Banaei, Shahram

Ghaemi, Mohammad Bagher

Saadati, Reza

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 83–94 DOI: 10.18514/MMN.2017.2086AN EXTENSION OF DARBO’S THEOREM AND ITSAPPLICATION TO SYSTEM OF NEUTRAL DIFFERENTIALEQUATIONS WITH DEVIATING ARGUMENTSH. BANAEI, M. B. GHAEMI, AND R. SAADATIReceived 12 September, 2016Abstract. In this paper we extend Darbo’s fixed point theorem in Banach spaces and obtain atripled fixed point theorem. The technique of measure of noncompactness is the main tool incarrying out our proof. Finally, as an application of our results, we analyze the existence ofsolutions for a system of neutral differential equations.2010 Mathematics Subject Classification: 47H08; 47H10Keywords: measure of noncompactness, fixed point theorem, tripled fixed point theorem, integ-ral equation1. INTRODUCTION AND PRELIMINARIESThe notion of a measure of non compactness (MNC) was introduced by Kur-atowski [9] in 1930. Darbo’s fixed point theorem [7] which ensures the existenceof fixed point is an important application of this measure, since it generalizes bothSchauder fixed point and Banach contraction principle. Thereafter, many investiga-tions and papers have been appeared in the MNC and it’s application, see for example[1, 2, 5, 8, 10]. The aim of this paper is to generalize the Darbo’s fixed point theoremvia the concept of the class of operatorsO.f I :/ and extend the results of the theorempresented by Samadi and Ghaemi in [10]. Further, we apply our extension to obtaina tripled fixed point and study the existence of solutions for the system8ˆˆˆ<ˆˆˆ:x0.t/D ft;x..t//;y..t//;´..t//;x0.ˇ.t//;y0.ˇ.t//;´0.ˇ.t//y0.t/D ft;y..t//;x..t//;´..t//;y0.ˇ.t//;x0.ˇ.t//;´0.ˇ.t//´0.t/D ft;´..t//;y..t//;x..t//;´0.ˇ.t//;y0.ˇ.t//;x0.ˇ.t// (1.1)where t 2 Œ0;T  with the initial conditionx.0/D x0; y.0/D y0; ´.0/D ´0:c 2017 Miskolc University Press84 SH. BANAEI, M. B. GHAEMI, AND R. SAADATIFirst, we introduce some notations and definitions which are used throughout thispaper. Let R be the set of real numbers, RC D Œ0;1/ and .E;k:k/ be a real Banachspace with the zero element 0. We write B.x;r/ to denote the closed ball centered atx with radius r . If X be a nonempty subset of E then the symbols X and ConvXstand for the closure and closed convex hull of X , respectively. Moreover,ME is thefamily of nonempty bounded subset of E and NE denote its subfamily consisting ofall relatively compact sets.Definition 1 ([6]). A mapping  WME  ! RC is said to be a measure of non-compactness in E if it satisfies the following conditions:1ı The family kerD fX 2ME W .X/D 0g is nonempty and kerNE ;2ı X  Y H) .X/ .Y /;3ı .X/D .X/;4ı .ConvX/D .X/;5ı .XC .1 /Y / .X/C .1 /.Y / for  2 Œ0;1;6ı If fXng is a sequence of closed sets from ME such that XnC1  Xn fornD 1;2;    , and if limn!1.Xn/D 0, then X1 D\1nD1Xn ¤¿.The following concept of O.f I :/ was given by Altun and Turkoglu [4].Let F.Œ0;1// be the class of all functions f W Œ0;1/! Œ0;1/ and let  be theclass of all operatorsO.I / W F.Œ0;1//! F.