x, \end{aligned}$$

(9)

where $(t,x) \in E$, $a_k \le t \le a$ and $\partial _q F_k = (\partial _{q_1} F_k, \ldots , \partial _{q_n} F_k)$. Let us denote by $g_{[k]}[z, u_{[k]}]( \, \cdot \,,t,x)$ the solution of (9). The function $g_{[k]}[z, u_{[k]}]( \, \cdot \,,t,x)$ is the $k$-th bicharactersitic of (1) corresponding to $(z, u).$ Write

$$\begin{aligned}&u_{(t,x)} = [(u_{ij})_{(t,x)}]_{i \in \mathbb {N}, 1 \le j \le n},\\&(u_{[k]})_{(t,x)} = ((u_{k1})_{(t,x)}, \ldots , (u_{kn})_{(t,x)}), \quad k \in \mathbb {N}, \end{aligned}$$

and $P[z, u_{[k]}](\tau , t, x) = S[z, u_{[k]}](\tau , g_{[k]}[z, u_{[k]}](\tau , t, x))$, $k \in \mathbb {N}$. For $P \in \Omega$ and for

$$\begin{aligned} {\tilde{w}} \in C(B, l^\infty ),\quad {\tilde{w}}=\{{\tilde{w}}_k\}_{k \in \mathbb {N}},\quad {\tilde{W}} \in C(B, M_{\infty \times n}),\quad {\tilde{W}}=[{\tilde{w}}_{ij}]_{i \in \mathbb {N}, 1 \le j \le n}, \end{aligned}$$

we put

$$\begin{aligned} \partial _v F_k (p) \diamond {\tilde{w}}&= \sum _{j = 1}^\infty \partial _{v_j} F_k(P) {\tilde{w}}_j,\\ \partial _v F_k(P) \star {\tilde{W}}&= \left[ \sum _{j = 1}^\infty \partial _{v_j} F_k (P) {\tilde{w}}_{j1}, \ldots , \sum _{j = 1}^\infty \partial _{v_j} F_k (P) {\tilde{w}}_{jn} \right] , \end{aligned}$$

where $k \in \mathbb {N}$. In similar way we define the expressions $\partial _w F_k(P) \diamond {\tilde{w}}, \partial _w F_k (P) \star {\tilde{W}}$ for $k \in \mathbb {N}$. Let us denote by $\mathbb {F}[z, u] = \{ \mathbb {F}_k[z, u]\}_{k \in \mathbb {N}}$ the function defined in the following way:

$$\begin{aligned} \mathbb {F}_k[z, u](t,x) = \psi _k(t, x)\quad \text {on }E_{0.k} \end{aligned}$$

and

$$\begin{aligned}&\mathbb {F}_k[z, u](t,x)=\psi _k(a_k, g_{[k]}[z, u_{[k]}](a_k, t, x)) + \int _{a_k}^t F_k (P[z, u_{[k]}](\tau , t, x))~d\tau \\&\quad -\int _{a_k}^t \partial _q F_k(P[z, u_{[k]}](\tau , t, x)) \diamond u_{[k]}(\tau , g_{[k]}[z, u_{[k]}](\tau , t, x))~d\tau \quad \text {on }E \cap ([a_k, c] \times \mathbb {R}^n). \end{aligned}$$

Moreover we put

$$\begin{aligned}&\mathbb {G}[z, u] = \{ \mathbb {G}_{ij}[z, u]\}_{i \in \mathbb {N}, 1 \le j \le n},\\&\mathbb {G}_{[k]}[z, u] = (\mathbb {G}_{k1}[z, u], \ldots , \mathbb {G}_{kn}[z, u]), \quad k \in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} \mathbb {G}_{[k]}[z, u](t,x) = \partial _x \psi _k (t,x)\quad \text {on } E_{0.k} \end{aligned}$$

and

$$\begin{aligned}&\mathbb {G}_{[k]}[z, u](t,x)=\partial _x \psi _k(a_k, g_{[k]}[z, u_{[k]}](a_k, t, x))+\int _{a_k}^t \partial _x F_k(P[z, u_{[k]}](\tau , t, x))~d\tau \\&\quad +\int _{a_k}^t \partial _v F_k(P[z, u_{[k]}](\tau , t, x)) \star u_{(\tau , g_{[k]}[z, u_{[k]}](\tau , t, x))}~d\tau \\&\quad +\int _{a_k}^t \partial _w F_k(P[z, u_{[k]}](\tau , t, x)) \star \left[ u_{\varphi (\tau , g_{[k]}[z, u_{[k]}](\tau , t, x))} \partial _x \phi (\tau , g_{[k]}[z, u_{[k]}](\tau , t, x)) \right] ~d\tau \end{aligned}$$

on $E \cap ([a_k, c] \times \mathbb {R}^n).$ The functions $u_{\varphi (\tau , y)} \partial _x \phi (\tau , y) :B \rightarrow M_{\infty \times n}$, $y = g_{[k]}[z, u_{[k]}](\tau , t, s)$, are defined by

$$\begin{aligned} u_{\varphi (\tau , y)} \partial _x \phi (\tau , y) = \left[ \sum _{\mu =1}^n (u_{i\mu })_{\varphi (\tau , y)} \partial _{x_j} \phi _{\mu }(\tau , y) \right] _{i \in \mathbb {N}, 1 \le j \le n}. \end{aligned}$$

We consider the system of functional integral equations

$$\begin{aligned} z = \mathbb {F}[z, u],\quad u = \mathbb {G}[z, u]. \end{aligned}$$

(10)

We show that under natural assumptions on given functions there exists a solution $({\bar{z}}, {\bar{u}}) :E_c \rightarrow l^\infty \times M_{\infty \times n}$ of (10) and there exist the derivatives $\partial _x {\bar{z}}_k = (\partial _{x_1} {\bar{z}}_k, \ldots ,\partial _{x_n} {\bar{z}}_k)$, $k \in \mathbb {N}$, and ${\bar{u}}_{[k]}=\partial _x {\bar{z}}_k$ for $k \in \mathbb {N}$.

We first give estimates of solutions to (10). For $z \in C(E_c, l^\infty )$, $\vartheta \in C(E_c, \mathbb {R}^n)$, $u \in C(E_c, M_{\infty \times n})$ we define the seminorms

$$\begin{aligned}&\Vert z\Vert _{(t, l^\infty )} = \max \{ \Vert z(\tau , s)\Vert _\infty \; : \; (\tau , s) \in E_t \},\\&\Vert \vartheta \Vert _{(t, \mathbb {R}^n)} = \max \{ \Vert \vartheta (\tau , s)\Vert _\infty \; : \; (\tau , s) \in E_t \},\\&\Vert v\Vert _{(t, M_{\infty \times n})}=\max \{\Vert u(\tau , s)\Vert _{\infty \times n} \; : \; (\tau , s) \in E_t\}, \end{aligned}$$

where $t \in [0,c]$.

Lemma 2.1

Suppose that Assumptions $H_0[F]$, $H[\varphi ]$ are satisfied and

• 1. $\psi \in \mathbb {X}$ and $\kappa < c \le a$,

• 2. the functions ${\bar{z}} :E_c \rightarrow l^\infty$, ${\bar{u}} :E_c \rightarrow M_{\infty \times n}$ satisfy (10) and ${\bar{z}} \in C(E_c, l^\infty )$, ${\bar{u}} \in C(E_c, M_{\infty \times n})$.

Then

$$\begin{aligned} \Vert {\bar{z}}\Vert _{(t, l^\infty )} \le \zeta (t),\quad \Vert {\bar{u}} \Vert _{(t, M_{\infty \times n})} \le \chi (t)\qquad \text {for } t \in [0,c], \end{aligned}$$

where

$$\begin{aligned}&\chi (t) = c_1 \exp \left\{ (1+Q_0) \int _0^t \beta (\xi )~d\xi \right\} + \int _0^t\beta (\xi ) \exp \left\{ (1+Q_0) \int _\xi ^t \beta (\tau )~d\tau \right\} ~d\xi ,\\&\zeta (t) = c_0 \exp \left\{ 2 \int _0^t \beta (\xi )~d\xi \right\} +\int _0^t {\bar{\gamma }} (\xi ) \exp \left\{ 2 \int _\xi ^t \beta (\tau )~d\tau \right\} ~d\xi ,\\&{\bar{\gamma }} (t)=\alpha (t)+(\beta (t)+\Vert L(t)\Vert )\chi (t). \end{aligned}$$

Proof

Write ${\bar{\zeta }}(t) = \Vert {\bar{z}} \Vert _{(t, l^\infty )}$, ${\bar{\chi }} (t) = \Vert {\bar{u}} \Vert _{(t, M_{\infty \times n})}$, $t \in [0,c]$. It follows from Assumptions $H_0[F]$ and $H[\varphi ]$ that the $({\bar{\zeta }}, {\bar{\chi }})$ satisfy the integral inequalities

$$\begin{aligned}&{\bar{\zeta }} (t) \le c_0 + \int _0^t \alpha (\xi )~d\xi + 2 \int _0^t \beta (\xi ) {\bar{\zeta }} (\xi )~d\xi + \int _0^t (\beta (\xi ) + \Vert L(\xi )\Vert ) {\bar{\chi }}(\xi )~d\xi ,\\&{\bar{\chi }} (t) \le c_1 + \int _0^t \beta (\xi )~d\xi + (1+ Q_0) \int _0^t \beta (\xi ) {\bar{\chi }}(\xi )~d\xi ,\quad t \in [0,c]. \end{aligned}$$

The functions $(\zeta , \chi )$ satisfy the integral equations corresponding to the above inequalities. This proves the lemma.

Suppose that $\psi \in \mathbb {X}$, $\kappa < c \le a$ and $d_0, r_0 \in \mathbb {R}_+$ and $d_0 \ge c_1$, $r_0 \ge c_2$. We denote by $C_{\psi .c}[\zeta , d_0]$ the class of all $z \in C_{\psi .c}$ such that

$$\begin{aligned} \Vert z \Vert _{(t, l^\infty )} \le \zeta (t)\quad \text {for }t \in [0,a] \end{aligned}$$

and

$$\begin{aligned} |z_k(t,x) - z_k(t, {\bar{x}}) | \le d_0 \Vert x - {\bar{x}}\Vert \quad \text {for }(t,x), (t, {\bar{x}}) \in E \cap ([a_k, a]\times \mathbb {R}^n),\ k \in \mathbb {N}. \end{aligned}$$

Let $C_{\partial \psi _k.c}[\chi , r_0]$, $k \in \mathbb {N}$, be the class of all $\vartheta \in C_{\partial \psi _k.c}$ satisfying the conditions:

$$\begin{aligned} \Vert \vartheta \Vert _{(t, \mathbb {R}^n)} \le \chi (t)\quad \text {for }t \in [a_k,c] \end{aligned}$$

and

$$\begin{aligned} \Vert \vartheta (t,x) - \vartheta (t, {\bar{x}}) \Vert \le r_0 \Vert x - {\bar{x}}\Vert \quad \text {on }E \cap ([a_k, c]\times \mathbb {R}^n). \end{aligned}$$

Write $d = \zeta (a)$, $r = \chi (a)$ and $\Omega [d, r] = E\times K_{C(B, l^\infty )}[d] \times K_{C(B, l^\infty )}[d] \times K_{\mathbb {R}^n}[r]$ where $K_{C(B, l^\infty )}[d] = \{ w \in C(B, l^\infty ): \, \Vert w\Vert _B \le d\}$, $K_{\mathbb {R}^n}[r] = \{ q \in \mathbb {R}^n : \, \Vert q \Vert \le r \}$.

