Nonlinear Pseudo-Differential Equations for Radial Real Functions on a Non-Archimedean Field

TL;DR

This paper extends the study of fractional differentiation operators on non-Archimedean fields to nonlinear equations, establishing conditions for their local and global solvability, thus creating a non-Archimedean analogue of classical differential equations.

Contribution

It introduces and analyzes nonlinear pseudo-differential equations in non-Archimedean settings, providing solvability conditions and expanding the framework beyond linear cases.

Findings

Established local and global solvability conditions for nonlinear equations

Extended the non-Archimedean fractional calculus to nonlinear contexts

Demonstrated the existence of solutions under specific conditions

Abstract

In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^\alpha$, $\alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^\alpha$ that the change of an unknown function $u=I^\alpha v$ reduces the Cauchy problem for a linear equation with $D^\alpha$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we study nonlinear equations of this kind, find conditions of their local and global solvability.

Equations

$$\left(D^{\alpha }u\right)(|t{|}_K)=f(|t{|}_K,u(|t{|}_K)),0\ne t\in K,$$

$$u(0)=u_0$$

$$\tilde{f}(\xi )=\underset{K}{\int }\chi (x\xi )f(x)dx,\xi \in K,$$

$$f(x)=\underset{K}{\int }\chi (-x\xi )\tilde{f}(\xi )d\xi .$$

$$\underset{|x{|}_K\le q^n}{\int }dx=q^n;\underset{|x{|}_K=q^n}{\int }dx=\left(1-\frac{1}{q}\right)q^n.$$

$$\underset{|x{|}_K\le q^n}{\int }|x{|}_K^{\alpha -1}dx=\frac{1-q^{-1}}{1-q^{-\alpha }}{q}^{\alpha n};\text{here and above n∈Z,α>0}.$$

$$f(x+x^{\prime })=f(x),\text{whenever ∣x′∣≤q−l}.$$

$$\mathcal{D}(K)=\underset{N\to \infty }{\underrightarrow{\lim }}\underset{l\to \infty }{\underrightarrow{\lim }}{\mathcal{D}}_N^l$$

$$\left(D^{\alpha }\phi \right)(x)={\mathcal{F}}^{-1}\left[|\xi {|}_K^{\alpha }(\mathcal{F}(\phi ))(\xi )\right](x).$$

$$\left(D^{\alpha }\phi \right)(x)=\frac{1-q^{\alpha }}{1-q^{-\alpha -1}}\underset{K}{\int }|y{|}_K^{-\alpha -1}[\phi (x-y)-\phi (x)]dy$$

$$\sum _{k=-\infty }^m{q}^k\left|u(q^k)\right|<\infty ,\sum _{l=m}^{\infty }{q}^{-\alpha l}\left|u(q^l)\right|<\infty ,$$

$$\begin{gathered} (D^{\alpha }u)(q^n)={\theta }_{\alpha }\left(1-\frac{1}{q}\right)q^{-(\alpha +1)n}\sum _{k=-\infty }^{n-1}{q}^{k}u(q^k)+q^{-\alpha n-1}\frac{q^{\alpha }+q-2}{1-q^{-\alpha -1}}u(q^n) \\ +{\theta }_{\alpha }\left(1-\frac{1}{q}\right)\sum _{l=n+1}^{\infty }{q}^{-\alpha l}u(q^l). \end{gathered}$$

$$(I^{\alpha }\phi )(x)=\frac{1-q^{-\alpha }}{1-q^{\alpha -1}}\underset{|y{|}_K\le |x{|}_K}{\int }\left(|x-y{|}_K^{\alpha -1}-|y{|}_K^{\alpha -1}\right)\phi (y)dy,\alpha \ne 1,$$

$$(I^1\phi )(x)=\frac{1-q}{q\log q}\underset{|y{|}_K\le |x{|}_K}{\int }\left(\log |x-y{|}_K-\log |y{|}_K\right)\phi (y)dy.$$

