ON MIXED NONLINEAR INTEGRAL EQUATIONS OF VOLTERRA-FREDHOLM TYPE WITH MODIFIED ARGUMENT

Transcription

1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LIV, Number 1, Mrch 29 ON MIXED NONLINEAR INTEGRAL EQUATIONS OF VOLTERRA-FREDHOLM TYPE WITH MODIFIED ARGUMENT Abstrct. In the present pper we consider the following mixed Volterr- Fredholm nonliner integrl eqution with modified rgument: ut, x) = g t, x, ut, x) ) b H t, x, s, us, x)) ds K t, x, s, y, us, y), u ϕ 1s, y), ϕ 2s, y) )) dyds For this eqution, we will study: the existence nd the uniqueness of the solution, the dt dependence of the solution nd the differentibility of the solution with respect to prmeters. 1. Introduction Let X, X ) be Bnch spce. In this pper we consider the following nonliner integrl eqution of Volterr- Fredholm type: ut, x) = g t, x, ut, x) ) b H t, x, s, us, x)) ds K t, x, s, y, us, y), u ϕ 1 s, y), ϕ 2 s, y) )) dyds 1) for ll t, x) [, T ] [, b] := D; u CD, R m ), where b > > nd T >. Volterr-Fredholm VF on short) integrl equtions often rise from the mthemticl modelling of the spreding, in spce nd time, of some contgious diseses, in the theory of nonliner prbolic boundry vlue problem nd in mny physicl nd Received by the editors: Mthemtics Subject Clssifiction. 45G1, 47H1. Key words nd phrses. Volterr-Fredholm integrl eqution, fixed point, Picrd opertor, dt dependence, differentibility of the solution. 29

2 biologicl models. Most results for VF eqution estblish numericl pproximtion of the solutions; e.g. [8], [9], [22], [2], [11], [3], [7]. In [21] H. R. Thieme considered model for the sptil spred of n epidemic consisting of nonliner integrl eqution of Volterr-Fredholm type which hs n unique solution. The uthor showed tht this solution hs temporlly symptotic limit which describes the finl stte of the epidemic nd is the miniml solution of nother nonliner integrl eqution. In [4] O. Diekmnn described, derived nd nlysed model of sptio-temporl development of n epidemic. The model considered leds see [13]) to the following nonliner integrl eqution of Volterr-Fredholm type: ut, x) = gt, x) gut τ, ξ))s ξ)aτ, x, ξ)dξdτ 2) Ω for ll t, x) [, ] Ω, where Ω is bounded domin in R n. In [13] B. G. Pchptte considered the integrl eqution ut, x) = gt, x) Ω gt, x, s, y, us, y))dyds 3) for ll t, x) [, T ] Ω = D, where Ω is bounded domin in R n. Using Contrction Principle, the uthor proved tht, under pproprite ssumptions, 3) hs unique solution in subset S of CD, R n ). The result ws then pplied to show the existence nd uniqueness of solutions to certin nonliner prbolic differentil equtions nd mixed Volterr-Fredholm integrl equtions occurring in specific physicl nd biologicl problems e.g. relible tretment of the Diekmnn s model mentioned bove is given). In [1], D. Mngeron nd L. E. Krivo sein obtined existence, uniqueness nd stbility conditions for the solutions of clss of boundry problems for liner nd nonliner het eqution with dely. Under certin conditions, this problem is equivlent with the following nonliner VF eqution: 3 ut, x) = nt, x) [Gx, ξ, t α)g ξ, α, uξ, α), u ξ, α r 1 α) ))

3 where VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT α nt, x) = Kξ, α, s, y)g s, y, us, y), u s, y r 2 s) )) ] dyds dξdα [ 2 i=1 πi e )2t sin πix ] sinπiξ ϕ ξ) dξ Applying Contrction Principle, n existence nd uniqueness theorem is obtined. In [14], the following problem is considered: u t t, x) = 2 u xx t, x) g ut, x), ux, [t]) ) ux, ) = ϕx) t R where [t] mens the integer prt of t. Using integrtion by prts twice for the eqution bove, in pproprite conditions, the problem is equivlent with VF eqution nd the successive pproximtion method is pplied. The purpose of the present pper is to give results concerning the following problems relted to eqution 1): the existence nd the uniqueness of the solution, the dt dependence of the solution nd the differentibility of the solution with respect to prmeters. Becuse the tool used in the present pper is the Picrd opertors theory, for the convenient of the reder, we present some bsic notions nd results concerning this importnt clss of opertors. 2. Picrd opertors Let X, d) be metric spce nd A : X X n opertor. In this pper we will use the following nottions: F A := {x X : Ax) = x}; A := 1 X, A n1 := A A n for ll n N. Definition 2.1. Rus [15]) The opertor A is sid to be: i) wekly Picrd opertor wpo) if A n x ) x for ny x X nd the limit x is fixed point of A, which my depend on x. 31

