\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 157, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/157\hfil Solvability of nonlocal boundary-value problems] {Solvability of nonlocal boundary-value problems for the Laplace equation in the ball} \author[M. A. Sadybekov, B. Kh. Turmetov, B. T. Torebek \hfil EJDE-2014/157\hfilneg] {Makhmud A. Sadybekov, Batirkhan Kh. Turmetov, Berikbol T. Torebek} % in alphabetical order \address{Makhmud A. Sadybekov \newline Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science Republic of Kazakhstan, 050010 Almaty, Kazakhistan} \email{makhmud-s@mail.ru} \address{Batirkhan Kh. Turmetov \newline Department of Mathematics, Akhmet Yasawi International Kazakh-Turkish University, 161200 Turkistan, Kazakhistan} \email{batirkhan.turmetov@iktu.kz} \address{Berikbol T. Torebek \newline Department of Mathematics, Akhmet Yasawi International Kazakh-Turkish University, 161200 Turkistan, Kazakhistan} \email{turebekb85@mail.ru} \thanks{Submitted March 18, 2014. Published July 10, 2014.} \subjclass[2000]{35J15, 35J25, 34B10, 26A33, 31A05, 31B05} \keywords{Riemann-Liouville operator; Caputo operator; \hfill\break\indent periodic problem; antiperiodic problem; nonlocal problem; Laplace equation; \hfill\break\indent Poisson kernel; harmonic function} \begin{abstract} In this article, we consider a class of nonlocal problems for the Laplace equation with boundary operators of fractional order. We prove the existence, uniqueness and a representation of the solutions. Also it is shown that the smoothness of solutions in Holder classes depends on the order of the boundary operators. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{problem}[theorem]{Problem} \allowdisplaybreaks \section{Introduction}\label{intr} Let $\Omega = \{ {x \in R^n :| x | < 1} \}$ be the unit ball, and $\partial \Omega = \{ {x \in R^n :| x| = 1} \}$ be a unit sphere. Further let, $u(x)$ be a harmonic function in the ball $ \Omega $, $r = | x |$, $\theta = x/| x |$, $$ \frac{d}{{dr}} = \sum_{j = 1}^n \frac{{x_j}}{{|x|}} \frac{\partial }{{\partial x_j }} . $$ For an arbitrary positive number $ \alpha > 0 $, the operator of fractional integration in the Riemann-Liouville sense of order $ \alpha $ is the expression \cite{kilbas-book}: $$ I^\alpha [u](x) = \frac{1}{{\Gamma (\alpha )}}\int_0^r {(r - \tau )^{\alpha - 1} u(\tau ,\varphi )} \,d\tau , \quad r > 0 . $$ Since $I^\alpha [u](x) \to u(x) $ almost everywhere as $ \alpha \to 0 $, by definition we suppose $ I^0 [u](x) =u(x) $. A fractional differentiation operator is naturally defined as the product of a fractional integration operator and differentiation operator of integer order. Thus, depending on the sequence of multiplication of operators their properties are changed. The most famous operator of the fractional differentiation is the Riemann-Liouville operator \cite{kilbas-book}, $$ _{RL} D^\alpha [u](x) = \frac{d}{{dr}}I^{1 - \alpha } [u](x),\quad 0 < \alpha \le 1 . $$ In another order of multiplication we obtain fractional differentiation operator in the sense of Caputo \cite{kilbas-book}, $$ _C D^\alpha [u](x) = I^{1 - \alpha } [ {u'} ](x),\quad 0 < \alpha \le 1. $$ Denote \begin{gather*} B^\alpha [u](x) = r^\alpha {}_{RL}D^\alpha [u](x) ,\\ B_*^\alpha [u](x) = r^\alpha {}_CD^\alpha [u](x) ,\\ B^{ -\alpha } [u](x) = \frac{1}{{\Gamma (\alpha )}}\int_0^1 {(1 - s)^{\alpha - 1} s^{ - \alpha } u(sx)\,ds} . \end{gather*} Note, that properties and some applications of the operators $B^\alpha $, $ B_*^\alpha $ and $B^{ - \alpha } $ to solvability questions of the local and nonlocal boundary-value problems were studied in \cite{karachik-sam,torebek-bvp}. The organization of this article is as follows. In Section \ref{FP} we give the formulation of the basic problems and some historical information about boundary-value problems with boundary operators of fractional order. In Section \ref{AS} we provide auxiliary statements. These statements are related with properties of the solutions of the Dirichlet problem and boundary-value problem with the boundary operators of fractional order. In Section \ref{USP} we prove theorems on the uniqueness of solution of the studied problems. Finally, Section \ref{ES} is devoted to study of the main problem, where we formulate and prove theorems on existence and smoothness of the solution. \section{Formulation of the problem}\label{FP} Denote \begin{gather*} \partial \Omega _ + = \partial \Omega \cap \{ {x \in R^n :x_1 \ge 0} \},\\ \partial \Omega _ - = \partial \Omega \cap \{ {x \in R^n :x_1 \le 0} \}, \\ I = \partial \Omega \cap \{ {x \in R^n :x_1 = 0}\} . \end{gather*} We associate each point $x = ( {x_1 ,x_2,\dots ,x_n } ) \in \Omega $ with its ``opposite'' point $$ x^* =( {a_1 x_1 ,a_2 x_2 ,\dots ,a_n x_n } ) \in \Omega , $$ where $a_1 = - 1$ , and $ a_j ,j = 2,\dots ,n $ take one of values $\pm 1$. Obviously, that if $x \in \partial \Omega _ + $, then $x^* \in\partial \Omega _ - $. In the domain $ \Omega $ we consider the following boundary-value problems. \begin{problem}\label{pr1} \rm Find a function $ u(x) \in C^2 (\Omega ) \cap C(\bar\Omega ) $ such that $ B^\alpha [u](x) $ is a continuous function in the domain $\bar \Omega $, and satisfies the following conditions: \begin{gather}\label{(2.1)} \Delta u(x) = 0,\quad x \in \Omega,\\ \label{(2.2)} u(x) - (-1)^k u( {x^*} ) = f( x ),\quad x \in \partial \Omega _ + ,\\ \label{(2.3)} {}_{RL}D^\alpha [u](x) + (-1)^k {}_{RL}D^\alpha [u](x^*) = g(x),\quad x \in \partial \Omega _ + . \end{gather} \end{problem} \begin{problem}\label{pr2} \rm Find a function $ u(x) \in C^2 (\Omega ) \cap C(\bar\Omega ) $ such that $ B_*^\alpha [u](x) $ is a continuous function in the domain $ \bar \Omega $, and satisfies equation \eqref{(2.1)}, equation \eqref{(2.2)} and $$ {}_CD^\alpha [u](x) + (-1)^k {}_CD^\alpha [u](x^*) = g(x),\quad x \in \partial \Omega _ + , $$ where $k = 1,2$, $ 0 <\alpha \le 1$, $ f(x) \in C^{\lambda + \alpha } (\partial \Omega_ + ),g(x) \in C^\lambda (\partial \Omega _ + ),0 < \lambda < 1$, $ \lambda + \alpha $ - non-integer. \end{problem} When $k = 1$, problems \ref{pr1} and \ref{pr2} are called antiperiodical boundary-value problems, and when $k = 2$ - periodical boundary-value problems. Necessary condition for existence of a solution of the problem \ref{pr1} (problem \ref{pr1}) with smoothness $ u(x) \in C^2 (\Omega ) \cap C(\bar \Omega ) $, $ B^\alpha [u](x) \in C(\bar \Omega ) $ ($B_*^\alpha [u](x) \in C(\bar \Omega ) $ ) is fulfillment of the matching conditions: \begin{gather}\label{(2.4)} f(0,x_2 ,\dots ,x_n) + (-1)^k f(0,a_2 x_2 ,\dots ,a_n x_n ) = 0,\quad (0,x_2 ,\dots ,x_n ) \in I, \\ \label{(2.5)} g(0,x_2 ,\dots ,x_n ) - (-1)^k g(0,a_2 x_2 ,\dots ,a_n x_n ) = 0,\quad (0,x_2 ,\dots ,x_n ) \in I,\\ \label{(2.6)} \frac{{\partial f(0,x_2 ,\dots ,x_n )}}{{\partial x_j }} + (-1)^k \frac{{\partial f(0,a_2 x_2 ,\dots ,a_n x_n )}}{{\partial x_j }} = 0,\quad (0,x_2 ,\dots ,x_n ) \in I, \end{gather} when $\lambda + \alpha > 1$. Furthermore we assume that these conditions are satisfied. Note that numerous publications were devoted to the questions of solvability of boundary-value problems for elliptic equations with boundary operators of fractional order, see \cite{berdyshev-cvee, berdyshev-smj, karachik-sam, kirane-smj1, kirane-smj2, muratbekova-bvp, torebek-bvp, torebek-kargu, turmetov-de, turmetov-sam, turmetov-kaznu, umarov-dm,umarov-fcaa}. In \cite{berdyshev-cvee,karachik-sam,kirane-smj1, kirane-smj2, muratbekova-bvp}, the Laplace equation nonlocal boundary-value problems with boundary operators of fractional order were investigated. It should also be noted that some questions of solvability of nonlocal problems for fractional order equations in the one-dimensional case were studied in \cite{ahmad-fcaa,ahmad-cma,chai-ade,liu-ejde,nyamoradi-ejde,tatar-ejde, wang-ade}. In these papers the natural generalizations of the Samarskii - Bisadze problem were studied, when nonlocal conditions were given in the relation form of the boundary values with values of the desired function within the domain. In this paper we consider the problems when non-local conditions are given in the form of periodic or antiperiodic conditions. Since $$ _{RL} D^1 [ u ]( x ) = {}_CD^1 [ u ]( x ) = \frac{{du( x)}}{{dr}}, $$ then when $\alpha = 1$ derivatives $ _CD^\alpha ,{}_{RL}D^\alpha $ coincides with derivative in the direction of the vector $r = | x |$. Note, that this case, i.e. when $\alpha = 1$, was studied in \cite{sadybekov-emj, sadybekov-de}. In particular, the following propositions were proved. \begin{theorem}\label{th1} Let $k = 1$, $ f(x) \in C^{1 + \lambda }(\partial \Omega _ + )$, $g(x) \in C^\lambda (\partial \Omega _ + )$, $ 0 < \lambda < 1 $ and the matching conditions \eqref{(2.4)}, \eqref{(2.5)}, \eqref{(2.6)} hold. Then a solution of problem \ref{pr1} (problem \ref{pr2}) exists, it is unique and represented in the form: $$ u(x) = - \int_{\partial \Omega _ + } {\frac{{\partial G_1 (x,y)}}{{\partial n_y }}f(y)\,ds_y } + \int_{\partial \Omega _ + } {G_1 (x,y)g(y)\,ds_y } , $$ where $ G_1 (x,y) $ is Green function of the anti-periodical problem \ref{pr1} (problem \ref{pr2}): $$ G_1 (x,y) = \frac{1}{2}[ {G_D (x,y) + G_D (x,y^*) + G_N (x,y) - G_N (x,y^*)} ] , $$ $ G_D (x,y) $ - Green function of the Dirichlet problem, $ G_N (x,y) $ - Green function of the Neumann