JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 222, 276᎐285Ž. 1998 ARTICLE NO. AY985941
Existence Theorems for Some Quadratic Integral Equations
Jozef´´ Banas* and Millenia Lecko†
Department of Mathematics, Technical Uni¨ersity of Rzeszow,´´ 35-959 Rzeszow, View metadata, citation and similar papers at core.ac.ukW. Pola 2, Poland brought to you by CORE
provided by Elsevier - Publisher Connector and
Wagdy Gomaa El-Sayed
Department of Mathematics, Uni¨ersity of Alexandria, Alexandria, Egypt
Submitted by W. L. Wendland
Received May 8, 1997
Using the theory of measures of noncompactness, we prove a few existence theorems for some quadratic integral equations. The class of quadratic integral equations considered below contains as a special case numerous integral equations encountered in the theories of radiative transfer and neutron transport, and in the kinetic theory of gases. In particular, the well-known Chandrasekhar integral equation also belongs to this class. ᮊ 1998 Academic Press
1. INTRODUCTION
In the theory of radiative transfer, one can encounter the following quadratic integral equation:
1 Ž.s xtŽ.s1qtx Ž. tH x Ž. s ds,1Ž. 0 tqs
where is a given continuous function defined on the intervalwx 0, 1 and x is an unknown function. This equation also plays an important role in the
*E-mail: [email protected]. † E-mail: [email protected].
276
0022-247Xr98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. EXISTENCE THEOREMS FOR INTEGRAL EQUATIONS 277 theory of neutron transport and in the kinetic theory of gasesŽ cf.w 5, 7, 8, 11x and references therein. . Equations of typeŽ. 1 and some of their generalizations were considered in several paperswx 2, 3, 6, 8, 9, 12, 13 . In those papers the authors proved thatŽ.Ž 1 or more general equations . is solvable in some classes of Banach spaces. Some other properties, such as uniqueness, location of solutions, and convergence of successive approximations, were also studied in the mentioned papers. Recently I. Argyroswx 3 investigated a class of quadratic equations of type Ž.1 with a nonlinear perturbation. Using the theory of measures of non- compactness, he proved that those equations have solutions in some Banach function algebras. Unfortunately, the investigations of the paperwx 3 contain several mis- prints and mistakes and are unclear in a few places. Moreover, some of the assumptions made inwx 3 seem to be a bit artificial. The aim of this paper is to improve, clarify, and generalize some results obtained in the mentioned paperwx 3 . We will also apply the theory of measures of noncompactness, but we restrict ourselves to the space of continuous functions on an interval. This is caused by the fact that this space seems to be sufficiently general for the study of the equations in question. Nevertheless, we review the possibility of studying quadratic integral equations with a perturbation in some Banach function algebras. The results obtained below generalize several obtained in the papers quoted above.
2. MEASURES OF NONCOMPACTNESS AND FIXED-POINT THEOREMS
Let Ž E, 55и .be a given Banach space and let ᑧ E denote the family of all nonempty and bounded subsets of E. By the symbol BxŽ.,r we will denote the closed ball centered at x and with radius r.
The function defined on the family ᑧ E by
Ž.XsinfÄ4 ) 0: X admits a finite -net in E is called the Hausdorff measure of noncompactness. In the literature there exist several other definitions of the notion of a measure of noncompact- nessŽ see, e.g.,wx 1, 4 , for example. . Nevertheless, the Hausdorff measure of noncompactness seems to be the most important and convenient in appli- cations. This is caused by the fact that in some Banach spaces it is possible to express this measure with the help of handy formulas. 278 BANAS´, EL-SAYED, AND LECKO
For example, let C s Cwx0, 1 be the Banach space consisting of all real continuous functions defined on the intervalwx 0, 1 and endowed with the maximum norm.
