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Öğe AN APPLICATION OF DARBO FIXED- POINT THEOREM TO A CLASS OF FUNCTIONAL INTEGRAL EQUATIONS(Taylor & Francis Inc, 2015) Cakan, Umit; Ozdemir, IsmetIn this article, using Darbo's fixed-point theorem associated with the measure of noncompactness, we prove a theorem on the existence of the solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. Our existence results include several existence results obtained earlier by Maleknejad et al. [7] and ozdemir et al. [8] as special cases under some weaker conditions. We give also some examples which show that our results are applicable.Öğe Applications of Measure of Noncompactness and Darbo's Fixed Point Theorem to Nonlinear Integral Equations in Banach Spaces(Taylor & Francis Inc, 2017) Cakan, Umit; Ozdemir, IsmetWe prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable.Öğe ASYMPTOTIC STABILITY OF SOLUTIONS FOR A CLASS OF NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS OF FRACTIONAL ORDER(Akademiai Kiado Rt, 2016) Cakan, Umit; Ozdemir, IsmetIn this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of asymptotically stable solutions of some nonlinear functional integral equations in the space of continuous and bounded functions on R+ = [ 0,infinity). We also give some examples satisfying the conditions our existence theorem.Öğe EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR SOME NONLINEAR INTEGRAL EQUATIONS ON AN UNBOUNDED INTERVAL(Texas State Univ, 2016) Ilhan, Bekir; Ozdemir, IsmetThe goal in this paper is to prove an existence theorem for the solutions of a class of functional integral equations which contain a number of classical nonlinear integral equations as special cases. Our investigations will be carried out in the space of continuous and bounded functions on an unbounded interval. The main tools are some techniques in analysis and the Schauder fixed point theorem via measures of noncompactness. Our results extend and improve some known results in the recent literature. Three nontrivial examples explain the generalizations and applications of our results.Öğe Existence of nondecreasing solutions of some nonlinear integral equations of fractional order(Int Scientific Research Publications, 2015) Cakan, Umit; Ozdemir, IsmetThe purpose of this paper is to examine the class of functional integral equations of fractional order in the space of continuous functions on interval [0, a]. Using Darbo's fixed point theorem associated with the measure of noncompactness, we present sufficient conditions for existence of nondecreasing solutions of some functional integral equations of fractional order. These existence results include several obtained from previous studies. Finally, we establish some examples to show that our results are applicable. (C) 2015 All rights reserved.Öğe On the bounded derivatives of the solutions of the linear Volterra integral equations(Taylor & Francis Ltd, 2009) Temizer, Oe Faruk; Ozdemir, IsmetThe boundaries for the solution of the linear Volterra integral equations of the second type of the form[image omitted] with unit source term and positive monotonically increasing convolution kernel were obtained as |f(t)|1, |f(t)|2 and |f(t)|4 in [R. Ling, Integral equations of Volterra type, J. Math. Anal. Appl. 64 (1978), pp. 381-397, R. Ling, Solutions of singular integral equations, Internat. J. Math. Math. Sci. 5 (1982), pp. 123-131.]. The sufficient conditions which are useful for finding the boundary such as |f(t)|2n of the solution of this equation were given, where 0t and n is a natural number, [I. Ozdemir and O. F. Temizer, The boundaries of the solutions of the linear Volterra integral equations with convolution kernel, Math. Comp. 75 (2006), pp. 1175-1199.]. In this paper, a method which ensures finding the boundaries of the derivative functions f', f'', ..., f(n+2) for n of the solution of the same equation has been developed.Öğe On the Existence and Uniform Attractivity of the Solutions of a Class of Nonlinear Integral Equations on Unbounded Interval(Mathematical Soc Rep China, 2017) Ozdemir, Ismet; Ilhan, Bekir. In this paper, we prove the existence and uniform attractivity of the solutions of a class of functional integral equations which contain a number of classical nonlinear integral equations as special cases. Our investigations will be carried out in the space of continuous and bounded functions on an unbounded interval. The main tools here are the measure of noncompactness and the suitable fixed point theorem. We introduce also some examples and remarks showing the di ff erence between our main result and some previous results.Öğe On the Existence of the Solutions for Some Nonlinear Volterra Integral Equations(Hindawi Publishing Corporation, 2013) Ozdemir, Ismet; Cakan, Umit; Ilhan, BekirWe present a theorem which gives sufficient conditions for existence of at least one solution for some nonlinear functional integral equations in the space of continuous functions on the interval [0, a]. To do this, we will use Darbo's fixed-point theorem associated with the measure of noncompactness. We give also an example satisfying the conditions of our main theorem but not satisfying the conditions described by Maleknejad et al. (2009).Öğe THE SOLVABILITY OF SOME NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS(Akademiai Kiado Rt, 2016) Ozdemir, Ismet; Cakan, UmitIn this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. We give also some examples which show that the obtained results are applicable.Öğe Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the Spaces H?,?,? ((a,b) x (a,b), X) and H?,?(a,b), X)(Hindawi Publishing Corp, 2013) Ozdemir, Ismet; Akhmedov, Ali M.; Temizer, O. FarukThe spaces H-alpha,H-delta,H-gamma ((a,b) x (a,b), R) and H-alpha,H-delta ((a,b), R) were defined in ((Huseynov (1981)), pages 271-277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419-2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the sets H-alpha,H-delta,H-gamma ((a,b) x(a,b), X) and H-alpha,H-delta ((a,b), X) by taking an arbitrary Banach space X instead of R, and we showthat these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms parallel to center dot parallel to(alpha,delta,gamma) and parallel to . parallel to(alpha,delta) Besides, the bounded linear integral operators on the spaces H-alpha,H-delta,H-gamma ((a,b) x (a,b), X) and H-alpha,H-delta ((a,b), X), some of which are singular, are derived, and the solutions of the linear Fredholmintegral equations of the form f(s) = phi(s) + lambda integral(b)(a)A(s,t) f(t)dt, f(s) = phi(s) + lambda integral(b)(a)A(t,s)f(t)dt and f(s, t) = phi(s, t) + lambda integral(b)(a) (s, t)f(t, s)dt are investigated in these spaces by analytical methods.