Œ0;1//; f !O.f I /satisfying the following conditions:.1/ O.f I t / > 0 for t > 0 and O.f I0/D 0..2/ O.f I t /O.f Is/ for t  s..3/ limn!1O.f I tn/DO.f I limn!1 tn/..4/ O.f Imaxft; sg/DmaxfO.f I t /;O.f Is/g for some f 2 F.Œ0;1//.Example 1. ([4] ) If f W Œ0;1/! Œ0;1/ is a nondecreasing, continuous functionsuch that f .0/D 0 and f .t/ > 0 for t > 0, then the operator defined byO.f I t /D f .t/1Cf .t/satisfies the above conditions.Definition 2 ([8]). A triple .x;y;´/ is called a triple fixed point of a mapping T WX X X ! X ifT .x;y;´/D x; T .y;x;´/D y; T .´;y;x/D ´:The following theorems are basic for our main results.Theorem 1 (Schauder [3]). Let C be a nonempty, bounded, closed and convexsubset of a Banach space E. Then every compact and continuous map T W C ! Chas at least one fixed point.AN EXTENSION OF DARBO’S THEOREM AND ITS APPLICATION 85Theorem 2 (Darbo [7]). Let C be a nonempty, bounded, closed and convex subsetof a Banach space E and T W C ! C be a continuous mapping. Assume that thereexists a constant K 2 Œ0;1/ such that .TX/ K.X/ for any nonempty subset Xof C , where  is a MNC defined in E. Then T has at least a fixed point in C .Theorem 3 ([10]). Suppose 1;2; : : : ;n are measures of noncompactness inBanach spaces E1;E2; : : : ;En respectively. Moreover assume that the function F WRnC  !RC is convex and F.x1; : : : ;xn/D 0 if and only if xi D 0 for i D 1;2; : : : ;n.ThenQ.X/D F.1.X1/;2.X2/; : : : ;n.Xn//;defines a measure of noncompactness in E1 E2  : : :En where Xi denotes thenatural projection of X into Ei , for i D 1;2; : : : ;n.Remark 1. Aghajani et al. [1] illustrated the Theorem 3 by the following example.Let F be as follows:F.x;y;´/Dmaxfx;y;´g; or F.x;y;´/D xCyC´; for any .x;y;´/ 2 Œ0;1/3:They showed thate.X/Dmax.1.X1/;2.X2/;3.X3//; or e.X/D1.X1/C2.X2/C3.X3/;defines an MNC in the space E1 E2 E3 where Xi , .i D 1;2;3/ are the naturalprojection of X into Ei .Samadi and Ghaemi [10] generalized Darbo’s fixed point theorem as fallow.Theorem 4. Let U be a nonempty, bounded, closed and convex subset of a Banachspace E. Assume F W U ! U be a continuous operator such that satisfying ..F.X/// . ..X/// ..X//for all nonempty subsetX of U , where  is an arbitrary MNC inE,  : RC!RC isnondecreasing function such that  .t/D 0 if and only if t=0 and  W Œ0;1/! Œ0;1/is a continuous function such that limsupt!r .t/ < 1 for all r  0. Then F has afixed point in U.The Darbo’s fixed point theorem is followed if  D I (identity map) and  D Kin the Theorem 4.2. MAIN RESULTSNow we state one of the main results in this paper which extends and generalizesDarbo’s fixed point theorem by using concept of O.f I :/.Theorem 5. Let C be a nonempty, bounded, closed and convex subset of a Banachspace E and T W C ! C be a continuous operator such that ŒO.f I.TX// ˚Œ .O.f I.X/// ..X//;86 SH. BANAEI, M. B. GHAEMI, AND R. SAADATIfor X of C , O.I / 2 and let  : RC!RC be a nondecreasing function such that .t/ D 0 if and only if t=0. Let ˚ W Œ0;1/! Œ0;1/ is a continuous function suchthat limn!1˚n.t/D 0 for each t  0, where  is an arbitrary MNC. Then T has atleast one fixed point in C .Proof. By induction, we construct a sequence fCng such that C0DC and CnC1DConv.TCn/ for n 1. We haveTC0 D TC  