Then it may be shownwx 4 that for any X g ᑧ C the following formula holds:
1 Ž.X s 2 0 Ž.X , where Ž.X limÄÄ sup Ž.x, : x X 44 and Ž.x, denotes the 0 s ª 0 g modulus of continuity of the function x, i.e.,
Ž.x,ssupÄ4< DEFINITION. Let M be a nonempty subset of a Banach space E, and let T: M ª E be a continuous operator that transforms bounded sets onto bounded ones. We will say that T satisfies the Darbo conditionŽ with a constant k G 0. if for any bounded subset X of M, we have Ž.TXŽ.Fk Ž.X. In the case k - 1, the operator T is said to be a contraction Žwith respect to .. THEOREM 110.wx Let Q be a nonempty bounded closed con¨ex subset of E and let T: Q ª Q be a contraction with respect to . Then T has at least one fixed point in the set Q. Now we are going to prove a theorem that allows us to indicate a large class of operators satisfying the Darbo condition in the case of Banach algebras. So, let us assume that E is a Banach algebra with the norm 55и and the zero element . THEOREM 2. Let M be a nonempty bounded subset of a Banach algebra E. Assume that T: M ª E satisfies the Darbo condition with a constant k and P: M ª E is completely continuous operatorŽ i.e., it is continuous and the set PŽ. M is relati¨ely compact .. Then the operator S s PT satisfies the Darbo condition with the constant kb, where b s supw5 PŽ. x 5: x g M x- ϱ. Proof. Fix an arbitrary nonempty subset X of M. Let Ž.X s r. Choose arbitrarily ) 0. Then there exists a finite collection of points EXISTENCE THEOREMS FOR INTEGRAL EQUATIONS 279 x12, x ,..., xngE such that n X Bx,r . ;DŽ.iq is1 Furthermore, let us denote Y s PXŽ.иTX Ž.. Since PX Ž.is compact, there exist points z12, z ,..., zmgE such that m PX Bz,. Ž.;D Ži . is1 In view of the assumption ŽŽTX ..Fk ŽX .skr, we can find a finite set Ä4u12, u ,...,uk;E such that k TX Bu,kr . Ž.;DŽ.jq js1 Then we have mk YPXTX Bz, Bu,kr sŽ.Ž.; DD Žij . Ž.q ž/i1 j1 s ž/s mk Bz,Bu,kr . ;DD Ž.ijŽ.q is1js1 Now, for fixed i g Ä4Ä41, 2, . . . , m and j g 1, 2, . . . , k let us consider the set Aijs BzŽ. i, иBuŽ.j,kr q . Obviously, zuijgA ij. Choose arbitrarily ¨ g Aij l Y. Then ¨ s xy, where x g BzŽ.i, l PXŽ.and y g Bu Žj,kr q .l TX Ž.. Hence we get 55555555¨yzuijs xy y zuijF xy y xujq xu jy zu ij F555xиyyujj 5q 5u 55иxyz i 5F 55xkrŽ.qq 5uj 5и. Ž.2 On the other hand, by virtue of the assumption, we have 55xFb.3Ž. 280 BANAS´, EL-SAYED, AND LECKO Moreover, without loss of generality, we can assume that Äuj: j s 1, 2, . . . , k4 ; BTŽŽ X .,kr q ., where the symbol BŽ.Z, t stands for the ‘‘ball’’ centered at the set Z and with radius t. Hence we infer that the constant L, defined as LssupÄ455w : w g BTŽ.Ž. X ,kr q , is finite. Now, taking into account the above assertion andŽ. 2 and Ž. 3 , we infer that 55xy y zuijFbkrŽ.q qL. Thus, keeping in mind the arbitrariness of , we deduce that mk Y PXTX A Y Ž.sŽ. Ž.Ž.F DD ij l ž/is1js1 smaxÄ4 Ž.Aij l Y : i s 1,2,...,m; js1,2,...,k Fbkr s bk Ž.X , which completes the proof. Observe that the above theorem generalizes Theorem 1 fromwx 3 . More- over, based on this theorem, we conclude the following fixed-point theorem. THEOREM 3. Let Q be a nonempty bounded closed con¨ex subset of a Banach algebra E and S s PT: Q ª Q, where P: Q ª E is a completely continuous operator and T: Q ª E satisfies the Darbo condition with a constant k. If bk - 1 Žwhere b s supÄ5 PŽ. x 5: x g Q4., then S has a fixed point in the set Q. 3. EXISTENCE THEOREMS In this section we shall deal with the quadratic integral equation of the form 1 xtŽ.s1q ŽTx .