$$\sum _{k=-\infty }^m\max \left(q^k,q^{\alpha k}\right)\left|u(q^k)\right|<\infty ,\text{if α=1},$$

$$\sum _{k=-\infty }^m|k|q^k\left|u(q^k)\right|<\infty ,\text{if α=1},$$

$$(I^{\alpha }u)(|x{|}_K)=q^{-\alpha }|x{|}_K^{\alpha }u(|x{|}_K)+\frac{1-q^{-\alpha }}{1-q^{\alpha -1}}\underset{|y{|}_K<|x{|}_K}{\int }\left(|x{|}_K^{\alpha -1}-|y{|}_K^{\alpha -1}\right)u(|y{|}_K)dy,\alpha \ne 1,$$

$$(I^{1}u)(|x{|}_K)=q^{-1}|x{|}_{K}u(|x{|}_K)+\frac{1-q}{q\log q}\underset{|y{|}_K<|x{|}_K}{\int }\left(\log |x{|}_K-\log |y{|}_K\right)u(|y{|}_K)dy.$$

$$\sum _{k=-\infty }^m\max \left(q^k,q^{\alpha k}\right)\left|v(q^k)\right|<\infty ,\sum _{l=m}^{\infty }\left|v(q^l)\right|<\infty ,$$

$$\sum _{k=-\infty }^m|k|q^k\left|v(q^k)\right|<\infty ,\sum _{l=m}^{\infty }l\left|v(q^l)\right|<\infty ,$$

$$I^{\alpha }1=0$$

$$|u(q^n)|\le C{q}^{dn},n\le 0;$$

$$|u(q^n)|\le C{q}^{hn},n\ge 0,$$

$$\sum _{k=-\infty }^m\max (q^k,q^{\alpha k})\left|w(q^k)\right|<\infty ,\sum _{l=m}^{\infty }\left|w(q^l)\right|<\infty ,$$

$$\sum _{k=-\infty }^m|k|\cdot q^k\left|w(q^k)\right|<\infty ,\sum _{l=m}^{\infty }|l|\cdot \left|w(q^l)\right|<\infty ,$$

$$I^{\alpha }{D}^{\alpha }u=u.$$

$$\sum _{k=-\infty }^{n-1}{q}^k\left|u(q^k)\right|\le C\sum _{k=-\infty }^0{q}^{(d+1)k}+C\sum _{k=1}^{n-1}{q}^{(h+1)k}\le C_1{q}^{(h+1)n},$$

$$\sum _{k=-\infty }^m{q}^{\alpha k}\left|S_1^{(k)}\right|\le C\sum _{k=-\infty }^m{q}^{dk}<\infty .$$

$$\left|S_3^{(n)}\right|\le \sum _{l=n+1}^{\infty }{q}^{(h-\alpha )l}=C_2{q}^{(h-\alpha )n},$$

$$\begin{gathered} \sum _{n=-\infty }^{-2}{q}^{\alpha n}\sum _{l=n+1}^{\infty }{q}^{-\alpha l}\left|u(q^l)\right|=\sum _{n=-\infty }^{-2}{q}^{\alpha n}\left(\sum _{l=n+1}^{-1}+\sum _{l=0}^{\infty }\right)q^{-\alpha l}\left|u(q^l)\right| \\ \le C\sum _{n=-\infty }^{-2}{q}^{\alpha n}\left(\sum _{l=n+1}^{-1}{q}^{(d-\alpha )l}+\sum _{l=0}^{\infty }{q}^{(h-\alpha )l}\right)=C\sum _{l=-\infty }^{-1}{q}^{(d-\alpha )l}\sum _{n=-\infty }^{l-1}{q}^{\alpha n}+C_3\sum _{n=-\infty }^{-2}{q}^{\alpha n} \\ =C_4\sum _{l=-\infty }^{-1}{q}^{dl}+C_3\sum _{n=-\infty }^{-2}{q}^{\alpha n}<\infty . \end{gathered}$$

Full text

**Key words: ** fractional differentiation operator; non-Archimedean local field; radial functions; Cauchy problem

MSC 2010. Primary: 11S80, 35S10.