C D C0, C1 D Conv.TC0/  C D C0; therefore by continuing thisprocess deduce thatC0  C1  ::: Cn  CnC1  ::::If there exists an integer N  0 such that .CN /D 0; then CN is relatively com-pact. In this case Theorem 1 implies that T has a fixed point. Now we suppose that.Cn/¤ 0 for n 0: We have ŒO.f I.CnC1//D  ŒO.f I.Conv.TCn///D  ŒO.f I.TCn// ˚Œ .O.f I.Cn/// ..Cn// ˚2Œ .O.f I.Cn 1/// ..Cn 1//::: ˚nŒ .O.f I.C0/// ..C0//D ˚nŒ .O.f I.C// ..C //(2.1)By letting n!1 in (1.2) we infer that limn!1 ŒO.f I.CnC1//D 0. Therefore,limn!1 ŒO.f I.CnC1//D  ŒO.f I limn!1.CnC1//D 0;we know that O.f I0/D 0. Thus,limn!1.CnC1/D 0. SinceCnCnC1 and TCnCn for all nD 1;2;3; ::: thenfrom 6ı of definition MNC, X1 D \1nD1Xn is a nonempty, closed and convex set,invariant under T and belongs to Ker. Consequently, from Theorem 1 we deducethat T has at least a fixed point. Remark 2. The Theorem 4 is followed if O.f I t /D t and f D I in Theorem 5.The following Corollary is immediate of Theorem 5.Corollary 1. LetC be a nonempty, bounded, closed and convex subset of a BanachspaceE, T W C ! C and : RC!RC are continuous functions. Suppose that thereexists a constant 0 <  < 1 such that for all X  C , ŒO.f I.TX// Œ .O.f I.X/// ..X/;where  is an arbitrary measure of noncompactness and O.I / 2. Then T has atleast one fixed point in C .AN EXTENSION OF DARBO’S THEOREM AND ITS APPLICATION 87Remark 3. The Darbo’s fixed point theorem is followed ifO.f I t /D t , f D I and D I in Corollary 1 .Theorem 6. Let C be a nonempty, bounded, closed and convex subset of a Banachspace E and T W C C C ! C be a continuous function such that ŒO.f I.T .x1x2x3/// 13˚Œ .O.f I.x1/C.x2/C.x3// Œ.x1/C.x2/C.x3/(2.2)for any subset x1;x2;x3 of C , where  is an arbitrary MNC, ˚ W Œ0;1/! Œ0;1/ isnondecreasing function such that limn!1˚n.t/D 0 for t  0 and  : RC!RC isa continuous function such that  .t/ D 0 if and only if t=0. Also, O.I / 2  andO.f I t C r C s/  O.f I t /CO.f Is/CO.f Ir/ for all t; s; r  0. Then T has atleast a tripled fixed point.Proof. From Remark 1, we havee.x/D .x1/C.x2/C.x3/. Now we definea mapping eT W C C C ! C C C byeT .x;y;´/D .T .x;y;´/;T .y;x;´/;T .´;y;x//:It is clear thateT is continuous. We prove thateT satisfies all the conditions of Theorem5. Let X  C C C be any nonempty subset by in (2.2) and 2ı we obtain ŒO.f Ie.eT .x//  ŒO.f Ie.T .x1x2x3//T .x2x1x3/T .x3x2x1/D  ŒO.f I.T .x1x2x3/C.T .x2x1x3/C.T .x3x2x1/D  ŒO.f I.T .x1x2x3///C ŒO.f I.T .x2x1x3///C ŒO.f I.T .x3x2x1/// 13˚Œ .O.f I.x1/C.x2/C.x3// Œ.x1/C.x2/C.x3/C 13˚Œ .O.f I.x2/C.x1/C.x3// Œ.x2/C.x1/C.x3/C 13˚Œ .O.f I.x3/C.x2/C.x1// Œ.x3/C.x2/C.x1/D ˚Œ .O.f I.x1/C.x2/C.x3// Œ.x1/C.x2/C.x3/:From Theorem 5 deduce that eT has at least a fixed point in C C C and T has atleast a tripled fixed point. As application of Theorem 6 we can get the following corollary.88 SH. BANAEI, M. B. GHAEMI, AND R. SAADATICorollary 2. LetC be a nonempty, bounded, closed and convex subset of a Banachspace E and T W C C C ! C be a continuous function such that.T .x1x2x3// 13˚..x1/C.x2/C.x3//;for any subset x1;x2;x3 ofC , where is an arbitrary MNC. Also,˚ W Œ0;1/! Œ0;1/is a nondecreasing