4 ii) Picrd opertor Po) if F A = {x } nd A n x ) x for ny x X. For wekly Picrd opertor A, the opertor A is defined s follows: A : X X, A x) := lim n An x). Notice tht A X) = F A. If A is Picrd opertor, then A x) = x for ll x X, where x is the unique fixed point of A. Exmple 2.1. Any α-contrction on complete metric spce X, d) is Picrd opertor. the solution: The following bstrct theorem is needed in the study of dt dependence of Theorem 2.1. Rus [17]) Let X, d) complete metric spce nd A, B : X X two opertors. Assume tht: i) there exists α [, 1[ such tht A is α-contrction; let F A = {x A } ii) F B ; let x B F B; iii) there exists η > such tht d Ax), Bx)) η for ll x X. Then dx A, x B) η 1 α. In order to study the differentibility of the solution with respect to prmeter, we need the following theorem, due to I. A. Rus: Theorem 2.2. Fiber Contrction Principle, Rus [16]) Let X, d), Y, ρ) be two metric spces nd B : X X, C : X Y Y two opertors such tht: i) Y, ρ) is complete; ii) B is Picrd opertor, F B = {x }; iii) C, y) : X Y is continuous for ll y Y ; iv) there exists α ], 1[ such tht the opertor Cx, ) : Y Y is α-contrction for ll x X; let y be the unique fixed point of Cx, ). Then A : X Y X Y, Ax, y) := Bx), Cx, y)) 32

5 VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT is Picrd opertor nd F A = {x, y )}. For Picrd opertors theory pplied in the study of differentil or integrl equtions see [19], [18], [17], [12], [2], [6], [5]. 3. Existence nd uniqueness theorem Consider the eqution 1). Theorem 3.1. If the following conditions re stisfied: c1) g CD X, X), H CD [, T ] X, X) K CD 2 X 2, X), ϕ 1 CD, [, T ]) nd ϕ 2 CD, [, b]); c2) there exists L g > such tht: gt, x, u) gt, x, v) X L g u v X 4) for ll t, x) D nd u, v X c3) there exists L H > such tht: Ht, x, s, u) Ht, x, s, v) X L H u v X 5) for ll t, x, s) D [, T ] nd u, v X c4) there exists L K > such tht: Kt, x, s, y, u, u) Kt, x, s, y, v, v) X L K u v X u v X ) 6) for ll t, x, s, y) D 2 nd u, v, u, v X c5) L g < 1 c6) there exists τ > such tht: α := L g 1 τ L H b { L K mx τ b Then 1) hs n unique solution u CD, X). } e τ[ϕ1s,y) t] dyds : t [, T ] L K < 1 7) norm Proof. Let the spce CD, X) be endowed with Bielecki-Chebysev suitble u BC := sup{ ut, x) X e τt : t [, T ], x [, b]}, τ > 8) 33

6 Consider the opertor A : CD, X) CD, X) defined by: Au)t, x) := gt, x) b for ll u CD), for ll t, x) D. For ny u, v CD, X) we hve see [1]): K t, x, s, y, u ϕ 1 s, y), ϕ 2 s, y) )) dyds 9) Au)t, x) Av)t, x) X b L g ut, x) vt, x) X L H us, x) vs, x) X ds L K u v τ BC e τt { b } L K mx e τ[ϕ1s,y) t] dyds : t [, T ] u v BC e τt so: Au) Av) BC α u v BC. From c6) there exists τ > such tht A : CD, X) CD, X) is α-contrction nd, by Contrction Principle, A is Picrd opertor, i.e. the eqution hs unique solution in CD, X). Remrk 3.1. Condition c6) from Theorem 3.1 cn be replced by the next simpler condition: c7) ϕ 1 t, x) t for ll t, x) D In this cse the opertor A given by 9) is α-contrction, with α = L g L H 2b )L K τ < 1 1) for suitble chosen τ. 4. Dt dependence of the solution In order to prove the dependence of the solution on dt, let us consider two mixed VF equtions: ) t ut, x) = g i t, x, ut, x) H i t, x, s, us, x)) ds 34 b K i t, x, s, y, us, y), u ϕ 1 s, y), ϕ 2 s, y) )) dyds 11)