1 Introduction

The basic linear operator defined on real- or complex-valued functions on a non-Archimedean local field, for example the field $\mathbb{Q}_{p}$ of $p$-adic numbers, is the Vladimirov pseudo-differential operator $D^{\alpha}$ of fractional differentiation; see [3, 5]; for further development of this and related subjects see [1, 2, 6].

It was found in [4] that properties of $D^{\alpha}$ become much simpler on radial functions. For this case, it resembles the Caputo-Djrbashian fractional differentiation of real analysis. In particular, this operator possesses a right inverse, which can be seen as a $p$-adic counterpart of the Riemann-Liouville fractional integral or, for $\alpha=1$, the classical anti-derivative. As an application, in [4] we considered linear equations for radial functions, in which the restriction of $D^{\alpha}$ plays the role of a derivative.

In this paper, we extend the above framework to nonlinear equations. We consider equations of the form

[TABLE]

with the initial condition

[TABLE]

where $K$ is non-Archimedean local field with the normalized absolute value $|\cdot|_{K}$ (see section 2 for the definitions).

Under certain Lipschitz type condition, we prove the existence and uniqueness of a local solution of the Cauchy problem (1.1)-(1.2) defined on a sufficiently small “interval” of $t$. While this property is similar to classical ones, our analytic technique is different, due to the different nature of our $p$-adic fractional integral. The analysis of the structure of (1.1), that is of the properties of $D^{\alpha}$ on radial functions, then shows that under some assumptions regarding the function $f$, the local solution is extended to a global one defined on the whole field $K$. While for linear equations [4] we used the theory of compact operators, for nonlinear equations studied in this paper, the above results are obtained by direct constructions based on iteration processes.

2 Preliminaries

2.1. Local fields. Let $K$ be a non-Archimedean local field, that is a non-discrete totally disconnected locally compact topological field. It is well known that $K$ is isomorphic either to a finite extension of the field $\mathbb{Q}_{p}$ of $p$-adic numbers (if $K$ has characteristic 0), or to the field of formal Laurent series with coefficients from a finite field, if $K$ has a positive characteristic. For a summary of main notions and results regarding local fields see, for example, [3].

Any local field $K$ is endowed with an absolute value $|\cdot|_{K}$, such that $|x|_{K}=0$ if and only if $x=0$, $|xy|_{K}=|x|_{K}\cdot|y|_{K}$, $|x+y|_{K}\leq\max(|x|_{K},|y|_{K})$. Denote $O=\{x\in K:\ |x|_{K}\leq 1\}$, $P=\{x\in K:\ |x|_{K}<1\}$. $O$ is a subring of $K$, and $P$ is an ideal in $O$ containing such an element $\beta$ that $P=\beta O$. The quotient ring $O/P$ is actually a finite field; denote by $q$ its cardinality. We will always assume that the absolute value is normalized, that is $|\beta|_{K}=q^{-1}$. The normalized absolute value takes the values $q^{N}$, $N\in\mathbb{Z}$. Note that for $K=\mathbb{Q}_{p}$ we have $\beta=p$ and $q=p$; the $p$-adic absolute value is normalized.

The additive group of any local field is self-dual, that is if $\chi$ is a fixed non-constant complex-valued additive character of $K$, then any other additive character can be written as $\chi_{a}(x)=\chi(ax)$, $x\in K$, for some $a\in K$. Below we assume that $\chi$ is a rank zero character, that is $\chi(x)\equiv 1$ for $x\in O$, while there exists such an element $x_{0}\in K$ that $|x_{0}|_{K}=q$ and $\chi(x_{0})\neq 1$.