function such that limn!1˚n.t/D 0 for t  0. Then T has atleast a tripled fixed point.3. APPLICATIONIn this section as an application of Theorem 6 and Corollary 2 we study the exist-ence of solutions for the system of equations (1.1). Let the space C Œ0;T  consistingof all real functions defined, bounded and continuous on the interval Œ0;T  equippedwith the standard normkxk D supfjx.t/j W t 2 Œ0;T g:Recall that the modulus of continuity of a function x 2 C Œ0;T  is defined by!.x;/D supfjx.t/ x.s/j W t; s 2 Œ0;T ; jt   sj  g:Since x is uniformly continuous on Œ0;T  then !.x;/! 0 as ! 0 and the Haus-dorff measure of noncompactness for all bounded sets ˝ of C Œ0;T  is equivalentto.˝/D lim!0nsupx2X!.x;/o:See more detail [3].Now we consider the following assumptions:.A1/ ;ˇ W Œ0;T ! Œ0;T  are continuous functions..A2/ The function f W Œ0;T R6 ! R is continuous and there exist continuousand nondecreasing functions '; W RC  ! RC such that limn!1˚n.t/D 0for t  0 andjf .t;x1;x2; : : : ;x6/ f .t;y1;y2; : : : ;y6/j  . max1i3 jxi  yi j/C (3.1)13'.jx4 y4jC jx5 y5jC jx6 y6j/.A3/ There exists a positive solution r0 of the inequalityM C.T r/C 13'.3r/ r (3.2)whereM D supfjf .t;x0;y0;´0;0;0;0/j W t 2 Œ0;T g:AN EXTENSION OF DARBO’S THEOREM AND ITS APPLICATION 89Now, let us put x0.t/DX.t/, y0.t/D Y.t/ and ´0.t/DZ.t/. Then the problem (1.1)can be replaced equivalently by the following functional-integral equation8ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ<ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ:X.t/D ft;x0CZ .t/0X.s/ds;y0CZ .t/0Y.s/ds;´0CZ .t/0Z.s/ds;X.ˇ.t/;Y.ˇ.t/;Z.ˇ.t/Y.t/D ft;y0CZ .t/0Y.s/ds;x0CZ .t/0X.s/ds;´0CZ .t/0Z.s/ds;Y.ˇ.t/;X.ˇ.t/;Z.ˇ.t/Z.t/D ft;´0CZ .t/0Z.s/ds;y0CZ .t/0Y.s/ds;x0CZ .t/0X.s/ds;X.ˇ.t/;Y.ˇ.t/;Z.ˇ.t/(3.3)where t 2 Œ0;T . In the sequel we will examine the equation (3.3).Theorem 7. Under the assumptions .A1/  .A3/ the equation .3:3/ has at leastone solution x 2 fC Œ0;T g3Proof. Let us consider the operator T defined on the space C Œ0;T  by the formulaT .x;y;´/.t/D ft;x0CZ .t/0X.s/ds;y0CZ .t/0Y.s/ds;´0CZ .t/0Z.s/ds;X.ˇ.t/;Y.ˇ.t/;Z.ˇ.t/ (3.4)for all t 2 Œ0;T . Observe that in view of assumptions .A1/ and .A2/, the functionT x is continuous on Œ0;T  for any x 2 C Œ0;T , i.e. T transforms the space C Œ0;T into itself. For arbitrarily fixed t 2 Œ0;T , we havejT .x;y;´/.t/j jft;x0CZ .t/0x.s/ds;y0CZ .t/0y.s/ds;´0CZ .t/0´.s/ds;x.ˇ.t//;y.ˇ.t//;´.ˇ.t// f .t;x0;y0;´0;0;0;0/jC jf .t;x0;y0;´0;0;0;0/jmax˚jZ .t/0x.s/dsj; jZ .t/0y.s/dsj; jZ .t/0´.s/dsj	C 13'.jx.ˇ.t//jC jy.ˇ.t//jC j´.ˇ.t//j/Cjf .t;x0;y0;´0;0;0;0/jT max˚kxk;kyk;k´k	C 13'.kxkCkykCk´k/CM90 SH. BANAEI, M. B. GHAEMI, AND R. SAADATIThus,kT .x;y;´/k  T max˚kxk;kyk;k´k	C 13'.kxkCkykCk´k/CM (3.5)and T .x;y;´/ 2 C Œ0;T  for any .x;y;´/ 2 .C Œ0;T /3 . Due to Inequality (3.2) thefunction T maps . NBr0/3 into NBr0 . Now we show that T is a continuous function on. NBr0/3. To do this, let us fix " > 0 and take arbitrary .x;y;´/; .u;v;w/ 2 . NBr0/3 suchthat max˚kx uk;ky vk;k´ wk	< ". Then, for t 2 Œ0;T , we haveˇˇˇT .x;y;´/.t/ T .u;v;w/.t/ˇˇˇˇˇˇft;x0CZ .t/0x.s/ds;y0CZ .t/0y.s/ds;´0CZ .t/0´.s/ds;x.ˇ.t//;y.ˇ.t//;´.