Assumption $H_\star [F]$. The function $f :\Omega \rightarrow \mathbb {S}$ satisfies Assumption $H_0[F]$ and there is $\gamma \in \mathbb {L}([0,a], \mathbb {R}_+)$ such that the terms

$$\begin{aligned}&\Vert \partial _x F(t, x, v, w, q) - \partial _x F (t, {\bar{x}}, {\bar{v}}, {\bar{w}}, {\bar{q}})\Vert _{\infty \times n},\quad \Vert \partial _q F(t, x, v, w, q)\nonumber \\&\qquad - \partial _q F (t, {\bar{x}}, {\bar{v}}, {\bar{w}}, {\bar{q}})\Vert _{\infty \times n} \end{aligned}$$

and

$$\begin{aligned}&\Vert \partial _v F(t, x, v, w, q) - \partial _v F (t, {\bar{x}}, {\bar{v}}, {\bar{w}}, {\bar{q}})\Vert _{\infty \times \infty ; \star },\quad \Vert \partial _w F(t, x, v, w, q)\nonumber \\&\qquad - \partial _w F (t, {\bar{x}}, {\bar{v}}, {\bar{w}}, {\bar{q}})\Vert _{\infty \times \infty ;\star } \end{aligned}$$

are bounded from above by $\gamma (t) [\Vert x - {\bar{x}}\Vert + \Vert v - {\bar{v}} \Vert _B + \Vert w - {\bar{w}}\Vert _B + \Vert q - {\bar{q}}\Vert ]$ on $\Omega [d, r]$.

Remark 2.2

It is important in our considerations that we have assumed the Lipschitz condition for $\partial _xF$, $\partial _vF$, $\partial _wF$, $\partial _qF$ with respect to $(x, v,w, q)$ and the estimates are local with respect to all variables. It is clear that there are differential systems with deviated variables and differential integral systems such that local estimates hold and global inequalities are not satisfied.

Lemma 2.3

Suppose that Assumptions $H_\star [F]$, $H[\varphi ]$ are satisfied and $\kappa < c \le a$ and

$$\begin{aligned}&\psi , {\bar{\psi }} \in \mathbb {X},\quad z \in C_{\psi .c}[\zeta , d_0], \quad {\bar{z}} \in C_{{\bar{\psi }}.c}[\zeta , d_0],\\&u, {\bar{u}} \in C(E_c,M_{\infty \times n}),\quad u = [u_{ij}]_{i \in \mathbb {N}, 1 \le j \le n},\quad {\bar{u}} = [{\bar{u}}_{ij}]_{i \in \mathbb {N}, 1 \le j \le n},\\&u_{[i]} = (u_{i1}, \ldots , u_{in}),\quad {\bar{u}}_{[i]} = ({\bar{u}}_{i1}, \ldots , {\bar{u}}_{in}),\qquad i \in \mathbb {N}, \end{aligned}$$

and $u_{[i]} \in C_{\partial \psi _i.c}[\chi , r_0]$, ${\bar{u}}_{[i]} \in C_{\partial {\bar{\psi }}_i.c}[\chi , r_0]$ for $i \in \mathbb {N}$.

Then for each $k \in \mathbb {N}$ we have:

• (i) the bicharacteristics $g_{[k]}[z, u_{[k]}](\, \cdot \,,t, x)$ and $g_{[k]}[{\bar{z}}, {\bar{u}}_{[k]}](\, \cdot \,,t, x)$, $(t,x) \in E\cap ([a_k, c] \times \mathbb {R}^n)$, exist on intervals $[a_k, \Delta [z, u_{[k]}](t,x)]$ and $[a_k, \Delta [{\bar{z}}, {\bar{u}}_{[k]}](t,x)]$ such that for $\tau = \Delta [z, u_{[k]}](t,x)$, $\bar{\tau }= \Delta [{\bar{z}}, {\bar{u}}_{[k]}](t,x)$ we have $(\tau ,g_{[k]}[z, u_{[k]}](\tau ,t, x) ) \in \partial E_c$, $(\bar{\tau },g_{[k]}[{\bar{z}}, {\bar{u}}_{[k]}](\bar{\tau },t, x) ) \in \partial E_c$, where $\partial E_c$ is the boundary of $E_c$,

• (ii) for each $k \in \mathbb {N}$ the solution of (9) is unique and we have the estimates

  $$\begin{aligned}&\Vert g_{[k]}[z, u_{[k]}](\tau ,t, x) - g_{[k]}[z, u_{[k]}](\tau ,{\bar{t}}, {\bar{x}})\Vert \nonumber \\&\quad \le \left[ \Vert x - {\bar{x}}\Vert +\left| \int _t^{{\bar{t}}}\Vert L(\xi ) \Vert ~d\xi \right| \right] \exp \left\{ {{\bar{C}}} \left| \int _\tau ^t \gamma (\xi )~d\xi \right| \right\} , \end{aligned}$$

  (11)

  and

  $$\begin{aligned}&\Vert g_{[k]}[z, u_{[k]}](\tau ,t, x) - g_{[k]}[{\bar{z}}, {\bar{u}}_{[k]}](\tau ,t, x)\Vert \nonumber \\&\quad \le \left| \int _\tau ^t \gamma (\xi ) [2 \Vert z - {\bar{z}} \Vert _{(\xi , l^\infty )} + \Vert u_{[k]} - {\bar{u}}_{[k]} \Vert _{(\xi , \mathbb {R}^n)}]~d\xi \right| \exp \left\{ {{\bar{C}}} \left| \int _t^\tau \gamma (\xi )~d\xi \right| \right\} \!,\qquad \quad \end{aligned}$$

  (12)

  where $(t, x), ({\bar{t}}, {\bar{x}}) \in E \cap ([a_k, c] \times \mathbb {R}^n)$ and ${{\bar{C}}} = 1 + d_0 (1+Q_0) + r_0$.

Proof

The local existence and uniqueness of the solution to (9) follows from classical theorems on Carethéodory solutions of ordinary differential equations. Suppose that $[t_0, t]$ is the interval on which the bicharacteristic $g_{[k]}[z, u_{[k]}](\, \cdot \,,t, x)$ is defined. Then

$$\begin{aligned} -L(\tau ) \le \frac{d}{d\tau } g_{[k]}[z, u_{[k]}](\tau ,t, x) \le L(\tau ),\quad \tau \in [t_0, t], \end{aligned}$$

and consequently

$$\begin{aligned} -b+M(\tau ) \le g_{[k]}[z, u_{[k]}](\tau ,t, x) \le b - M(\tau ), \quad \tau \in [t_0, t]. \end{aligned}$$

We conclude that $(\tau , g_{[k]}[z, u_{[k]}](\tau ,t, x)) \in E_c$ for $\tau \in [t_0, t]$ and the bicharacteristic $g_{[k]}[z, u_{[k]}](\, \cdot \,,t, x)$ is defined on $[a_k, t]$ and the assertion (i) follows.

Now we prove that for each $k \in \mathbb {N}$ the function $g_{[k]}[z, u_{[k]}](\, \cdot \,,t, x) - g_{[k]}[z, u_{[k]}](\, \cdot \,,{\bar{t}}, {\bar{x}})$ satisfies a linear integral inequality. Note that the functions $z_{(\tau , y)}$ and $z_{(\tau , {\bar{y}})}$ where $(\tau , y), (\tau , {\bar{y}}) \in E \cap ([0,c] \times \mathbb {R}^n)$, $y \ne {\bar{y}}$, have different domains. Hence we need the following construction. Write $B_\star = [-b_0, c] \times [-2b + 2b^-, 2b + 2b^+]$. There is $z_\star \in C(B_\star , l^\infty )$ such that

• (i) $z_\star (t,x) = z(t,x)$ on $E_c$ and $\Vert (z_\star )_{(t,x)} \Vert _B \le d$ on $E\cap ([0,c] \times \mathbb {R}^n)$,

• (ii) $\Vert z_\star (t,x) - z_\star (t,{\bar{x}}) \Vert _\infty \le d_0 \Vert x - {\bar{x}}\Vert$ on $E \cap ([0,c] \times \mathbb {R}^n)$.

Then the functions $(z_\star )_{(\tau , y)}$ and $(z_\star )_{(\tau , {\bar{y}})}$ where $(\tau , y), (\tau , {\bar{y}}) \in E \cap ([0,c] \times \mathbb {R}^n)$ are defined on $B$. It follows form (9) that

$$\begin{aligned}&g_{[k]}[z, u_{[k]}](\tau ,t, x) - g_{[k]}[z, u_{[k]}](\tau ,{\bar{t}}, {\bar{x}}) = x - {\bar{x}}\\&\quad + \int _\tau ^t \partial _q F_k(P[z_\star , u_{[k]}](\xi , t, x)) ~d\xi - \int _\tau ^{{\bar{t}}} \partial _q F_k(P[z_\star , u_{[k]}](\xi , {\bar{t}}, {\bar{x}}))~d\xi \end{aligned}$$

and consequently

$$\begin{aligned}&\Vert g_{[k]}[z, u_{[k]}](\tau ,t, x) - g_{[k]}[z, u_{[k]}](\tau ,{\bar{t}}, {\bar{x}}) \Vert \\&\quad \!\le \! \Vert x- {\bar{x}}\Vert \!+\! \left| \int _t^{{\bar{t}}} \Vert L(\xi ) \Vert ~d\xi \right| \!+\! {{\bar{C}}} \left| \int _\tau ^t \gamma (\xi ) \Vert g_{[k]}[z, u_{[k]}](\xi ,t, x) - g_{[k]}[z, u_{[k]}](\xi ,{\bar{t}}, {\bar{x}})\Vert d\xi \right| . \end{aligned}$$

From the Gronwall inequality we deduce (11). It follows from Assumptions $H[\varphi ]$, $H_0[F]$ and from (9) that

$$\begin{aligned}&\Vert g_{[k]}[z, u_{[k]}](\tau ,t, x) - g_{[k]}[{\bar{z}}, {\bar{u}}_{[k]}](\tau ,t, x) \Vert \nonumber \\&\quad \le \left| \int _\tau ^t \gamma (\xi ) [2 \Vert z - {\bar{z}}\Vert _{(\xi , l^\infty )} + \Vert u_{[k]} - {\bar{u}}_{[k]} \Vert _{(\xi , \mathbb {R}^n)} ]~d\xi \right| \\&\qquad + {{\bar{C}}} \left| \int _\tau ^t \gamma (\xi ) \Vert g_{[k]}[z, u_{[k]}](\xi ,t, x) - g_{[k]}[{\bar{z}}, {\bar{u}}_{[k]}](\xi ,t, x) \Vert ~d\xi \right| . \end{aligned}$$

Then we obtain (12) form the Gronwall inequality.