Ž. tH k Ž t, s . Ž.sxsds Ž. ,4Ž. 0 where t g I s wx0, 1 and T is an operator to be described below. Notice that Eq.Ž. 1 is a particular case of Eq. Ž.Ž 4 cf. Example 1 . . EXISTENCE THEOREMS FOR INTEGRAL EQUATIONS 281 In what follows we shall discuss Eq.Ž. 4 , assuming that the following hypotheses are satisfied: Ž.ik:I=I_ÄŽ.0, 0 4ª ޒ is continuous, and for each t g I there exists the integral 1 H< Ž.ii g C s CIŽ.. Ž.iii The operator T: C ª C is continuous and satisfies the Darbo condition with a constant a. Ž. ޒޒŽ ϱ.. iv There exists a bounded function w: I ª qqsw0, with the property wŽ.0 lim qwtŽ. 0 and such that s t ª 0 s 1 H<< 11 1 HH< 1 FwtŽ.qH< 1 qssup H< 5555TxŽ.Fbx for each x g C. Finally, let us denote by Q the quantity 1 Qssup H<5 Since Q F q55 , this quantity is finite. 282 BANAS´, EL-SAYED, AND LECKO Then we can formulate our main existence result. THEOREM 4. Let the assumptionsŽ. i ᎐ Ž¨ . be satisfied and bQ - 1r4. Moreo¨er, assume that either 1Њ a - 2b or 2 2Њ a G 2b and a y b G a Q. Then Eq.4Ž.is sol¨able in the space C s CI Ž .. Proof. For an arbitrary x g C let us denote by Ax the function defined by the right-hand side of Eq.Ž. 4 , i.e., 1 ŽAx .Ž. t s 1 q ŽTx .Ž. tH k Ž t, s . Ž.sxsds Ž. , tgI. 0 In view of assumptionsŽ.Ž ii ᎐ iv . , we infer that Ax g C. Moreover, we have 1 <<<<<<<<ŽAx .Ž. t F 1 q ŽTx .Ž. tH k Ž t, s . Ž.sxsds Ž. 0 2 F1qbQ55 x . Since bQ - 1r4, it is easily seen that A transforms the ball BŽ., r into itself for r01F r F r , where 1r21r2 r01s1yŽ.1y4bQ r2bQ, r s 1 q Ž.1 y 4bQ r2bQ. For further reasons, we assume that r s r0. Next, let us take a nonempty subset X of the ball BŽ., r . Fix x g X. Then, for arbitrarily chosen t12, t g I, we have Ž.Ž.Ž.Ž.Ax t21y Ax t 11 FŽTx .Ž. t22HH k Ž t , s . Ž.sxsds Ž.y ŽTx .Ž. t11 k Ž t , s . Ž.sxsds Ž. 00 11 FŽTx .Ž. t22HH k Ž t , s . Ž.sxsds Ž.y ŽTx .Ž. t21 k Ž t , s . Ž.sxsds Ž. 00 11 qŽTx .Ž. t21HH k Ž t , s . Ž.sxsds Ž.y ŽTx .Ž. t11 k Ž t , s . Ž.sxsds Ž. 00 1 FŽTx .Ž. t22H k Ž t , s .y kt Ž 1,s .<<<< Ž.sxsds Ž. 0 1 q<<ŽTx .Ž. t211y ŽTx .Ž. tH<< k Ž t , s . Ž.sxsds< Ž.< 0 2 Fbx5555<<wtŽ.Ž21yt qTx, <<55t21y tQx . 2 Fb55rwŽ.Ž<< t21yt qrQ Tx, < Hence, in view of assumptionŽ. iv , we get 00Ž.AX F rQ Ž.TX . Consequently, Ž.AX F rQ Ž.TX F rQa Ž.X Ž.cf. Section 2 . Observe that the assumptions of our theorem yield rQa s rQa0 -1, which implies that A is a contraction with respect to on the ball BŽ., r . Thus, applying Theorem 1, we infer that there exists a function x g BŽ., r that is a solution of Eq.Ž. 4 . This completes the proof. Now we provide a few examples illustrating the applicability of Theo- rem 4. EXAMPLE 1. Consider Eq.Ž. 1 , which we write here in the following form: 1 t xtŽ.s1qxt Ž.H Ž.sxsds Ž. . 0 tqs Obviously this equation is a particular case of Eq.Ž. 1 , where kt Ž,s .s trŽ.tqsand Tx Ž.sx. It is easy to check that in this situation the assumptions of Theorem 4 are satisfies for a s 1 and b s 1. Thus case 1Њ of Theorem 4 is fulfilled. Moreover, it may be verified that 1t<<Ž.s Qssup H ds : t g I F 55 ln 2. ½50tqs Thus, to ensure that the inequality bQ s Q - 1r4 will be valid, it is enough to take g CIŽ.such that 55 - Ž.1r4ln2. The above calculations show that a result fromwx 3 is an easy corollary of Theorem 4. EXAMPLE 2. Let us take the following integral equation: 2 t xtŽ. 1 1 xtŽ.s1q H2expŽ.y1rŽt q s . Ž.sxsds Ž..5 Ž. 1q< txŽ. t t Ž.