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Öğe Approximating Functions in the Power-Type Weighted Variable Exponent Sobolev Space by the Hardy Averaging Operator(Univ Nis, Fac Sci Math, 2022) Ayazoglu, Rabil; Ekincioglu, Ismail; Sener, S. SuleWe investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space W-alpha(.)(r,p(.)) (0, 1), (r = 1, 2,...), by the Hardy averaging operator A (f) (x) = 1/x integral(x)(0) f (t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space W-alpha(.)(r,p(.)) (0, 1), it is shown that parallel to A(f) - f parallel to(p(.),alpha(.)-rp(.)) <= C parallel to f((r))parallel to(p(.),alpha(.)') where C is a positive constant. Moreover, we consider the problem of boundedness of Hardy averaging operator A in power-type weighted variable exponent grand Lebesgue spaces Lp(alpha(.))(p(.),theta) (0, 1). The sufficient criterion established on the power-type weight function alpha(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.Öğe B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces(De Gruyter Poland Sp Z O O, 2020) Hasanov, Javanshir J.; Ayazoglu, Rabil; Bayrakci, SimtenIn this article, we consider the Laplace-Bessel differential operator Delta(Bk,n) = Sigma(k)(i=1) (partial derivative(2)/partial derivative x(i)(2) + gamma(i)/x(i) partial derivative/partial derivative x(i)) + Sigma(n)(i=k+1) partial derivative(2)/partial derivative x(i)(2), gamma(1) > 0,..., gamma(k) > 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator M-b,M-gamma and the commutator [b, A(gamma)] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator [b, I-alpha,I-gamma] of the B-Riesz potential on B-Morrey spaces L-p,L-lambda,L-gamma, when b is an element of BMO gamma.Öğe Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with variable logistic source(Springer Int Publ Ag, 2022) Ayazoglu, Rabil; Akkoyunlu, EbubekirThis paper deals with the higher dimension quasilinear parabolic-parabolic chemotaxis model involving a source term of logistic type u(t) = del . (phi(u)del u) - V . (psi(u)del upsilon) + g(x, u), tau upsilon(t) = del upsilon - upsilon + u, in (x, t) is an element of Omega x (0, T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Omega is a smooth and bounded domain in R-N, N >= 1 and psi, phi, g are smooth, positive functions satisfying nu s(q) <= psi <= chi s(q), phi >= sigma s(p), p,q is an element of R, nu, chi, sigma > 0 when s >= s(0) >1, g(x, s) <= eta s(k(x))-mu s(rn)(x) for s > 0, eta >= 0, mu > 0 constants and g(x, 0) >= 0, x is an element of Omega, where k, m are measurable functions with 0 <= k(-) := ess(x is an element of Omega )infk (x) <= k(x) <= m(+) := ess(x is an element of Omega)sup k(x) < +infinity, 1 < m(-) := ess(x is an element of Omega)infm (x) <= m(x) <= m(+) := ess(x is an element of Omega)sup m(x)< +infinity. We extend the constant exponents k = {0, 1} , m > 1 which in logistic source term g(s) < eta s(k) - mu s(m) for s > 0, eta >= 0 , > 0 as variable exponents k(.) >= 0, m(.) > 1 with k(+) < m(-) . It is proved that if q = m(-) -1 (critical case) with mu properly large that mu > mu(0) for some mu(0) > 0, then there exists a classical solution which is global in time and bounded. Furthermore, if q < m(-) - 1, we prove that the classical solutions to the above system are uniformly in-time-bounded without restriction on mu.Öğe Bounds for the blow-up time a class of integro-differential problem of parabolic type with variable reaction term(Acad Sciences, 2023) Ayazoglu, Rabil; Akkoyunlu, EbubekirThis paper is concerned with the blow-up time of the solutions to an integro-differential problem of parabolic type with variable growth if blow-up occurs. By using the differential inequality technique, we obtain lower bounds for the blow-up time and some global existence results under some conditions to variable exponent of reaction, memory kernel, and initial value.