Remark 2.4

Set ${\bar{\Omega }} = E \times C(B, l^\infty ) \times \mathbb {R}^n$ and suppose that ${\bar{F}} :{\bar{\Omega }} \rightarrow \mathbb {S}$, ${\bar{F}} = \{ {\bar{F}}_k \}_{k \in \mathbb {N}}$, is a given function of the variables $(t, x, v, q)$. Let us consider the functional differential system

$$\begin{aligned} \partial _t z_k(t,x) = {\bar{F}}_k (t, x, z_{(t,x)}, \partial _x z_k(t,x)),\quad k \in \mathbb {N}, \end{aligned}$$

(13)

which is a particular case of (1).

There are the following motivation for investigations of (1), (2) instead of (13), (2). Differential equations with deviated variables are obtained from (13) in the following way. Suppose that $G :E\times l^\infty \times l^\infty \times \mathbb {R}^n \rightarrow \mathbb {S}$, $G = \{ G_k\}_{k \in \mathbb {N}}$, is a given function. Write

$$\begin{aligned} {\bar{F}}(t, x, v, q) = G(t, x, v(0, 0_{[n]}), v(\varphi (t,x) - (t,x)), q). \end{aligned}$$

(14)

Then system (13) is equivalent to (4).

Note that Assumption $H_0[F]$ is not satisfied in this case for ${\bar{F}}$ given by (14). More precisely, the derivatives

$$\begin{aligned} \partial _x {\bar{F}} (t, x, v, q) = [\partial _{x_j}{\bar{F}}_i(t, x, v, q) ]_{i \in \mathbb {N}, 1 \le j \le n} \end{aligned}$$

(15)

do not exist on ${\bar{\Omega }}$. It is clear that under natural assumptions on $G$ the function $F$ given by (3) satisfies Assumption $H_0[F]$.

Let us consider the second example. Suppose that $E$ and $E_0$ are given by (5) and (6) respectively. For the above $G$ we put

$$\begin{aligned} {\bar{F}} (t, x, v, q) = G(t, x, \int _{-h}^h v(\mu , y)~dy, v(\varphi (t,x) - (t,x)), q)\quad \text {on }{\bar{\Omega }}. \end{aligned}$$

(16)

Then system (13) is equivalent to (8). Note that Assumption $H_0[F]$ is not satisfied for ${\bar{F}}$ given by (16) because the derivatives (15) do not exist on ${\bar{\Omega }}$. It is clear that under natural assumptions on $G$ the function $F$ given by (7) satisfies Assumption $H_0[F]$.

With the above motivation we have considered problem (1), (2).

3 Existence of solutions to initial problems

For $\psi \in \mathbb {X},\ \psi = \{\psi _k\}_{k\in \mathbb {N}},$ we put

$$\begin{aligned}&\Vert \psi _k\Vert _{E_{0.k}}=\max \{|{\psi _{k}(t,x)}| : (t,x)\in E_{0.k}\},\\&\Vert \partial _x\psi _k\Vert _{E_{0.k}}=\max \{\Vert \partial _x\psi _k(t,x)\Vert :(t,x)\in E_{0.k}\} \end{aligned}$$

where $k\in \mathbb {N}$ and

$$\begin{aligned} \Vert \psi \Vert _{\mathbb {X}} = \sup \{\Vert \psi _k\Vert _{E_{0.k}} + \Vert \partial _x\psi _k\Vert _{E_{0.k}}:k\in \mathbb {N}\}. \end{aligned}$$

Write

$$\begin{aligned}&{{\hat{\Gamma }}}(t)=c_1+(1+d_0+d_0Q_0)\int _0^t\beta (\xi )~d\xi + r{{\bar{C}}}\int _0^t\gamma (\xi )~d\xi + 2r_0\int _0^t\Vert L(\xi )\Vert ~d\xi ,\\&\Gamma (t) = {\hat{\Gamma }}(t)\exp \left[ {\bar{C}}\int _0^t\gamma (\xi )~d\xi \right] ,\\&\Lambda (t)=\left\{ c_2+B_0\int _0^t\gamma (\xi )~d\xi + B_1\int _0^t\beta (\xi )~d\xi \right\} \exp \left[ {\bar{C}} \int _0^t\gamma (\xi )~d\xi \right] ,\\&B_0 = {\bar{C}}(1+r+rQ_0),\quad B_1 = r_0+r_0Q_0^2+rQ. \end{aligned}$$

Assumption $H[F]$. The function $F:\Omega \rightarrow \mathbb {S}$ satisfies Assumption $H_{\star }[F]$ and the constant $c\in (\kappa ,a]$ is small enough to satisfy the conditions

$$\begin{aligned} \Gamma (c) \le d_0,\quad \Lambda (c)\le r_0. \end{aligned}$$

(17)

Remark 3.1

If we assume that $d_0>c_1$ and $r_0>c_2$ then there is $c\in (0,a]$ such that condition (17) is satisfied.

Theorem 3.2

If Assumption $H[\varphi ],\ H[F]$ are satisfied and $\psi \in \mathbb {X}$ then there exists a solution ${\hat{z}}:E_c\rightarrow l^{\infty }$ to (1), (2) and

$$\begin{aligned} \Vert {\hat{z}}\Vert _{(t,l^{\infty })} \le d,\quad \Vert \partial _x{\hat{z}}\Vert _{(t,M_{\infty \times n})} \le r\qquad {\text {for}}\ t\in [0,c], \end{aligned}$$

(18)

and

$$\begin{aligned} \Vert \partial _x{\hat{z}}(t,x) - \partial _x{\hat{z}}(t,{\bar{x}})\Vert _{\infty \times n}\le r_0\Vert x-{\bar{x}}\Vert \quad \text {on}\ E_c. \end{aligned}$$

(19)

If $\tilde{\psi }\in \mathbb {X},\ \tilde{\psi }=\{\tilde{\psi }_k\}_{k\in \mathbb {N}},$ and ${\tilde{z}}:E_c\rightarrow l^{\infty }$ is a solution to (1) with the initial conditions

$$\begin{aligned} z_k(t,x) = \tilde{\psi }_k(t,x)\quad \text {on}\ E_{0.k}\ \quad \text {for}\ k\in \mathbb {N}, \end{aligned}$$

(20)

then

$$\begin{aligned} \Vert {\hat{z}}-{\tilde{z}}\Vert _{(t,l^{\infty })} + \Vert \partial _x{\hat{z}} - \partial _x{\tilde{z}}\Vert _{(t,M_{\infty \times n})}\le \Vert \psi -\tilde{\psi }\Vert _{\mathbb {X}}\exp \left[ \int _0^t\Gamma _{\star }(\tau )~d\tau \right] ,\quad t\in [0,c],\qquad \end{aligned}$$

(21)

where

$$\begin{aligned} \Gamma _{\star }(\tau ) = B_{\star }\gamma (\tau )+2\beta (\tau )+2\Vert L(\tau )\Vert ,\quad B_{\star }=2[\Gamma (c)+\Lambda (c)+1+r+rQ_0]. \end{aligned}$$

Proof

We have divided the proof into a sequence of steps. We use a method of successive approximations.

I. We consider the sequences $\{z^{(m)}\}$ and $\{u^{(m)}\}$ where

$$\begin{aligned}&z^{(m)}:E_c\rightarrow l^{\infty },\quad u^{(m)}:E_c\rightarrow M_{\infty \times n},\\&z^{(m)} = \{z^{(m)}_k\}_{k\in \mathbb {N}},\quad u^{(m)} = \{u^{(m)}_{[k]}\}_{k\in \mathbb {N}},\quad u^{(m)}_{[k]} = (u^{(m)}_{k1},\ldots ,u^{(m)}_{kn}),\qquad k\in \mathbb {N}. \end{aligned}$$

We put first

$$\begin{aligned} z^{(0)}_k(t,x) = \psi _k(t,x)\ \mathrm{on} \ E_{0.k}\ \mathrm{and} \ z^{(0)}_k=\psi _k(a_k,x)\ \mathrm{on} \ E\cap ([a_k,c]\times \mathbb {R}^n)\quad \mathrm{for} \ k\in \mathbb {N}, \end{aligned}$$

and

$$\begin{aligned} u^{(0)}_k(t,x)=\partial _x\psi _k(a_k,x)\ \mathrm{on} \ E\cap ([a_k,c]\times \mathbb {R}^n)\quad \mathrm{for} \ k\in \mathbb {N}. \end{aligned}$$

If $z^{(m)}:E_c\rightarrow l^{\infty }$ and $u^{(m)}:E_c\rightarrow M_{\infty \times n}$ are known functions then for each $k\in \mathbb {N}$ the function $u^{(m+1)}_{[k]}$ is a solution of the equation

$$\begin{aligned} \vartheta (t,x) = \mathbb {G}^{(m)}_k[\vartheta ](t,x) \end{aligned}$$

(22)

where $\vartheta =(\vartheta _1,\ldots ,\vartheta _n)$ and $\mathbb {G}^{(m)}_k[\vartheta ] = (\mathbb {G}^{(m)}_{k1}[\vartheta ],\ldots ,\mathbb {G}^{(m)}_{kn}[\vartheta ])$ and $\mathbb {G}^{(m)}_k[\vartheta ]$ is defined by

$$\begin{aligned} \mathbb {G}^{(m)}_k[\vartheta ](t,x) = \partial _x\psi _k(t,x)\quad \mathrm{on} \ E_{0.k} \end{aligned}$$

and

$$\begin{aligned}&\mathbb {G}^{(m)}_k[\vartheta ](t,x) = \partial _x\psi _k(a_k,g_{[k]}[z^{(m)},\vartheta ](a_k,t,x))\nonumber \\&\quad +\int _{a_k}^t\partial _xF_k(P[z^{(m)},\vartheta ](\tau ,t,x))~d\tau \!+\! \int _{a_k}^t\partial _vF_k(P[z^{(m)},\vartheta ](\tau ,t,x))\star u^{(m)}_{(\tau ,g_{[k]}[z^{(m)},\vartheta ](\tau ,t,x))}~d\tau \nonumber \\&\quad +\int _{a_k}^t\partial _wF_k(P[z^{(m)},\vartheta ](\tau ,t,x)) \star [u^{(m)}_{\varphi (\tau ,g_{[k]}[z^{(m)},\vartheta ](\tau ,t,x))} \partial _x\phi (\tau ,g_{[k]}[z^{(m)},\vartheta ](\tau ,t,x))]~d\tau \nonumber \\ \end{aligned}$$

