The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equations have been studied by J. Banas. Here we are concerned with a singular quadratic functional integral equations. The existence of positive monotonic solutions $x \in L_1[0,1]$ will be proved. The fractional order nonlinear functional differential equation will be given as a special case

El-Sayed, Ahmed

Hashem, H.H.G.

Electronic Journal of Qualitative Theory of Differential Equations

Ahmed El-Sayed

H. H. G. Hashem

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Electronic journal of qualitative theory of differential equations

Monotonic solutions of functional integral  and differential equations of fractional order

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Electronic Journal of Qualitative Theory of Differential Equations2009, No. 7, 1-8; http://www.math.u-szeged.hu/ejqtde/Monotonic solutions of functional integral and differentialequations of fractional orderEl-Sayed, A.M.A and Hashem, H.H.GFaculty of Science, Alexandria University, Alexandria, Egypte.mails : amasayed@hotmail.com and hendhghashem@yahoo.comAbstractThe existence of positive monotonic solutions, in the class of continuous functions,for some nonlinear quadratic integral equations have been studied in [5]-[8]. Here weare concerned with a singular quadratic functional integral equations. The existence ofpositive monotonic solutions x ∈ L1[0, 1] will be proved. The fractional order nonlinearfunctional differential equation will be given as a special case.Keywords: Quadratic functional integral equations, Positive monotonic solutions, Measureof noncompactness fractional order integral.1 IntroductionNonlinear quadratic integral equations appear very often, in many applications of real worldproblem. For example, some problems considered in vehicular traffic theory, biology andqueuing theory lead to the quadratic integral equations of this type (see [5] -[8]). J. Banas(see ([5]-[8]) proved the existence of solution of some equations in the class of continuousfunctions.Here, we shall be concerned with the quadratic functional integral equationx(t) = a(t) + f(t, x(t))∫ t0k (t, s) g(s, x(φ(s))) ds t ∈ [0, 1]. (1)We prove, under certain condition, the existence of positive monotonic solutions of Eq. (1)in the space of integrable functions.The singular (fractional order) quadratic functional integral equation of the formx(t) = a(t) + f(t, x(t))∫ t0(t− s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1], β ∈ (0, 1), (2)will be studied as an application.The existence of a monotonic positive integrable solution of the nonlinear functional differ-ential equation of fractional orderDβx(t) = g(s, x(φ(s))), t ∈ (0, 1] and I1−βx(t)|t=0 = a, (3)EJQTDE, 2009 No. 7, p. 1where Dβ is the Riemann-Liouville fractional order derivative, will be given as anotherapplication.Also the results concerning the existence of monotonic positive integrable solution of thenonlinear functional equationx(t) = g(t, x(φ(t))), (4)which was proved in [3], will be given as a special case.2 PreliminariesLet L1 = L1[0, 1] be the class of Lebesgue integrable functions on I = [0, 1] with thestandard norm. In this section we collect some definitions and results needed in our furtherinvestigations.Assume that the function f : (0, 1) × R → R satisfies Caratheodory conditions i.e., it ismeasurable in t for any x and continuous in x for almost all t. Then to every function x(t)being measurable on the interval (0, 1) we may assign the function(Fx)(t) = f(t, x(t)), t ∈ (0, 1).The operator F defined in such a way is called the superposition