The above duality is used in the definition of the Fourier transform over $K$. Denoting by $dx$ the Haar measure on the additive group of $K$ (normalized in such a way that the measure of $O$ equals 1) we write

[TABLE]

where $f$ is a complex-valued function from $L_{1}(K)$. As usual, the Fourier transform $\mathcal{F}$ can be extended from $L_{1}(K)\cap L_{2}(K)$ to a unitary operator on $L_{2}(K)$. If $\mathcal{F}f=\widetilde{f}\in L_{1}(K)$, we have the inversion formula

[TABLE]

Working with functions on $K$ and operators upon them we often use standard integration formulas; see [3, 5]. The simplest of them are as follows:

[TABLE]

[TABLE]

A function $f:\ K\to\mathbb{C}$ is said to be locally constant, if there exists such an integer $l$ that for any $x\in K$

[TABLE]

The smallest number $l$ with this property is called the exponent of local constancy of the function $f$.

Let $\mathcal{D}(K)$ be the set of all locally constant functions with compact supports; it is a vector space over $\mathbb{C}$ with the topology of double inductive limit

[TABLE]

where $\mathcal{D}_{N}^{l}$ is the finite-dimensional space of functions supported in the ball B_{N}=\big{\{}x\in K: |x|\leq q^{N}\big{\}} and having the exponents of local constancy $\leq l$. The strong conjugate space $\mathcal{D}^{\prime}(K)$ is called the space of Bruhat-Schwartz distributions.

The Fourier transform preserves the space $\mathcal{D}(K)$. Therefore the Fourier transform of a distribution defined by duality acts continuously on $\mathcal{D}^{\prime}(K)$. As in the case of $\mathbb{R}^{n}$, there exists a well-developed theory of distributions over local fields; it includes such topics as convolution, direct product, homogeneous distributions etc (see [1, 3, 5]).

2.2. Vladimirov’s operator. On a test function $\varphi\in\mathcal{D}(K)$, the fractional differentiation operator $D^{\alpha}$, $\alpha>0$, is defined as

[TABLE]

Note that $D^{\alpha}$ does not preserve $\mathcal{D}(K)$; see [1] regarding the spaces of test functions and distributions preserved by this operator.

The operator $D^{\alpha}$ can also be represented as a hypersingular integral operator:

[TABLE]

[3, 5]. In contrast to (2.1), the expression in the right of (2.2) makes sense for wider classes of functions. In particular, $D^{\alpha}$ is defined on constant functions and annihilates them. Denote for brevity $\theta_{\alpha}=\dfrac{1-q^{\alpha}}{1-q^{-\alpha-1}}$.

Below we consider the operator $D^{\alpha}$ on a radial function $u=u(|x|_{K})$; here we identify the function $x\mapsto u(|x|_{K})$ on $K$ with the function $|x|_{K}\mapsto u(|x|_{K})$ on $q^{\mathbb{Z}}$. This abuse of notation does not lead to confusion.

The explicit expression of $D^{\alpha}u$ for a radial function $u$ satisfying some growth restrictions near the origin and infinity was found in [4]. If $u=u(|x|_{K})$ is such that

[TABLE]

for some $m\in\mathbb{Z}$, then for each $n\in\mathbb{Z}$ the expression in the right-hand side of (2.2) with $\varphi(x)=u(|x|_{K})$ exists for $|x|_{K}=q^{n}$, depends only on $|x|_{K}$, and

[TABLE]

Under the conditions (2.3), the expression (2.4) agrees also with the definition of $D^{\alpha}$ in terms of Bruhat-Schwartz distributions (see Chapter 2 of [5]).

2.3. The regularized integral. The fractional integral mentioned in Introduction, was defined in [4] initially for $\varphi\in\mathcal{D}(K)$ as follows:

[TABLE]

and

[TABLE]

Note that the integrals are taken, for each fixed $x\in K$, over bounded sets, and $(I^{\alpha}\varphi)(0)=0$. These properties are different from those of the anti-derivatives $D^{-\alpha}$ studied in [5].