(23)

on $E\cap ([a_k,c]\times \mathbb {R}^n)$. The function $z^{(m+1)}$ is given by

$$\begin{aligned} z^{(m+1)}(t,x) = \mathbb {F}[z^{(m)},u^{(m+1)}](t,x)\quad \mathrm{on} \ E_c. \end{aligned}$$

(24)

II. We prove that

$(A_m)$ the sequences $\{z^{(m)}\}$ and $\{u^{(m)}\}$ are defined on $E_c$ and for $m\ge 0$ we have

$$\begin{aligned} z^{(m)}\in C_{\psi .c}[\zeta ,d_0],\quad u^{(m)}_k\in C_{\partial \psi _k.c}[\chi ,r_0]\quad \mathrm{for} \ k\in \mathbb {N}, \end{aligned}$$

$(B_m)$ there are $\lambda ,\ \lambda _0\in \mathbb {L}([0,c],\mathbb {R}_+)$ such that for any $m\ge 0$ we have

$$\begin{aligned} \Vert z^{(m)}(t,x) \!-\! z^{(m)}(\tilde{t},x)\Vert _{\infty }\!\le \! \left| \int _t^{\tilde{t}}\lambda _0(\tau )~d\tau \right| ,\quad \Vert u^{(m)}(t,x) \!-\! u^{(m)}(\tilde{t},x)\Vert _{\infty \times n}\!\le \! \left| \int _t^{\tilde{t}}\lambda (\tau )~d\tau \right| , \end{aligned}$$

where $(t,x),\ (\tilde{t},x)\in E_c, \ 0\le t,\tilde{t}\le c,$

$(C_m)$ there exists the sequence $\{\partial _xz^{(m)}\}$ and for $m\ge 0$ we have: $\partial _xz^{(m)}(t,x) = u^{(m)}(t,x)$ on $E_c$.

We prove $(A_m)-(C_m)$ by induction. It is clear that conditions $(A_0)-(C_0)$ are satisfied. Supposed now that $(A_m)-(C_m)$ hold for a given $m\ge 0$, we will prove that there exists $u^{(m+1)}:E_c\rightarrow M_{\infty \times n}$ and $u^{(m+1)}_{[k]}\in C_{\partial \psi _k.c}[\chi ,r_0]$ for $k\in \mathbb {N}$. We first prove that

$$\begin{aligned} \mathbb {G}_k^{(m)}: C_{\partial \psi _k.c}[\chi ,r_0]\rightarrow C_{\partial \psi _k.c}[\chi ,r_0],\quad k\in \mathbb {N}. \end{aligned}$$

(25)

It follows from Assumptions $H[\varphi ]$ and $H[F]$ that for $\vartheta \in C_{\partial \psi _k.c}[\chi ,r_0],\ k\in \mathbb {N}$, we have

$$\begin{aligned} \Vert \mathbb {G}_k^{(m)}[\vartheta ](t,x)\Vert \le c_1+(1+Q_0)\int _0^t\beta (\xi )\chi (\xi )~d\xi +\int _0^t\beta (\xi )~d\xi = \chi (t) \end{aligned}$$

and

$$\begin{aligned} \Vert \mathbb {G}_k^{(m)}[\vartheta ](t,x)-\mathbb {G}_k^{(m)}[\vartheta ](t,\tilde{x})\Vert \le \Lambda (t)\Vert x-\tilde{x}\Vert \le r_0\Vert x-\tilde{x}\Vert \quad \mathrm{on} \ E_c\cap ([a_k,c]\times \mathbb {R}^n). \end{aligned}$$

From the above estimates we deduce (25). It follows easily that for $\vartheta ,\ \tilde{\vartheta }\in C_{\partial \psi _{k}.c}[\chi ,r_0]$ we have

$$\begin{aligned} \Vert \mathbb {G}_k^{(m)}[\vartheta ](t,x)-\mathbb {G}_k^{(m)}[\tilde{\vartheta }](t,x)\Vert \le \Lambda (c)\int _{a_k}^t\gamma (\xi )\Vert \vartheta - \tilde{\vartheta }\Vert _{(\xi ,\mathbb {R}^n)}~d\xi \quad \mathrm{on} \ E\cap ([a_k,c]\times \mathbb {R}^n). \end{aligned}$$

For the above $\vartheta ,\ \tilde{\vartheta }$ we put

$$\begin{aligned}{}[|\vartheta -\tilde{\vartheta }|] = \max \left\{ \Vert \vartheta - \tilde{\vartheta }\Vert _{(t,\mathbb {R}^n)}\exp \left[ -2\Lambda (c) \int _{a_k}^t\gamma (\xi )~d\xi \right] :t\in [a_k,c]\right\} . \end{aligned}$$

We deduce from Assumption $H[F]$ that

$$\begin{aligned}{}[|\mathbb {G}_k^{(m)}[\vartheta ]-\mathbb {G}_k^{(m)}[\tilde{\vartheta }]|]\le \frac{1}{2}[|\vartheta - \tilde{\vartheta }|]. \end{aligned}$$

It follows from the Banach fixed point theorem that for each $k\in \mathbb {N}$ there exists exactly one solution of Eq. (22). Then there exists $u^{(m+1)}:E_c\rightarrow M_{\infty \times n}$ and $u^{(m+1)}_k\in C_{\partial \psi _k.c}[\chi ,r_0]$ for $k\in \mathbb {N}$.

We deduce from Assumption $H[\varphi ],\ H[F]$ and from (24) that

$$\begin{aligned} \Vert z^{(m+1)}\Vert _{(t,\infty )}\le c_0 +\int _0^t\alpha (\tau )~d\tau + 2\int _0^t\beta (\tau )\zeta (\tau )~d\tau + 2\int _0^t\Vert L(\tau )\Vert \chi (\tau )~d\tau = \zeta (t) \end{aligned}$$

where $t\in [0,c]$ and

$$\begin{aligned} |z^{(m+1)}_k(t,x)-z^{(m+1)}_k(t,\tilde{x})|\le \Gamma (t)\Vert x-\tilde{x}\Vert \quad \mathrm{on} \ E\cap ([a_k,c]\times \mathbb {R}^n). \end{aligned}$$

The above relations and Assumption $H[c]$ show that $z^{(m+1)}\in C_{\psi .c}[\zeta ,d_0].$

An easy computation shows that condition $(B_{m+1})$ is satisfied with

$$\begin{aligned}&\lambda _0(\tau ) = (\Gamma (c)+2r)\Vert L(\tau )\Vert +2d\beta (\tau )+\alpha (\tau ),\\&\lambda (\tau ) = \Lambda (c)\Vert L(\tau )\Vert +(1+r+rQ_0)\beta (\tau ),\quad \tau \in [0,c]. \end{aligned}$$

Now we prove $(C_{m+1})$. Write

$$\begin{aligned} \mathbb {D}_k(t,x,y)= z^{(m+1)}_k(t,y) - z^{(m+1)}_k(t,x) - u^{(m+1)}_{[k]}(t,x)\circ (y-x),\quad k\in \mathbb {N}, \end{aligned}$$

where $(t,x),\ (t,y)\in E\cap ([a_k,c]\times \mathbb {R}^n)$. We prove that there is $C_{\star }\in \mathbb {R}_+$ such that

$$\begin{aligned} |\mathbb {D}_k(t,x,y)|\le C_{\star }\Vert x-y\Vert ^2,\quad k\in \mathbb {N}. \end{aligned}$$

(26)

Set $g^{(m)}_{[k]}(\tau ,t,x)= g_{[k]}[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x)$. Then we have

$$\begin{aligned} \mathbb {D}_k(t,x,y)&= \mathbb {F}_k[z^{(m)},u^{(m+1)}](t,y) -\mathbb {F}[z^{(m)},u^{(m+1)}](t,x) - \mathbb {G}_{[k]}[u^{(m+1)}_{[k]}](t,x)\circ (y-x)\\&= \psi _k(a_k,g^{(m)}_{[k]}(a_k,t,y)) - \psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x))\\&+\int _{a_k}^t[F_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,y)) - F_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))]~d\tau \\&-\int _{a_k}^t\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,y))\circ u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]}(\tau ,t,y))~d\tau \\&+\int _{a_k}^t\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\circ u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]}(\tau ,t,x))~d\tau \\&-\,\mathbb {G}_{[k]}[u^{(m+1)}_{[k]}](t,x)\circ (y-x),\quad k\in \mathbb {N}. \end{aligned}$$

We transform the above expressions in the following way. We apply the Hadamard mean value theorem to the differences

$$\begin{aligned} F_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,y)) - F_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x)),\quad k\in \mathbb {N}, \end{aligned}$$

and we denote by

$$\begin{aligned}&Q^{(m)}(\xi ,\tau ,t,x,y) = \xi P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,y)\nonumber \\&\quad + (1-\xi )P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x),\quad \xi \in [0,1],\ k\in \mathbb {N}, \end{aligned}$$

suitable intermediate points. Let us denote by $\mathbb {D}_{k.0}(t,x,y),$ $\mathbb {D}_{k.1}(t,x,y)$, $\mathbb {D}_{k.2}(t,x,y),$ $\mathbb {D}_{k.3}(t,x,y)$ the expressions defined by

$$\begin{aligned}&\mathbb {D}_{k.0}(t,x,y) = \psi _k(a_k,g^{(m)}_{[k]}(a_k,t,y)) - \psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x))\\&\qquad -\,\partial _x\psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x)) \circ [g^{(m)}_{[k]}(a_k,t,y)-g^{(m)}_{[k]}(a_k,t,x)],\\&\mathbb {D}_{k.1}(t,x,y) = \int _{a_k}^t\int _0^1\left[ \partial _xF_k(Q^{(m)}(\xi ,\tau ,t,x,y))\right. \\&\qquad \left. -\,\partial _xF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\right] \circ [g^{(m)}_{[k]}(\tau ,t,y) - g^{(m)}_{[k]}(\tau ,t,x)]~d\xi ~d\tau \\&\qquad +\int _{a_k}^t\int _0^1[\partial _vF_k(Q^{(m)}(\xi ,\tau ,t,x,y)) - \partial _vF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))]\\&\qquad \star \left[ z^{(m)}_{(\tau ,g^{(m)}_{[k]}(\tau ,t,y))} - z^{(m)}_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\right] ~d\xi ~d\tau \\&\qquad +\int _{a_k}^t\int _0^1[\partial _wF_k(Q^{(m)}(\xi ,\tau ,t,x,y)) -\,\partial _wF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))]\\&\quad \star \left[ z^{(m)}_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,y))} - z^{(m)}_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\right] ~d\xi ~d\tau \\&\qquad +\int _{a_k}^t\int _0^1[\partial _qF_k(Q^{(m)}(\xi ,\tau ,t,x,y)) -\,\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))]\\&\qquad \circ [u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]}(\tau ,t,y)) - u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]}(\tau ,t,x))]~d\xi ~d\tau \end{aligned}$$

and

$$\begin{aligned}&\mathbb {D}_{k.2}(t,x,y) =\int _{a_k}^t\partial _vF_k(P[z^{(m)}, u^{(m+1)}_{[k]}](\tau ,t,x))\star \left[ z^{(m)}_{(\tau ,g^{(m)}_{[k]} (\tau ,t,y))}\right. \\&\quad \left. -\,z^{(m)}_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))} - (u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}(g^{(m)}_{[k]} (\tau ,t,y)-g^{(m)}_{[k]}(\tau ,t,x))^T\right] ~d\tau \\&\quad +\int _{a_k}^t\partial _wF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))\star \left[ z^{(m)}_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,y))} - z^{(m)}_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\right. \\&\quad \left. -\,(u^{(m)})_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))} \partial _x\phi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))(g^{(m)}_{[k]} (\tau ,t,y)-g^{(m)}_{[k]}(\tau ,t,x))^T\right] ~d\tau . \end{aligned}$$