ˇ.t// ft;x0CZ .t/0u.s/ds;y0CZ .t/0v.s/ds;´0CZ .t/0w.s/ds;u.ˇ.t//;v.ˇ.t//;w.ˇ.t//ˇˇˇ max˚Z .t/0jx.s/ u.s/jds;Z .t/0jy.s/ v.s/jds;Z .t/0j´.s/ w.s/jds	C 13'jx.ˇ.t// u.ˇ.t//jC jy.ˇ.t// v.ˇ.t//jC j´.ˇ.t// w.ˇ.t//j .T "/C 13'.3"/So we have .T "/C 13'.3"/  ! 0 as "  ! 0. Thus T is a continuous function on. NBr0/3. Now we show that T satisfies all the conditions of Corollary 2. Let X1, X2and X3 be nonempty and bounded subsets of NBr0 . Assume that " > 0 is arbitraryconstant. Also we take t1; t2 2 Œ0;T , with jt2   t1j  " and x 2 X1, y 2 X2 and´ 2X3. Then we haveAN EXTENSION OF DARBO’S THEOREM AND ITS APPLICATION 91ˇˇˇT.x;y;´/.t1/ T.x;y;´/.t2/ˇˇˇˇˇˇft1;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t1//;y.ˇ.t1//;´.ˇ.t1// ft1;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2//ˇˇˇCˇˇˇft1;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2// ft2;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2//ˇˇˇCˇˇˇft2;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2// ft2;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2//ˇˇˇCˇˇˇft2;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2// ft2;x0CZ .t1/0x.s/ds;y0CZ .t1/0y.s/ds;´0CZ .t1/0´.s/ds;x.ˇ.t2//;y.ˇ.t2//;´.ˇ.t2//ˇˇˇ 13'jx.ˇ.t1// x.ˇ.t2//jC jy.ˇ.t1// y.ˇ.t2//jC j´.ˇ.t1// ´.ˇ.t2//jC!.f;"/Cmax˚Z .t1/.t2/jx.s/jds;Z .t1/.t2/jy.s//jds;Z .t1/.t2/j´.s/jds	 13'!.x;!.ˇ;"//C!.y;!.ˇ;"//C!.´;!.ˇ;"//C!.f;"/Cmax˚r0!.;"/	;92 SH. BANAEI, M. B. GHAEMI, AND R. SAADATIwhere!.ˇ;"/D supnjˇ.t2/ ˇ.t1/j W jt1  t2j  "; t1; t2 2 Œ0;T o!.;"/D supnj.t2/  .t1/j W t1; t2 2 Œ0;T ; jt1  t2j  "o!.x;!.ˇ;"//D supnjx.t2/ x.t1/j W jt1  t2j  !.ˇ;"/; t1; t2 2 Œ0;ˇ.T /o;!.f;"/D supnjf .t1;x1;x2; :::x6/ f .t2;x1;x2; :::x6/j W jt1  t2j  ";t1; t2 2 Œ0;T ;x4;x5;x6 2 Œ r0; r0;x1;x2;x3 2 Œ ˇ.T /r0;ˇ.T /r0o:We obtainedˇˇˇT .x;y;´/.t1/ T .x;y;´/.t2/ˇˇˇ13'!.x;!.ˇ;"//C!.y;!.ˇ;"//C!.´;!.ˇ;"//C!.f;"/Cmax˚r0!.;"/	Therefore, we have!T .X1X2X3/;"13'!.X1;!.ˇ;"//C!.X2;!.ˇ;"//C!.X3;!.ˇ;"//C!.f;"//Cmax˚r0!.;"/	:On the other hand by the uniform continuity of f , ˇ and  on Œ0;T Œ ˇ.T /r0;ˇ.T /r03Œ r0; r03, Œ0;T  and Œ0;T  respectively, we have!.f;"/  ! 0, !.ˇ;"/ and !.;"/  ! 0 as "  ! 0. Therefore we obtainT .X1X2X3 13'.X1/C.X2/C.X3/:By applying Corollary 2, T has a fixed point.Example 2. Consider the following system of functional differential equations8<:x0.t/D t2C 3px.t/C 5py.t/C 7p´.t/C 13ln.1Cjx0.t/Cy0.t/C´0.t/j/y0.t/D t2C 3py.t/C 5px.t/C 7p´.t/C 13ln.1Cjx0.t/Cy0.t/C´0.t/j/´0.t/D t2C 3p´.t/C 5py.t/C 7px.t/C 13ln.1Cjx0.t/Cy0.t/C´0.t/j/(3.6)where t 2 Œ0;3 with the initial conditionx.0/D 3; y.0/D 1; ´.0/D 2:Eq. (3.6) is a special case of Eq. (1.1) whereˇ.t/D .t/D t;AN EXTENSION OF DARBO’S THEOREM AND ITS APPLICATION 93f .t;x1; : : : ;x6/D t2C 3px1C 5px2C 7px3C 13ln.1Cjx4Cx5Cx6j/:From the definitions of ˇ and , hypothesis .A1/ are obviously satisfied. Supposethat t 2 Œ0;3. Now, by taking '.t/D ln.1C t /, .t/DmaxiD3;5;7f iptg we havejf .t;x1; : : : ;x6/ f .t;y1; : : : ;y6/j  j 3px1  3py1jC j 5px2  5py2jC j 7px3  7py3jC 13j ln.1Cjx4Cx5Cx6j/  ln.1Cjy4Cy5Cy6j/ j 3px1  3py1jC j 5px2  5py2jC j 7px3  7py3jC 13ln.1C jx4Cx5Cx6j  jy4Cy5Cy6j1Cjy4Cy5Cy6j / 3pjx1 y1jC 5pjx2 