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2019 | Book

Advances on Fixed Point Results on Partial Metric Spaces Read chapter Read first chapter

Mathematical Methods in Engineering

Theoretical Aspects

Editors: Kenan Taş, Dumitru Baleanu, J. A. Tenreiro Machado

Publisher: Springer International Publishing

Book Series : Nonlinear Systems and Complexity

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book collects chapters dealing with some of the theoretical aspects needed to properly discuss the dynamics of complex engineering systems. The book illustrates advanced theoretical development and new techniques designed to better solve problems within the nonlinear dynamical systems. Topics covered in this volume include advances on fixed point results on partial metric spaces, localization of the spectral expansions associated with the partial differential operators, irregularity in graphs and inverse problems, Hyers-Ulam and Hyers-Ulam-Rassias stability for integro-differential equations, fixed point results for mixed multivalued mappings of Feng-Liu type on Mb-metric spaces, and the limit q-Bernstein operators, analytical investigation on the fractional diffusion absorption equation.

Table of Contents

Frontmatter

Fixed Point Theory and Applications

Frontmatter
Chapter 1. Advances on Fixed Point Results on Partial Metric Spaces
Abstract
In this note, we shall consider recent advances and improvements on fixed point theory in the setting of partial metric spaces. We investigate the existence and uniqueness of several distinct type contractive mapping in the context of complete partial metric space. We also recollect sum existing results to give complete survey for this topic.
Erdal Karapınar, Kenan Taş, Vladimir Rakočević
Chapter 2. Fixed Point Results for Mixed Multivalued Mappings of Feng-Liu Type on M b -Metric Spaces
Abstract
In this paper, first we present a more suitable definition of M b-metric than existing in the literature. Our approach includes the concepts of both M-metric and b-metric on a nonempty set X. Then, we established the topological structure of M b-metric space. After that by taking into account the set X ∪ C(X), where C(X) the class of all nonempty closed subsets of X, we introduce the concept of mixed multivalued mapping on M b-metric space and then we present a general fixed point result for mixed multivalued mapping. Our result certainly contains the well-known Feng-Liu fixed point theorem. Consequently, to show the validity of our results we provided some illustrative examples.
Hakan Şahin, Ishak Altun, Duran Türkoğlu
Chapter 3. Hyers-Ulam and Hyers-Ulam-Rassias Stability for a Class of Integro-Differential Equations
Abstract
We analyse different kinds of stabilities for a class of very general nonlinear integro-differential equations of Volterra type within appropriate metric spaces. Sufficient conditions are obtained in view to guarantee Hyers-Ulam stability and Hyers-Ulam-Rassias stability for such a class of integro-differential equations. We will consider the different situations of having the integrals defined on finite and infinite intervals. Among the used techniques, we have fixed point arguments and generalizations of the Bielecki metric. Concrete examples will be also described in view to illustrate the obtained results.
L. P. Castro, A. M. Simões

New Mathematical Ideas

Frontmatter
Chapter 4. Exact Solutions, Lie Symmetry Analysis and Conservation Laws of the Time Fractional Diffusion-Absorption Equation
Abstract
A three-dimensional Lie algebra of the time fractional diffusion-absorption (TFDA) equation, spanned by vector fields, is obtained. One of the generators is singled out in order to extract an invariant solution in a special domain. Conservation laws of TFDA equation are considered by a developed version of Ibragimov’s method. Then the invariant subspace method is used to construct its exact solutions.
Mir Sajjad Hashemi, Zahra Balmeh, Dumitru Baleanu
Chapter 5. Integral Balance Approach to 1-D Space-Fractional Diffusion Models
Abstract
This chapter summarizes the recent results on approximate analytical integral-balance solutions of initial-boundary value problems of spatial-fractional partial differential diffusion equation with Riemann–Liouville fractional derivative in space. The approximate method is based on two principal steps: the integral-balance method and a series expansion of an assumed parabolic profile with undefined exponent. The spatial correlation of the superdiffusion coefficient in two power-law forms has been discussed. The law of the spatial and temporal propagation of the solution is the primary issue. Approximate solutions based on assumed parabolic profile with unspecified exponent have been developed.
Jordan Hristov
Chapter 6. Fractional Order Filter Discretization by Particle Swarm Optimization Method
Abstract
Fractional-order filter functions are generalization of rational filter functions, which includes integer order filter functions. Fractional-order filters present advantages of more options in frequency selectivity properties of filters compared to integer order counterparts. This study presents an application of Particle Swarm Optimization (PSO) for IIR filter discretization of fractional-order continuous filter functions. The proposed method enforces particles to search in stable filter search regions and ensures the stability of optimized IIR filter functions that approximate to amplitude response of continues fractional-order filter functions. In this chapter, illustrative filter discretization examples are demonstrated to show results of PSO algorithm and these results are compared with results of Continued Fraction Expansion (CFE) approximation method. Stop band approximation performance is very substantial for band reject filter design. We observed that proposed discretization method can provide better approximation to the amplitude response of fractional-order filter functions at the stop bands compared to CFE approximation method.
Ozlem Imik, Baris Baykant Alagoz, Abdullah Ates, Celaleddin Yeroglu
Chapter 7. On the Existence of Solution for a Sum Fractional Finite Difference Inclusion
Abstract
By using a recent fixed point result for α-admissible α-Ψ multifunctions, we investigate the existence of solutions for the fractional finite difference inclusion \(\varDelta _{\nu -2}^\nu x(t)+\varDelta _{\nu -2}^{\nu -1} x(t)+\varDelta _{\nu -2}^{\nu -2} x(t+1)\in F\big (t,x(t),\varDelta x(t),\varDelta ^2 x(t),\varDelta ^\mu x(t),\varDelta ^\gamma x(t)\big )\) with the boundary conditions x(ν) = 0 and x(ν + b + 2) = 0, where 1 < γ ≤ 2, 0 < μ ≤ 1, 3 < ν ≤ 4 and \(F:\mathbb {N}_{\nu -2}^{b+\nu }\times \mathbb {R}^5\to 2^{\mathbb {R}} \) is a compact valued multifunction.
Vahid Ghorbanian, Shahram Rezapour, Saeid Salehi
Chapter 8. Comparison on Solving a Class of Nonlinear Systems of Partial Differential Equations and Multiple Solutions of Second Order Differential Equations
Abstract
We apply the reproducing kernel Hilbert space method to a class of nonlinear systems of partial differential equations and to get multiple solutions of second order differential equations. We have reached meaningful results. These results have been depicted by figures. This method is a very impressive technique for solving nonlinear systems of partial differential equations and second order differential equations.
Ali Akgül, Esra Karatas Akgül, Yasir Khan, Dumitru Baleanu
Chapter 9. Effect of Edge Deletion and Addition on Zagreb Indices of Graphs
Abstract
Edge deletion and addition to a graph is an important combinatorial method in Graph Theory which enables one to calculate some properties of a graph by means of similar graphs. The effect of edge addition on the first and second Zagreb indices was recently investigated by the authors. In this sequel paper, we consider the change in the first and second Zagreb indices of any simple graph G when an arbitrary edge is deleted. Further, we calculate the change in the first Zagreb index when an arbitrary number of edges are deleted. This method can be used to calculate the first and second Zagreb indices of larger graphs in terms of the Zagreb indices of smaller graphs. As some examples, we give some inequalities for the change of Zagreb indices for path, cycle, star, complete, complete bipartite, and tadpole graphs.