Moreover we set

$$\begin{aligned}&\mathbb {D}_{k.3}(t,x,y) = \partial _x\psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x)) \circ [[g^{(m)}_{[k]}(a_k,t,y)-g^{(m)}_{[k]}(a_k,t,x)] - (y-x)]\\&\qquad +\int _{a_k}^t[\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))-\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,y))]\circ u^{(m+1)}_{[k]}\nonumber \\&\qquad \times (\tau ,g^{(m)}_{[k]}(\tau ,t,y))~d\tau \\&\qquad +\int _{a_k}^t\partial _xF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))\circ [[g^{(m)}_{[k]}(\tau ,t,y)-g^{(m)}_{[k]} (\tau ,t,x)] - (y-x)]~d\tau \\&\qquad +\int _{a_k}^t\partial _vF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))\star (u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}] \circ [[g^{(m)}_{[k]}(\tau ,t,y)\nonumber \\&\qquad -\,g^{(m)}_{[k]}(\tau ,t,x)] - (y-x)]~d\tau \\&\qquad +\int _{a_k}^t\{\partial _wF_k(P[z^{(m)}, u^{(m+1)}_{[k]}](\tau ,t,x))\star [(u^{(m)})_{\varphi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\partial _x\phi (\tau ,g^{(m)}_{[k]} (\tau ,t,x))]\}\\&\qquad \circ [[g^{(m)}_{[k]}(\tau ,t,y)-g^{(m)}_{[k]}(\tau ,t,x)] - (y-x)]~d\tau . \end{aligned}$$

We put $k\in \mathbb {N}$ in the above definitions. Then we have

$$\begin{aligned} \mathbb {D}_{k}(t,x,y)=\mathbb {D}_{k.0}(t,x,y)+\mathbb {D}_{k.1}(t,x,y)+\mathbb {D}_{k.2}(t,x,y)+\mathbb {D}_{k.3}(t,x,y),\quad k\in \mathbb {N}. \end{aligned}$$

(27)

Since $\psi \in \mathbb {X}$, there is $C_0\in \mathbb {R}_+$ such that

$$\begin{aligned} |\mathbb {D}_{k.0}(t,x,y)|\le C_0\Vert x-y\Vert ^2,\quad k\in \mathbb {N}, \end{aligned}$$

(28)

where $(t,x),\ (t,y)\in E\cap ([a_k,c]\times \mathbb {R}^n).$ It is easily seen that

$$\begin{aligned} |\mathbb {D}_{k.1}(t,x,y)|\le C_1\Vert x-y\Vert ^2,\quad k\in \mathbb {N}, \end{aligned}$$

(29)

where

$$\begin{aligned} C_1=(\tilde{c}{\bar{C}})^2\int _0^c\gamma (\xi )~d\xi ,\quad \tilde{c}=\exp \left[ {\bar{C}}\int _0^c\gamma (\xi )~d\xi \right] . \end{aligned}$$

It follows from $(C_m)$ that there is $c_{\star }\in \mathbb {R}_+$ such that

$$\begin{aligned} |z^{(m)}_k(t,y) - z^{(m)}_k(t,x) -u^{(m)}_{[k]}(t,x)\circ (y-x)|\le c_{\star }\Vert x-y\Vert ^2, \end{aligned}$$

(30)

where $(t,x),\ (t,y)\in E\cap ([a_k,c]\times \mathbb {R}^n).$ We conclude from Assumptions $H[F]$, $H[\varphi ]$ and from (11), (30) that there is $C_2\in \mathbb {R}_+$ such that

$$\begin{aligned} |\mathbb {D}_{k.2}(t,x,y)|\le C_2\Vert x-y\Vert ^2,\ k\in \mathbb {N}, \end{aligned}$$

(31)

where $(t,x),\ (t,y)\in E\cap ([a_k,c]\times \mathbb {R}^n).$

We transform the expressions $\mathbb {D}_{k.3}(t,x,y),\ k\in \mathbb {N}$, in the following way. Write

$$\begin{aligned}&\mathbb {V}_k(\xi ,\tau ,t,x,y)=\left\{ \partial _xF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x)) \right. \\&\qquad +\,\partial _vF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\star (u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\\&\qquad \left. +\,\partial _wF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\star [(u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}\partial _x\phi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))] \right\} \\&\quad \circ [\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,y))-\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,x))] \end{aligned}$$

and

$$\begin{aligned}&\mathbb {W}_k(\xi ,t,x) = \partial _x\psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x))\\&\quad +\int _{a_k}^\xi [\partial _xF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))+\partial _vF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\nonumber \\&\quad \star (u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))}]~d\tau \\&\quad +\int _{a_k}^\xi \partial _wF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,x))\star [(u^{(m)})_{(\tau ,g^{(m)}_{[k]}(\tau ,t,x))} \partial _x\phi (\tau ,g^{(m)}_{[k]}(\tau ,t,x))]~d\tau ,\quad k\!\in \! \mathbb {N}, \end{aligned}$$

and

$$\begin{aligned}&\mathbb {D}_{k.4}(t,x,y) = \int _{a_k}^t\int _{\tau }^t\mathbb {V}_k(\xi ,\tau ,t,x,y)~d\xi ~d\tau \nonumber \\&\quad +\partial _x\psi _k(a_k,g^{(m)}_{[k]}(a_k,t,x))\circ \int _{a_k}^t[\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,y))\nonumber \\&\quad -\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,x))]~d\xi \end{aligned}$$

(32)

where $k\in \mathbb {N}.$ Then we have

$$\begin{aligned} \int _{a_k}^t\int _{\tau }^t\mathbb {V}_k(\xi ,\tau ,t,x,y)~d\xi ~d\tau =\int _{a_k}^t\int _{a_k}^{\xi }\mathbb {V}_k(\xi ,\tau ,t,x,y)~d\tau ~d\xi \end{aligned}$$

and consequently

$$\begin{aligned}&\mathbb {D}_{k.4}(t,x,y)\nonumber \\&\quad \!=\!\int _{a_k}^t\mathbb {W}_k(\xi ,t,x)\circ [\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,y))\!-\! \partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\xi ,t,x))]~d\xi .\nonumber \\ \end{aligned}$$

(33)

It is clear that the bicharacteristics satisfy the relations

$$\begin{aligned} g^{(m)}_{[k]}(\tau ,\xi ,g^{(m)}_{[k]}(\xi ,t,x)) = g^{(m)}_{[k]}(\tau ,t,x),\quad k\in \mathbb {N}, \end{aligned}$$

where $(t,x)\in E\cap ([a_k,c]\times \mathbb {R}^n).$ This gives

$$\begin{aligned} u^{(m+1)}_{[k]}(\xi ,g^{(m)}_{[k]}(\xi ,t,x)) = \mathbb {W}_k(\xi ,t,x),\quad k\in \mathbb {N}. \end{aligned}$$

We conclude from (32), (33) that

$$\begin{aligned}&\mathbb {D}_{k.3}(t,x,y) = \int _{a_k}^t\left[ \partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}](\tau ,t,x))\right. \\&\quad \left. -\partial _qF_k(P[z^{(m)},u^{(m+1)}_{[k]}] (\tau ,t,y))\right] \circ [u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]} (\tau ,t,y))\nonumber \\&\quad -u^{(m+1)}_{[k]}(\tau ,g^{(m)}_{[k]}(\tau ,t,x))]~d\tau , \quad k\in \mathbb {N}. \end{aligned}$$

Hence, there is $C_3\in \mathbb {R}_+$ such that

$$\begin{aligned} |\mathbb {D}_{k.3}(t,x,y)|\le C_3\Vert x-y\Vert ^2,\quad k\in \mathbb {N}, \end{aligned}$$

(34)

where $(t,x),\ (t,y)\in E\cap ([a_k,c]\times \mathbb {R}^n).$

It follows from (27)–(29), (31), (34) that estimates (26) are satisfied with $C^{\star } = C_0+C_1+C_2+C_3$. Hence, for each $k\in \mathbb {N}$ there exists $\partial _xz^{(m+1)}(t,x)$ for $(t,x)\in E\cap ([a_k,c]\times \mathbb {R}^n)$ and $\partial _xz^{(m+1)} = u^{(m+1)}_{[k]}$. This proves $(C_{m+1})$. Thus $(A_m)-(C_m)$ follow by induction.

III. Now we prove that the sequences $\{z^{(m)}\}$ and $\{u^{(m)}\}$ are uniformly convergent on $E_c$. It follows from Assumptions $H[F]$, $H[\varphi ]$ and from (23) that there are $\Upsilon _0,\ \Upsilon _1\in \mathbb {L}([0,c],\mathbb {R}_+)$ such that

$$\begin{aligned}&\Vert u^{(m+1)}_{[k]} - u^{(m)}_{[k]}\Vert _{(t,\mathbb {R}^n)}\le \int _0^t\Upsilon _0(\tau )\Vert u^{(m+1)}_{[k]}-u^{(m)}_{[k]} \Vert _{(\tau ,\mathbb {R}^n)}~d\tau \\&\int _0^t\Upsilon _1(\tau )[\Vert z^{(m)}-z^{(m-1)} \Vert _{(\tau ,\mathbb {R}^n)}+\Vert u^{(m)}-u^{(m-1)}\Vert _{(\tau ,M_{\infty \times n})}]~d\tau ,\quad k\in \mathbb {N}. \end{aligned}$$

By using the Gronwall inequality we get

$$\begin{aligned}&\Vert u^{(m+1)}_{[k]} - u^{(m)}_{[k]}\Vert _{(t,\mathbb {R}^n)}\nonumber \\&\quad \le \exp \left[ \int _0^t\Upsilon _0(\tau )~d\tau \right] \int _0^t\Upsilon _1(\tau )[\Vert z^{(m)}-z^{(m-1)} \Vert _{(\tau ,\mathbb {R}^n)}+\Vert u^{(m)}-u^{(m-1)} \Vert _{(\tau ,M_{\infty \times n})}]~d\tau ,\nonumber \\ \end{aligned}$$