Öğe DYNAMICS IN A PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH LOGISTIC SOURCE INVOLVING EXPONENTS DEPENDING ON THE SPATIAL VARIABLES(Amer Inst Mathematical Sciences-Aims, 2024) Ayazoglu, Rabil; Kadakal, Mahir; Akkoyunlu, EbubekirWe consider the parabolic-elliptic chemotaxis system with the exponents depending on the spatial variables logistic source and nonlinear signal production: ut = Delta u-chi del (u del upsilon)+f (x, u), (x, t) is an element of Omega x (0, T), 0 = Delta upsilon - upsilon +u(gamma) in a bounded domain Omega subset of R-N (N > 1) with smooth boundary, subject to non negative initial data and homogeneous Neumann boundary conditions, where chi > 0, gamma >= 1 and partial derivative/partial derivative nu denotes the outward normal derivative on partial derivative Omega. The logistic function f fulfilling f (x, s) <= eta s - mu s(alpha(x)+1), eta >= 0, mu > 0 for all s > 0 with f (x, 0) >= 0 for all x is an element of Omega, where alpha : Omega -> [1, infinity) is a measurable function. It is proved that if 1 <= alpha (x) < infinity for all x is an element of Omega such that ess inf(x is an element of Omega) alpha (x) > gamma or ess inf(x is an element of Omega) alpha (x) = gamma with mu > chi, then there exists a nonnegative classical solution (u, upsilon) that is global-in-time and bounded. In addition, under the particular conditions gamma = 1 and f (x, s) = mu (s - s(alpha(x)+1)), if mu is sufficiently large, the global bounded solution (u, upsilon) satisfies IIu (, t) - 1II(L)infinity(Omega) + II upsilon (, t) - 1II(L)infinity(Omega) <= Ce (- k/N+2t) for all t > 0 with k = min{ chi 2/4 , 1/2 } , C > 0. The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameter conditions, which improves the known results.Öğe Existence and multiplicity of solutions for a Schrodinger-Kirchhoff type equation involving the fractional p(.,.)-Laplacian operator in RN(Springer-Verlag Italia Srl, 2021) Ayazoglu, Rabil; Sarac, Yesim; Sener, S. Sule; Alisoy, GulizarIn this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii's genus theory, we show the existence and multiplicity of solutions for a Schrodinger-Kirchhoff type equation involving the fractional p-Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch-Wang type compact embedding theorem for fractional Sobolev space with variable exponent.Öğe Existence of solutions for a resonant problem under landesman-lazer type conditions involving more general elliptic operators in divergence form(Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 2020) Ayazoglu, Rabil; Şule Şener, Sidika; Aydın, Tuba AğırmanThe present paper is concerned with the resonant problem ?div (a (x, ?u)) = ?1|u|p ?2 u + f (x, u) ? g(x) in ?, where ? is a bounded domain with smooth boundary in RN (N ? 2), p ? (1, ?) and div(a (x, ?u)) is a general elliptic operators in divergence form. By assuming a Landesman-Lazer type condition and using a variational method based on the Minimum Principle, we show the existence of a weak solution in the Sobolev space W0 1,p (?). © 2020, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. All rights reserved.Öğe Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity(Taylor & Francis Ltd, 2024) Ayazoglu, Rabil; Akkoyunlu, Ebubekir; Naghizadeh, ZohrehWe consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.Öğe Global attractors in a two-species chemotaxis system with two chemicals and variable logistic sources(Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 2024) Ayazoglu, Rabil; Salmanova, Kamala A.This paper deals with the higher dimension quasilinear parabolic-parabolic chemotaxis model involving a source term of logistic type (formula presented), subject to the homogeneous Neumann boundary conditions in a ? ? RN (N ? 1) with smooth boundary. It is shown that for the attractionrepulsion case with ?2 ? 0, the global boundedness of solutions can be ensured by µ1, µ2 > 0 without any other assumptions, due to the contribution of the logistic sources included in addition to the repulsion mechanism. While for the attraction-attraction case with ?2 > 0, the global boundedness of solutions has to require logistic coefficients µ1, µ2 > 0 such that µ2 properly large. © 2024, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. All rights reserved.