Let $u=u(|x|_{K})$ be a radial function, such that

[TABLE]

and

[TABLE]

for some $m\in\mathbb{Z}$. Then [4] $I^{\alpha}u$ exists, it is a radial function, and for any $x\neq 0$,

[TABLE]

and

[TABLE]

Another result from [4] shows that $I^{\alpha}$ is indeed a right inverse of $D^{\alpha}$ on an appropriate class of radial functions. Namely, suppose that for some $m\in\mathbb{Z}$,

[TABLE]

if $\alpha\neq 1$, and

[TABLE]

if $\alpha=1$. Then there exists $\left(D^{\alpha}I^{\alpha}v\right)(|x|_{K})=v(|x|_{K})$ for any $x\neq 0$.

Note that the decay conditions at infinity in (2.9) and (2.10) cannot be dropped. This follows from the important identity

[TABLE]

proved in [4].

Let us prove that under some decay conditions near the origin and infinity $I^{\alpha}$ is also a left inverse of $D^{\alpha}$. First we need an estimate for $D^{\alpha}u$. Below $C$ will denote various positive constants.

Proposition**.**

Suppose that $u(0)=0$,

[TABLE]

[TABLE]

where $d>\max(0,\alpha-1)$, $0\leq h<\alpha$, and $h<\alpha-1$, if $\alpha>1$. Then the function $w=D^{\alpha}u$ satisfies, for any $m\in\mathbb{Z}$, the inequalities

[TABLE]

if $\alpha\neq 1$, or

[TABLE]

if $\alpha=1$. Moreover,

[TABLE]

Proof. Let $\alpha\neq 1$. Under (2.12) and (2.13), the conditions (2.3) are satisfied, and we may use the expression (2.4). Denote the three terms in (2.4) by $S_{1}^{(n)}$, $S_{2}^{(n)}$ and $S_{3}^{(n)}$ respectively. We have for $n\geq 1$ that

[TABLE]

so that $\left|S_{1}^{(n)}\right|\leq Cq^{(h-\alpha)n}$ and $\sum\limits_{n=m}^{\infty}\left|S_{1}^{(n)}\right|<\infty$.

If $n\leq 0$, then $\left|S_{1}^{(n)}\right|\leq Cq^{(d-\alpha)n}$, whence

[TABLE]

Similar estimates for $S_{2}^{(k)}$ are obvious.

For $n\geq 1$,

[TABLE]

so that $\sum\limits_{n=m}^{\infty}\left|S_{3}^{(n)}\right|<\infty$. Next, let $0<\alpha<1$, so that $\max(q^{k},q^{\alpha k})=q^{\alpha k}$ for $k\leq 0$. We have

[TABLE]

Therefore

[TABLE]

which implies the first inequality in (2.14).

If $\alpha>1$, then $\max(q^{k},q^{\alpha k})=q^{k}$ for $k\leq 0$, and we perform similar calculations assuming that $d>\alpha-1$.

Finally, the proofs of the inequalities (2.15) are similar to the above ones, since the factor $|k|$ does not influence the convergence of series with estimates exponential in $k$.

Let us prove (2.16). Denote $v=I^{\alpha}D^{\alpha}u$. This is a legitimate object, since $D^{\alpha}u$ satisfies the conditions, under which $I^{\alpha}$ can be applied. Moreover, $D^{\alpha}v=D^{\alpha}I^{\alpha}(D^{\alpha}u)=D^{\alpha}u$, with a possible exception of the origin. Therefore $D^{\alpha}(v-u)$ is concentrated at the origin, which means (see e.g. Theorem 1.9 from [3] or Section 6.3 in [5]) that $D^{\alpha}(v-u)=a\delta$, $a\in\mathbb{R}$. By Theorem 1, Section 9.3 of [5],

[TABLE]

where $c\in\mathbb{R}$ and ([1], Example 9.2.1)

[TABLE]

[TABLE]

We need to prove that $a=c=0$.