(35)

where $k\in \mathbb {N}$. We conclude from (10) and from Assumption $H[F]$, $H[\varphi ]$ that there is $\Upsilon _2\in \mathbb {L}([0,c],\mathbb {R}_+)$ such that

$$\begin{aligned}&\Vert z^{(m+1)}-z^{(m)}\Vert _{(t,l^{\infty })}\nonumber \\&\quad \le \int _0^t\Upsilon _2(\tau )[\Vert z^{(m)}-z^{(m-1)} \Vert _{(\tau ,l^{\infty })}+\Vert u^{(m+1)}-u^{(m)}\Vert _{(\tau , M_{\infty \times n})}]~d\tau . \end{aligned}$$

(36)

Write

$$\begin{aligned} K^{(m)}(t) = \Vert z^{(m+1)}-z^{(m)}\Vert _{(t,l^{\infty })} + \Vert u^{(m+1)}_{[k]} - u^{(m)}_{[k]}\Vert _{(t,\mathbb {R}^n)},\quad t\in [0,c]. \end{aligned}$$

We deduce from (35), (36) that there is $\Upsilon \in \mathbb {L}([0,c],\mathbb {R}_+)$ such that

$$\begin{aligned} K^{(m)}(t) \le \int _0^t\Upsilon (\tau )K^{(m-1)}(\tau )~d\tau ,\quad m\ge 1. \end{aligned}$$

Set

$$\begin{aligned}{}[|K^{(m)}|] = \max \left\{ K^{(m)}(t)\exp \left[ -2\int _0^t\Upsilon (\tau )~d\tau \right] :t\in [0,c]\right\} . \end{aligned}$$

Then we have

$$\begin{aligned} K^{(m)}(t)&\le [|K^{(m-1)}|]\int _0^t\Upsilon (\tau ) \exp \left[ 2\int _0^{\tau }\Upsilon (\xi )~d\xi \right] ~d\tau \\&\le [|K^{(m-1)}|]\exp \left[ 2\int _0^t\Upsilon (\tau )~d\tau \right] \end{aligned}$$

and consequently

$$\begin{aligned}{}[|K^{(m)}|]\le \frac{1}{2}[|K^{(m-1)}|]\quad \mathrm{for} \ m\ge 1. \end{aligned}$$

Then $\lim _{m\rightarrow \infty }[|K^{(m)}|] = 0$ and consequently there are the limits

$$\begin{aligned} {\tilde{z}}(t,x) = \lim _{m\rightarrow \infty }z^{(m)}(t,x),\quad \tilde{u}(t,x) = \lim _{m\rightarrow \infty }u^{(m)}(t,x)\quad \mathrm{uniformly on} \ E_c, \end{aligned}$$

where ${\tilde{z}} = \{{\tilde{z}}_k\}_{k\in \mathbb {N}},$ $\tilde{u} = [\tilde{u}_{ij}]_{i\in \mathbb {N},\ 1\le j\le n}$ and $\tilde{u}_{[k]} = (\tilde{u}_{k1},\ldots ,\tilde{u}_{kn})$ for $k\in \mathbb {N}.$

It follows from $(C_m)$ that there exist the derivatives $\partial _x{\tilde{z}}$, $k\in \mathbb {N}$ and $\partial _x{\tilde{z}}_k = \tilde{u}_{[k]}$ for $k\in \mathbb {N}$.

IV. Now we prove that ${\tilde{z}}:E_c\rightarrow \mathbb {S}$ is a solution to (1), (2). Write $\tilde{g}_{[k]}(\cdot ,t,x) = g_{[k]}[{\tilde{z}},\partial _x{\tilde{z}}](\cdot ,t,x),$ $(t,x)\in E\cap ([a_k,c]\times \mathbb {R}^n)$, $k\in \mathbb {N}$. It follows from (III) that ${\tilde{z}}_k(t,x) = \psi _k(t,x)$ on $E_{0.k}$ for $k\in \mathbb {N}$ and

$$\begin{aligned} \partial _t{\tilde{z}}_k(t,x)&= \psi _k(a_k, \tilde{g}_{[k]}(a_k,t,x)) + \int _{a_k}^tF_k(P[{\tilde{z}},\partial _x{\tilde{z}}_k] (\tau ,t,x))~d\tau \nonumber \\&-\int _{a_k}^t\partial _qF_k(P[{\tilde{z}}, \partial _x{\tilde{z}}_k](\tau ,t,x)))\circ \partial _x{\tilde{z}}_k(\tau ,\tilde{g}_{[k]}(\tau ,t,x))~d\tau , \end{aligned}$$

(37)

where $(t,x)\in E\cap ([a_k,c]\times \mathbb {R}^n)$, $k\in \mathbb {N}$. Suppose that $k\in \mathbb {N}$ is fixed. For given $(t,x)\in E\cap ([a_k,c]\times \mathbb {R}^n)$, let us put $y=\tilde{g}_{[k]}(a_k,t,x)$. It follows that the relations $x=\tilde{g}_{[k]}(t,a_k,y)$ and $y=\tilde{g}_{[k]}(a_k,t,x)$ are equivalent. We conclude from (37) that

$$\begin{aligned}&{\tilde{z}}_k(t,\tilde{g}_{[k]}(t,a_k,x)) = \psi _k(a_k,y)\nonumber \\&\quad + \int _{a_k}^tF_k(\tilde{T}_k(\tau ,y))~d\tau - \int _{a_k}^t\partial _qF_k(\tilde{T}_k(\tau ,y))\circ \partial _x{\tilde{z}}_k(\tau ,\tilde{g}_{[k]}(\tau ,a_k,y))~d\tau \end{aligned}$$

(38)

where

$$\begin{aligned} \tilde{T}_k(\tau ,y) = (\tau ,\tilde{g}_{[k]}(\tau ,a_k,y), {\tilde{z}}_{(\tau ,\tilde{g}_{[k]}(\tau ,a_k,y))}, {\tilde{z}}_{\varphi (\tau ,\tilde{g}_{[k]}(\tau ,a_k,y))}, \partial _x{\tilde{z}}_k(\tau ,\tilde{g}_{[k]}(\tau ,a_k,y))). \end{aligned}$$

By differentiating (38) with respect to $t$ and by putting again $x=\tilde{g}_{[k]}(t,a_k,y)$, we obtain that ${\tilde{z}}$ is a weak solution of (1), (2).

V. It follows form $(A_m)-(C_m)$ that the sequences $\{z^{(m)}\}$ and $\{\partial _xz^{(m)}\}$ satisfy the conditions

$$\begin{aligned} \Vert z^{(m)}\Vert _{(t,l^{\infty })} \le d,\quad \Vert \partial _xz^{(m)}\Vert _{(t,M_{\infty \times n})} \le r \end{aligned}$$

and

$$\begin{aligned} \Vert \partial _xz^{(m)}(t,x) - \partial _xz^{(m)}(t,\bar{x})\Vert _{(t,M_{\infty \times n})} \le r_0\Vert x-\bar{x}\Vert \end{aligned}$$

where $m\in \mathbb {N}$, $(t,x),\ (t,\bar{x}) \in E_c$. From the above inequalities we obtain in the limit, letting $m$ tend to $\infty$, estimates (18), (19).

VI. Now we prove (21). It follows from Assumption $H[F]$ that

$$\begin{aligned}&\Vert {\hat{z}}-{\tilde{z}}\Vert _{(t,l^{\infty })} + \Vert \partial _x{\hat{z}}^{(m)}-\partial _x{\tilde{z}}^{(m)} \Vert _{(t,M_{\infty \times n})}\\&\quad \le \Vert \psi -\tilde{\psi }\Vert _{\mathbb {X}} + \int _0^t\Gamma _{\star }(\tau )[\Vert {\hat{z}} -{\tilde{z}}\Vert _{(\tau ,l^{\infty })} + \Vert \partial _x{\hat{z}}^{(m)}-\partial _x{\tilde{z}}^{(m)} \Vert _{(\tau ,M_{\infty \times n})}]~d\tau ,\quad t\in [0,c]. \end{aligned}$$

Then we obtain (21) from the Gronwall inequality. This completes the proof of the theorem.

4 The main result

Suppose that Assumptions $H[\varphi ]$, $H[F]$ are satisfied and $\psi \in \mathbb {X}$. Let us denote by $\Xi [\psi ]$ the solution to (1), (2). It follows from Theorem 3.2 that $\Xi [\psi ]$ exists on $E_c$ and it is unique. Then we have: $\Xi :\mathbb {X}\rightarrow C(E_c,l^{\infty })$. We will denote by $\mathbb {Y}$ the class of all $\psi \in \mathbb {X}$ satisfying the conditions:

$$\begin{aligned}&\sup \,\{\Vert \psi _k\Vert _{\,E_{0.k}}:\;k\in \mathbb {N}\}<c_0,\quad \sup \,\{\Vert \partial _x\psi _k\Vert _{E_{0.k}}:\;k\in \mathbb {N}\}<c_1,\\&\sup \,\left\{ \frac{1}{\Vert x-\bar{x}\Vert } \Vert \partial _x\psi _k(t,x)-\partial _x\psi _k(t,\bar{x})\Vert :\; (t,x),(t,\bar{x})\in E_{0.k},\;x\ne \bar{x},\;k\in \mathbb {N}\right\} <c_1. \end{aligned}$$

We prove that for each $\psi \in \mathbb {Y}$ there exists the Fréchet derivative $\partial \Xi [\psi ]$ of $\Xi$ at the point $\psi$. Moreover, if $\psi \in \mathbb {Y}$, $\pi \in \mathbb {X}$ and $\bar{{\mathbf {z}}}=\partial \Xi [\psi ]\pi$ then $\bar{{\mathbf {z}}}$ is a solution of a linear system of integral functional equations generated by (1), (2).