operator. This operator isone of the simplest and most important operators investigated in the nonlinear functionalanalysis. One of the most important result concerning the superposition operators containedin the below given theorem due to Krasnosel’skii [3].Theorem 2.1 The superposition operator F maps L1 into itself if and only if|f(t, x)| ≤ |c(t)|+ k|x| for all t ∈ (0, 1)and x ∈ R, where c(t) is a function from L1 and k is a nonnegative constant.Now let E be a Banach space with zero element 0 and X be a nonempty bounded sub-set of E, moreover denote by Br = B(0, r) the closed ball in E centered at 0 and withradius r.In the sequel we shall need some criteria for compactness in measure. The complete de-scription of compactness in measure was given by Fre’chet [11] , but the following sufficientcondition will be more convenient for our purposes (see[3]).Theorem 2.2 Let X be a bounded subset of L1. Assume that there is a family of subsets(Ωc)o≤c≤b−a of the interval (a,b) such that meas Ωc = c for every c ∈ [0, b − a], and forevery x ∈ X, x(t1) ≤ x(t2) , (t1 ∈ Ωc, t2 6∈ Ωc), then the set X is compact in measure.The measure of weak noncompactness defined by De Blasi [5]-[6] is given by,β(X) = inf(r > 0 : there exists a weakly copmact subset Y of E such that X ⊂ Y +Kr)The function β(X) possess several useful properties which may be found in [5].The convenient formula for the function β(X) in L1 was given by Appell and De Pascale( see [1])β(X) = lim→0(supx∈X(sup[∫D|x(t)|dt : D ⊂ [a, b],meas D ≤ ])), (5)EJQTDE, 2009 No. 7, p. 2where the symbol meas D stands for Lebesgue measure of the set D.Next, we shall also use the notion of the Hausdorff measure of noncompactness χ (see[2])defined byχ(X) = inf(r > 0 : there exists a finite subset Y of E such that X ⊂ Y +Kr)In the case when the set X is compact in measure, the Hausdorff and De Blasi measures ofnoncompactness will be identical. Namely we have (see[10])Theorem 2.3 Let X be an arbitrary nonempty bounded subset of L1. If X is compact inmeasure then β(X) = χ(X).Finally we will recall the fixed point theorem du to Darbo [5]-[6].Theorem 2.4 Let G be a nonempty, bounded, closed and convex subset of E and letH : G → G be a continuous transformation which is a contraction with respect to theHausdorff measure of noncompactness χ, i.e.,there exists a constant α ∈ [0, 1) such thatχ(HX) ≤ α χ(X) for any nonempty subset X of G. Then H has at least one fixed pointin the set G.2.1 Fractional order integral operatorsThis section is devoted to study the definitions and some properties of the fractional orderintegral operators. Let β be a positive real numberDefinition 1 The fractional order integral of order β of f ∈ L1 is defined by (see [12])Iβf(t) =∫ t0(t− s)β−1Γ(β)f(s) ds, (6)where (it is proved that)(i) Iγ Iβ f(t) = Iγ+β f(t), γ, β > 0 .(ii) The fractional order integral operator Iβ maps L1 into itself continuously.(iii) limβ→1 Iβ f(t) =∫ t0 f(s)ds.(iv) limβ→0 Iβ f(t) = f(t).And the following Lemma was proved in ([9])Lemma 2 Let the operator Iβ be defined on L1, then it maps the monotonic nondecrasingpositive functions into functions of the same type.Definition 2 The Riemann-Liouville fractional order derivative of order β ∈ (0, 1) of thefunction f is given byDβf(t) =ddtI1−βf(t)3 Existence of solutions of a quadratic functional integralequationAssume that the following assumptions are satisfiedEJQTDE, 2009 No. 7, p. 3(i) a : I → R+ = [ 0,+ ∞ ) is integrable and nondecreasing function on I;(ii) f(t, x) = f : I × R+ → R+ satisfy Carathe´odory conditions (i.e. it is measurablein t for all x ∈ R+ and