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1
Content available Potential method in the coupled linear quasi-static theory of thermoelasticity for double porosity materials
Mikelashvili M.
Archives of Mechanics
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2023
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Vol. 75, nr 5
559--590
EN
This paper concerns the coupled linear quasi-static theory of thermoelasticity for materials with double porosity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy’s law of the flow of a fluid through a porous medium, Fourier’s law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. By virtue of Green’s identity the uniqueness theorems for classical solutions of the internal and external quasi-static boundary value problems (BVPs) are proved. The fundamental solution of the system of steady vibration equations in the considered theory is constructed and its basic properties are established. Then, the surface and volume potentials are presented and their basic properties are given. Finally, on the basis of these results the existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.
2
Content available Problems of steady vibrations in the coupled linear theory of double-porosity viscoelastic materials
Svanadze M. M.
Archives of Mechanics
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2021
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Vol. 73, nr 4
365--390
EN
In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green’s identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations.
3
Content available remote Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes
Bandura Andriy, Skaskiv Oleh
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
82--87
EN
We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variablesand has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2→C.
4
Content available remote Potential method in the theory of thermoelasticity for materials with triple voids
Svanadze M.
Archives of Mechanics
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2019
|
Vol. 71, nr 2
113--136
EN
In the present paper the linear theory of thermoelasticity for isotropic and homogeneous solids with macro-, meso- and microporosity is considered. In this theory the independent variables are the displacement vector field, the changes of the volume fractions of pore networks and the variation of temperature. The fundamental solution of the system of steady vibrations equations is constructed explicitly by means of elementary functions. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations.
5
Content available remote Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure
Svanadze M.
Archives of Mechanics
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2017
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Vol. 69, nr 4-5
347--370
EN
The purpose of the present paper is to develop the classical potential method in the linear theory of thermoelasticity for materials with a double porosity structure based on the mechanics of materials with voids. The fundamental solution of the system of equations of steady vibrations is constructed explicitly by means of elementary functions and its basic properties are established. The Sommerfeld-Kupradze type radiation conditions are established. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method and the theory of singular integral equations.
6
Content available remote The existence of a unique solution of the hyperbolic functional differential equation
Karpowicz A.
Demonstratio Mathematica
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2014
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Vol. 47, nr 4
866--877
EN
We consider the Z. Szmydt problem for the hyperbolic functional differential equation. We prove a theorem on existence of a unique classical solution and the Carathéodory solution of the hyperbolic equation.
7
Existence results for second-order impulsive neutral functional differential inclusions in Banach spaces
Bahaj M., Cherti I.
Journal of Applied Analysis
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2012
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Vol. 18, nr 2
225-242
EN
In this paper, we investigate the existence of mild solutions on a compact interval to second-order impulsive neutral functional differential inclusions in Banach spaces. The results are obtained by using the theory of continuous cosine families and a fixed point theorem due to Dhage.
8
Content available remote The existence theory for the differential equation xm(t)=f(t,x) in Banach spaces and Henstock-Kurzweil integral
Sikorska-Nowak A.
Demonstratio Mathematica
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2007
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Vol. 40, nr 1
115-124
EN
Using the properties of the Henstock–Kurzweil integral and corresponding theorems, we prove the existence theorem for the equation x(m)(t) = f(t, x) in a Banach space, where f is HL integrable and satis.es certain conditions. Our fundamental tool is the measure of noncompactness developed by Kuratowski and Hausdorff.
9
Content available remote Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex space
Shkarin S. S.
Demonstratio Mathematica
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2003
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Vol. 36, nr 3
611--626
EN
According to Mickael's selection theorem any surjective continuous linear operator from one Prechet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if E is a Frechet space and T : E -> E is a continuous linear operator such that the Cauchy problem x = T x, x(0) = X0 is solvable in [0,1] for any X06 E, then for anyf zawiera się C([0, 1],E), there exists a continues map S : [0,1] x E -> E, (t x) ->o StX such that for any X0 zawiera się w E, the function x(t) = StX0 is a solution of the Cauchy problem x(t) = Tx(t) +- f(t), x(0) = X0 (they call S a fundamental system of solutions of the equation x = Tx + f). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Frechet spaces and strong duals of Frechet-Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.
10
Content available remote Continuous solutions of some fractional order integral equations
Salem H. A. H., El-Sayed A. M. A., Moustafa O. L.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2002
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[Z] 42(2)
209-220
EN
In this paper, Schauder fixed point theorem is used to prove an existence of positive continuous solutions for the nonlinear fractional order integral equation x(t) = h(t) + λ Iα (ƒ(x(t)) + g(x(t))), t ϵ [0, 1], α > 0 (E), where ƒ and g are nonlinear continuous functions and ƒ is nondecreasing while g is nonincreasing. Also the existence of maximal and minimal solutions of (E) will be proved. Some fractional order differential equations will be considered.
11
Content available remote Existence theorems for nonlinear functional integral equations of fractional orders
Cichoń M., El-Sayed A.M., Hussien A.H.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2001
|
[Z] 41
59-67
12
Content available remote On the existence of weak solutions of the Darboux problem for the hyperbolic partial differential equations in Banach spaces
Kubiaczyk I., Mostafa Ali N.
Fasciculi Mathematici
|
1998
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Nr 28
93-99
EN
In this paper we prove an existence theorem for the hyperbolic partial differential equation zx.y = f(x,y,z,zxy), z(x,O)=o, z(O,y)=O for x,y>O, where Zxy means the second mixed derivative in the weak sence. The continuity of the xy function f is replaced by the weak continuity and the compactness condition is expressed in ternls of the measuresa of weak noncompactness. This paper extends some previous results for our equation.
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