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Mathematical Methods in Engineering and Economics Report.

Published on Oct View Download 0. All papers of the present volume were peer reviewed by no less than two independent reviewers.

### Mathematical Methods in Engineering and Economics

However the computational power doubles in 18 months. Engineeringproblems solved will be more complicated, complex and will lead to a numerically illconditioned problems especially in the perspective of today available floating pointrepresentationandformulationintheEuclideanspace. Homogeneous coordinates and projective geometry are mostly connected with geometrictransformations only.

However the projective extension of the Euclidean system allowsreformulation of geometrical problems which can be easily solved. In many cases quitecomplicated formulaearebecomingsimple from thegeometricalandcomputationalpointofview. In this short tutorial we will introduce “practical theory” of the projective space andhomogeneous coordinates.

We will show that a solution of linear system of equations isequivalenttogeneralizedcrossproductandhowthis influencesbasicgeometricalalgorithms. He has been a member of several international program posniku of prestigiousconferences and workshops.

Vaclav Skala has published over research papers in scientific journal and atinternational research conferences. His current research interests are computer graphics,visualizationandmathematics,especiallygeometricalalgebra,algorithmsanddatastructures. Duringand,hewasa visiting scholarat theUniversityofWestBohemia inPlzenunder a program supported by the international exchange scholarship between China andCzech governments.

His research interests include3D shapemodelingandanalysis,computergraphicsandvision, imageprocessing. Some related generalizations are also given for non self-mappings of the form nBAT: The convergence of the s equences in the domain and the image sets of the non self-mapping as well as the existence and uniquenes s of the best proximity points are als o investigated. KeywordsBest proximity point, proximal contrac tion, weak dediuchov contractionset theoretic limit.

See also [] on fixed point theory and so me of its ap plications to stab ility pr oblems and [] on char acterizations of stab ility o f d ynamic systems. Some related ge neralizations a re a lso gi ven fo r no n self-mappings nBAT: The convergence of the sequences in the domain and the image sets of the n on self-mapping as well as th e ex istence and un iqueness of t he best proximity po ints are al so in vestigated fo r t he dif ferent restrictions and the given extension.

Let B,A be n onempty cl osed su bsets o f a metric sp podnkku d,X. A non self-mapping BAT: The set A is approximatively c ompact wi th dedouchovv spect to B if any Some further results on weak proximal contractions including the case of iteration-dependent image sets M.

The following result holds from Definition 1: Let B,A be a pa ir of n onempty c losed subsets o f a m etric sp ace d,X.

A ssume th at BAT: Dx,Txdx,xdx,xd nnnnnn ; 0Zn 3 Then, nx is bounded, if 0x is bounded, 01 nn x,xd and DTx,xd nn as n and 0 1n nn x,xd. Note that th e giv en con dition 3 fo llows f rom 2 with the first condition of t he lo gic im plication b eing tr ue ; 0Zn with th e rep lacements nxx1 nxuy2 nxv. No te that the im plying inequality of 2 h olds since: On t he ot her ha nd since DTx,xd nn 1Dx,xdx,xdTx,xdTx,xd nnnnnnnn 1 so that the im plying co ndition of 2 al ways h olds an d then 3.

Under all devouchov he a ssumptions of Proposition 1, assume also t hat d,X is co mplete and that B is approximatively co mpact wi th resp ect to Aor the weak er condition t hat 0A is clo sed.

Si nce 01 nn x,xd as n xxn from Proposition 1. Since Axn and A is closed then Ax.

Also, 0Bclz with DTx,xd kn as k. T hen, DTx,xdinflim nnn an d taking lim its as strateyie in the equation: Assume now th at th ere are 0Axy,x such that DTy,ydTx,xd.

Thus, one gets from 2 for xu and vy that for any real constant ,0: Also, the following weakened result holds under conditions which guarantee that the implied logic proposition 2 always holds: Let B,A be a of nonempty closed subsets of a metric sp ace d,X. Assume t hat BAT: Then, Dy,Txdy,xdv,udy,xdv,ud 6 Dedoucnov Th en, w e g et that the logic implication pr oposition 6 ho lds since it s eq uivalent contrapositive logic proposition holds, since: Dy,Txdy,xdv,ud Dx,Txdy,xd 10 Lemma 1 establishes that the contractive w eak pr oximal condition can b e satisfied un der stro nger contractive conditions irrespective o f th e am ount Dy,xdTx,xd 1 bein g p ositive, n egative or null.

Thus, Lemma 1 adopts the following equivalent form: Then, Dy,Txdy,xd,y,xdminv,ud 11 In summary, note from triangle inequality that the condition y,xdDTx,yd guarantees that the implying condition of 2 holds since Dy,xdDTx,ydy,xdTx,xd 1 while the implied one adopts the form y,xdv,ud Lemma 1 leads t o th e fo llowing d istance co nvergence result: Let B,A be a pair of n onempty c losed subsets o f a m etric space shrategie.

As sume that BAT: Since th e p arabola p i s convex then 0eq uivalently 1if 21, a nd podni,u ince 10and ,0 strattegie constraints ass ociated with BA: T be ing a generalized weak p roximal contraction, 10 r equires 10since 10. Thus, since x,xdx,xdx,xd nnnnn ; Zn then, 01 nn x,xd as n and nx i s bo unded i f 0x is bounded.