continuous in x for all t ∈ [0, 1] ) and there exists afunction m(t) ∈ L1 such thatf(t, x) ≤ |m(t)|.Moreover f is nondecreasing with respect to both variables separately;(iii) g(t, x) = g : I × R+ → R+ satisfy Carathe´odory conditions and there exists afunction a1(t) ∈ L1 and a constant b such thatg(t, x) ≤ |a1(t)| + b | x | ∀ (t, x) ∈ I × R+.Moreover, g is nondecreasing with respect to both variables separately;(iv) φ : I → I is increasing, absolutely continuous on I and there exists a constantM1 > 0 such that φ′(t) ≥ M1 on I ;(v) k : I × I → R+ is measurable in both variables and the operator K defined by(Ky)(t) =∫ t0k(t, s) y(s) ds, t ∈ [0, 1]maps nondecreasing positive function x ∈ L1 into function of the same type and∫ 1s| k(t, s) | m(t) dt < M, s ∈ [0, 1].Now, we are ready to prove the existence theorem.Theorem 3.1 Let the assumptions (i)-(v) be satisfied. If b M < M1, then thequadratic functional integral equation (1) has at least one positive nondecreasing solutionx ∈ L1(I).Proof. Let the operator H be defined by the formula(Hx)(t) = a(t) + f(t, x(t))∫ t0k (t, s) g(s, x(φ(s))) ds.Let x ∈ L1, then by assumptions (i),(ii) and (iii) we find that| (Hx)(t) | ≤ | a(t) | + | f(t, x(t)) |∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds≤ | a(t) | + m(t)∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds.This implies that|| Hx || =∫ 10| (Hx)(t) | dtEJQTDE, 2009 No. 7, p. 4≤ || a || +∫ 10m(t)∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds dt= || a || +∫ 10∫ 1sk(t, s) m(t) dt ( |a1(s)| + b | x(φ(s)) | ) ds≤ || a || + M∫ 10( |a1(s)| + b | x(φ(s)) | ) ds≤ || a || + M∫ 10|a1(s)| ds + M b∫ 10| x(φ(s)) | ds≤ || a || + M || a1 || + M b M1−1∫ φ(1)φ(0)| x(u) | du≤ || a || + M || a1 || + M b M1−1∫ 10| x(u) | duwhich gives|| Hx || ≤ || a || + M || a1 || + M b M1−1 || x || (7)and proves that Hx ∈ L1.Moreover the estimate (7) shows that the operator H maps Br into it self, wherer = [ || a || + M || a1 || ] [ 1 − M b M1−1 ]−1.Let Qr ⊂ Br consisting of all functions positive and nondecreasing on I.Clearly Qr is nonempty, bounded, closed and convex ([3]). Now Qr is a bounded subsetof L1 consisting of all functions positive and nondecreasing on I, then Theorem 2.2shows that Qr is compact in measure [3].Now, we shall show that the operator H transforms positive and nondecreasing functioninto function of the same type.First if x ∈ Qr is an arbitrary function, then x(φ(t)) is positive and nondecreasingfunction on I. Therefore∫ t0k(t, s) g(s, x(φ(s))) ds, t ∈ (0, 1)is positive and nondecreasing functions on I.Thus the operator H maps Qr into itself. By using assumption (ii),(iii) and (v), theoperator H is continuous on Qr (see [13] and [4]).Let X be a nonempty subset of Qr. Fix  > 0 and take a measurable subset D ⊂ Isuch that meas. D ≤ . Then for any x ∈ X, we get|| Hx ||D =∫D| (Hx)(t) | dt ≤∫D| a(t) | dt+∫Dm(t)∫ t0k(t, s)( a1(s) + b | x(φ(s)) | ) ds dt≤ || a ||D + M || a1 ||D + b M1−1 M∫φ(D)| x(u) | ds.EJQTDE, 2009 No. 7, p. 5Butlim → 0{ sup{∫D| a(t) | dt : D ⊂ I, meas.D <  } } = 0.andlim → 0{ sup{∫D| a1(t) | dt : D ⊂ I, meas.D <  } } = 0.Then we obtainβ(Hx(t)) ≤ M b M1−1 β(x(t))andβ(HX) ≤ M b M1−1 β(X).which by Theorem 2.3 implies thatχ (HX) ≤ M b M1−1 χ(X).Since M b M1−1 < 1, it follows, from Theorems 2.3 and 2.4, that H is contraction andhas at least one fixed point in Qr which proves that the nonlinear quadratic functionalintegral equation (1) has at least one positive nondecreasing solution x ∈ L1[0, 1].4 Fractional order quadratic functional integral equationConsider now the quadratic functional integral equation (2). Now we have the