Suppose that Assumptions $H[\varphi ]$, $H[F]$ are satisfied. Write

$$\begin{aligned}&\hat{\zeta }(t)=c_0\;\exp \left[ 2\int _0^t\beta (\tau )\,d\tau \right] ,\\&\hat{\vartheta }(t)=\hat{d}\left\{ \exp \,\left[ \hat{d} \int _0^t \beta (\tau )\,d\tau \right] +\int _0^t\,\gamma (\mu )\,\exp \left[ \hat{d}\int _{\mu }^t \beta (\tau )\,d\tau \right] \,d\mu \right\} , \end{aligned}$$

where $t\in [0,c]$ and

$$\begin{aligned} \hat{C}=\hat{\zeta }(c),\quad \hat{d}=\exp \left[ {\bar{C}} \int _0^c\gamma (\tau )\,d\tau \right] \; \max \{c_1,\;2\bar{c}\hat{C},\;1+Q_0\}. \end{aligned}$$

(39)

Suppose that $\pi \in \mathbb {X}$, $\pi =\{\pi _k\}_{k\in \mathbb {N}}$. Let us denote by $C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$ the class of all $z:E_c\rightarrow l^{\infty }$, $z=\{z_k\}_{k\in \mathbb {N}}$, satisfying the conditions:

• (i) $z\in C(E_c,l^{\infty })$ and $z_k(t,x)=\pi _k(t,x)$ for $(t,x)\in E_{0.k}$ and $k\in \mathbb {N}$,

• (ii) $\Vert z\Vert _{(t,l^{\infty })}\le \hat{\zeta }(t)$ for $t\in [0,c]$ and

  $$\begin{aligned} \sup \left\{ \frac{1}{\Vert x-\bar{x}\Vert }\,\Vert z(\tau ,x) \!-\!z(\tau ,\bar{x})\Vert _{\infty }:\, (\tau ,x),(\tau ,\bar{x})\in E_c,\,\tau \!\le \! t\right\} \!\le \! \hat{\vartheta }(t)\quad \mathrm{for}\ t\!\in \![0,c]. \end{aligned}$$

In this section we denote by $z(\cdot \,;\psi )=\{z_k(\cdot \,;\psi )\}_{k\in \mathbb {N}}$ the solution to (1), (2). Let us consider the Cauchy problem

$$\begin{aligned} \omega '(\tau )\!=\!-\partial _qF_k\big (\tau ,\omega (\tau ),(z(\cdot \,;\psi ))_ {(\tau ,\omega (\tau ))},\,(z(\cdot \,;\psi ))_{\varphi (\tau ,\omega (\tau ))},\, \partial _xz(\tau ,\omega (\tau );\psi )\big ),\quad \omega (t)\!=\!x,\nonumber \\ \end{aligned}$$

(40)

where $(t,x)\in E\cap \big ([a_k,c]\times \mathbb {R}^n\big )$ and $k\in \mathbb {N}$. The solution to (40) will be denote by $g_{[k]}[\psi ](\cdot \,,t,x)$. If Assumptions $H[\varphi ]$, $H[F]$ are satisfied and $\psi \in \mathbb {X}$ then for each $k\in \mathbb {N}$ the solution $g_{[k]}[\psi ](\cdot \,,t,x)$ is defined on $[a_k,t]$. For $k\in \mathbb {N}$ we put

$$\begin{aligned}&T_k[\psi ](\tau ,t,x)\\&\quad =\big (\tau ,g_{[k]}[\psi ](\tau ,t,x),(z(\cdot \,;\psi ))_ {(\tau ,g_{[k]}[\psi ](\tau ,t,x))}, (z(\cdot \,;\psi ))_{\varphi (\tau ,g_{[k]}[\psi ](\tau ,t,x))},\nonumber \\&\qquad \times \partial _xz(\tau ,g_{[k]}[\psi ](\tau ,t,x);\psi )\big ). \end{aligned}$$

Suppose that $\pi \in \mathbb {X}$, $\pi =\{\pi _k\}_{k\in \mathbb {N}}$, and $z\in C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$, $z=\{z_k\}_{k\in \mathbb {N}}$. Let us denote by $W[z]=\{W_k[z]\}_{k\in \mathbb {N}}$ the function defined by

$$\begin{aligned} W_k[z](t,x)=\pi _k(t,x)\quad \mathrm{for}\ (t,x)\in E_{0.k} \end{aligned}$$

(41)

and

$$\begin{aligned}&W_k[z](t,x)=\pi _k(a_k,g_{[k]}[\psi ](a_k,t,x))+ \int _{a_k}^t \partial _vF_k\big (T_k[\psi ](\tau ,t,x)\big )\diamond z_{(\tau ,g_{[k]}[\psi ](\tau ,t,x))}\,d\tau \nonumber \\&\quad +\int _{a_k}^t \partial _wF_k\big (T_k[\psi ](\tau ,t,x)\big )\diamond z_{\varphi (\tau ,g_{[k]}[\psi ](\tau ,t,x))}\,d\tau \quad \mathrm{for}\ (t,x)\in E\cap \big ([a_k,c]\times \mathbb {R}^n\big ),\qquad \quad \end{aligned}$$

(42)

where $k\in \mathbb {N}$. We consider the linear system of integral functional equations

$$\begin{aligned} z=W[z]. \end{aligned}$$

(43)

Lemma 4.1

Suppose that Assumptions $H[\varphi ]$, $H[F]$ are satisfied and $\pi \in \mathbb {X}$. Then there exists exactly one solution $\bar{{\mathbf {z}}}:E_c\rightarrow l^{\infty }$ of (43) and

$$\begin{aligned}&\Vert \bar{{\mathbf {z}}}\Vert _{\infty }\le \hat{C}\quad \text {on}\ E_c,\end{aligned}$$

(44)

$$\begin{aligned}&\Vert \bar{{\mathbf {z}}}(t,x)-\bar{{\mathbf {z}}}(t,\bar{x})\Vert _{\infty }\le \hat{L} \Vert x-\bar{x}\Vert \quad \text {on}\ E_c \end{aligned}$$

(45)

where $\hat{C}$ is given by (39) and $\hat{L}=\hat{\vartheta }(c)$.

Proof

We prove that

$$\begin{aligned} W:C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]\rightarrow C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]. \end{aligned}$$

(46)

It follows from Assumptions $H[\varphi ]$, $H[F]$ that for $z\in C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$, $(t,x)\in E_c$, $\tau \le t$, we have

$$\begin{aligned} |W_k[z](\tau ,x)|\le c_0+2\int _0^t\beta (\mu )\,\hat{\zeta }(\mu )\,d\mu =\hat{\zeta }(t) \end{aligned}$$

and consequently

$$\begin{aligned} \Vert W[z]\Vert _{(t,l^{\infty })}\le \hat{\zeta }(t)\quad \mathrm{for}\ t\in [0,c]. \end{aligned}$$

(47)

For the above $z$ and for $(\tau ,x), (\tau ,\bar{x})\in E_c$, $\tau \le t$, we have

$$\begin{aligned} |W_k[z](\tau ,x)-W_k[z](\tau ,\bar{x})|&\le \hat{d}\left[ 1+\int _0^t\gamma (\mu )\,d\mu + \int _0^t\beta (\mu )\,\hat{\vartheta }(\mu )\,d\mu \right] \Vert x-\bar{x}\Vert \\&= \hat{\vartheta }(t)\Vert x-\bar{x}\Vert ,\quad k\in \mathbb {N}. \end{aligned}$$

This gives

$$\begin{aligned} \Vert W[z](\tau ,x)-W[z](\tau ,\bar{x})\Vert _{\infty }\le \hat{\vartheta }(t)\Vert x-\bar{x}\Vert , \end{aligned}$$

(48)

where $(\tau ,x), (\tau ,\bar{x})\in E_c$, $\tau \le t$. From (47), (48) we deduce (46).

For $z,z_{\star }\in C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$ and for $k\in \mathbb {N}$ we have

$$\begin{aligned} W_k[z](t,x)-W_k[z_{\star }](t,x)=0\quad \mathrm{on}\ E_{0.k}, \end{aligned}$$

(49)

and

$$\begin{aligned} |W_k[z](t,x)-W_k[z_{\star }](t,x)|\le 2\int _0^t\beta (\tau )\,\Vert z-z_{\star }\Vert _{(\tau ,l^{\infty })}\,d\tau \quad \mathrm{on}\ E\cap \big ([a_k,c]\times \mathbb {R}^n\big ).\qquad \end{aligned}$$

(50)

Write

$$\begin{aligned}{}[|z-z_{\star }|]=\max \,\left\{ \Vert z-z_{\star } \Vert _{(\tau ,l^{\infty })}\, \exp \left[ -4\int _0^t\beta (\tau )\,d\tau \right] :\,t\in [0,c]\right\} . \end{aligned}$$

It follows from (49), (50) that for $z,z_{\star }\in C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$ we have

$$\begin{aligned}{}[|W[z]-W[z_{\star }]|]\le \frac{1}{2}[|z-z_{\star }|]. \end{aligned}$$

From the Banach fixed point theorem we deduce that there is exactly one solution $\bar{{\mathbf {z}}}\in C_{\pi .c}[\hat{\zeta },\hat{\vartheta }]$ of (43). We conclude from (46) that conditions (44), (45) are satisfied. This completes the proof.

Theorem 4.2

Suppose that Assumptions $H[\varphi ]$, $H[F]$ are satisfied. Then

• 1. for each $\psi \in \mathbb {Y}$ there exists the Fréchet derivative $\partial \Xi [\psi ]$,

• 2. if $\psi \in \mathbb {Y}$, $\pi \in \mathbb {X}$ and $\bar{{\mathbf {z}}}=\partial \Xi [\psi ]\pi$, $\bar{{\mathbf {z}}}=\{\bar{{\mathbf {z}}}_k\}_{k\in \mathbb {N}}$, then $\bar{{\mathbf {z}}}$ is a solution to (43) with $W$ given by (41), (42).

Proof

The proof will be divided into three steps.

I. Let $\psi \in \mathbb {Y}$ and $\pi \in \mathbb {X}$ be fixed. There is $\varepsilon _0>0$ such that for $\xi \in I_0=(-\varepsilon _0,\varepsilon _0)$ we have $\psi +\xi \pi \in \mathbb {X}$. Let $\Delta _{\xi }:E_c\rightarrow l^{\infty }$, $\xi \in I_0$, $\xi \ne 0$, be defined by

$$\begin{aligned} \Delta _{\xi }=\big \{\Delta _{\xi .k}\big \}_{k\in \mathbb {N}},\;\; \Delta _{\xi .k}(t,x)=\frac{1}{\xi }\big [z_k(t,x;\psi +\xi \pi )- z_k(t,x;\psi )\big ],\quad k\in \mathbb {N}. \end{aligned}$$

It follows from Lemma 4.1 that there is exactly one solution $\bar{{\mathbf {z}}}=\{\bar{{\mathbf {z}}}_k\}_{k\in \mathbb {N}}$ to (43), $\bar{{\mathbf {z}}}\in C(E_c,l^{\infty })$ and $\bar{{\mathbf {z}}}$ satisfies (44), (45). We prove that

$$\begin{aligned} \lim _{\xi \rightarrow 0} \Vert \Delta _{\xi }(t,x)-\bar{{\mathbf {z}}}(t,x)\Vert _{\infty }=0\quad \mathrm{uniformly\;on}\ E_c. \end{aligned}$$

(51)

II. It follows from Theorem 3.2 that for $k\in \mathbb {N}$, $\xi \in I_0$, $\xi \ne 0$ we have

$$\begin{aligned}&\partial _t\Delta _{\xi .k}(t,x)\nonumber \\&\quad =\frac{1}{\xi }\big \{F_k\big (t,x,(z(\cdot \,;\psi +\xi \pi ))_{(t,x)}, (z(\cdot \,;\psi +\xi \pi ))_{\varphi (t,x)},\partial _xz_k(t,x;\psi +\xi \pi )\big )\nonumber \\&\qquad -F_k\big (t,x,(z(\cdot \,;\psi ))_{(t,x)}, (z(\cdot \,;\psi ))_{\varphi (t,x)},\partial _xz_k(t,x;\psi )\big )\quad \mathrm{on}\ E\cap \big ([a_k,c]\times \mathbb {R}^n\big ),\qquad \quad \end{aligned}$$