Also, since Dx,Txd nn 1 ; 0Znone gets: Its proof is based on an induction argument combined with an associate strict contraction for sequences of distances. U nder al l the assumptions of Steategie oposition 2, assume also t hat d,X i s c omplete and that B is approximatively c ompact w ith resp ect t o Ao r the weaker condition that 0A is clo sed.

The convergence to a best proximity point is proved as in Theorem 1. T strqtegie u niqueness of such a point follows from the co ntradiction of t he fo rm 00 arising fr om c ontractive condition with 1xuvy and yx: As sume t hat BAT: Note from 5 that the implying condition of 2 always holds p rovided that nnnn x,xdx,xd ; 0Zn since then one gets: Dx,xdx,xdTx,xdTx,xd nnnnnnnn Dx,xdx,xdmax nnnn Dx,xd nn 11 14 Assume that 14 holds for 0npodnikj at is x,xdx,xd wi th Ax,x 100x fin ite and Ax 2 b eing su ch that DTx,xd Th en, pr oceed by complete induction by assuming that it also holds for any fixed 0Zn Then, since ,max, which is eq uivalent to and which i s gua ranteed i fso that th e m aximum zer o o f t he con vex par abola q is r eached at 22so gu aranteeing q on 20, podnikuu ch together with leads to: T hen 14 also h olds for the replacement 1 nn by th e com plete ind uction m ethodthen for any 0Znsin ce T hus, one ge ts from 15 after r eplacing nn 2 that x,xdx,xd nnn so that nn x,xd as n.

As a result, 01 nn x,xd as n from 15DTx,xd nn as n fr om 14nx i s bo unded, if 0x is f inite, strtaegie d 0 1n nn x,xd i s stratwgie bou nded from a closed proof to that of i ts c ounterpart of Pr oposition 1. T he following result, whose proof is omitted, can be p roved in a similar way as those of Theorems Under all t he a ssumptions of Proposition 3, assume also t hat d,X is co mplete and that Strahegie is approximatively co mpact with resp ect to Ao r the strtegie eaker condition t hat 0A is clo sed.

Another further result, which generalizes Proposition 3, while it guarantees that the implying logic proposition o f 2 always hol ds for c ertain a pproximation s equences, t hen guaranteeing that the implied logic condition now follows by using a sequence of distances nB,Ad for 0Zn. The next result is concerned with booindedness an d co nvergence o f weak proximal sequences. If, furthermore, th e set th eoretic l imit nnBlimB: Let A and nB ; 0Zn be none mpty subsets o f a m etric space d,X.

A ssume that nBAT: Also, if follows fr om 1 9 a nd 17 with 1 that 18 holds since nnnnnnnnn Dx,xdx,Txdx,xdx,xd kknk knn DDx,xd ; Zn 29 for any giv en Ax 0. If 1 and 00 kknk kn DD then th e seq uence 1nn x,xd i s bounded. T hus, P roperty i ha s b een proved.

Th e relations 20 and 21 o f P roperty ii f ollow directly from 18 of Property i if 1. On the other hand, the triangle inequality and 19 lead to 25 since 11 nnnnnn x,xdTx,xdTx,xd The relation 23 follows from 21 and To prove 24note th at if 00 nnnDDlimt hen for a ny given Rthere i s 0Zmm su ch that 01kkmnkmDDsup for a ny mn Z a nd t hen, fr om 2411 nnnx,xdsuplim. Si nce is arbitrary, th e l imit 1 nnn x,xdlim e xists and 01 nnn x,xdlim.

Th is property a nd 22 y ield directly 01 nnnnn Tx,xdTx,xdlim and th en Property ii has been fully proved. Also, 19 is identical to 10 1 nnnnn x,xdDTx,xd ; 0Zn which leads to 26 by taking into account The relation 27 follows from 2126 and the relation: Pr operty iv is a direct co nsequence o f Pr operty iii fo r the case when 01 nn x,xd and 00 nn DD as n including its particular sub-case when DDn 0 and DDn.

Under all the assumptions of Proposition 3. As sume al so that t he no n-self mapping restriction BAA: T 1for some nonempty subset AA 1which c ontains 0Ais a gen eralized weak proximal contraction.

## Mathematical Methods in Engineering and Economics

Sin ce 01 nn x,xd as n xxn from 2 4. Since Axn and A stratege clos ed t hen Ax. Since B is approximatively compact w ith r espect to A and Axnthen th ere are 0Ay an d a seq uence Axnsu ch that 0 nn xxthen xxn since xxnand BTxn such that as n: T n1 for some no nempty AA 1.

A ssume not so derouchov 1A. If 1A then 0A is also empty which is impossible t hen 1A. It i s now pro ved b y c ontradiction that t he be st p roximity p oint i s uni que.

A ssume not so that there are t wo be st pr oximity po ints y,x su ch th at t here are two sequences xxn and yyn contained in A. T n1 is a geenralized w eak proximal contraction, one gets from the stratwgie logic proposition of 2 with xu and yv and DDn that Dy,xdx,TxdDy,Txdy,xd 1 y,xd which fails for 10 if dedouchob. S adiq Bas ha, Bes t proxim ity points: E ldred and P. Veeraman i, Existence and convergence of best proximity pointsJournal of Mathematical Analysis with Applications, Vol.

Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlin. Gabeleh, Dis crete optimizarion v ia various proximal non-self mappings in press. D e la Sen, S ome c ombined relations betw een c ontractive mappings, Kannan mappingsreasonable expas nsive mappings and T- stability, Fixed Point Theor y and Applications, Ar ticle ID, doi: