<p>Doktorska disertacija je posvećena re&scaron;avanju nelinearnih diferen&shy; cijalnih jednačina, kao i linearnih diferencijalnih jednačina sa singularite-tim a u okviru prostora Kolomboovih uop&scaron;tenih funkcija. U osnovi, dis&shy;ertacija se može podeliti na dva dela. Prvi deo disertacije je posvećen re&scaron;avanju determinističkih parcijalnih diferencijalnih jednačina primenom teorije polugrupa operatora definisanih na prostorima Kolomboa. Drugi deo disertacije posvećen je re&scaron;avanju stohastičkih običnih i parcijalnih dife&shy;rencijalnih jednačina. Ove jednačine sadrže Kolomboove uop&scaron;tene slučajne procese kao nelinearni deo, ili kao početne uslove.</p> / <p>Doctoral thesis is devoted to nonlinear differential equations, as well as linear differential equations with singularities in the framework of Colombeau generalized function spaces. Basically, the thesis can be devided into two parts. The first part is devoted to solving deterministic partial differential equations applaying semigroup theory where those semigroups are defined on Colombeau spaces. The second part of the thesis is devoted to stochastic ordinary and partial differential equations. Those equations contain Colombeau generalized stochastic processes as nonlinear part, or as initial data.</p>

<p>Doktorska disertacija je posvećena re&scaron;avanju Ko&scaron;ijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uop&scaron;tenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 &lt; &alpha; &lt; 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno re&scaron;avana, tako &scaron;to je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za re&scaron;avanje smo koristili dobro poznate uop&scaron;tene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su re&scaron;avane nehomogene frakcione evolucione jednačine sa Kaputovim<br />frakcionim izvodom reda 0 &lt; &alpha; &lt; 2, linearnim, zatvorenim i gusto definisanim<br />operatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zavise<br />od x. Odgovarajuća aproksimativna jednačina sadrži uop&scaron;teni operator asociran sa polaznim operatorom, dok su re&scaron;enja dobijena primenom, za tu svrhu&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />u disertaciji konstruisanih, uop&scaron;tenih uniformno neprekidnih operatora re&scaron;enja.<br />U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenost<br />re&scaron;enja Ko&scaron;ijevog problema na odgovarajućem Kolomboovom prostoru.</p> / <p>Colombeau spaces of generalized functions. In the firs part, we studied inhomogeneous evolution equations with space fractional differential operators of order 0 &lt; &alpha; &lt; 2 and variable coefficients depending on x and t. This class of equations is solved&nbsp; approximately, in such a way that instead of the originate equation we considered the corresponding approximate equation given by regularized fractional derivatives, i.e. their&nbsp; regularized multipliers. In the solving procedure we used a well-known generalized uniformly continuous semigroups of operators. In the second part, we solved approximately inhomogeneous fractional evolution equations with Caputo fractional derivative of order 0 &lt; &alpha; &lt; 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation&nbsp;&nbsp; is a given by the generalized operator associated to the originate&nbsp; operator, while the solutions are obtained by using generalized uniformly continuous solution operators, introduced and developed for that purpose. In both cases, we provided the conditions that ensure the existence and uniqueness solutions of the Cauchy problem in some Colombeau spaces.</p>

<p>Doktorska disertacija je posvećena primeni teorije polugrupa operatora na re&scaron;avanje dve klase Cauchy-jevih početnih problema. U prvom delu smo<br />ispitivali parabolične stohastičke parcijalne diferencijalne jednačine (SPDJ-ne), odredjene sa dva tipa operatora: linearnim zatvorenim operatorom koji<br />generi&scaron;e <em>C</em>₀&minus;polugrupu i linearnim ograničenim operatorom kombinovanim<br />sa Wick-ovim proizvodom. Svi stohastički procesi su dati Wiener-It&ocirc;-ovom<br />haos ekspanzijom. Dokazali smo postojanje i jedinstvenost re&scaron;enja ove klase<br />SPDJ-na. Posebno, posmatrali smo i stacionarni slučaj kada je izvod po<br />vremenu jednak nuli. U drugom delu smo konstruisali kompleksne stepene<br /><em>C</em>-sektorijalnih operatora na sekvencijalno kompletnim lokalno konveksnim<br />prostorima. Kompleksne stepene operatora smo posmatrali kao integralne<br />generatore uniformno ograničenih analitičkih <em>C</em>-regularizovanih rezolventnih<br />familija, i upotrebili dobijene rezultate na izučavanje nepotpunih Cauchy-jevih problema vi&scaron;3eg ili necelog reda.</p> / <p>The doctoral dissertation is devoted to applications of the theory<br />of semigroups of operators on two classes of Cauchy problems. In the first<br />part, we studied parabolic stochastic partial differential equations (SPDEs),<br />driven by two types of operators: one linear closed operator generating a<br /><em>C</em>₀&minus;semigroup and one linear bounded operator with Wick-type multipli-cation. All stochastic processes are considered in the setting of Wiener-It&ocirc;<br />chaos expansions. We proved existence and uniqueness of solutions for this<br />class of SPDEs. In particular, we also treated the stationary case when the<br />time-derivative is equal to zero. In the second part, we constructed com-plex powers of <em>C</em>&minus;sectorial operators in the setting of sequentially complete<br />locally convex spaces. We considered these complex powers as the integral<br />generators of equicontinuous analytic <em>C</em>&minus;regularized resolvent families, and<br />incorporated the obtained results in the study of incomplete higher or frac-tional order Cauchy problems.</p>