followingtheorem.Theorem 4.1 Let 1Γ(β)∫ 1s (t − s)β−1 m(t) dt < M. Then under the assumptions ofTheorem 3.1. The nonlinear quadratic functional integral equation of fractional orderx(t) = a(t) + f(t, x(t))∫ t0(t− s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1], β ∈ (0, 1),has at least one positive nondecreasing solution x ∈ L1.Proof. Let k(t, s) = 1Γ(β)(t − s)β−1. Then from the properties of the fractional orderintegral operator (subsection 2.1) we deduce that the operatorKx(t) =∫ t0(t− s)β−1Γ(β)x(s) ds, β ∈ (0, 1)satisfies the assumption (v) of Theorem 3.1 and the results follows from the results of The-orem 3.1.Corollary 4.1 Under the assumptions of Theorem 3.1, with a(t) = 0 and f(t, x(t)) = 1,the nonlinear functional equation (4) has at least one positive nondecreasing solutionx ∈ L1.Proof. Putting a(t) = 0 and f(t, x) = 1. Letting β → 0 the quadratic functional in-tegral equation (1) will be the functional equation (4) and the results follows from Theorem3.1.EJQTDE, 2009 No. 7, p. 65 Fractional order functional differential equationsFor the initial value problem of the nonlinear fractional order differential equation (3) wehave the following theorem.Theorem 5.1 Under the assumptions of Theorem 3.1, with a(t) = a tβ−1Γ(β) andf(t, x(t) = 1, the Cauchy type problem (3) has at least one positive nondecreasingsolution x ∈ L1.Proof. Integrating (3) we obtain the integral equationx(t) = atβ−1Γ(β)+∫ t0(t− s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1] (8)which by Theorem 3.1 has the desired solution.Operating by Dβ on (8) we obtain the problem (3). So the equivalence between problem(3) and integral equation (8) is proved and then the results follow from Theorem 3.1.References[1] J. Appel and E. De Pascale, Su alcuni parameteri connssi con la misura dinoncompat-tezza di Hausdorff in spazi di functioni misurabili, Boll. Un. Mat. Ital. (6), 3-B (1984),497-515.[2] J. Banas and K. Goebel, Measure of noncompactness in Banach spaces, Lect. Notes inpure and applied Math. 60, M. Dekker, New York and Basel 1980.[3] J. Banas, On the superposition operator and integrable solutions of some functionalequations, Nonlin. Analysis T.M.A. Vol. 12(1988), 777-784.[4] J. Banas, Integrable solutions of Hammerstein and Urysohn integral equations J. Aus-tral Math. Soc. (series A) 46 (1989), 51-68.[5] J. Banas, M. Lecko, W. G. El-Sayed, Existence theorems of some quadratic integralequation, J.Math. Anal. Appl. 227 (1998), 276 - 279.[6] J. Banas, A. Martininon, Monotonic solutions of a quadratic integral equation ofVolterra type, Comput. Math. Appl. 47 (2004), 271 - 279.[7] J. Banas, J. R. Martin and K. Sadarangani, On the solution of a quadratic integralequation of Hammerstein tupe, Mathematical and Computer Modelling, 43 (2006),97-104.[8] J. Banas, B. Rzepka, Monotonic solutions of a quadratic integral equations of fractionalorder J.Math. Anal. Appl. 332 (2007), 1371 -1378.[9] A. M. A El-Sayed, W. G. El-Sayed and O. L Moustafa On some fractional functionalequations Pure Math. and Appl. Vol. 6,No.4 (1995), 321-332.EJQTDE, 2009 No. 7, p. 7[10] F. S. De Blasi, On a property of the unit sphere in Banach space, Bull. Math. Soc.Sci. Math. R.S. Roum.(N.S.) 21 (1977), 259-262.[11] N. Dunford and J. Schwartz, Linear operators I, Int. Publ., Leyden (1963)[12] I. Podlubny, Fractional differential equations, Academic Press, (1999).[13] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnoselskii, S.G.Mikhlin, L.S. Rakovshchik andV.J. Stecenko Integral equations Noordhoff, Leyden (1975).(Received June 18, 2008)EJQTDE, 2009 No. 7, p. 8

El-Sayed A. M. A.

Hashem H. H. G.

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2009, No. 7, 1-8; http://www.math.u-szeged.hu/ejqtde/Monotonic solutions of functional integral and differentialequations of fractional orderEl-Sayed, A.M.A and Hashem, H.H.GFaculty of Science, Alexandria University, Alexandria, Egypte.mails : amasayed@hotmail.com and hendhghashem@yahoo.comAbstractThe existence of positive monotonic solutions, in the class of continuous functions,for some nonlinear quadratic integral equations have been studied in [5]-[8]. Here weare concerned with a singular quadratic functional integral equations. The existence ofpositive monotonic solutions x ∈ L1[0, 1] will be proved. The fractional order nonlinearfunctional differential equation will be given as a special case.Keywords: Quadratic functional integral equations, Positive monotonic solutions, Measureof noncompactness fractional order integral.1 IntroductionNonlinear quadratic integral equations appear very often, in many applications of real worldproblem. For example, some problems considered in vehicular traffic theory, biology andqueuing theory lead to the quadratic integral equations of this type (see [5] -[8]). J. Banas(see ([5]-[8]) proved the existence of solution of some equations in the class of continuousfunctions.Here, we shall be concerned with the quadratic functional integral equationx(t) = a(t) + f(t, x(t))∫ t0k (t, s) g(s, x(φ(s))) ds t ∈ [0, 1]. (1)We prove, under certain condition, the existence of positive monotonic solutions of Eq. (1)in the space of integrable functions.The singular (fractional order) quadratic functional integral equation of the formx(t) = a(t) + f(t, x(t))∫ t0(t − s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1], β ∈ (0, 1), (2)will be studied as an application.The existence of a monotonic positive integrable solution of the nonlinear functional differ-ential equation of fractional orderDβx(t) = g(s, x(φ(s))), t ∈ (0, 1] and I1−βx(t)|t=0 = a, (3)EJQTDE, 2009 No. 7, p. 1where Dβ is the Riemann-Liouville fractional order derivative, will be given as anotherapplication.Also the results concerning the existence of monotonic positive integrable solution of thenonlinear functional equationx(t) = g(t, x(φ(t))), (4)which was proved in [3], will be given as a special case.2 PreliminariesLet L1 = L1[0, 1] be the class of Lebesgue integrable functions on I = [0, 1] with thestandard norm. In this section we collect some definitions and results needed in our furtherinvestigations.Assume that the function f : (0, 1) × R → R satisfies Caratheodory conditions i.e., it ismeasurable in t for any x and continuous in x for almost all t. Then to every function x(t)being measurable on the interval (0, 1) we may assign the function(Fx)(t) = f(t, x(t)), t ∈ (0, 1).The operator F defined in such a way is called the superposition operator. This operator isone of the simplest and most important operators investigated in the nonlinear functionalanalysis. One of the most important result concerning the superposition operators containedin the below given theorem due to Krasnosel’skii [3].Theorem 2.1 The superposition operator F maps L1 into itself if and only if|f(t, x)| ≤ |c(t)| + k|x| for all t ∈ (0, 1)and x ∈ R, where c(t) is a function from L1 and k is a nonnegative constant.Now let E be a Banach space with zero element 0 and X be a nonempty bounded sub-set of E, moreover denote by Br = B(0, r) the closed ball in E centered