problem. \end{theorem} \begin{theorem}\label{th2} Let $k = 2$, $ f(x) \in C^{1 + \lambda } ( {\partial \Omega _ + })$, $ g(x) \in C^\lambda ( {\partial \Omega _ + }) $, $ 0 < \lambda < 1 $ and the matching conditions \eqref{(2.4)}, \eqref{(2.5)}, \eqref{(2.6)} hold. Then for solvability of problem \ref{pr1} (problem \ref{pr2}) it is necessary and sufficient fulfillment of the condition $$ \int_{\partial \Omega _ + } {g(y)\,ds_y } = 0. $$ If a solution exists, then it is unique with up to a constant and can be represented as $$ u(x) = - \int_{\partial \Omega _ + } {\frac{{\partial G_2 (x,y)}}{{\partial n_y }}f(y)\,ds_y + \int_{\partial \Omega _ + } {G_2 (x,y)g(y)\,ds_y } } + {\rm const}, $$ where $ G_2 (x,y) $ is Green function of the periodical problem, that is defined by the equality: $$ G_2 (x,y) = \frac{1}{2}[ {G_D (x,y) - G_D (x,y^*) + G_N (x,y) + G_N (x,y^*)} ] + {\rm const}. $$ \end{theorem} Later we will conduct a full investigation of the questions of existence, uniqueness and smoothness of solutions of problems \ref{pr1} and \ref{pr2}, depending on the order of the boundary operators within $ 0 < \alpha \le 1 $. Moreover, for completeness of investigation, we give proofs of some statements in the case $\alpha = 1$. \section{Auxiliary statements}\label{AS} To investigate questions on solvability of the problems \ref{pr1} and \ref{pr2} we have to provide some properties of the operators $ B^\alpha $, $ B_*^\alpha $ and $B^{ - \alpha } $. The following propositions have been proved for the case $ 0 < \alpha < 1 $ in \cite{karachik-sam}, and for $\alpha = 1$ in \cite{bavrin-de}. \begin{lemma}\label{lem1} Let function $ u(x) $ be harmonic in the domain $\Omega $. Then (1) for any $\alpha \in ( {0,1} ]$ functions $ B^\alpha [u](x)$, $B_*^\alpha [u](x)$ are also harmonic in $\Omega $; (2) if $\alpha \in ( {0,1} )$, then the function $B^{ - \alpha } [ u ]( x )$ is harmonic in $\Omega$; (3) if $\alpha = 1$, then for $u( 0 ) = 0$ the function $B^{ - 1} [ u ]( x )$ is harmonic $\Omega $. \end{lemma} \begin{lemma}\label{lem2} Let $ 0 < \alpha \le 1 $, function $ u(x) $ be harmonic in the domain and continuous in $ \bar \Omega $. Then, if $ B^\alpha [u](x) $, $B_*^\alpha [u](x)$ are continuous in the domain $ \bar \Omega $, Then the following equalities are true: (1) for any $\alpha \in ( {0,1} )$, \begin{equation}\label{(3.1)} B^{ - \alpha } [ {B^\alpha [u]} ](x) = B^\alpha [ {B^{ - \alpha } [u]} ](x) = u( x ),\quad x \in \overline \Omega, \end{equation} (2) for any $\alpha \in ( {0,1} )$, $$ B_*^\alpha [u ](x) = B[ u ]( x ) - \frac{{u( 0 )}}{{\Gamma ( {1 - \alpha } )}},\quad x \in \overline \Omega . $$ (3) if $\alpha = 1$, then \begin{equation}\label{(3.2)} B^{ - 1}[ {B^1 [ u ]} ]( x ) = u( x) - u( 0 ),\quad x \in \overline \Omega \end{equation} (4) if $\alpha = 1$ and $u( 0 ) = 0$, then $$ B^1 [{B^{ - 1} [ u ]} ]( x ) = u( x ),x \in \overline \Omega . $$ \end{lemma} Let $v( x )$ and $w( x )$ be solutions of the following problems: \begin{equation}\label{(3.3)} \begin{gathered} \Delta v( x ) = 0,\quad x \in \Omega , \\ v( x ) = \tau ( x ),\quad x \in \partial \Omega , \end{gathered} \end{equation} and \begin{equation}\label{(3.4)} \begin{gathered} \Delta w( x ) = 0,x \in \Omega , \\ B^\alpha [ w ]( x ) = \mu ( x ),x \in \partial \Omega . \end{gathered} \end{equation} The following propositions refer to the smoothness of the solution of the Dirichlet problem \eqref{(3.3)} (see\cite{alimov-de}). \begin{lemma}\label{lem3} Let $\lambda> 0$, $\lambda $ - non-integer and $\tau ( x ) \in C^\lambda ({\partial \Omega } )$. Then a solution of problem \eqref{(3.3)} exists, belongs to the class $C^\lambda ({\overline \Omega } )$ and for any multi-index $\beta =( {\beta _1 ,\beta _2 ,\dots ,\beta _n } )$ with $|\beta | > \lambda $ the following estimate is true: \begin{equation}\label{(3.5)} | {\partial ^\beta v( x )} | \le C( {1 - | x |} )^{\lambda - | \beta|}, \end{equation} where $\partial ^\beta v( x )= \frac{{\partial ^{| \beta |} v( x )}}{{\partial x_1^{\beta _1 } \dots \partial x_n^{\beta _n } }}$. \end{lemma} And the converse is also true. \begin{lemma}\label{lem4} Let $\lambda > 0$, $v( x ) \in C^2 ( \Omega ) \cap C( {\overline \Omega } )$ and for any multi-index $\beta = ( {\beta _1 ,\beta _2 ,\dots ,\beta _n } )$ with $| \beta | > \lambda $ the inequality \eqref{(3.5)} holds. Then $v( x ) \in C^\lambda ( {\overline \Omega } )$. \end{lemma} The following statement defines smoothness of a solution of probelem \eqref{(3.4)} (see \cite{turmetov-kaznu}). \begin{lemma}\label{lem5} Let $\lambda > 0$, $ 0 < \alpha < 1$, $\mu ( x ) \in C^\lambda ( {\partial \Omega })$, $\lambda $ and $\lambda + \alpha $ - non-integer. Then a solution