Ž.Tx t s , ktŽ.,ss 2expŽ.y1rŽ.t q s . 1q< Then it is easy to check that the assumptions of Theorem 4 are satisfied for b s 1 and a s 2. Indeed, taking an arbitrary function x g CIŽ.,we have txt22Ž. txt 11 Ž. <<Ž.Ž.Ž.Ž.Tx t21y Tx t sy 1q< F<< Ž.Tx, F q 2 Ž.x, , and consequently, 00Ž.TX F 2 Ž.X , for any bounded subset X of the space C. Now observe that in view of a s 2 G 2b s 2, case 2Њ of Theorem 4 comes into play. Thus the function g C has to be taken in such a way that bQ s Q - 1r4. This implies immediately that the second part of case 2 2Њ, i.e., a y b G a Q, is satisfied. Let us notice that the existence result for Eq.Ž. 5 deduced with the help of Theorem 4 cannot be obtained as a consequence of results given inwx 3 . 4. FINAL REMARKS In this section we summarize the paper with a few comments and remarks. At first let us observe that the existence result contained in Theorem 4 can be obtained with help of the fixed-point principle formulated in Theorem 3. A similar approach has been realized in paperwx 3 , but it seems that the method determined by Theorem 1 is easier and more natural. On the other hand, it is clear that this method is based on the fact that the space CIŽ.is a Banach algebra. Second, let us mention that we can investigate equations with a more complicated form than Eq.Ž. 4 . For example, the method of Theorem 4 Ž or Theorem 3. can be adopted for the study of the Chandrasekhar integral equation with nonlinear perturbation of the form 11t xtŽ.s1qxt Ž.HH Ž.sxsds Ž. q FtŽ.,s,xtŽ.,xs Ž. ds.6 Ž. 00tqs This equation was considered inwx 6 , for example. EXISTENCE THEOREMS FOR INTEGRAL EQUATIONS 285 Based on Example 1, we will assume that g C s CIŽ.and that satisfies the following condition: Ž.␣55F1r4ln2. Furthermore, we assume the following hypotheses: 22 Ž.F:I=ޒªޒis a continuous function such that <<< REFERENCES 1. R. R. Akhmerow, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, ‘‘Measures of Noncompactness and Condensing Operators,’’ Nauka, Novosibirsk, 1986 ŽEnglish translation Operator Theory, Advances and Applications, Vol. 55, Birkhauser,¨ Basel, 1992. . 2. I. K. Argyros, Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc. 32 Ž.1985 , 275᎐292. 3. I. K. Argyros, On a class of quadratic integral equations with perturbations, Funct. Approx. 20 Ž.1992 , 51᎐63. 4. J. Banas´ and K. Goebel, ‘‘Measures of noncompactness in Banach spaces,’’ Lecture Notes in Pure and Appl. Math. 60 Ž.1980 . 5. L. W. Busbridge, ‘‘The Mathematics of Radiative Transfer,’’ Cambridge Univ. Press, Cambridge, England, 1960. 6. B. Cahlon and M. Eskin, Existence theorems for an integral equation of the Chan- drasekhar H-equation with perturbation, J. Math. Anal. Appl. 83 Ž.1981 , 159᎐171. 7. K. M. Case and P. F. Zweifel, ‘‘Linear Transport Theory,’’ Addison-Wesley, Reading, MA 1967. 8. S. Chandrasekhar, ‘‘Radiative Transfer,’’ Oxford Univ. Press, London, 1950. 9. M. Crum, On an integral equation of Chandrasekhar, Quart. J. Math. Oxford Ser.2Ž.18 Ž.1947 , 244᎐252. 10. G. Darbo, Punti uniti in transformazioni a codominio non compatto, Rend. Sem. Mat. Uni¨. Pado¨a 24 Ž.1955 , 84᎐92. 11. C. T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Eq. 4 Ž.1982 , 221᎐237. 12. R. W. Leggett, A new approach to the H-equation of Chandrasekhar, SIAM J. Math. 7 Ž.1976 , 542᎐550. 13. C. A. Stuart, Existence theorems for a class of nonlinear integral equations, Math. Z. 137 Ž.1974 , 49᎐66.