Let be a sequence in a -metric space such thatįor some, , and each. Thenįrom Lemma 5, we deduce the following lemma. Let be a -metric space and let be a sequence in. The following results are useful for some of the proofs in the paper. We note that if are two nonnegative real numbers, then Is a -metric with constant and so is a -metric space. Next, we give some examples of -metric spaces.Įxample 2. So the notions of convergent sequence, Cauchy sequence, and complete space are defined as in metric spaces. We observe that a -metric space is a metric space if. A function is said to be a -metric if and only if for all the following conditions are satisfied: (1) if and only if (2) (3). Let be a nonempty set and let be a given real number. Let be the set of positive integers, the set of real numbers, and the set of nonnegative real numbers.ĭefinition 1. The aim of this section is to collect some relevant definitions and results for our further use. Clearly, these results are a generalization of the results recently obtained from Khojasteh et al. The last part contains a fixed point result for multivalued mappings on complete -metric spaces. Then we use this result for establishing existence of fixed points for single-valued mappings. In the first part of this paper we consider a class of Picard sequences with a property of regularity and prove that these sequences are Cauchy. For some results of fixed and common fixed point in the setting of -metric spaces, see. In dealing with the fixed point theory for single-valued and multivalued mappings in -metric spaces, seven years later, Khamsi and Khamsi and Hussain reintroduced such spaces under the name of metric-type spaces. The concept of -metric space appeared in some works, such as Bakhtin and Czerwik. The study of fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers see, for example. Also its significance lies in its vast applicability in a number of branches of mathematics. It is widely considered as the source of metric fixed point theory. dissertation of Banach in 1920, which was published in 1922, is one of the most important theorems in classical functional analysis. It is well known that the contraction mapping principle, formulated and proved in the Ph.D.
These results are generalizations of the analogous ones recently proved by Khojasteh, Abbas, and Costache. So the derivative operator on this set of functions we denote by L, and its inverse is a bounded integral operator.We obtain some fixed point results for single-valued and multivalued mappings in the setting of a -metric space. To restrict its output to a single one, we consider the differential operator on the set of functions (which becomes a vector space only when the differential equation and the initial condition are all homogeneous) with a specified initial condition f( x 0) = y 0.
\dot, \) called in mathematical literature as «antiderivative», is not an operator because it assigns to every input-function infinite many outputs. In this section, we widen this procedure for systems of first orderĭifferential equations written in normal form \( Historically, Picard's iteration scheme was the first method to solve analytically nonlinear differential equations, and it was discussed in the first part of the course (see introductory section xv Picard). Introduction to Linear Algebra with Mathematica Glossary Return to the main page for the second course APMA0340 Return to the main page for the first course APMA0330 Return to Mathematica tutorial for the second course APMA0340 Return to Mathematica tutorial for the first course APMA0330 Return to computing page for the second course APMA0340 Return to computing page for the first course APMA0330
Boundary Value Problems for heat equation.Part VI: Partial Differential Equations.Part III: Non-linear Systems of Ordinary Differential Equations.Part II: Linear Systems of Ordinary Differential Equations.
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