(52)

and

$$\begin{aligned} \Delta _{\xi .k}(t,x)=\pi _k(t,x)\quad \mathrm{on}\ E_{0.k}. \end{aligned}$$

(53)

Set

$$\begin{aligned} \overline{Q}_k(t,x;\psi )= \big (t,x,(z(\cdot \,;\psi ))_{(t,x)}, (z(\cdot \,;\psi ))_{\varphi (t,x)},\partial _xz_k(t,x;\psi )\big ),\quad k\in \mathbb {N}, \end{aligned}$$

and

$$\begin{aligned} \Theta _k[\psi ,\pi ](t,x;\xi ,\eta )=(1-\eta )\overline{Q}_k(t,x;\psi )+ \eta \overline{Q}_k(t,x;\psi +\xi \pi ),\quad k\in \mathbb {N}, \end{aligned}$$

where $\eta \in [0,1]$. Then we obtain from (52) that

$$\begin{aligned}&\partial _t\Delta _{\xi .k}(t,x)= \int _0^1\partial _vF_k(\Theta _k[\psi ,\pi ](t,x;\xi ,\eta )) \diamond (\Delta _{\xi })_{(t,x)}\,d\eta \nonumber \\&\quad +\int _0^1\partial _wF_k(\Theta _k[\psi ,\pi ](t,x;\xi ,\eta )) \diamond (\Delta _{\xi })_{\varphi (t,x)}\,d\eta \nonumber \\&\quad +\int _0^1\partial _qF_k(\Theta _k[\psi ,\pi ](t,x;\xi ,\eta )) \circ \partial _x\Delta _{\xi .k}(t,x)\,d\eta ,\quad (t,x)\in E\cap \big ([a_k,c]\times \mathbb {R}^n\big ),\qquad \quad \end{aligned}$$

(54)

where $k\in \mathbb {N}$. Let us denote by $g_{[k]}[\psi ,\pi ;\xi ](\cdot \,,t,x)$ the solution of the Cauchy problem

$$\begin{aligned} \omega '(\tau )=- \int _0^1\partial _qF_k(\Theta _k[\psi ,\pi ](\tau ,\omega (\tau ); \xi ,\eta ))\,d\eta ,\quad \omega (t)=x, \end{aligned}$$

where $(t,x)\in E\cap \big ([a_k,c]\times \mathbb {R}^n\big )$. Write

$$\begin{aligned} Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)= \Theta _k[\psi ,\pi ](\tau ,g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x); \xi ,\eta ),\quad k\in \mathbb {N}. \end{aligned}$$

It follows from (54) that $\Delta _{\xi }$ satisfies relations

$$\begin{aligned}&\Delta _{\xi .k}(t,x)= \pi _k(a_k,g_{[k]}[\psi ,\pi ;\xi ](a_k,t,x))\nonumber \\&\quad +\int _{a_k}^t \int _0^1\partial _vF_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond (\Delta _{\xi })_{(\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))}\,d\eta \,d\tau \nonumber \\&\quad +\int _{a_k}^t \int _0^1\partial _wF_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond (\Delta _{\xi })_{\varphi (\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))}\,d\eta \,d\tau \end{aligned}$$

(55)

where $(t,x)\in E\cap \big ([a_k,c]\times \mathbb {R}^n\big )$ and $k\in \mathbb {N}$.

III. We construct an integral functional inequality for $\Delta _{\xi }-\bar{{\mathbf {z}}}$. Write

$$\begin{aligned}&A_k(t,x)=\pi _k(a_k,g_{[k]}[\psi ,\pi ;\xi ](a_k,t,x))- \pi _k(a_k,g_{[k]}[\psi ](a_k,t,x)),\\&B_k(t,x)= \int _0^t \int _0^1\partial _vF_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond \big (\Delta _{\xi }-\bar{{\mathbf {z}}}\big )_{(\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))}\,d\eta \,d\tau \\&\quad +\int _0^t \int _0^1\partial _wF_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond \big (\Delta _{\xi }-\bar{{\mathbf {z}}}\big )_{\varphi (\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))}\,d\eta \,d\tau \end{aligned}$$

and

$$\begin{aligned}&C_k(t,x)= \int _{a_k}^t \int _0^1\partial _v F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond [\bar{{\mathbf {z}}}_{(\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))}\nonumber \\&\quad \times -\bar{{\mathbf {z}}}_{(\tau , g_{[k]}[\psi ](\tau ,t,x))}]\,d\eta \,d\tau \\&\quad +\int _{a_k}^t \int _0^1\partial _w F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \diamond [\bar{{\mathbf {z}}}_{\varphi (\tau , g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x))} -\bar{{\mathbf {z}}}_{\varphi (\tau , g_{[k]}[\psi ](\tau ,t,x))}]\,d\eta \,d\tau ,\\&D_k(t,x)= \int _{a_k}^t\int _0^1 \big [\partial _v F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) -\partial _v F_k\big (T_k[\psi ](\tau ,t,x)\big )\big ]\nonumber \\&\quad \times \diamond \bar{{\mathbf {z}}}_{(\tau ,g_{[k]}[\psi ](\tau ,t,x))} \,d\eta \,d\tau \\&\quad +\int _{a_k}^t\int _0^1 \big [\partial _w F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) \!-\!\partial _w F_k\big (T_k[\psi ](\tau ,t,x)\big )\big ] \diamond \bar{{\mathbf {z}}}_{\varphi (\tau ,g_{[k]}[\psi ](\tau ,t,x))} \,d\eta \,d\tau \end{aligned}$$

where $k\in \mathbb {N}$. Then we have

$$\begin{aligned} \Delta _{\xi .k}(t,x)-\bar{{\mathbf {z}}}_k(t,x)= A_k(t,x)+B_k(t,x)+C_k(t,x)+D_k(t,x),\quad k\in \mathbb {N}. \end{aligned}$$

(56)

It follows from Assumption $H[F]$ and from (21) that

$$\begin{aligned}&\Vert g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x) -g_{[k]}[\psi ](\tau ,t,x)\Vert \\&\quad \le {\bar{C}}\left| \int _{\tau }^t \gamma (\mu )\Vert g_{[k]}[\psi ,\pi ;\xi ](\mu ,t,x) -g_{[k]}[\psi ](\mu ,t,x)\Vert \,d\mu \right| +|\xi |A_{\star },\quad k\in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} A_{\star }=3\Vert \pi \Vert _{\mathbb {X}}\,\exp \left[ \int _0^c\Gamma _{\star } (\tau )\,d\tau \right] \int _0^c\gamma (\tau )\,d\tau . \end{aligned}$$

This gives

$$\begin{aligned} \Vert g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x) -g_{[k]}[\psi ](\tau ,t,x)\Vert \le |\xi |\hat{A},\quad k\in \mathbb {N}, \end{aligned}$$

(57)

where

$$\begin{aligned} \hat{A}= A_{\star }\exp \left[ {\bar{C}}\int _0^c\gamma (\tau )\,d\tau \right] , \end{aligned}$$

and consequently

$$\begin{aligned} \big |A_k(t,x)\big |\le c_1\hat{A}|\xi |,\quad k\in \mathbb {N}. \end{aligned}$$

(58)

It is clear that

$$\begin{aligned} \big |B_k(t,x)\big |\le 2\int _0^t\beta (\tau ) \Vert \Delta _{\xi }-\bar{{\mathbf {z}}}\Vert _{(\tau ,l^{\infty })}\,d\tau , \quad k\in \mathbb {N}. \end{aligned}$$

(59)

We conclude from Assumption $H[F]$ and from (45) that

$$\begin{aligned} \big |C_k(t,x)\big |&\le \hat{L}(1+Q_0) \int _{a_k}^t\beta (\tau ) \Vert g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x) -g_{[k]}[\psi ](\tau ,t,x)\Vert \,d\tau \nonumber \\&\le |\xi |\hat{L}(1+Q_0) \int _0^c\beta (\tau )\,d\tau \,\hat{A},\quad k\in \mathbb {N}. \end{aligned}$$

(60)

It follows from Assumption $H[F]$ and from (21) that the expressions

$$\begin{aligned}&\int _{a_k}^t\int _0^1 \big \Vert \partial _v F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) -\partial _v F_k\big (T_k[\psi ](\tau ,t,x)\big )\big \Vert _{\infty ;\star }\,d\eta \,d\tau ,\\&\int _{a_k}^t\int _0^1 \big \Vert \partial _w F_k\big (Q_k[\psi ,\pi ;\xi ,\eta ](\tau ,t,x)\big ) -\partial _w F_k\big (T_k[\psi ](\tau ,t,x)\big )\big \Vert _{\infty ;\star }\,d\eta \,d\tau ,\quad k\in \mathbb {N}, \end{aligned}$$

can be estimated from above by

$$\begin{aligned} {\bar{C}}\int _{a_k}^t\gamma (\tau ) \Vert g_{[k]}[\psi ,\pi ;\xi ](\tau ,t,x) -g_{[k]}[\psi ](\tau ,t,x)\Vert \,d\tau +|\xi |\hat{B},\quad k\in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} \hat{B}=3\int _0^c\gamma (\tau )\,d\tau \,\Vert \pi \Vert _{\mathbb {X}}\, \exp \left[ \int _0^c\Gamma _{\star }(\tau )\,d\tau \right] . \end{aligned}$$

We conclude from the above relations and from (44), (57) that

$$\begin{aligned} \big |D_k(t,x)\big |\le |\xi |\hat{D},\quad k\in \mathbb {N}, \end{aligned}$$

(61)

where

$$\begin{aligned} \hat{D}=2\hat{C}\left[ {\bar{C}}\hat{A} \int _0^c\gamma (\tau )\,d\tau +\hat{B}\right] . \end{aligned}$$

It follows from (56) and (58)–(61) that there is $\hat{Q}\in \mathbb {R}_+$ such that

$$\begin{aligned} \big \Vert \Delta _{\xi }-\bar{{\mathbf {z}}}\big \Vert _{(t,l^{\infty })}\le |\xi |\hat{Q}+2\int _0^c \beta (\tau ) \big \Vert \Delta _{\xi }-\bar{{\mathbf {z}}}\big \Vert _{(\tau ,l^{\infty })}\,d\tau ,\;\;t\in [0,c]. \end{aligned}$$

By the Gronwall inequality we obtain

$$\begin{aligned} \big \Vert \Delta _{\xi }-\bar{{\mathbf {z}}}\big \Vert _{(t,l^{\infty })}\le |\xi |\hat{Q}\,\exp \left[ 2\int _0^c \beta (\tau )\,d\tau \right] , \quad t\in [0,c]. \end{aligned}$$

This completes the proof of (51).

The assertion of the theorem follows from (51).

Remark 4.3

It is easy to see that Theorems 3.2 and 4.2 can be applied to problems (4), (2) and (8), (2).