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TAIWANESE JOURNAL OF MATHEMATICSVol. 21, No. 2, pp. 385–402, April 2017 DOI: 10.11650/tjm/7810This paper is available online at  http://journal.tms.org.tw On the Existence and Uniform Attractivity of the Solutions of a Class of Nonlinear Integral Equations on Unbounded Interval ˙Ismet ¨Ozdemir* and Bekir ˙Ilhan Abstract. In this paper, we prove the existence and uniform attractivity of the solu-tions of a class of functional integral equations which contain a number of classicalnonlinear integral equations as special cases. Our investigations will be carried out inthe space of continuous and bounded functions on an unbounded interval. The maintools here are the measure of noncompactness and the suitable ﬁxed point theorem.We introduce also some examples and remarks showing the diﬀerence between ourmain result and some previous results. 1. Introduction It is well known that integral equations have wide application in engineering, mechan-ics, physics, economics, optimization, queing theory and so on. The theory of integralequations is rapidly developing with the help of tools in functional analysis, topology andﬁxed-point theory.Agarwal and O’Regan  gave the existence of the solutions for the nonlinear integralequation(1.1)  x ( t ) =    ∞ 0 k ( t,s ) f  ( s,x ( s )) ds, t ∈ R + , in the space C l [0 , ∞ ), where C l [0 , ∞ ) denotes the space of bounded and continuous func-tions on  R + which have limit at inﬁnity, in 2004.Meehan and O’Regan [10, 11] discussed both the existence of the solutions for the nonlinear integral equation(1.2)  x ( t ) =  h ( t ) + µ    ∞ 0 k ( t,s ) f  ( s,x ( s )) ds, t ∈ R + , in the space C l [0 , ∞ ) and the existence of the solutions for the nonlinear integral equation(1.3)  x ( t ) =  h ( t ) +    ∞ 0 k ( t,s )[ f  ( x ( s )) + g ( x ( s ))] ds, t ∈ R + , Received May 16, 2016; Accepted October 13, 2016.Communicated by Eiji Yanagida.2010  Mathematics Subject Classiﬁcation  . Primary: 45G10, 47H10; Secondary: 47H08, 45M99. Key words and phrases  . Nonlinear integral equation, Measure of noncompactness, Fixed-point theorem.*Corresponding author. 385  386 ˙Ismet ¨Ozdemir and Bekir ˙Ilhan in the space BC( R + , R ), where BC( R + , R ) denotes the space of bounded and continuousfunctions on R + , in 1999 and 2000, respectively. Later in  they established the existenceof at least one positive solution of nonlinear integral equation(1.4)  x ( t ) =  h ( t ) +    ∞ 0 k ( t,s ) f  ( s,x ( s )) ds, t ∈ R + , in the space  L  p ( R + ) in 2001.In 2004, Bana´s and Poludniak  investigated the monotonic solutions for the nonlinear integral equation(1.5)  x ( t ) =  f  ( t ) +    ∞ 0 u ( t,s,x ( s )) ds, t ∈ R + , in the space of Lebesque integrable functions on unbounded interval by using the Darboﬁxed point theorem and the measure of noncompactness deﬁned in Deﬁnition 2.1.Bana´s and Martin  studied the existence and asymptotic stability of the solutionsfor the nonlinear integral equation(1.6)  x ( t ) =  g ( t ) + f  ( t,x ( t ))    ∞ 0 K  ( t,s ) h ( s,x ( s )) ds, t ∈ R + , in the Banach space BC( R + , R ), in 2006.In 2004, Cabellaro and others , in 2008, Bana´s and Olszowy  and more recently in 2013, Darwish and others  studied the existence of the solutions for the Urysohnintegral equation deﬁned on unbounded interval(1.7)  x ( t ) =  a ( t ) + f  ( t,x ( t ))    ∞ 0 u ( t,s,x ( s )) ds, t ∈ R + , with the help of measure of noncompactness and a ﬁxed point theorem in the spaceBC( R + , R ). Of course authors studied integral equation (1.7) under diﬀerent assumptionsand measure of noncompactness, also they have given rather diﬀerent existence theorems.Olszowy [13–15] studied (1.7) in the Fr´echet space of real functions being deﬁned and continuous on R + and has given results about monotonicity of the solutions of the integralequation (1.7).In 2010, Karoui and others  studied (1.7) in the space  L  p ( R + ) by means of Schauder’sﬁxed point theorem.Motivated by recent researches in this ﬁeld, we study the more general nonlinearintegral equation,(1.8)  x ( t ) = ( T  1 x )( t ) + ( T  2 x )( t )    ∞ 0 u ( t,s,x ( s )) ds, t ∈ R + , where the functions  u ( t,s,x ) and the operators  T  i , ( i  = 1 , 2) appearing in (1.8) are given,while  x  =  x ( t ) is an unknown function. It is clear that (1.8) includes (1.1)–(1.7) as special  The Solutions of a Class of Nonlinear Integral Equations 387 cases. Using the technique of a suitable measure of noncompactness, we prove an existencetheorem for (1.8). We give some examples satisfying the conditions given in this paper.The approach applied in this paper depends on extending and generalizing of the methodsand tools used in the study of some nonlinear integral equations which are presented in thepapers [4–7,9]. It is worthwhile mentioning that the class of integral equations considered in this paper are more general then those investigated up to now. 2. Auxiliary facts and notations In this section, we give a collection of auxiliary facts which will be needed in the sequel.Assume that ( E, · ) is a real Banach space with zero element  θ . Let  B ( x,r ) denote theclosed ball centered at  x  and with radius  r . The symbol  B r  stands for the ball  B ( θ,r ).If   X   is a subset of   E  , then  X   and Conv X   denote the closure and convex closure of   X  ,respectively. With the symbols  λX   and  X  + Y  , we denote the standard algebraic operationson sets. Moreover, we denote by  M E   the family of all nonempty and bounded subsets of  E   and N E   its subfamily consisting of all relatively compact subsets. The deﬁnition of theconcept of a measure of noncompactness presented below comes from . Deﬁnition 2.1.  A function  µ :  M E   →  R + = [0 , ∞ ) is said to be a measure of noncom-pactness in  E   if it satisﬁes the following conditions:(1) The family ker µ  = { X   ∈ M E   :  µ ( X  ) = 0 }  is nonempty and ker µ ⊂ N E  ;(2)  X   ⊂ Y   ⇒ µ ( X  ) ≤ µ ( Y  );(3)  µ ( X  ) =  µ (Conv X  ) =  µ ( X  );(4)  µ ( λX   + (1 − λ ) Y  ) ≤ λµ ( X  ) + (1 − λ ) µ ( Y  ) for  λ ∈ [0 , 1];(5) if   { X  n }  is a sequence of nonempty, bounded, closed subsets of the set  E   such that X  n +1  ⊂  X  n , ( n  = 1 , 2 ,... ) and lim n →∞ µ ( X  n ) = 0, then the set  X  ∞  =   ∞ n =1 X  n  isnonempty.In the sequel, we will work in the Banach space BC( R + , R ). The space BC( R + , R ) isfurnished with the standard norm   x  = sup {| x ( t ) | :  t ∈ R + } .We will use a measure of noncompactness in the space BC( R + , R ). In order to deﬁnethis measure let us ﬁx a nonempty and bounded subset  X   of BC( R + , R ). For  x  ∈  X  , ε ≥ 0 and  L >  0 denoted by  w L ( x,ε ) the modulus of continuity of function  x , i.e., w L ( x,ε ) = sup {| x ( s ) − x ( t ) | :  t,s ∈ [0 ,L ] and  | t − s |≤ ε } .  388 ˙Ismet ¨Ozdemir and Bekir ˙Ilhan Further let us put w L ( X,ε ) = sup  w L ( x,ε ) :  x ∈ X   ,w L 0  ( X  ) = lim ε → 0 w L ( X,ε )and(2.1)  w 0 ( X  ) = lim L →∞ w L 0  ( X  ) . Moreover, if   t ∈ R + is a ﬁxed number, let us denote X  ( t ) = { x ( t ) :  x ∈ X  } anddiam X  ( t ) = sup {| x ( t ) − y ( t ) | :  x,y  ∈ X  } . With help of the above mappings we deﬁne the following measure of noncompactness inBC( R + , R ), :(2.2)  µ ( X  ) =  w 0 ( X  ) + limsup t →∞ diam X  ( t ) . The kernel of this measure consists of all nonempty and bounded subsets  X   of BC( R + , R )such that functions from  X   are locally equicontinuous on  R + and the thickness of thebundle formed by functions from  X   tends to zero at inﬁnity.Now we recall deﬁnitions of the concepts of local attractivity and asymptotic stabilityof the solutions of operator equations. Let us assume that Ω is a nonempty subset of thespace BC( R + , R ) and  F   is an operator deﬁned on Ω with values in BC( R + , R ). Let usconsider the operator equation of the form(2.3)  x ( t ) = ( Fx )( t ) , t ∈ R + . Deﬁnition 2.2.  We say that solutions of (2.3) are locally attractive if there exist an x 0  ∈  BC( R + , R ) and an  r >  0 such that for all solutions  x  =  x ( t ) and  y  =  y ( t ) of (2.3)belonging to  B ( x 0 ,r ) ∩ Ω we have thatlim t →∞ ( x ( t ) − y ( t )) = 0 . In the case when limit is uniform with respect to the set  B ( x 0 ,r ) ∩ Ω, that is, when foreach  ε ≥ 0 there exists  L >  0 such that | x ( t ) − y ( t ) |≤ ε for all  x,y  ∈ B ( x 0 ,r ) ∩ Ω being solutions of (2.3) for any  t ≥ L , we will say that solutionsof (2.3) are uniformly locally attractive (or equivalently asymptotically stable) on  R + , .

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