at 0 and withradius r.In the sequel we shall need some criteria for compactness in measure. The complete de-scription of compactness in measure was given by Fre’chet [11] , but the following sufficientcondition will be more convenient for our purposes (see[3]).Theorem 2.2 Let X be a bounded subset of L1. Assume that there is a family of subsets(Ωc)o≤c≤b−a of the interval (a,b) such that meas Ωc = c for every c ∈ [0, b − a], and forevery x ∈ X, x(t1) ≤ x(t2) , (t1 ∈ Ωc, t2 6∈ Ωc), then the set X is compact in measure.The measure of weak noncompactness defined by De Blasi [5]-[6] is given by,β(X) = inf(r > 0 : there exists a weakly copmact subset Y of E such that X ⊂ Y + Kr)The function β(X) possess several useful properties which may be found in [5].The convenient formula for the function β(X) in L1 was given by Appell and De Pascale( see [1])β(X) = limε→0(supx∈X(sup[∫D|x(t)|dt : D ⊂ [a, b],meas D ≤ ε])), (5)EJQTDE, 2009 No. 7, p. 2where the symbol meas D stands for Lebesgue measure of the set D.Next, we shall also use the notion of the Hausdorff measure of noncompactness χ (see[2])defined byχ(X) = inf(r > 0 : there exists a finite subset Y of E such that X ⊂ Y + Kr)In the case when the set X is compact in measure, the Hausdorff and De Blasi measures ofnoncompactness will be identical. Namely we have (see[10])Theorem 2.3 Let X be an arbitrary nonempty bounded subset of L1. If X is compact inmeasure then β(X) = χ(X).Finally we will recall the fixed point theorem du to Darbo [5]-[6].Theorem 2.4 Let G be a nonempty, bounded, closed and convex subset of E and letH : G → G be a continuous transformation which is a contraction with respect to theHausdorff measure of noncompactness χ, i.e.,there exists a constant α ∈ [0, 1) such thatχ(HX) ≤ α χ(X) for any nonempty subset X of G. Then H has at least one fixed pointin the set G.2.1 Fractional order integral operatorsThis section is devoted to study the definitions and some properties of the fractional orderintegral operators. Let β be a positive real numberDefinition 1 The fractional order integral of order β of f ∈ L1 is defined by (see [12])Iβf(t) =∫ t0(t − s)β−1Γ(β)f(s) ds, (6)where (it is proved that)(i) Iγ Iβ f(t) = Iγ+β f(t), γ, β > 0 .(ii) The fractional order integral operator Iβ maps L1 into itself continuously.(iii) limβ→1 Iβ f(t) =∫ t0 f(s)ds.(iv) limβ→0 Iβ f(t) = f(t).And the following Lemma was proved in ([9])Lemma 2 Let the operator Iβ be defined on L1, then it maps the monotonic nondecrasingpositive functions into functions of the same type.Definition 2 The Riemann-Liouville fractional order derivative of order β ∈ (0, 1) of thefunction f is given byDβf(t) =ddtI1−βf(t)3 Existence of solutions of a quadratic functional integralequationAssume that the following assumptions are satisfiedEJQTDE, 2009 No. 7, p. 3(i) a : I → R+ = [ 0,+ ∞ ) is integrable and nondecreasing function on I;(ii) f(t, x) = f : I × R+ → R+ satisfy Carathéodory conditions (i.e. it is measurablein t for all x ∈ R+ and continuous in x for all t ∈ [0, 1] ) and there exists afunction m(t) ∈ L1 such thatf(t, x) ≤ |m(t)|.Moreover f is nondecreasing with respect to both variables separately;(iii) g(t, x) = g : I × R+ → R+ satisfy Carathéodory conditions and there exists afunction a1(t) ∈ L1 and a constant b such thatg(t, x) ≤ |a1(t)| + b | x | ∀ (t, x) ∈ I × R+.Moreover, g is nondecreasing with respect to both variables separately;(iv) φ : I → I is increasing, absolutely continuous