DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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VARIATION-OF-PARAMETERS FORMULAE AND LIPSCHITZ STABILITY CRITERIA FOR NONLINEAR MATRIX DIFFERENTIAL EQUATIONS WITH INITIAL TIME <strong>DIFFERENCE</strong> COSKUN YAKAR This paper investigates the relationship between an unperturbed matrix differential equation and a perturbed matrix differential system which both have different initial positions and an initial time difference. Variation-of-parameter techniques are employed to obtain integral formulae and to establish Lipschitz stability criteria for nonlinear matrix differential systems and make use of the variational system associated with the unperturbed differential system. Copyright © 2006 Coskun Yakar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The method of variation-of-parameters formulae (VPF) has been a very useful technique in the qualitative theory of system of differential equations and nonlinear matrix differential equations since it is a practical tool in the investigation of the properties of solutions. Recently in [1–3], the study of nonlinear matrix initial value problems with an initial time difference (ITD) has been initiated and the corresponding theory of differential inequalities has been investigated. Below, we will derive VPF showing the relationship between unperturbed matrix differential systems with different initial conditions and unperturbed and perturbed systems with different initial conditions. The qualitative behavior of matrix differential equations has been explored extensively and the investigation of initial value problems with a perturbation in the space variable is well known when the perturbation is restricted to the space variable with the initial time unchanged [1–4, 6, 7]. Recently, several investigations have been initiated to explore the qualitative behavior of matrix differential systems that have a different initial position and a different initial time. We call this type of stability analysis ITD stability analysis. In Section 3, variation-of-parameter formulae were used to investigate the relationship between (1) unperturbed matrix equations with different initial conditions and (2) unperturbed and perturbed matrix equations with ITD. ITD Lipschitz stability criteria for matrix differential systems are established by employing the variational system associated with the unperturbed matrix differential system. Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1201–1216
MONOTONE TECHNIQUE FOR NONLINEAR SECOND-ORDER PERIODIC BOUNDARY VALUE PROBLEM IN TERMS OF TWO MONOTONE FUNCTIONS COSKUN YAKAR AND ALI SIRMA This paper investigates the fundamental theorem concerning the existence of coupled minimal and maximal solutions of the second-order nonlinear periodic boundary value problems involving the sum of two different functions. We have such a problem in applied mathematics, which has several applications to the theory of monotone iterative techniques for periodic problems, as has been pointed in the several results and theorems made in the paper. Copyright © 2006 C. Yakar and A. Sırma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The method of upper and lower solutions has been effectively used, an interesting and fruitful technique, for proving the existence results for a wide variety of nonlinear periodic boundary value problems. When coupled with monotone iterative technique, it manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by lower and upper solutions. One obtains a constructive procedure for obtaining the solutions of the nonlinear problems besides enabling the study of the qualitative properties of the solutions of periodic boundary value problems. The concept embedded in these techniques has proved to be of enormous value and has played an important role consolidating a wide variety of nonlinear problems [1–4]. Moreover, iteration schemes are also useful for the investigation of qualitative properties of the solutions, particularly, in unifying a variety of periodic nonlinear boundary value problems. We have to refer [1, 2] for an excellent and comprehensive introduction to the monotone iterative techniques for nonlinear periodic boundary value problems. This method has further been exploited in combination with the method of quasilinearization to obtain concurrently the lower and upper bounding monotone sequences, whose elements are solutions of linear boundary value problems which converge unifomly and monotonically to the unique solution of (2.1) and the convergence is quadratic. Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1217–1229
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Proceedings of the Conference on Di
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Proceedings of the Conference on Di
Page 6 and 7:
CONTENTS Preface...................
