scholarly journals Existence and uniqueness regions for solutions of nonlinear equations

1978 ◽  
Vol 19 (2) ◽  
pp. 277-282 ◽  
Author(s):  
A.L. Andrew
Keyword(s):  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Kantorovich Theorem ◽  
Potential Applications ◽  
Uniqueness Theory

A refinement of the Newton-Kantorovich Theorem, which has many-potential applications in existence - uniqueness theory, is used to strengthen a result of Lancaster and Rokne concerning existence and uniqueness regions for zeros of operator polynomials.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation

2020 ◽  
Vol 23 (5) ◽  
pp. 1381-1400 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Integral Operator ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Formal Solution ◽  
Type Equation ◽  
Multiplier Theorem ◽  
Problem Of Time

Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

scholarly journals CREDIT RISK PREMIA AND QUADRATIC BSDEs WITH A SINGLE JUMP

2010 ◽  
Vol 13 (07) ◽  
pp. 1103-1129 ◽  
Author(s):  
STEFAN ANKIRCHNER ◽  
CHRISTOPHETTE BLANCHET-SCALLIET ◽  
ANNE EYRAUD-LOISEL
Keyword(s):  
Credit Risk ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Risk Premia ◽  
Contingent Claims ◽  
Quadratic Growth ◽  
Utility Preferences ◽  
Random Times ◽  
Uniqueness Theory

This paper is concerned with the determination of credit risk premia of defaultable contingent claims by means of indifference valuation principles. Assuming exponential utility preferences we derive representations of indifference premia of credit risk in terms of solutions of Backward Stochastic Differential Equations (BSDE). The class of BSDEs needed for that representation allows for quadratic growth generators and jumps at random times. Since the existence and uniqueness theory for this class of BSDEs has not yet been developed to the required generality, the first part of the paper is devoted to fill that gap. By using a simple constructive algorithm, and known results on continuous quadratic BSDEs, we provide sufficient conditions for the existence and uniqueness of quadratic BSDEs with discontinuities at random times.

A generalized Kantorovich theorem on the solvability of nonlinear equations

2009 ◽  
Vol 77 (1-2) ◽  
pp. 99-105 ◽  
Author(s):  
Livinus U. Uko ◽  
Ioannis K. Argyros
Keyword(s):  
Nonlinear Equations ◽  
Kantorovich Theorem

scholarly journals Optimal existence and uniqueness theory for the fractional heat equation

2017 ◽  
Vol 153 ◽  
pp. 142-168 ◽  
Author(s):  
Matteo Bonforte ◽  
Yannick Sire ◽  
Juan Luis Vázquez
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Fractional Heat Equation ◽  
Uniqueness Theory

scholarly journals SOME LINEAR AND NONLINEAR EQUATIONS OF HIGHER ORDER

2016 ◽  
pp. 46-52
Author(s):  
Nadezda Chuesheva ◽  
Nadezda Chuesheva ◽  
Aleksandr Chueshev ◽  
Aleksandr Chueshev
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Uniqueness Of Solutions ◽  
Partial Derivatives ◽  
Petviashvili Equation ◽  
De Vries ◽  
Linear And Nonlinear

In this article we consider boundary value problems for some linear and nonlinear differential equations with partial derivatives of the sixth, fifth, fourth and third orders. We write out conditions on equation coefficients for which existence and uniqueness of solutions from Sobolev's space occur. If these conditions on equation coefficients are not valid, then there are given examples when solution is not unique, or is not unstable, or does not belong to Sobolev's space from existence and uniqueness theorem even for analytical coefficients and analytical right side of differential equation. After S.P. Novikov’s fundamental study in 1974 the interest to the nonlinear Korteweg-de Vries equation, Kadomtsev-Petviashvili equation and other nonlinear equations significantly grew. In this study of such equations we used methods of algebraic geometry integration and expansion method. In these studies exact solutions of special nonlinear equations series in partial derivatives play a big role. Solvability of similar equations was also studied in articles of A.I. Kozhanov, N.A. Larkin and other authors. The aim of this article is to find some exact solutions for special series partial differential equations. Solution graphs of such problems for linear equations and for the Korteweg-de Vries, Burgers-Korteweg-deVries, and Kadomtsev-Petviashvili equations are constructed.

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