Let $0<\alpha<1$. We saw above that $\left|S_{1}^{(n)}\right|,\left|S_{2}^{(n)}\right|\leq Cq^{(d-\alpha)n}$, $n\leq 0$. We also have that

[TABLE]

so that

[TABLE]

Substituting (2.18) into the expression (2.7) for $I^{\alpha}w$ we find by a straightforward calculation that

[TABLE]

Now it suffices to compare the assumption (2.12), the estimate (2.19) for $v=I^{\alpha}w$, and the relation (2.17), to come to the conclusion that $a=c=0$, so that $v=u$.

The same argument works for $\alpha=1$. In this case the inequality (2.18) remains valid, so that again $(I^{1}w)(q^{n})\to 0$, as $n\to-\infty$, while $\log q^{n}\to-\infty$, as $n\to-\infty$.

Let $\alpha>1$. It follows from the above estimates of $S_{1}^{(n)},S_{2}^{(n)},S_{3}^{(n)}$ for $n\geq 1$ that

[TABLE]

This implies the inequality

[TABLE]

(see the remark after Lemma 2 in [4]). Comparing this with (2.17) and our assumption (2.13) we find that $a=0$. Then we consider as above the behavior of $u,v$ near the origin and find that $c=0$. $\qquad\blacksquare$

Corollary**.**

Let $v=v_{0}+u$ where $v_{0}$ is a constant, $u$ satisfies the conditions of the above Proposition. Then $I^{\alpha}D^{\alpha}v=v-v_{0}$.

Proof. We have $D^{\alpha}v=D^{\alpha}u$, $I^{\alpha}D^{\alpha}v=u=v-v_{0}$. $\qquad\blacksquare$

3 The Cauchy problem

3.1. Local solvability. Let us consider the problem (1.1)-(1.2), that is

[TABLE]

with the initial condition

[TABLE]

where the function $f:\ q^{\mathbb{Z}}\times\mathbb{R}\to\mathbb{R}$ satisfies the conditions

[TABLE]

[TABLE]

for all $t\in K,x,y\in\mathbb{R}$.

With the problem (3.1)-(3.2) we associate the integral equation

[TABLE]

Note that, by the definition of $I^{\alpha}$, in order to compute $(I^{\alpha}\varphi)(|t|_{K})$ for $|t|_{K}\leq q^{m}$ ($m\in\mathbb{Z}$), one needs to know the function $\varphi$ in the same ball $|t|_{K}\leq q^{m}$. Therefore local solutions of the equation (3.5) make sense, in contrast to solutions of (3.1).

We will call a solution $u$ of (3.5), if it exists, a mild solution of the Cauchy problem (3.1)-(3.2). By the above Corollary, a solution $u$ of (3.1)-(3.2), such that $u-u_{0}$ satisfies the conditions of Proposition, is a mild solution.

Theorem 1**.**

Under the assumptions (3.3),(3.4), the problem (3.1)-(3.2) has a unique local mild solution, that is the integral equation (3.5) has a solution $u(|t|_{K})$ defined for $|t|_{K}\leq q^{N}$ where $N\in\mathbb{Z}$ is sufficiently small, and another solution $\overline{u}(|t|_{K})$, if it exists, coincides with $u$ for $|t|_{K}\leq q^{\overline{N}}$ where $\overline{N}\leq N$.

Proof. We look for a solution of the equation (3.5) as a limit of the sequence $\{u_{k}\}$ where $u_{0}$ is the initial value from (3.5),

[TABLE]

We will use the following inequality proved in [4]. Let

[TABLE]

and

[TABLE]

Then

[TABLE]

where $A>0$ does not depend on $m$.

Note in particular the case $m=0$, which implies the following inequality: if $\varphi$ is a bounded continuous function, then

[TABLE]

The meaning of this property is different from its classical counterparts, in view of the identity (2.11).