on I and there exists a constantM1 > 0 such that φ′(t) ≥ M1 on I ;(v) k : I × I → R+ is measurable in both variables and the operator K defined by(Ky)(t) =∫ t0k(t, s) y(s) ds, t ∈ [0, 1]maps nondecreasing positive function x ∈ L1 into function of the same type and∫ 1s| k(t, s) | m(t) dt < M, s ∈ [0, 1].Now, we are ready to prove the existence theorem.Theorem 3.1 Let the assumptions (i)-(v) be satisfied. If b M < M1, then thequadratic functional integral equation (1) has at least one positive nondecreasing solutionx ∈ L1(I).Proof. Let the operator H be defined by the formula(Hx)(t) = a(t) + f(t, x(t))∫ t0k (t, s) g(s, x(φ(s))) ds.Let x ∈ L1, then by assumptions (i),(ii) and (iii) we find that| (Hx)(t) | ≤ | a(t) | + | f(t, x(t)) |∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds≤ | a(t) | + m(t)∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds.This implies that|| Hx || =∫ 10| (Hx)(t) | dtEJQTDE, 2009 No. 7, p. 4≤ || a || +∫ 10m(t)∫ t0k(t, s)( |a1(s)| + b | x(φ(s)) | ) ds dt= || a || +∫ 10∫ 1sk(t, s) m(t) dt ( |a1(s)| + b | x(φ(s)) | ) ds≤ || a || + M∫ 10( |a1(s)| + b | x(φ(s)) | ) ds≤ || a || + M∫ 10|a1(s)| ds + M b∫ 10| x(φ(s)) | ds≤ || a || + M || a1 || + M b M1−1∫ φ(1)φ(0)| x(u) | du≤ || a || + M || a1 || + M b M1−1∫ 10| x(u) | duwhich gives|| Hx || ≤ || a || + M || a1 || + M b M1−1 || x || (7)and proves that Hx ∈ L1.Moreover the estimate (7) shows that the operator H maps Br into it self, wherer = [ || a || + M || a1 || ] [ 1 − M b M1−1 ]−1.Let Qr ⊂ Br consisting of all functions positive and nondecreasing on I.Clearly Qr is nonempty, bounded, closed and convex ([3]). Now Qr is a bounded subsetof L1 consisting of all functions positive and nondecreasing on I, then Theorem 2.2shows that Qr is compact in measure [3].Now, we shall show that the operator H transforms positive and nondecreasing functioninto function of the same type.First if x ∈ Qr is an arbitrary function, then x(φ(t)) is positive and nondecreasingfunction on I. Therefore∫ t0k(t, s) g(s, x(φ(s))) ds, t ∈ (0, 1)is positive and nondecreasing functions on I.Thus the operator H maps Qr into itself. By using assumption (ii),(iii) and (v), theoperator H is continuous on Qr (see [13] and [4]).Let X be a nonempty subset of Qr. Fix ε > 0 and take a measurable subset D ⊂ Isuch that meas. D ≤ ε. Then for any x ∈ X, we get|| Hx ||D =∫D| (Hx)(t) | dt ≤∫D| a(t) | dt+∫Dm(t)∫ t0k(t, s)( a1(s) + b | x(φ(s)) | ) ds dt≤ || a ||D + M || a1 ||D + b M1−1 M∫φ(D)| x(u) | ds.EJQTDE, 2009 No. 7, p. 5Butlimε → 0{ sup{∫D| a(t) | dt : D ⊂ I, meas.D < ε } } = 0.andlimε → 0{ sup{∫D| a1(t) | dt : D ⊂ I, meas.D < ε } } = 0.Then we obtainβ(Hx(t)) ≤ M b M1−1 β(x(t))andβ(HX) ≤ M b M1−1 β(X).which by Theorem 2.3 implies thatχ (HX) ≤ M b M1−1 χ(X).Since M b M1−1 < 1, it follows, from Theorems 2.3 and 2.4, that H is contraction andhas at least one fixed point in Qr which proves that the nonlinear quadratic functionalintegral equation (1) has at least one positive nondecreasing solution x ∈ L1[0, 1].4 Fractional order quadratic functional integral equationConsider now the quadratic functional integral equation (2). Now we have the followingtheorem.Theorem 4.1 Let 1Γ(β)∫ 1s (t − s)β−1 m(t) dt < M. Then under the assumptions ofTheorem 3.1. The nonlinear quadratic functional integral equation of fractional orderx(t) = a(t) + f(t, x(t))∫ t0(t − s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1], β ∈ (0, 1),has at least one positive nondecreasing solution x ∈ L1.Proof. Let k(t, s) = 1Γ(β)(t − s)β−1. Then from the properties of the fractional orderintegral operator (subsection 2.1) we deduce that the operatorKx(t) =∫ t0(t − s)β−1Γ(β)x(s) ds, β ∈ (0, 1)satisfies the assumption (v) of Theorem 3.1 and the results follows from the results of The-orem 3.1.Corollary 4.1 Under the assumptions of Theorem 3.1, with a(t) = 0 and f(t, x(t)) = 1,the nonlinear functional equation (4) has at least one positive nondecreasing solutionx ∈ L1.Proof. Putting a(t) = 0 and f(t, x) = 1. Letting β → 0 the quadratic functional in-tegral equation (1) will be the functional equation (4) and the results follows from Theorem3.1.EJQTDE, 2009 No. 7, p. 65 Fractional order functional differential equationsFor the initial value problem of the nonlinear fractional order differential equation (3) wehave the following theorem.Theorem 5.1 Under the assumptions of Theorem 3.1, with a(t) = a tβ−1Γ(β) andf(t, x(t) = 1, the Cauchy type problem (3) has at least one positive nondecreasingsolution x ∈ L1.Proof. Integrating (3) we obtain the integral equationx(t) = atβ−1Γ(β)+∫ t0(t − s)β−1Γ(β)g(s, x(φ(s))) ds t ∈ [0, 1] (8)which by Theorem 3.1 has the desired solution.Operating by Dβ on (8) we obtain the problem (3). So the equivalence between problem(3) and integral equation (8) is proved and then the results follow from Theorem 3.1.References[1] J. Appel and E. De Pascale, Su alcuni parameteri connssi con la misura dinoncompat-tezza di Hausdorff in spazi di functioni misurabili, Boll. Un. Mat. Ital. (6), 3-B (1984),497-515.[2] J. Banas and K. Goebel, Measure of noncompactness in Banach spaces, Lect. Notes inpure and applied Math. 60, M. Dekker, New York and Basel 1980.[3] J. Banas, On the superposition operator and integrable solutions of some functionalequations, Nonlin. Analysis T.M.A. Vol. 12(1988), 777-784.[4] J. Banas, Integrable solutions of Hammerstein and Urysohn integral equations J. Aus-tral Math. Soc. (series A) 46 (1989), 51-68.[5] J. Banas, M. Lecko, W. G. El-Sayed, Existence theorems of some quadratic integralequation, J.Math. Anal. Appl. 227 (1998), 276 - 279.[6] J. Banas, A. Martininon, Monotonic solutions of a quadratic integral equation ofVolterra type, Comput. Math. Appl. 47 (2004), 271 - 279.[7] J. Banas, J. R. Martin and K. Sadarangani, On the solution of a quadratic integralequation of Hammerstein tupe, Mathematical and Computer Modelling, 43 (2006),97-104.[8] J. Banas, B. Rzepka, Monotonic solutions of a quadratic integral equations of fractionalorder J.Math. Anal. Appl. 332 (2007), 1371 -1378.[9] A. M. A El-Sayed, W. G. El-Sayed and O. L Moustafa On some fractional functionalequations Pure Math. and Appl. Vol. 6,No.4 (1995), 321-332.EJQTDE, 2009 No. 7, p. 7[10] F. S. De Blasi, On a property of the unit sphere in Banach space, Bull. Math. Soc.Sci. Math. R.S. Roum.(N.S.) 21 (1977), 259-262.[11] N. Dunford and J. Schwartz, Linear operators I, Int. Publ., Leyden (1963)[12] I. Podlubny, Fractional differential equations, Academic Press, (1999).[13] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnoselskii, S.G.Mikhlin, L.S. Rakovshchik andV.J. Stecenko Integral equations Noordhoff, Leyden (1975).(Received June 18, 2008)EJQTDE, 2009 No. 7, p. 8

Monotonic solutions of functional integral and differential equations of fractional order

Monotonic solutions of functional integral and differential equations of fractional order

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