The real functions satisfying the inequality Ф (uv) ≤ KФ (u) Ф (v) for some positive K which occur among others in [5], [3], [4], and referred there as submultiplicative, are discussed. A simplifying remark that Ф satisfies this inequality iff KФ is submultiplicative in the standard sense, is done. It is shown that, under general conditions, the standard submultiplicativity of Ф and the inequality Ф (u) Ф (1/u) ≤ 1 imply that Ф must be multi-plicative. Applying a result of Bhatt [1], we observe that if p is a nontrivial seminorm on a Banach algebra X such that the set { [formula] .. : ∈ G X, p (x) ≠ 0} is a singleton {λ}, then s = λp is a submultiplicative seminorm on X.
Here we present Conformable fractional Iyengar type inequalities with respect to $$L_{p}$$ norms, with $$1
In this paper, we establish some fixed point results of a mapping satisfying certain rational type contractive conditions in the frame work of a metric space endowed with partial order. Our results generalize and extend the result of Singh and Chatterjee (1988) [7] in partially ordered metric spaces and some existing results in the literature. Few illustrative examples are given to support our results.
Some oscillation criteria for the second order neutral delay differential equationsx(t)±∑i=1lci(t)x(t-τi)″+∑i=1mpi(t)x(t-δi)-∑i=1nqi(t)x(t-σi)=0,t>0are established. New oscillation criteria are different from one recently established in the sense that the boundedness of the solution in the results of Parhi and Chand [Oscillation of second order neutral delay differential equations with positive and negative coefficients, J. Indian Math. Soc., 66 (1999) 227–235.] has been erased., i.e. we give sufficient conditions for the oscillation of all solutions.
In this paper a general fixed point theorem for two pairs of subsequentially mappings compatible of type E is proved, which generalize the results by [2]-[4], [6] and other results. As applications, new results for mappings satisfying contractive conditions of integral type, φ-contractive conditions and weak contractive conditions are obtained.
In [11], the author discussed a new class of nearly weak uniformly L-Lipschitzian mappings and prove some strong convergence results of the modified Ishikawa iteration with errors in real Banach spaces. And the author has given the open problem as follows: Are there any difference on convergence between the Mann iteration and Ishikawa iteration? Can we prove the equivalence on convergence between these two iterations? In this paper, we given an affirmative answer to the open problem.
The purpose of the present paper is to introduce and investigate a new class of functions, namely αδs -irresolute functions. Several properties of these functions and a decomposition of αδs -irresoluteness are also given. References [1] M. Caldas, D. N. Georgiou, S. Jafari and T. Noiri, More on δ-semiopen sets, Note Mat., 22 (2003), 113-126. [2] M. Caldas and S. Jafari, On semi δs-irresolute functions, Fasc. Math., 58 (2017), 47-55. [3] G. I. Chae, T. Noiri and D. W. Lee, On na-continuous functions, Kyungpook Math. J., 26 (1986), 73-79. [4] B. Y. Lee, M. J. Son and J. H. Park, δ-semiopen sets and its applications, Far East J. Math. Sci. (FJMS), 3 (2001), 745-759. [5] N. Levine, Semi open sets and semi continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41. [6] S. N. Maheshwari and S. S. Thakur, On α-irresolute mappings, Tamkang J. Math., 11 (1980), 209-214. [7] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53. [8] A. S. Mashhour, I. A. Hasanein and S. N. El-Deeb, α-continuous and α-open mappings, Acta Math. Hungar, 41 (1983), 213-218. [9] O. Nj̊astad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961-970. [10] T. Noiri, Remarks on δ-semi-open sets and δ-preopen sets, Demonstratio Math., 36 (2003), 1007-1020. [11] J. H. Park, B. Y. Lee and M. J. Son, On δ-semiopen sets in topological spaces, J. Indian Acad. Math., 19 (1997), 59-67. [12] N. V. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl. (2), 78 (1968), 103-118. Received 19 February 2019 1 Department of Mathematics, Faculty of Sciences, Selçuk University, Selçuklu, Konya, 42130 Turkey. 2 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan. E-mail address: 1 ybeceren@selcuk.edu.tr, 2 t.noiri@nifty.com 2010 Mathematics Subject Classification. 54C08, 54C10, 54A05.
In this paper we obtain some new reverses of Hölder vector inequality for positive operators on Hilbert spaces.
