<abstract xmlns="http://www.w3.org/1999/xhtml">

The aim of this paper is to study the superstability problem of the d’Alembert type functional equation
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0001_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math>
<tex-math>$$f(x + y + z) + f(x + y + \sigma (z)) + f(x + \sigma (y) + z) + f(\sigma (x) + y + z) = 4f(x)f(y)f(z)$$</tex-math>
</alternatives>
</disp-formula>
for all <italic>x, y, z</italic> ∈ <italic>G</italic>, where <italic>G</italic> is an abelian group and σ : <italic>G</italic> → <italic>G</italic> is an endomorphism such that <italic>σ</italic>(<italic>σ</italic>(<italic>x</italic>)) = <italic>x</italic> for an unknown function <italic>f</italic> from <italic>G</italic> into ℂ or into a commutative semisimple Banach algebra.
</abstract>

On the superstability of generalized d’Alembert harmonic functions

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.
</abstract>

Strong ideals and horizontal ideals in pseudo-BCH-algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">

We consider the eigenvalue problem for the <italic>p</italic>(<italic>x</italic>)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.
</abstract>

An integro-differential inequality related to the smallest positive eigenvalue of <italic>p</italic>(<italic>x</italic>)-Laplacian Dirichlet problem

<abstract xmlns="http://www.w3.org/1999/xhtml">

For two-dimensional manifold <italic>M</italic> with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle <italic>TM</italic> ⊕ <italic>E</italic>, where <italic>E</italic> is a trivial bundle. The metrizable case, when <italic>M</italic> is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: <bold>36</bold> (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on <italic>TM</italic> ⊕ (ℝ × <italic>M</italic>) and two on <italic>TM</italic> ⊕ (ℝ2 × <italic>M</italic>) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝ<italic>N</italic>, after suitable local affine embedding of (<italic>M,</italic>∇) into ℝ<italic>N</italic>.
</abstract>

Affine analogues of the Sasaki-Shchepetilov connection

<abstract xmlns="http://www.w3.org/1999/xhtml">

For an arbitrary triangle <italic>ABC</italic> and an integer <italic>n</italic> we define points <italic>Dn</italic>, <italic>En</italic>, <italic>Fn</italic> on the sides <italic>BC</italic>, <italic>CA</italic>, <italic>AB</italic> respectively, in such a manner that
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0005_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AB</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">CD</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">BD</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AB</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">BC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">AE</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">CE</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">BC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow><mrow><msup><mrow><mrow><mo stretchy="false">|</mo><mrow><mi mathvariant="italic">AC</mi></mrow><mo stretchy="false">|</mo></mrow></mrow><mi>n</mi></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">BF</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi mathvariant="italic">AF</mi><mi>n</mi></msub></mrow><mo stretchy="false">|</mo></mrow></mrow></mfrac><mo>.</mo></mrow></mtd></mtr></mtable></math>
<tex-math>$$\matrix{{{{\left| {AC} \right|^n } \over {\left| {AB} \right|^n }} = {{\left| {CD_n } \right|} \over {\left| {BD_n } \right|}},} \hfill &amp; {{{\left| {AB} \right|^n } \over {\left| {BC} \right|^n }} = {{\left| {AE_n } \right|} \over {\left| {CE_n } \right|}},} \hfill &amp; {{{\left| {BC} \right|^n } \over {\left| {AC} \right|^n }} = {{\left| {BF_n } \right|} \over {\left| {AF_n } \right|}}.}} $$</tex-math>
</alternatives>
</disp-formula>
Cevians <italic>ADn</italic>, <italic>BEn</italic>, <italic>CFn</italic> are said to be the <italic>Maneeals of order n</italic>. In this paper we discuss some properties of the Maneeals and related objects.
</abstract>

Department of Mathematics, Kesava Reddy Educational Instutions, Machilipatnam, Kurnool, Andhra Pradesh,

Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Kraków,

Several observations about Maneeals - a peculiar system of lines

<abstract xmlns="http://www.w3.org/1999/xhtml">

A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0006_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi mathvariant="italic">du</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></math>
<tex-math>$\int_a^b {f(t)\;du(t)} $</tex-math>
</alternatives>
</inline-formula>, where <italic>f</italic> is assumed to be of <italic>r</italic>-<italic>H</italic>-Hölder type on [<italic>a, b</italic>] and <italic>u</italic> is of bounded variation on [<italic>a, b</italic>], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.
</abstract>

A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we propose and analyse an iterative algorithm for the approximation of a common solution for a finite family of <italic>k</italic>-strict pseudocontractions and two finite families of generalized equilibrium problems in the setting of Hilbert spaces. Strong convergence results of the proposed iterative algorithm together with some applications to solve the variational inequality problems are established in such setting. Our results generalize and improve various existing results in the current literature.
</abstract>

Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

The investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator.
</abstract>

Institute of Mathematics, Pedagogical University, ul. Podchorążych 2, 30-084 Kraków,

Properties of two variables Toeplitz type operators

<abstract xmlns="http://www.w3.org/1999/xhtml">

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.
</abstract>

New method to solve certain differential equations

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we study the superstablity problem of the cosine and sine type functional equations:
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0010_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mi>σ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="italic">xya</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math>
<tex-math>$$f(x\sigma (y)a) + f(xya) = 2f(x)f(y)$$</tex-math>
</alternatives>
</disp-formula>
and
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_aupcsm-2016-0010_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mi>σ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="italic">xya</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></math>
<tex-math>$$f(x\sigma (y)a) - f(xya) = 2f(x)f(y),$$</tex-math>
</alternatives>
</disp-formula>
where <italic>f</italic> : <italic>S</italic> → ℂ is a complex valued function; <italic>S</italic> is a semigroup; <italic>σ</italic> is an involution of <italic>S</italic> and <italic>a</italic> is a fixed element in the center of <italic>S</italic>.
</abstract>

On the superstability of the cosine and sine type functional equations

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Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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