7 VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT for ll u CD, X) nd t, x) D, with g i CD X, X), H i CD [, T ] X, X) nd K i CD 2 X 2, X) for i = 1, 2. Theorem 4.1. Assume tht the first eqution from 11) stisfies conditions c1)- c5) nd c7); let u be its unique solution. Assume tht the second eqution from 11) hs t lest one solution; let v be such solution. If there exist η 1, η 2, η 3 > such tht: g 1 t, x, u) g 2 t, x, u) X η 1 H 1 t, x, s, u) H 2 t, x, s, u) X η 2 for ll t, x, u) D X for ll t, x, s, u) D [, T ] X K 1 t, x, s, y, u) K 2 t, x, s, y, u) X η 3 for ll t, x, s, y, u) D 2 X Then: u v BC η 1 T η 2 T b )η 3 1 α where α = L g L H 2b )L K < 1 for suitble chosen τ. τ Proof. Consider the opertors A 1, A 2 : CD, X) CD, X) given by: ) t A i u)t, x) := g i t, x, ut, x) H i t, x, s, us, x)) ds b for ll u CD) nd t, x) D, i = 1, 2. For ny u CD) we hve: K i t, x, s, y, us, y), u ϕ 1 s, y), ϕ 2 s, y) )) dyds A 1 u)t, x) A 2 u)t, x) X η 1 T η 2 T b )η 3 for ll t, x) D Applying sup t,x) D, we obtin: A 1 u) A 2 u) C η 1 T η 2 T b )η 3 where C is Chebysev norm: u C := sup{ ut, x) X : t, x) D}, for ll u CD, X) But BC C, so: A 1 u) A 2 u) BC η 1 T η 2 T b )η 3 12) 35

8 Consider the opertors A 1 nd A 2 defined bove, on the spce CD, X), BC ). By Theorem 3.1, A 1 is α-contrction for suitble chosen τ, so F A1 = {u }. Tking ccount of 12), we re in the conditions of Theorem 2.1 nd the conclusion follows. 5. Differentibility of the solution with respect to prmeters In order to study the differentibility of the solution with respect to prmeters nd b, let us consider the sme eqution 1): ut, x) = g t, x, ut, x) ) b H t, x, s, us, x)) ds K t, x, s, y, us, y), u ϕ 1 s, y), ϕ 2 s, y) )) dyds for ll t [, T ], for ll x [α, β], where < α < < b < β. Theorem 5.1. Assume tht: i) g C[, T ] [α, β] R), H C[, T ] [α, β] [, T ] R), K C[, T ] [α, β] [, T ] [α, β] R 2 ), ϕ 1 C[, T ] [α, β], [, T ]) nd ϕ 2 C[, T ] [α, β], [α, β]); ii) gt, x, ) C 1 R) for ll t, x) [, T ] [α, β] nd there exists M g > such tht: gt, x, u) M g 13) for ll t, x, u) [, T ] [α, β] R; iii) Ht, x, s, ) C 1 R) for ll t, x, s) [, T ] [α, β] [, T ] nd there exists M H > such tht: Ht, x, s, u) M H 14) for ll t, x, s, u) [, T ] [α, β] [, T ] R; iv) Kt, x, s, y,, ) C 1 R 2 ) for ll t, x, s, y) [, T ] [α, β] [, T ] [α, β] nd there exists M K > such tht: Kt, x, s, y, u, u) M K nd for ll t, x, s, y, u, u) [, T ] [α, β] [, T ] [α, β] R 2 ; v) M g < 1; vi) ϕ 1 t, x) t 36 for ll t, x) [, T ] [α, β]. Kt, x, s, y, u, u) M K 15)