Öğe Global boundedness of solutions to a quasilinear parabolic-parabolic Keller-Segel system with variable logistic source(Academic Press Inc Elsevier Science, 2022) Ayazoglu, RabilThis paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of variable logistic type u(t) = del . (phi(u)del u) - del. (phi(u)del upsilon) + g(x, u), -Delta upsilon = -upsilon + u, in (x, t) is an element of Omega x (0, T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Omega is a smooth and bounded domain in R-N(N >= 1) and g is an element of C-1 ((Omega) over bar x [0, infinity)) function satisfies g(x, s) <= eta(sk(x)) - mu s(m(x)), s > 0 with g(x, 0) >= 0, x is an element of Omega and eta >= 0, mu > 0 are constants, k, m are measurable functions with 1 < k(-) := ess(x is an element of Omega) infk (x) <= k(x) <= k(+) := ess sup(x is an element of Omega) k(x)< +infinity, 2 <= m(-) := ess(x is an element of Omega) infm (x) <= m(x) <= m(+) := ess sup(x is an element of Omega) m(x)< +infinity. Positive functions psi, phi is an element of C-2(inverted right perpendicular0, infinity)) satisfy nu s(q) <= phi <= chi(sq) and phi >= sigma s(p) with p, q is an element of R, nu, chi, sigma > 0 when s >= 0. It is proved that if q = m(-) - 1 (critical case) and mu > chi (1 - 2/N (m(-) - p - 1)(+), then there exists a classical solution which is global in time and bounded. Moreover, if q is an element of (k(+) - 1, m(-) - 1), we obtain that there is a classical solution of the above system uniformly in time bounded without any restrictions on m(-) and mu. (c) 2022 Elsevier Inc. All rights reserved.Öğe Infinitely many solutions for the stationary fractional p-Kirchhoff problems in RN(Springer India, 2019) Akkoyunlu, Ebubekir; Ayazoglu, RabilIn the present paper, we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Kirchhoff equation M(integral integral R2N vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+ps)dxdy + integral V-RN(x)vertical bar u vertical bar p dx) x((-Delta)(p)(s) u + V(x) vertical bar u vertical bar(p-2) u) = f (x, u) in R-N, where (-Delta)(p)(s) is the fractional p-Laplacian operator, 0 < s < 1 < p < infinity with sp < N, M : R-0(+) -> R-0(+) is a nonnegative, continuous and increasing Kirchhoff function, the nonlinearity f : R-N x R -> R is a Caratheodory function that obeys some conditions which will be stated later and V is an element of C(R-N, R+) is a non-negative potential function. We first establish the Bartsch-Pankov-Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the variational method, (S+) mapping theory and Krasnoselskii's genus theory.Öğe Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width(Walter De Gruyter Gmbh, 2021) Aydin, Tuba Agirman; Ayazoglu, Rabil; Kocayigit, HuseyinThe curves of constant width are special curves used in engineering, architecture and technology. In the literature, these curves are considered according to different roofs in different spaces and some integral characterizations of these curves are obtained. However, in order to examine the geometric properties of curves of constant width, more than characterization is required. In this study, firstly differential equations characterizing quaternionic space curves of constant width are obtained. Then, the approximate solutions of the differential equations obtained are calculated by the Morgan-Voyce polynomial approach. The geometric properties of this curve type are examined with the help of these solutions.Öğe TIMELIKE HELICES IN THE SEMI-EUCLIDEAN SPACE E24(Honam Mathematical Soc, 2022) Aydin, Tuba Agirman; Ayazoglu, Rabil; Kocayigit, HuseyinIn this paper, we define timelike curves in R-2(4) and characterize such curves in terms of Frenet frame. Also, we examine the timelike helices of R-2(4), taking into account their curvatures. In addition, we study timelike slant helices, timelike B-1-slant helices, timelike B-2-slant helices in four dimensional semi-Euclidean space, R-2(4). And then we obtain an approximate solution for the timelike B-1 slant helix with Taylor matrix collocation method.
