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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 Earthquake convexity and some new related inequalities(Walter De Gruyter Gmbh, 2024) Kadakal, Mahir; Iscan, Imdat; Kadakal, HuriyeUnfortunately, eleven of our provinces were severely affected due to two severe earthquakes that occurred in our country, the Republic of Turkey, on February 6, 2023. As a result, thousands of buildings were destroyed and tens of thousands of our citizens lost their lives. From past to present, such disasters have occurred in many parts of our world and will continue to happen. In order to raise awareness for researchers and academicians reading our article, we will give a new definition of convexity in this article, and we will call it earthquake convexity. In this paper, we study some algebraic properties of the earthquake convexity. Then we compare the results obtained with both Holder, Holder-Iscan inequalities and power-mean, improved power-mean integral inequalities and show that the results obtained with Holder-Iscan and improved power-mean inequalities are better than the others. Some applications to special means of real numbers are also given.Öğe Exponential Convex Functions with Respect to s(Univ Maragheh, 2024) Kadakal, MahirIn this paper, we study the concept of exponential convex functions with respect to s and prove Hermite-Hadamard type inequalities for the newly introduced this class of functions. In addition, we get some refinements of the Hermite-Hadamard (HH) inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential convex with respect to s. Our results coincide with the results obtained previously in special cases.Öğe Generalized strongly n-polynomial convex functions and related inequalities(Springer, 2024) Ozcan, Serap; Kadakal, Mahir; Iscan, Imdat; Kadakal, HuriyeThis paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite-Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite-Hadamard type are derived for this class of functions using the Holder-Iscan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.Öğe (m1, m2) Geometric Arithmetically Convex Functions and Related Inequalities(2022) Kadakal, MahirIn this manuscript, we introduce and study the concept of $\\left( m_{1},m_{2}\\right) $-geometric arithmetically (GA) convex functions and their some algebric properties. In addition, we obtain Hermite-Hadamard type inequalities for the newly introduced this type of functions whose derivatives in absolute value are the class of $\\left( m_{1},m_{2}\\right) $ -GA-convex functions by using both well-known power mean and Hölder's integral inequalities.Öğe MULTIPLICATIVELY PREINVEX P-FUNCTIONS(Editura Bibliotheca-Bibliotheca Publ House, 2023) Kadakal, Huriye; Kadakal, MahirIn this study, using both Holder's integral inequality and Holder-Iccan integral inequality, some inequalities of Hermite Hadamard type obtained for multiplicatively preinvex P -functions are given. Afterwards, the result obtained with Holder-Iccan inequality has been shown to be better than that obtained with Holder's integral inequality.Öğe Semi P-geometric-arithmetically functions and some new related inequalities(Univ Nis, Fac Sci Math, 2023) Kadakal, Mahir; Kadakal, Huriye; Iscan, ImdatIn this manuscript, the authors introduce the concept of the semi P-geometric-arithmetically functions (semi P-GA functions) and give their some algebraic properties. Then, they get Hermite-Hadamard's integral inequalities for semi P-GA-functions (geometric-arithmetically convex). In addition, the authors obtain new inequalities by using Holder and Holder-Iscan integral inequalities with the help of an identity. Then, the aouthors compare the results obtained with both Holder, Holder-Iscan integral inequalities and prove that the Holder-Iscan integral inequality gives a better approximation than the Holder integral inequality. Also, some applications to special means of real numbers are also given.Öğe STRONGLY HYPERBOLIC TYPE CONVEXITY AND SOME NEW INEQUALITIES(Editura Bibliotheca-Bibliotheca Publ House, 2023) Kadakal, Huriye; Kadakal, MahirIn this study, we introduce and study the concept of strongly hyperbolic type convexity functions and their some algebraic properties. We obtain Hermite-Hadamard type inequalities for the strongly hyperbolic type convex functions. After that, by using an identity, we get some inequalities for strongly hyperbolic type convex functions. In addition, we compare the results obtained with both Holder, Holder-Iscan inequalities and power-mean, improved power-mean integral inequalities.