9 VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT Then: ) for ll < b [α, β], the eqution 1) hs unique solution u,,, b) C[, T ] [α, β]); b) for ll u C[, T ] [α, β]), the sequence u n ) n defined by: u n t, x,, b) = gt, x, u n 1 t, x,, b)) b H t, x, s, u n 1 s, x)) ds K t, x, s, y, u n 1 s, y,, b), u n 1 ϕ1 s, y), ϕ 2 s, y),, b )) dyds converges uniformly to u on [, T ] [α, β] [α, β] [α, β]; c) u C[, T ] [α, β] [α, β] [α, β]); d) u t, x,, ) C 1 [α, β] [α, β]), for ll t, x) [, T ] [α, β]. by: Proof. Let X := C[, T ] [α, β] [, T ] [α, β]) nd B : X X defined Bu)t, x,, b) := gt, x, ut, x,, b)) b H t, x, s, us, x)) ds K t, x, s, y, us, y,, b), u ϕ 1 s, y), ϕ 2 s, y),, b )) dyds. The boundedness conditions 13) nd 15) implies tht f nd K re Lipschitz, with Lipschitz constnts M g nd M K. B stisfies c1)-c5) nd c7), so ), b) nd c) result. Let u CX) be the unique fixed point of B. Obviously we hve: u t, x,, b) = gt, x, u t, x,, b)) b Let us prove tht t, x,, b) 1. Assume tht t, x,, b) H t, x, s, u s, x,, b)) ds K t, x, s, y, u s, y,, b ), u ϕ 1 s, y), ϕ 2 s, y),, b )) dyds. 16) nd t, x,, b) b exist nd re continuous. exists. Differentite 16) with respect to we hve: g ) t, x, u t, x,, b) t, x,, b) = H t, x, s, u s, x,, b) ) t, x,, b) s, x,, b) ds 37

10 b Kt, x, s,, u s,,, b), u ϕ 1 s, ), ϕ 2 s, ),, b ) )ds K t, x, s, y, u s, y,, b ), u ϕ 1 s, y), ϕ 2 s, y),, b )) b K t, x, s, y, u s, y,, b ), u ϕ 1 s, y), ϕ 2 s, y),, b )) ) ϕ 1 s, y), ϕ 2 s, y),, b) dyds. s, y,, b) dyds This lst reltionship suggests us to consider the opertor C : X X X defined by: g ) t, x, ut, x,, b) Cu, v)t, x,, b) := vt, x,, b) H t, x, s, us, x,, b) ) vs, x,, b)ds K t, x, s,, us,,, b), u ϕ 1 s, ), ϕ 2 s, ),, b )) ds b b K t, x, s, y, us, y,, b), u ϕ 1 s, y), ϕ 2 s, y),, b )) vs, y,, b)dyds K t, x, s, y, us, y,, b), u ϕ 1 s, y), ϕ 2 s, y),, b )) v ϕ 1 s, y), ϕ 2 s, y),, b ) dyds. From the hypotheses, the opertor Cu, ) is contrction, for ny u X. Let v be the unique fixed point of Cu, ). Now consider the opertor A : X X X X defined by Au, v)t, x,, b) := Bu)t, x,, b), Cu, v)t, x,, b)), which is in the hypotheses of Theorem 2.2. So A is Picrd opertor nd F A = {u, v )}. Consider the sequences u n ) n nd v n ) n defined by: 38 u n t, x,, b) := Bu n 1 t, x,, b))

11 VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT b for ll n 1 nd = gt, x, u n 1 t, x,, b)) H t, x, s, u n 1 s, x)) ds K t, x, s, y, u n 1 s, y,, b), u n 1 ϕ1 s, y), ϕ 2 s, y),, b )) dyds v n t, x,, b) := Cu n 1 t, x,, b), v n 1 t, x,, b)) g ) t, x, u n 1 t, x,, b) = v n 1 t, x,, b) H t, x, s, u n 1 s, x,, b) ) v n 1 s, x,, b)ds K t, x, s,, u n 1 s,,, b), u n 1 ϕ1 s, ), ϕ 2 s, ),, b )) ds b b K t, x, s, y, u n 1 s, y,, b), u n 1 ϕ1 s, y), ϕ 2 s, y),, b )) v n 1 s, y,, b)dyds K t, x, s, y, u n 1 s, y,, b), u n 1 ϕ1 s, y), ϕ 2 s, y),, b )) v n 1 ϕ1 s, y), ϕ 2 s, y),, b ) dyds, for ll n 1. Obviously, we hve: u n u for n nd v n v for n uniformly with respect to t, x,, b) [, T ] [α, β] [α, β] [α, β], for ny u, v C[, T ] [α, β] [α, β] [α, β]). Choosing u = v := we hve v 1 = 1. By induction we cn prove tht v n = n n v for n for ny positive integer n, so From Weierstrss theorem, it follows tht t, x,, b) exists nd = v t, x,, b). 39