Page 8 and 9:
Inequalities for positive solutions
Page 10 and 11:
Modelling the effect of surgical st
Page 12:
Maximum principles for elliptic equ
Page 16 and 17:
A UNIFIED APPROACH TO MONOTONE METH
Page 18 and 19:
IMPULSIVE CONTROL OF THE POPULATION
Page 20 and 21:
BOUNDARY ESTIMATES FOR BLOW-UP SOLU
Page 22 and 23:
SECOND-ORDER DIFFERENTIAL GRADIENT
Page 24 and 25:
ON STRUCTURE OF FRACTIONAL SPACES G
Page 26 and 27:
ON THE STABILITY OF THE DIFFERENCE
Page 28 and 29:
TWO-WEIGHTED POINCARÉ-TYPE INTEGRA
Page 30 and 31:
PRODUCT DIFFERENCE EQUATIONS APPROX
Page 32 and 33:
QUASIDIFFUSION MODEL OF POPULATION
Page 34 and 35:
FIXED-SIGN EIGENFUNCTIONS OF TWO-PO
Page 36 and 37:
MULTIPLE POSITIVE SOLUTIONS OF SUPE
Page 38 and 39:
SPECTRAL STABILITY OF ELLIPTIC SELF
Page 40 and 41:
HO CHARACTERISTIC EQUATIONS SURPASS
Page 42 and 43:
ASYMPTOTIC STABILITY IN DISCRETE MO
Page 44 and 45:
CONTINUOUS AND DISCRETE NONLINEAR I
Page 46 and 47:
ON PARABOLIC SYSTEMS WITH DISCONTIN
Page 48 and 49:
INEQUALITIES FOR POSITIVE SOLUTIONS
Page 50 and 51:
NONOSCILLATION OF ONE OF THE COMPON
Page 52 and 53:
ANNULAR JET AND COAXIAL JET FLOW JO
Page 54 and 55:
JUSTIFICATION OF QUADRATURE-DIFFERE
Page 56 and 57:
VLASOV-ENSKOG EQUATION WITH THERMAL
Page 58 and 59:
SUBDIFFERENTIAL OPERATOR APPROACH T
Page 60 and 61:
A DISCRETE EIGENFUNCTIONS METHOD FO
Page 62 and 63:
ON NONLINEAR SCHRÖDINGER EQUATIONS
Page 64 and 65:
NUMERICAL SOLUTION OF A TRANSMISSIO
Page 66 and 67:
THE SEMILINEAR EVOLUTION EQUATION F
Page 68 and 69:
ON DOUBLY PERIODIC SOLUTIONS OF QUA
Page 70 and 71:
DYNAMICS OF A DISCONTINUOUS PIECEWI
Page 72 and 73:
PROBABILISTIC SOLUTIONS OF THE DIRI
Page 74 and 75:
MONOTONICITY RESULTS AND INEQUALITI
Page 76 and 77:
WELL-POSEDNESS AND BLOW-UP OF SOLUT
Page 78 and 79:
REARRANGEMENT ON THE UNIT BALL FOR
Page 80 and 81:
CONVERGENCE OF SERIES OF TRANSLATIO
Page 82 and 83:
ON MINIMAL (MAXIMAL) COMMON FIXED P
Page 84 and 85:
BOUNDEDNESS OF SOLUTIONS OF FUNCTIO
Page 86 and 87: MODELLING THE EFFECT OF SURGICAL ST
Page 88 and 89: LIMIT CYCLES OF LIÉNARD SYSTEMS M.
Page 90 and 91: MAXIMUM PRINCIPLES AND DECAY ESTIMA
Page 92 and 93: LIPSCHITZ REGULARITY OF VISCOSITY S
Page 94 and 95: ON AN ELASTIC BEAM FULLY EQUATION W
Page 96 and 97: BOUNDARY VALUE PROBLEMS ON THE HALF
Page 98 and 99: EXISTENCE AND UNIQUENESS OF AN INTE
Page 100 and 101: STABILITY OF SOME MODELS OF CIRCULA
Page 102 and 103: LIPSCHITZ CLASSES OF A-HARMONIC FUN
Page 104 and 105: A TWO-DEGREES-OF-FREEDOM HAMILTONIA
Page 106 and 107: QUANTIZATION OF LIGHT FIELD IN PERI
Page 108 and 109: CONSERVATION LAWS IN QUANTUM SUPER
Page 110 and 111: ON SETS OF ZERO HAUSDORFF s-DIMENSI
Page 112 and 113: ON THE GLOBAL ATTRACTOR IN A CLASS
Page 114 and 115: SANDWICH PAIRS MARTIN SCHECHTER Sin
Page 116 and 117: EXISTENCE RESULTS FOR EVOLUTION INC
Page 118 and 119: AVERAGING DOMAINS: FROM EUCLIDEAN S
Page 120 and 121: CONCENTRATION-COMPACTNESS PRINCIPLE
Page 122 and 123: NONLINEAR VARIATIONAL INCLUSION PRO
Page 124 and 125: MAXIMUM PRINCIPLES FOR ELLIPTIC EQU
Page 126 and 127: GLOBAL CONVERGENT ALGORITHM FOR PAR
Page 128 and 129: SECOND-ORDER NONLINEAR OSCILLATIONS
Page 130 and 131: ANALYTIC SOLUTIONS OF UNSTEADY CRYS
Page 132 and 133: UNBOUNDEDNESS OF SOLUTIONS OF PLANA
Page 134 and 135: QUENCHING OF SOLUTIONS OF NONLINEAR
Page 138: DIFFERENCE EQUATIONS AND THE ELABOR
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DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

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