Muge Togan, Aysun Yurttas, Ahmet Sinan Cevik, Ismail Naci Cangul
Chapter 10. The Limit q-Bernstein Operators with Varying q
Abstract
In this paper, the continuity of the limit q-Bernstein operator with respect to parameter q is investigated. It is proved that the map q↦B q is continuous in the strong operator topology on C[0, 1] for q ∈ [0, 1]. Meanwhile, in the uniform operator topology, this map is discontinuous at every q ∈ [0, 1].
Manal Mastafa Almesbahi, Sofiya Ostrovska, Mehmet Turan
Chapter 11. Localization of the Spectral Expansions Associated with the Partial Differential Operators
Abstract
In this paper we discuss precise conditions of the summability and localization of the spectral expansions associated with various partial differential operators. In this we study the problems in the spaces of both smooth functions and singular distributions. We study spectral expansions of the distributions with the compact support and classify the distributions with the Sobolev spaces. All theorems are formulated in terms of the smoothness and degree of the regularizations.
Abdumalik Rakhimov
Chapter 12. Energy Decay in a Quasilinear System with Finite and Infinite Memories
Abstract
In this paper, we consider the following quasilinear system of two coupled nonlinear equations with both finite and infinite memories
$$\displaystyle \left \{ \begin {array}{l} \left \vert u_{t}\right \vert ^{\rho }u_{tt}-\Delta u-\Delta u_{tt}+\int _{0}^{t}g_{1}(s)\Delta u(t-s)ds+f_{1}(u,v)=0 \\ \left \vert v_{t}\right \vert ^{\rho }v_{tt}-\Delta v-\Delta v_{tt}+\int _{0}^{\infty }g_{2}(s)\Delta v(t-s)ds+f_{2}(u,v)=0 \end {array} \right . $$
and investigate the asymptotic behavior of this system. We use the multiplier method to establish an explicit energy decay formula. Our result allows a wider class of relaxation functions and provides more general decay rates for which the usual exponential and polynomial rates are only special cases. AMS Classification35B40, 74D99, 93D15, 93D20
Muhammad I. Mustafa
Chapter 13. A Modified and Enhanced Ant Colony Optimization Algorithm for Traveling Salesman Problem
Abstract
In this paper an effective modification has been performed on the Ant Colony Optimization algorithm and used for solving traveling salesman problem (TSP). The traveling salesman problem is one of the famous and important problems and it has been used in the algorithms to analyze its performance in solving the discreet problems. The modified and enhanced ACO has been used for solving this problem and it is called MEACO. In MEACO the modification has been performed by taking effect of mutation on the global best and personal best of each ant. The personal best is stored for each ant same as the PSO algorithm. Original ACO for discrete problems mostly trap in the local solutions, but the proposed method has been designed to cover this deficiency and make it more suitable for optimization of discrete problems. The experiment on the set of benchmark problems for Traveling salesman was performed and obtained results showed that MEACO is an effective method in finding the path for TSP.
Leila Eskandari, Ahmad Jafarian, Parastoo Rahimloo, Dumitru Baleanu
Chapter 14. A Note on the Upper Bound of Average Distance via Irregularity Index
Abstract
In this paper, we use a technique introduced by Dankelmann and Entringer (J Graph Theory 33:1–13, 2000) to obtain a strengthening of an old classical theorem by Kouider and Winkler (J Graph Theory 25:95–99, 1997) on mean distance and minimum degree. It is known that the mean distance and the average distance are the same parameters. To be more detailed, we will prove that if G is a connected graph of order n with minimum degree δ, then the average distance of G does not exceed
$$\displaystyle \frac {n-t+1}{\delta +1}+1, $$
where t is the irregularity index (that is the number of distinct terms of the degree sequence of G). We note that the irregularity index has been recently defined and studied in the paper (Mukwembi, Appl Math Lett 25:175–178, 2012).
Nihat Akgunes, Ismail Naci Cangul, Ahmet Sinan Cevik
Backmatter
Metadata
Title
Mathematical Methods in Engineering
Editors
Kenan Taş
Dumitru Baleanu
J. A. Tenreiro Machado
Copyright Year
2019
Electronic ISBN
978-3-319-91065-9
Print ISBN
978-3-319-91064-2
DOI
https://doi.org/10.1007/978-3-319-91065-9

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