Similarly, if $|\varphi(|s|_{K})|\leq\mu|s|_{K}^{\alpha m}$, then

[TABLE]

Using (3.3),(3.4),(3.7) and (3.8),(3.9) we find that

[TABLE]

[TABLE]

where

[TABLE]

so that

[TABLE]

By induction, we find that

[TABLE]

This shows that the sequence $\{u_{k}\}$ converges uniformly on the ball $\{|t|_{K}\leq T\}$, if $T$ is sufficiently small, to a limit $u(|t|_{K})$. By (2.5) and (2.6), we obtain that

[TABLE]

so that $u(|t|_{K})$ is indeed a local solution of the equation (3.5).

Suppose we have another local solution $\operatorname{\overline{u}}(|t|_{K})$. We have by (3.8) that

[TABLE]

On the other hand,

[TABLE]

and by (3.4) and (3.9),

[TABLE]

Repeating these arguments we obtain by induction that

[TABLE]

For sufficiently small $|t|_{K}$, this iteration with $m\to\infty$ implies the equality $u(|t|_{K})=\operatorname{\overline{u}}(|t|_{K})$. $\qquad\blacksquare$

3.2. Extension of solutions. Let us study the possibility to continue the local solution constructed in Theorem 1 to a solution of the integral equation (3.5) defined for all $t\in K$.

Suppose that the conditions of Theorem 1 are satisfied, and we obtained a local solution $u(|t|_{K})$, $|t|_{K}\leq q^{N}$, $N\in\mathbb{Z}$. Let $\alpha\neq 1$. In order to find a solution for $|t|_{K}=q^{N+1}$, we have, by virtue of (2.7), to solve the equation

[TABLE]

where

[TABLE]

is a known constant. A similar equation can be written for $\alpha=1$. Then the above procedure, if it is successful, is repeated for all $l>N$.

The following result is an immediate consequence of Banach’s fixed point theorem.

Theorem 2**.**

Suppose that the conditions of Theorem 1 are satisfied, as well as the following Lipschitz condition:

[TABLE]

where $0<F_{l}<q^{-\alpha l}$ for each $l\in\mathbb{Z}$. Then a local solution of the equation (3.5) admits a continuation to a global solution defined for all $t\in K$.

3.3. From an integral equation to a differential one. Let us study conditions, under which the above continuation procedure leads to a solution of the problem (3.1)-(3.2). As before, we assume the conditions (3.3),(3.4) and (3.12). In addition, we will assume that

[TABLE]

where $\beta>\alpha$.

Let us estimate the mild solution $u(|t|_{K})$ obtained by iterations for $|t|_{K}\leq q^{N}$, and then extended like in (3.10). This solution is automatically continuous at the origin. For each $l\geq N$, the solution satisfies the equation similar to (3.10):

[TABLE]

For $l\geq 1$, let us write $v_{0}^{(l)}=v_{0,1}^{(l)}+v_{0,2}^{(l)}$ where the summands correspond to the integration over the sets $\{|y|_{K}\leq 1\}$ and $\{|y|_{K}\geq q\}$ respectively. Then

[TABLE]

[TABLE]

It follows from (3.10) and (3.13) that the conditions (2.3) of the existence of $D^{\alpha}u$ are satisfied. In addition, it follows from (3.13) that $f(|t|_{K},u(|t|_{K}))$ satisfies the condition (2.9) (or (2.10), for $\alpha=1$), under which $D^{\alpha}I^{\alpha}f=f$. We come to the following result.

Theorem 3**.**

Under the assumptions (3.3),(3.4), (3.12) and (3.13), the mild solution obtained by the iteration process with subsequent continuation, satisfies the equation (3.1).

Acknowledgments

This work was funded in part under the budget program of Ukraine No. 6541230 “Support to the development of priority research trends” and under the research work ”Markov evolutions in real and p-adic spaces” of the Dragomanov National Pedagogical University of Ukraine.