12 2. The differentibility with respect to b cn be proved in the sme wy. Acknowledgments. The uthor would like to express her grtitude to Professor Ion A. Rus for some very importnt suggestions. References [1] Bcoţiu, C., Volterr-Fredholm Nonliner Integrl Equtions Vi Picrd Opertors Theory, Mthemtic, to pper. [2] Brunner, H., nd Messin, E., Time-stepping methods for Volterr-Fredholm integrl equtions, Rend. Mt., 2323), [3] Crdone, A., Messin, E., Russo, E., A fst itertive method for discretized Volterr- Fredholm integrl equtions, J. Comp. nd Appl. Mth., 18926), [4] Diekmnn, O., Thresholds nd trveling wves for the geogrphicl spred of infection, J. Mth. Biol., 61978), [5] Dobriţoiu, M., An Integrl Eqution with Modified Argument, Studi Univ. Bbeş-Bolyi Mthemtic), ), 3, [6] Dobriţoiu, M., Rus, I.A., Şerbn, M.A., An Integrl Eqution Arising from Infectious Diseses, Vi Picrd Opertors, Studi Univ. Bbeş-Bolyi Mthemtic), ), 3, [7] Hci, L., On integrl equtions in spce-time, Demonstr. Mth., ), 4, [8] Hdizdeh, M., Posteriori Error Estimtes for the Nonliner Volterr-Fredholm Integrl Equtions, Comp. nd Mth. Appl., 4523), [9] Mleknejd, K., Hdizdeh, M., A New Computtionl Method for Volterr-Fredholm Integrl Equtions, Comp. nd Mth. Appl., ), 1-8. [1] Mngeron, D., Krivo sein, L.E., Sistemi policlorici rimnenz ed rgomento ritrdto; problemi l contorno per le equzioni integro-differenzili con opertore clorico ed rgomento ritrdto, Rend. Sem. Mt., Univ. Pdov, 1965), [11] Mleknejd, K., Fdei Ymi, M.R., A computtionl method for system of Volterr- Fredholm integrl equtions, Appl. Mth. nd Comput., 18326), [12] Mureşn, V., Existence, uniqueness nd dt dependence for the solution of Fredholm integrl eqution with liner modifiction of the rgument, Act Sci. Mth. Szeged), 6822), [13] Pchptte, B.G., On mixed Volterr-Fredholm type integrl equtions, Indin J. Pure Appl. Mth., ), [14] Poorkrimi, H., Wiener, J., Bounded solutions of nonliner prbolic equtions with time dely, Electron. J. Diff. Eq., 21999),

13 VOLTERRA-FREDHOLM EQUATIONS WITH MODIFIED ARGUMENT [15] Rus, I.A., Generlized Contrctions nd Applictions, Cluj University Press, Cluj- Npoc, 21. [16] Rus, I.A., A dely integrl eqution from biomthemtics, Preprint Nr.3, 1989, [17] Rus, I.A., Picrd opertors nd pplictions, Scientie Mthemtice Jponice, 58,123), [18] Rus, I.A., Wekly Picrd opertors nd pplictions, Seminr on Fixed Point Theory Cluj Npoc, 221), [19] Rus, I.A., Fiber Picrd opertors nd pplictions, Studi Univ. Bbeş-Bolyi Mthemtic), ), [2] Tămăşn, A., Differentibility with respect to lg for nonliner pntogrph eqution, Pure Mth. Appl., 91998), [21] Thieme, H.R., A Model for the Sptil Spred of n Epidemic, J. Mth. Biol., 41977), [22] Wzwz, A.M., A relible tretment for mixed Volterr-Fredholm integrl equtions, Appl. Mth. nd Comput., 12722), Smuel Brssi High School, Bd 21 Decembrie 1989 No 9, 415, Cluj-Npoc, Romni E-mil ddress: Cludi.Bcotiu@clujnpoc.ro 41