y2jC 7pjx3 y3jC 13ln.1Cjx4Cx5Cx6  .y4Cy5Cy6/j/ 3pjx1 y1jC 5pjx2 y2jC 7pjx3 y3jC 13ln.1Cjx4 y4jC jx5 y5jC jx6 y6j/D 13'.jx4 y4jC jx5 y5jC jx6 y6j/C. maxiD1;2;3fjxi  yi jg/:(3.7)Thus, from (3.7) we infer that condition .A2/ holds. Furthermore,M D supfjf .t;x0;y0;´0;0;0;0/j W t 2 Œ0;T gD supft2C 3p3C 5p1C 7p2 W t 2 Œ0;3g 12:It is easy to see that each number r  20 satisfies the inequality in condition .A3/,i.e.,12C.3r/C 13'.3r/ r:Thus, as the number r0 we can take r0 D 20. Consequently, all the conditions ofTheorem 7 are satisfied. Hence the system equations 3.6 has at least one solutionwhich belong to the space fC Œ0;T g3.REFERENCES[1] A. Aghajani, R. Allahyari, and M. Mursaleen, “A generalization of darbo’s theorem with applic-ation to the solvability of systems of integral equations.” Journal of Computational and AppliedMathematics, vol. 260, no. 2, pp. 68–77, 2014, doi: 10.1016/j.cam.2013.09.039.94 SH. BANAEI, M. B. GHAEMI, AND R. SAADATI[2] A. Aghajani, J. Banas´ , and N. Sabzali, “Some generalizations of Darbo fixed point theorem andapplications.” Bulletin of the Belgian Mathematical Society, vol. 20, no. 2, pp. 345–358, 2013.[3] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measuresof Noncompactness and Condensing Operators. Basel: Birkha¨user, 1992. doi: 10.1007/978-3-0348-5727-7.[4] I. Altun and D. Turkoglu, “A fixed point theorem for mappings satisfying a general contractivecondition of operator type.” Journal of Computational Analysis and Applications, vol. 9, no. 1,pp. 9–14, 2007.[5] J. Banas´ and R. Nalepa, “On a measure of noncompactness in the space of functions withtempered increments.” Journal of Mathematical Analysis and Applications, vol. 435, no. 2, pp.1634–1651, 2016, doi: 10.1016/j.jmaa.2015.11.033.[6] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, ser. Lecture Notes inPure and Applied Mathematics. New York, Basel: Marcel Dekker, Inc., 1980, vol. 60.[7] G. Darbo, “Punti uniti in trasformazioni a codominio non compatto,” Rendiconti del SeminarioMatematico della Universita di Padova, vol. 24, pp. 84–92, 1955.[8] V. Karakaya, N. E. H. Bouzara, and Y. Atalan, “Existence of tripled fixed points for aclass of condensing operators in banach spaces,” The Scientific World Journal, 2014, doi:10.1155/2014/541862.[9] C. Kuratowski, “Sur les espaces complets,” Fundamenta Mathematicae, vol. 15, no. 1, pp. 301–309, 1930.[10] A. Samadi and M. B. Ghaemi, “ An extension of Darbo fixed point theorem and its applicationsto coupled fixed point and integral equations.” Filomat, vol. 28, no. 4, pp. 879–886, 2014, doi:10.2298/FIL1404879S.Authors’ addressesSh. BanaeiDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, IranE-mail address: sh.banaei@kiau.ac.irM. B. GhaemiDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, IranE-mail address: mghaemi@iust.ac.irR. SaadatiDepartment of Mathematics, Iran University of Science and Technology, IranE-mail address: rsaadati@iust.ac.ir

An extension of Darbo's theorem and its application to system of neutral differential equation with‎ ‎deviating argument‎

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