In this thesis, first we examine the existence of a sequence of weak solution for the non-local equation of the following fraction\\\\ ‎‎\\begin{cases}‎‎‎M\\Big(\\iint‎‎‎‎_{\\mathbb{R}^{2N}}\\frac{\\vert u(x)‎ -u (y)\\vert^{p}}{\\vert x‎ - ‎y\\vert^{N+ps}}dxdy\\Big)‎(-\\Delta)_{p} ‎^{s}u(x)=f(x,u)‎\\qquad‎\\qquad ‎in‎\\qquad‎\\Omega,\\\\‎  ‎‎‎ ‎u=0‎\\qquad‎\\qquad‎\\qquad‎\\qquad‎\\qquad‎\\qquad‎\\qquad‎\\qquad‎‎\\qquad‎‎\\qquad‎ ‎in‎\\qquad‎‎‎\\mathbb{R‎^{N}‎}‎‎\\backslash\\Omega, ‎‎‎\\end{cases}‎‎‎‎\\\\‎where ‎$ ‎\\Omega ‎$‎ ‎is ‎an ‎open ‎bounded ‎subset ‎of‎ ‎$ ‎\\mathbb{R‎^{N}‎} ‎$‎ ‎with ‎Lipshcitz ‎boundary‎ ‎$ ‎\\partial‎\\Omega‎ $‎,‎‎ ‎$ (-‎\\Delta)‎_{p}‎^{s}‎‎‎ $ ‎is ‎the ‎fractional ‎p-Laplacian ‎operator ‎with‎ ‎$ 0<  lt;1<  lt;N $ ‎such ‎that ‎‎‎$   lt;N $‎,‎‎ ‎‎$ M $ ‎is a‎ ‎continuous ‎function ‎and‎ ‎$ f $ ‎is a ‎‎Carat‎‎heodory ‎function ‎satisfying ‎the ‎Ambrosetti-Rabinowitz-type ‎condition‎. When ‎$ f $ ‎satisfies ‎the ‎suplinear ‎growth ‎condition,we ‎obtain ‎the‎ existence of a sequence of nontrivial solutions by using the symmetric mountan pass theorem; when ‎$ f $ ‎satisfies ‎the ‎sublinear ‎growth ‎condition, ‎we ‎obtain ‎infinitely ‎many ‎pairs ‎of ‎nontrivial ‎solutions ‎by ‎applying ‎the ‎Krasnoselskii ‎genus ‎theory. ‎Our ‎results ‎cover ‎the ‎degenerate ‎case ‎in ‎the ‎fractional ‎setting: ‎the ‎Kirchhoff ‎function‎ ‎$ M $ can be zero at zero. ‎Using Mountain pass theorem we prove the existence of infinitely many solutions for the above problem. Then we discuss the existence of infinitely many solutions for fractional equations with sign changing nonlinear terms via varational methods in fact if the nonlinear terms are sign changing and satisfy p-supper growth, we obtain the existence of infinitely many solutions for boundary value problems driven by fractional p-Laplacian.‎‎‎‎‎‎‎‎‎‎\\\\‎‎\\textbf{Keywords:} \\\\‎‎\\textit{Fractinal p-Laplacian‎, Kirchhoff problems‎, Differential operator‎, Mountain pass Theorem‎, Nonlocal operator‎, Variational methods‎, Weak solution‎ }‎\\end{latin}‎

  AbstractIn this thesis, we investigate the existence of two nontrivial solutions in weighted Sobolev spaces, for a class of fourth-order elliptic equations, with assuming that, nonlinear parts is continuous with a quasicritical growth and it’s potential vanish at infinity, byusingthevariationalmethodandMountainPassTheorem. Therefore, we study the existence of two nontrivial solutions for fourth-order elliptic equations by settingtheAmbrosetti-Rabinwitsconditiononnonlinearpartsandsteepedpotential, by utilizing critical point theory, Mountain Pass Theorem and local minimization. Finally, as an application we will make report the similar results and cocentration phenomenona for second elliptic equations with concave and convex nonlinearities. Keyword: Fourth-order elliptic equation, Mixed nonlinearity, Variational method, cocentration phenomenona, concave-convex nonlinearity, quasicritical growth .

   In this thesis first the existence and multiplicity of positive solutions for a class of Kirchhoff type equations with concav and convex nonlinearities are investi- gated. Next, the existence of positive solutions for a class of Kirchhoff type equations involving a crieical growth nonlinearity is studied. To prove the mentivned results, the Nehari manifold method and the Ekland Variational Principle are used. First it is proved that local minima of the energy functional are critical points of the energy functional. Next using critical points of fibring maps, the nehari manifold is divided into three sets, and it is shown that the energy functional has two local minima in these sets.

  In this thesis, we study American option problem under di?erent methods and conditions. First, we consider American put option pricing under regime switching model (based on fron-?xing transformation and the calculation of optimal stopping boundary) and by calculating the optimal stopping boundary, we obtain a stable so- lution. In fact, this solution is the best price in the shortest possible time and has the better consistency with other methods. Then, by inserting rational parameter under regime switching model and employing a wieghted ?nite di?erence method, the problem would be discretized and we check the stability and positivity condition of American option problem, again. By having rational parameters and Thomas algorithm, we simplify the calculations and show that numerical analysis is e?ective in the stability and consistency of the solution. Finally, using a ?nite di?erence method for di?erential equation, we consider both time and space fractional deriva- tives. For this, introduce an implicit direction scheme and minimal residual method, we also propose a preconditioner and we calculate the accuracy of method up to the second order.

In this thesis, we study numerical schems for solving American put option pricing problem and for this purpose present efficient numerical methods. Unlike an European option, the value of an American option satisfies in a linear complementarity problem. We first approximate the linear complementarityproblemwithanonlinearfractionalpartialdifferentialequationbyapenaltyterm, thenweobtainsolutionsofthisequationbyFiniteDifferenceMethodandfinallywestudyanother linear complementarity problem by Laplace Transform Method and Finite Difference Method, and compare these methods by giving examples. So the purpose of this thesis is to provide methods for American put option pricing.   

One of the continuity conditions identified by Utumi on self-injective rings isthe $C_3$-condition, where a module $M$ is called a $C_3$-module if whenever $A$ and $B$ are directsummands of $M$ and $A \\cap B = 0$, then $A \\oplus B$ is a summand of $M$. In addition to injectiveand direct-injective modules, the 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">indecomposable and regular modules. Indeed, every commutative ring is a $C_3$-ring. Inthis thesis provide a general and unified treatment of the above mentioned 0px; TEXT-INDENT: 0px; -qt-block-indent: 0">modules in terms of the $C_3$-condition, and establish new characterizations of several wellknown classes of rings.

  This thesis concerns the stability and bifurcations in two planer systems of ODE’s,which are models of a prey - predator system and a SIR epidemic model.It is proved that the predator - prey model exhibits several bifurcations.These bifurcations are ecologically important and the saddle - node bifurcation andcodimension 2 Bogdanov - Takens bifurcation especially will lead to the potentiallydramatic variation of the system dynames.In the SIR model, it is proved that under some conditions the system exhibits backward bifurcation and Hopf birfurcation.

پايان نامه ارشد(6واحدي)