of the problem \eqref{(3.4)} exists, it is unique, belongs to the class $C^{\lambda + \alpha } ( {\overline \Omega } )$ and can be represented as $$ w( x ) = \int_{\partial \Omega } {P_\alpha ( {x,y} )\mu( y )\,ds_y } , $$ where $$ P_\alpha ( {x,y} ) = \frac{1}{{\Gamma ( \alpha )}}\int_0^1 {( {1 - s} )^{\alpha - 1} s^{ - \alpha } P(sx,y)ds} , $$ $P(x,y) = \frac{1}{{\omega _n }}\frac{{1 - | x |^2 }}{{| {x - y} |^n }}$ - Poisson kernel of the Dirichlet problem \eqref{(3.3)}. \end{lemma} Let us give a proposition about smoothness of the fractional derivative of the Dirichlet problem. \begin{lemma}\label{lem6} Let $\lambda > \alpha $, $ 0 < \alpha \le 1 $, $\lambda $ and $\lambda - \alpha $ non-integer. Further, let $\tau ( x ) \in C^\lambda ( {\partial \Omega })$, $v( x )$ be a solution of the Dirichlet problem \eqref{(3.3)}. Then $B^\alpha [ v ]( x ) \in C^{\lambda - \alpha } ( {\overline \Omega })$. \end{lemma} \begin{proof} Let $v( x )$ be a solution of the problem \eqref{(3.3)}. Introduce the function $B^\alpha [ v ]( x )$ in the form \begin{align*} B^\alpha [ v ]( x ) &= \frac{{r^\alpha }}{{\Gamma (1 - \alpha )}}\frac{d}{{dr}}\int_0^r {(r - \tau )^{ - \alpha } v( {\tau \theta } )} d\tau \mathop = _{\tau = r\xi }\\ &= \frac{{r^\alpha }}{{\Gamma (1 - \alpha )}}\frac{d}{{dr}}r^{1 - \alpha } \int_0^1 {( {1 - \xi } )^{ - \alpha }v( {\xi x} )} d\xi \\ &= \frac{1}{{\Gamma (1 -\alpha )}}r^\alpha [ {( {1 - \alpha } )r^{ - \alpha } + r^{1 - \alpha } \frac{d}{{dr}}} ]\int_0^1 {( {1 - \xi } )^{ - \alpha } v( {\xi x} )} d\xi \\ &= \frac{1}{{\Gamma (1 - \alpha )}}( {r\frac{d}{{dr}} + 1 - \alpha } )\int_0^1 {( {1 - \xi } )^{ - \alpha } v( {\xi x} )} d\xi . \end{align*} Denote $$ v_1 ( x ) = \int_0^1 {( {1 - \xi })^{ - \alpha } v( {\xi x} )} d\xi . $$ Let $\beta$ be a multi-index $\beta = ( {\beta _1 ,\beta _2 ,\dots ,\beta_n } )$ with $| \beta | > \lambda + 1 - \alpha$. Since $\tau ( x ) \in C^\lambda ( {\partial \Omega } )$, due to the lemma \ref{lem3} $v( x ) \in C^\lambda ( {\overline \Omega } )$ and $$ | {\partial ^\beta v( x )} | \le C({1 - | x |} )^{\lambda - | \beta |}. $$ Then $$ | {\partial ^\beta v_1 ( x )} | \le C\int_0^1 {( {1 - \xi } )^{ - \alpha } } \xi ^{| \beta |} ( {1 - \xi | x |} )^{\lambda - | \beta |} d\xi . $$ Represent the last integral in the form $$ \int_0^1 { =\int_0^{| x |} + \int_{| x|}^1 = I_1 + I_2 } $$ and estimate $I_1 $. We consider two cases: (a) Let $1/2 \le | x | \le 1$. Since for any $\xi \in [ {0,| x |} ]$ inequalities $1 -\xi | x | \ge 1 - \xi $ and $| \xi |^{| \beta |} \le 1$ are true, it follows that \begin{align*} I_1 &\le C\int_0^{| x |} {( {1 - \xi })^{\lambda - \alpha - | \beta |} } d\xi = {\frac{{( {1 - \xi } )^{\lambda + 1 - \alpha - | \beta |} }}{{| \beta | - \lambda - 1 +\alpha }}} \big|_0^{| x |} \\ &= \frac{1}{{| \beta | - \lambda - 1 + \alpha }}[ {( {1 - | x |} )^{\lambda + 1 - \alpha - | \beta |} - 1} ] \le C( {1 - | x |} )^{\lambda + 1 - \alpha - | \beta |}. \end{align*} (b) Let $| x | \le 1/2$. In this case $1 - \xi | x | \ge 1 - | x |^2 \ge 1 - \frac{1}{4} =\frac{3}{4}$. Consequently, $I_1 \le C$, i.e. $I_1 $ is bounded. Thus, in general case $$ I_1 \le C( {1 - | x |} )^{\lambda + 1 - \alpha - | \beta |} . $$ Next we estimate integral $I_2 $. In this case for all $\xi \in [ {| x |,1} ]$ inequality $1 - \xi | x | \ge 1 - | x |$ holds, and, thus $$ ( {1 - \xi| x |} )^{\lambda - | \beta |} \le ( {1 - | x |} )^{\lambda - | \beta |} . $$ Then \begin{align*} I_2 & \le ( {1 - | x |})^{\lambda - | \beta |} \int_0^1 {( {1 - \xi } )^{ - \alpha } } d\xi \\ & = ( {1 - | x |} )^{\lambda - | \beta |} \frac{{( {1 - \xi } )^{1 - \alpha } }}{{1 - \alpha}} \big| _{| x |}^1 \\ &= \frac{{( {1 - | x |})^{\lambda + 1 - \alpha - | \beta |} }}{{1 -\alpha }}. \end{align*} Hence, for any multi-index $\beta = ( {\beta _1 ,\beta _2 ,\dots ,\beta _n } )$ with $| \beta | > \lambda + 1 - \alpha $ the inequality $$ | {\partial ^\beta v_1 ( x )} | \le C( {1 - | x |} )^{\lambda + 1 - \alpha - | \beta |} $$ holds. The by lemma \ref{lem4} $v_1 ( x ) \in C^{\lambda +1 - \alpha } ( {\overline \Omega } )$. Since $$ B^\alpha [ v ]( x ) = \frac{1}{{\Gamma ( {1 - \alpha } )}}\Big( {r\frac{d}{{dr}} + 1 - \alpha } \Big)v_1 ( x ), $$ then obviously, $B^\alpha [ v ]( x ) \in C^{\lambda - \alpha } ( {\overline \Omega } )$. The proof is complete. \end{proof} \begin{lemma}\label{lem7} Let $\tau ( x ) \in C( {\partial \Omega } )$ and $v( x )$ be a solution of \eqref{(3.3)}. If $\tau ( x )$ has the property $$ \tau ( x ) = \pm \tau ( {x^*} ), \quad x \in \partial \Omega _ + , $$ then for any $x \in \overline \Omega $, we have $v( x ) = \pm v( {x^*} )$. \end{lemma} \begin{proof} If $\tau ( x ) \in C( {\partial \Omega } )$, then a solution of \eqref{(3.3)} exists and can be represented as a Poisson integral: \begin{equation}\label{(3.6)} v( x ) = \int_{\partial \Omega } {P(x,y)\tau ( y )\,ds_y } . \end{equation} Using property of the function $\tau ( x )$, the function \eqref{(3.6)} can be represented as follows: \begin{align*} v( x ) &= \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y } + \frac{1}{{\omega _n }}\int_{\partial \Omega _ - } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y} \\ & = \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y } + \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y^*} |^n }}\tau ( {y^*})\,ds_y } \\ &= \frac{1}{{\omega _n}}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y } \pm \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y^*} |^n }}\tau ( y )\,ds_y } \\ &= \frac{1}{{\omega _n}}\int_{\partial \Omega _ + } {[ {\frac{{1 - | x |^2 }}{{| {x - y} |^n }} \pm \frac{{1 - | x |^2 }}{{| {x - y^*} |^n }}} ]\tau ( y)\,ds_y } . \end{align*} Then for $v( {x^*} )$ we have $$ v( {x^*} ) = \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {[ {\frac{{1 - | {x^*} |^2 }}{{| {x^* - y} |^n }} \pm \frac{{1 - | {x^*} |^2 }}{{| {x^* - y^*} |^n }}} ]\tau ( y )\,ds_y } . $$ Further, since $| x | = | {x^*} |$ and \begin{align*} | {x^* - y} |^2 &= \sum_{j = 1}^n ( {\alpha _j x_j -y_j } )^2 = \sum_{j = 1}^n ( {\alpha _j^2 x_j^2 - 2\alpha _j x_j y_j + y_j^2 } )^2 \\ &= \sum_{j = 1}^n ( {x_j^2 - 2\alpha _j x_j y_j + \alpha _j^2 y_j^2 } )^2 \\ &= \sum_{j = 1}^n ({x_j - \alpha _j y_j } )^2 \\ &= | {x - y^*} |^2 \end{align*} then \begin{align*} v( {x^*} ) &= \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } [ {\frac{{1 - | {x^*} |^2 }}{{| {x^* - y} |^n }} \pm \frac{{1 - | {x^*} |^2 }}{{| {x^* - y^*} |^n }}}]\tau ( y )\,ds_y \\ &= \pm \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {[ {\frac{{1 - | x |^2 }}{{| {x - y} |^n }} \pm \frac{{1 - | x |^2 }}{{| {x - y^*} |^n }}} ]\tau ( y )\,ds_y } = v( x ). \end{align*} The proof is complete. \end{proof} \section{Uniqueness of a solutions to problems \ref{pr1} and \ref{pr2}}\label{USP} \begin{theorem}\label{th3} If a solution of problem \ref{pr1} exists, then (1) when $k = 1,2$ for all $\alpha \in ( {0,1} )$ the solution is unique; (2) in the case $\alpha = 1$ when $k = 1$ the solution is unique, and when $k = 2$ it is unique up to a constant value. \end{theorem} \begin{proof} Suppose that $u( x )$ is a solution of the homogenous problem. Then due to \eqref{(2.2)} we obtain \begin{equation}\label{(4.1)} u( x) = (-1)^k u( {x^*} ),\quad x \in \partial \Omega _ + \end{equation} Let $\alpha \in ( {0,1} )$. Apply the operator $B^\alpha $ to the function $u( x )$. Then by the lemma \ref{lem1} function $B^\alpha [ u ]( x)$ is harmonic in the domain $ \Omega $, and since $$ {B^\alpha [ u ]( x )} \big|_{\partial\Omega } = {_{RL} D^\alpha [ u ]( x )} \big|_{\partial \Omega } , $$ and due to the boundary condition \eqref{(2.3)}, it follows that $$ B^\alpha [ u ]( x ) = - (-1)^k B^\alpha [ u ]( {x^*} ),\quad x \in \partial \Omega _ + . $$ Further, since $B^\alpha [ u ]( x ) \in C( {\overline \Omega } )$, then from lemma \ref{lem7} for any $x \in \overline \Omega $ it follows that \begin{equation}\label{(4.2)} B^\alpha [ u ]( x) = - (-1)^k B^\alpha [ u]( {x^*} ) \end{equation} Applying the operator $B^{ - \alpha } $ to the equality \eqref{(4.2)}, and taking account equality \eqref{(3.1)}, we obtain \begin{align*} u( x )& = B^{ - \alpha } [ {B^\alpha [ u]} ]( x ) \\ & = - (-1)^k B^{ - \alpha } [ {B^\alpha [ u ]}]( {x^*} ) \\ &= - (-1)^k u({x^*} ),\quad x \in \overline \Omega . \end{align*} i.e. for all $x \in\overline \Omega $, we have $u( x ) = - ( { -1} )^k u( {x^*} )$. In particular, we get \begin{equation}\label{(4.3)} u( x) = - ( - 1 )^k u( x^* ),\quad x \in \partial \Omega . \end{equation} Comparing equalities \eqref{(4.1)} and \eqref{(4.3)} we have $u( x ) = 0$ for $x \in \partial \Omega _ + $; thus $$ u( x ) = 0,\quad x \in\partial \Omega . $$ Then due to maximum principle for harmonic functions: $$ u( x )\equiv 0,\quad x \in \overline \Omega . $$ Now let $\alpha = 1$ and $k= 1$. Then from the boundary condition \eqref{(2.2)} we have $u( x ) = - u( {x^*} )$, and from \eqref{(3.2)} and condition \eqref{(2.3)}, $$ u( x) - u( 0 ) = u( {x^*} ) - u( 0). $$ Consequently, $u( x ) = u( {x^*} )$; thus, $u( x) = 0,\quad x \in\partial \Omega $. Then $$ u( x )\equiv 0,\quad x \in \overline \Omega . $$ If $k = 2$, then from condition \eqref{(2.2)} we obtain $u( x ) = u( {x^*})$, and from the conditions \eqref{(3.2)} and \eqref{(2.3)}: $$ u( x ) - u( 0 ) = - [ {u( {x^*}) - u( 0 )} ]. $$ Then $$ u( x ) = 2u(0) \equiv {\rm const},\quad x \in\partial \Omega , $$ hence $u( x) \equiv C,x \in \overline \Omega$. The proof is complete. \end{proof} The following result can be proved analogously, as the above theorem. \begin{theorem}\label{th4} If a solution of problem \ref{pr2} exists, then (1) when $k = 1$ for all $\alpha \in ( {0,1} ]$ the solution is unique; (2) when $k = 2$ for all $\alpha \in ( {0,1} ]$ the solution is unique up to constant item. \end{theorem} \section{Existence of a solution}\label{ES} Let a function $P_\alpha (x,y)$ be defined by \begin{equation}\label{(5.1)} P_\alpha (x,y) = \begin{cases} \frac{1}{{\Gamma (\alpha )}}\int_0^1 {(1 - s)^{\alpha - 1} s^{ - \alpha } } P(sx,y)ds, & 0 < \alpha < 1 \\[4pt] \int_0^1 {[ {P(sx,y) - 1} ]\frac{{ds}}{s}} , &\alpha = 1 \end{cases} \end{equation} \begin{theorem}\label{th5} Assume that in problem \ref{pr1}: $f( x ) \in C^{\lambda + \alpha } ( {\partial \Omega _ + } )$ and $g( x ) \in C^\lambda ( {\partial \Omega _ + })$, where $0 < \lambda < 1$, $0 < \alpha \le 1$, $\lambda $ and $\lambda + \alpha $ - non-integer. Then (1) if $\alpha \in ( {0,1} )$ and $k = 1,2$, then a solution of the problem exists and is unique; (2) if $\alpha = 1$ , then for $k = 1$ a solution of the problem exists and is unique, and for $k = 2$, for existence of a solution of the problem it is necessary any sufficient the fulfillment of the condition: \begin{equation}\label{(5.2)} \int_{\partial \Omega _ + } {g( y )\,ds_y = 0}, \end{equation} If a solution of the problem exists, then it is unique up to constant term; (3) if a solution of the problem exists, then it belongs to the class $C^{\lambda + \alpha } ( {\bar \Omega } )$, and can be represented as follows: \begin{equation} \label{(5.3)} \begin{aligned} u(x) &= \frac{1}{2}\int_{\partial \Omega _ + } [ {P(x,y) - (-1)^k P(x,y^*)} ]f( y )\,ds_y \\ &\quad + \frac{1}{2}\int_{\partial \Omega _ + } {[ {P_\alpha ( {x,y} ) + (-1)^k P_\alpha ( {x,y^*} )} ]g( y )} \,ds_y. \end{aligned} \end{equation} \end{theorem} \begin{proof} We introduce the auxiliary functions: $$ v(x) =\frac{1}{2}( {u(x) - (-1)^k u(x^* )}),\quad w(x) = \frac{1}{2}( {u(x) + (-1)^k u(x^* )} ). $$ It is obvious, that $u( x) = v( x ) + w( x )$. Assuming, that $u( x )$ is a solution of \ref{pr1}, we find two problems, satisfied by $v( x )$ and $w( x )$. The function $v( x )$ is a solution of Dirichlet problem \eqref{(3.3)}, and the function $w( x )$ is a solution of the problem \eqref{(3.4)}, where \begin{equation}\label{(5.4)} \tau ( x ) = \begin{cases} \frac{1}{2}f( x ),&\text{if } x \in \partial \Omega _ + \\[4pt] - \frac{{(-1)^k }}{2}f( {x^*} ),&\text{if } x \in \partial \Omega _ - \end{cases} \end{equation} and \begin{equation}\label{(5.5)} \mu ( x ) = \begin{cases} \frac{1}{2}g( x ),&\text{if } x \in \partial \Omega _ + \\[4pt] \frac{{(-1)^k }}{2}f( {x^*} ),&\text{if } x \in \partial \Omega _ - \end{cases} \end{equation} Indeed, if $x \in \partial \Omega _ + $, then $$ \tau ( x) \equiv v( x )\big|_{\partial \Omega _ + } = \frac{1}{2}[ {u( x ) - (-1)^k u( {x^*} )} ]\big| _{\partial \Omega_ + } = \frac{{f( x )}}{2}, $$ And if $x \in \partial \Omega _ - $, then in this case $x^* \in\partial \Omega _ + $ and \begin{align*} \tau ( x ) &\equiv v( x )|_{\partial \Omega _ - } = [ {u( x ) -(-1)^k u( {x^*} )} ] \\ &= - \frac{{(-1)^k }}{2}[ {u( {x^*}) - (-1)^k u( x )} ] = -(-1)^k \frac{{f( {x^*} )}}{2}. \end{align*} Thus, a function $\tau ( x )$ is defined by equality \eqref{(5.4)}. Further, $$ B^\alpha [ w ]( x ) = \frac{1}{2}[ {B^\alpha [ u ]( x ) + (-1)^k B^\alpha [ u ]( {x^*})} ], $$ and, hence \begin{gather*} B^\alpha [ w ](x )| _{\partial \Omega _ + } = \frac{1}{2}g( x ), \\ B^\alpha [ w ](x )| _{\partial \Omega _ - } = ( { -1} )^k \frac{{g( {x^*} )}}{2}. \end{gather*} i.e. for the function $\mu ( x )$ we obtain equality \eqref{(5.5)}. If $\tau ( x ) \in C^{\lambda + \alpha } ({\partial \Omega } )$, then for any $\alpha \in (0,1]$ a solution of the Dirichlet problem \eqref{(3.3)} exists, belongs to the class $v( x ) \in C^{\lambda + \alpha } ({\overline \Omega } )$ and is represented as \eqref{(3.6)}. Further, since for $\tau ( y )$ equality \eqref{(5.4)} holds, then \begin{align*} v( x ) &= \frac{1}{{\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y } + \frac{1}{{\omega _n }}\int_{\partial \Omega _ - } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}\tau ( y )\,ds_y } \\ &= \frac{1}{{2\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}f( y )\,ds_y } - \frac{{(-1)^k }}{{2\omega _n }}\int_{\partial \Omega _ - } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}f( {y^*} )\,ds_y} \\ &= \frac{1}{{2\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y} |^n }}f( y )\,ds_y } - \frac{{(-1)^k }}{{2\omega _n }}\int_{\partial \Omega _ + } {\frac{{1 - | x |^2 }}{{| {x - y^*} |^n }}f( y)\,ds_y } \\ &= \frac{1}{2}\int_{\partial \Omega _+ } {[ {P(x,y) - (-1)^k P(x,y^*)} ]f( y )\,ds_y } . \end{align*} Let $0 < \alpha < 1$. By lemma \ref{lem5} when $\mu ( y ) \in C^\lambda ({\partial \Omega } )$ a solution of problem \eqref{(3.4)} exists, belongs to the class $C^{\lambda + \alpha } ( {\overline \Omega } )$ and is represented in the form $$ w( x ) = \int_{\partial \Omega } {P_\alpha ( {x,y} )\mu ( y )\,ds_y } . $$ Then, using the representation of the function $\mu ( y )$, we have \begin{align*} w( x )& = \int_{\partial \Omega } {P_\alpha ( {x,y} )\mu ( y )\,ds_y } \\ &= \frac{1}{2}\int_{\partial \Omega _ + } {P_\alpha ( {x,y} )g( y )\,ds_y } + \frac{{(-1)^k }}{2}\int_{\partial \Omega _ - } {P_\alpha ( {x,y} )g( {y^*} )\,ds_y } \\ & = \frac{1}{2}\int_{\partial \Omega _ + } {[ {P_\alpha ( {x,y} ) + (-1)^k P_\alpha ({x,y^*} )} ]g( y )} \,ds_y . \end{align*} Thus, in the case $0 < \alpha < 1$, $k = 1,2$ for a solution of the problem \ref{pr1} representation \eqref{(5.3)} holds. Now let $\alpha = 1$. In this case the problem \eqref{(3.4)} is the Neumann problem and for the existence of a solution of this problem it is necessary and sufficient fulfillment of the condition: \begin{equation}\label{(5.6)} \int_{\partial \Omega } {\mu (y )} \,ds_y = 0 . \end{equation} If $k = 1$, then due to the equality \eqref{(5.5)}, \begin{align*} \int_{\partial \Omega } {\mu ( y )} \,ds_y &=\frac{1}{2}\int_{\partial \Omega _ + } {g( y )} \,ds_y - \frac{1}{2}\int_{\partial \Omega _ - } {g({y^*} )} \,ds_y \\ &= \frac{1}{2}\int_{\partial\Omega _ + } {g( y )} \,ds_y - \frac{1}{2}\int_{\partial \Omega _ + } {g( y )} \,ds_y = 0 ; \end{align*} i.e. in this case condition of solvability \eqref{(5.6)} always holds, hence a solution of problem \eqref{(3.4)} exists. If $k = 2$, then $$ \int_{\partial \Omega } {\mu ( y )} \,ds_y = \frac{1}{2}\int_{\partial \Omega _ + } {g( y )} \,ds_y + \frac{1}{2}\int_{\partial \Omega _ - } {g( {y^*} )} \,ds_y = \int_{\partial \Omega _ + } {g( y )} \,ds_y , $$ and then condition on solvability of Neumann problem \eqref{(5.6)} can be rewritten in the form \eqref{(5.3)}. It is known \cite{bitsadze-dan,bitsadze-de}, that a solution of the Neumann problem is represented as follows: \begin{equation}\label{(5.7)} w(x) = \int_{\partial \Omega} {P_1 (x,y)\mu ( y )} \,ds_y + C \end{equation} where $$ P_1 (x,y) = \int_0^1 {[ {P(sx,y) - 1} ]\frac{{ds}}{s}} . $$ Further, using representation of the function $\mu (x)$, function \eqref{(5.7)} is easy reduced to the form \begin{equation}\label{(5.8)} w(x) = \int_{\partial \Omega _+ } {[ {P_1 (x,y) - P_1 (x^*,y)} ]g( y )} \,ds_y + C . \end{equation} Note, that if $x^* = ( - x_1 ,\alpha _2 x_2 ,\dots ,\alpha _n x_n)$, then $$ (x^*)^* = (x_1 ,x_2 ,\dots ,x_n ) = x. $$ Then \begin{align*} w(x^*) &= \frac{1}{2}( {u(x^*) + (-1)^k u({x^*}^*)}) \\ &= \frac{1}{2}( {(-1)^k u(x) +u(x^*)} ) \\ &= \frac{{(-1)^k}}{2}( {u(x) + (-1)^k u(x^*)} ) =(-1)^k w(x). \end{align*} Thus, when $k = 1$ the function $w(x)$ has the symmetric property $$ w(x) = - w(x^*),x \in \Omega . $$ For the function \eqref{(5.8)} this is possible, only when $ C = 0 $. Hence, when $k = 1$ for a solution of problem \ref{pr1} we obtain representation \eqref{(5.3)}. If $k = 2$, then $w(x) = w(x^*)$, $x\in \Omega$. In this case the solution of Problem \ref{pr1} is unique up to a constant term and the representation \eqref{(5.3)} holds. The theorem is proved. \end{proof} Let a function $P_\alpha ^* (x,y)$ be defined by \begin{equation}\label{(5.9)} P_\alpha ^* (x,y) = \begin{cases} \frac{1}{{\Gamma (\alpha )}}\int_0^1 {(1 - s)^{\alpha - 1} s^{ - \alpha } } [ {P(sx,y) - 1} ]ds, & 0 < \alpha < 1 \\ \int_0^1 {s^{ - 1} } [ {P(sx,y) - 1} ]ds, & \alpha = 1 \end{cases} \end{equation} The following proposition can be proved analogously to the above theorem. \begin{theorem}\label{th6} In problem \ref{pr2} let $0 < \alpha \le 1$, $f( x ) \in C^{\lambda + \alpha } ( {\partial\Omega _ + } )$, $g( x ) \in C^\lambda ({\partial \Omega _ + } )$, $0 < \lambda < 1$, $\lambda $ and $\lambda + \alpha $ - non-integer. Then (1) if $k = 1$ a solution of the problem exists and unique; (2) if $k = 2$ then for solvability of the problem it is necessary any sufficient fulfillment of the condition \eqref{(5.2)}. If a solution exists, then it is unique up to constant term; (3) if a solution of the problem exists, then it belongs to the class $C^{\lambda + \alpha } ( {\bar \Omega } )$, and can be represented as: \begin{equation}\label{(5.10)} \begin{aligned} u(x) &= \frac{1}{2}\int_{\partial \Omega _ + } {[ {P(x,y) - (-1)^k P(x,y^*)} ]f( y )\,ds_y } \\ &\quad + \frac{1}{2}\int_{\partial \Omega _ + } {[ {P_\alpha ^* ( {x,y} ) + (-1)^k P_\alpha ^* ( {x,y^*} )} ]g( y )} \,ds_y. \end{aligned} \end{equation} \end{theorem} \section{Examples} \begin{example}\label{ex1} \rm Let $n = 2$, $a_2 = - 1$, $k = 1$. Then in problem \ref{pr1} we obtain the boundary value conditions: \begin{gather*} u( {1,\varphi } ) + u( {1,\varphi + \pi }) = f( \varphi ),\quad 0 \le \varphi \le \pi ,\\ B^\alpha [ u ]( {1,\varphi } ) - B^\alpha[ u ]( {1,\varphi + \pi } ) = g( \varphi ),\quad 0 \le \varphi \le \pi . \end{gather*} By the theorem \ref{th5}, problem \ref{pr1} has a unique solution, which can be represented as follows \begin{align*} u(x) &= \frac{1}{{4\pi}}\int_0^\pi [ {P( {r,\varphi - \theta }) + P( { - r,\varphi - \theta } )}]f( \theta )\,d\theta \\ &\quad + \frac{1}{{4\pi }}\int_0^\pi {[ {P_\alpha ( {r,\varphi - \theta } ) - P_\alpha ( { - r,\varphi - \theta } )} ]g( \theta )\,d\theta } . \end{align*} In \cite{torebek-kargu}, an explicit form of the function \eqref{(5.1)} was obtained: $$ P_\alpha ( {r,\gamma } ) = 2\Gamma ( {1 - \alpha } )\Big( {\frac{{\cos [ {( {1 - \alpha } )\arctan \frac{{r\sin \gamma }}{{1 - r\cos \gamma }}} ]}}{{( {1 - 2r\cos \gamma + r^2 } )^{\frac{{1 - \alpha }}{2}} }} - \frac{1}{2}} \Big). $$ Then a solution of the problem has the form: \begin{align*} u(x) & = \frac{1}{{2\pi }}\int_0^\pi \frac{{1 - r^4 }}{{1 - 2r^2 \cos 2( {\varphi - \theta } ) + r^4 }}f( \theta )\,d\theta \\ &\quad + \frac{{\Gamma ( {1 - \alpha } )}}{{2\pi }}\int_0^\pi {\frac{{\cos [ {( {1 - \alpha } )\arctan\frac{{r\sin ( {\varphi - \theta } )}}{{1 - r\cos ( {\varphi - \theta } )}}} ]}}{{( {1 - 2r\cos ( {\varphi - \theta } ) + r^2 } )^{\frac{{1 - \alpha }}{2}} }}g( \theta )\,d\theta} \\ &\quad - \frac{{\Gamma ( {1 - \alpha } )}}{{2\pi }}\int_0^\pi {\frac{{\cos [ {( {1 - \alpha } )\arctan\frac{{r\sin ( {\varphi - \theta } )}}{{1 + r\cos ( {\varphi - \theta } )}}} ]}}{{( {1 + 2r\cos ( {\varphi - \theta } ) + r^2 } )^{\frac{{1 - \alpha }}{2}} }}g( \theta )\,d\theta } . \end{align*} \end{example} \begin{example}\label{ex2} \rm Let $n = 2$, $a_2 = 1$, $k = 2$. In this case, the boundary conditions of the problem \ref{pr2} have the form \begin{align*} u( {1,\varphi } ) - u( {1,2\pi - \varphi }) = f( \varphi ),\quad 0 \le \varphi \le \pi ,\\ B_*^\alpha [ u ]( {1,\varphi } ) - B_*^\alpha [ u ]( {1,2\pi - \varphi } ) = g( \varphi ),0 \le \varphi \le \pi . \end{align*} By Theorem \ref{th6} for the solvability of the considered problem it is necessary and sufficient fulfillment of the condition $\int_0^\pi {g( \theta )\,d\theta } = 0$. The problem has a unique solution up to a constant, which is represented in the form \eqref{(5.10)}. As in Example \ref{ex1} one can construct the explicit form of the function \eqref{(5.9)}, and then the solution has the form: \begin{align*} u(x) &= \frac{1}{{2\pi }}\int_0^\pi \Big[ {\frac{{1 - r^2 }}{{1 - 2r\cos ( {\varphi - \theta } ) + r^2 }} - \frac{{1 - r^2 }}{{1 - 2r\cos ( {\varphi + \theta } ) + r^2 }}} \big]f( \theta )\,d\theta \\ &\quad + \frac{{\Gamma ( {1 - \alpha } )}}{{2\pi }}\int_0^\pi {\frac{{\cos [ {( {1 - \alpha } )\arctan\frac{{r\sin ( {\varphi - \theta } )}}{{1 - r\cos ( {\varphi - \theta } )}}} ]}}{{( {1 - 2r\cos ( {\varphi - \theta } ) + r^2 } )^{\frac{{1 - \alpha }}{2}} }}g( \theta )\,d\theta} \\ &\quad + \frac{{\Gamma ( {1 - \alpha } )}}{{2\pi }}\int_0^\pi {\frac{{\cos [ {( {1 - \alpha } )\arctan\frac{{r\sin ( {\varphi + \theta } )}}{{1 - r\cos ( {\varphi + \theta } )}}} ]}}{{( {1 - 2r\cos ( {\varphi + \theta } ) + r^2 } )^{\frac{{1 - \alpha }}{2}} }}g( \theta )\,d\theta } . \end{align*} \end{example} \subsection*{Conclusion} In this paper questions about solvability of some nonlocal boundary-value problems for the Laplace equation are studied. Boundary conditions are given in the form of periodic or anti-periodic conditions, i.e. values of the function and values of the fractional derivative in the upper part of the boundary are associated with the values of these functions in the bottom part of the boundary. Theorems on existence and uniqueness of solutions are proved, and conditions for solvability of the investigated problems are established. Moreover, in the Holder class the order of smoothness of the solution are studied depending on the order of the boundary operator. \subsection*{Acknowledgements} This research is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No. 0743/GF). The authors would like to thank the editor and referees for their valuable comments and remarks, which led to a great improvement of the article. \begin{thebibliography}{00} \bibitem{ahmad-fcaa} B. Ahmad, J. J. 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