GLASNIK MATEMATIČKI Vol. 37(57)(22), 32 33 DELAY INTEGRO DIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES Tadeusz Jankowski Technical University of Gdańsk, Poland Abstract. This paper contains sufficient conditions under which there exist extremal solutions of initial value problems for delay integro differential equations of mixed type in Banach spaces. We use the monotone iterative technique for proving existence results. Some comparison results are also established.. Let B denote a real Banach space with a norm, and B be a cone in B which defines a partial ordering in the space B by relation x y iff y x B. By θ we denote the zero element in B. The cone B is said to be regular if every nondecreasing and bounded in order sequence in B has a limit, i.e., x x 2 x n y implies x n x as n for some x B (for details, see for example [2, 7, 8, ]). In this paper we consider the following initial value problem (.) x (t) = f(t, x(t), x(α(t)), T x(t), Sx(t)), t J = [, b], b >, x() = x where f C(J B 4, B), α C(J, J), α(t) t, t J, and the operators T and S are defined by T x(t) = β(t) k(t, s)x(s)ds, Sx(t) = b l(t, s)x(s)ds, t J with β C(J, J), β(t) t, t J, k C(D, R + ), l C(D 2, R + ), D = {(t, s) J J : t s}, D 2 = J J, R + = [, ). Indeed, T, S : C(J, B) C(J, B). The method of lower and upper solutions is very useful for proving existence results to differential problems 2 Mathematics Subject Classification. 45J5, 34G2. Key words and phrases. Delay integro-differential equations, lower and upper solutions, monotone sequences, convergence, extremal solutions. 32
322 T. JANKOWSKI (for example, see for details []). In this paper we use this technique proving existence of extremal solutions to problem (.). In section 3, we study a delay linear integro differential equation of Volterra type giving sufficient conditions under which such problem has a unique solution. To apply the monotone method we need some comparison results from section 4. The main result is given in section 5. We construct monotone sequences giving sufficient conditions under which they are convergent to extremal solutions of problem (.). Some existence and comparison results for corresponding linear delay problems are needed but the method of this paper is similar to that of [6]. If f(t, u, v, w, z) does not depend on v and z, and β(t) = t, t J, then we have the problem from [6], and if f does not depend on the last three arguments, then we have problem considered in [3]; see also [, 4, 9, 2]. This paper generalizes the results of [6]. Periodic boundary value problems for second order integro differential equations are considered, for example, in [5, 3]. if 2. A function u C (J, B) is said to be a lower solution of problem (.) u (t) f(t, u(t), u(α(t)), T u(t), Su(t)), t J, u() x, and an upper solution of (.) if the inequalities are reversed. Let us introduce the following assumptions for later use: (H ) f C(J B 4, B), l C(D 2, R + ), (H 2 ) α, β C(J, J), α(t) t, β(t) t, t J, k C(D, R + ), (H 3 ) y, z are lower and upper solutions of () and y (t) z (t) on J, (H 4 ) there exist nonnegative constants M, N, P such that f(t, ū, v, w, z) f(t, u, v, w, z) M(ū u) N( v v) P ( w w) for y (t) u ū z (t), y (α(t)) v v z (α(t)), T y (t) w w T z (t), Sy (t) z Sz (t), t J, (H 5 ) function f is nondecreasing in the last argument, (H 6 ) bne Mb + P k b ( e Mb ) if M >, and Nb+P k b 2 if M =, M where k = max{k(t, s) : (t, s) D }. 3. Now we consider a delay linear integro differential problem. Lemma 3.. Let Assumption H 2 hold. Let M, N, P, f C(J, B). Then the problem (3.) y (t) = f (t) My(t) Ny(α(t)) P T y(t), t J, y() = x has a unique solution.
DELAY INTEGRO DIFFERENTIAL EQUATIONS 323 Proof. Replace (3.) by t } y(t) = e {x Mt + e Ms [f (s) Ny(α(s)) P T y(s)] ds Ay(t), t J. where Let y = max t J [ y(t) e Kt ], where K > N + k P b. Then Ay Aȳ = t max t J e (K+M)t e Ms [ Ne Kα(s) e Kα(s) [y(α(s)) ȳ(α(s))] ] β(s) P k(s, r)[y(r) ȳ(r)]e Kr e Kr dr ds Note that Q = max t J e (K+M)t k = max k(t, s). (t,s) D Q [N + k P b] max t J y ȳ Q, t e Ms [Ne Kα(s) + k P {e (M+K)t t β(s) e Kr dr ] ds, } e (M+K)s ds < e (M+K)b Q. By the Banach fixed point theorem, problem (3.) has a unique solution because Ay Aȳ < Q y ȳ and Q <. It ends the proof. 4. To apply the monotone iterative technique we need some comparison results. Lemma 4.. Let Assumptions H 2 and H 6 hold. Assume that M, N, P and (4.) p (t) Mp(t) Np(α(t)) P T p(t), p() θ. Then p(t) θ on J. Proof. Let B be the set of all continuous linear functionals g on B such that g(x) for all x B. For any g B, let m(t) = g(p(t)). Then m C (J, IR), and m (t) = g(p (t)), g(t p(t)) = T m(t), g(sp(t)) = Sm(t). By (4.), we have (4.2) m (t) Mm(t) Nm(α(t)) P T m(t), t J, m().
324 T. JANKOWSKI Let v(t) = e Mt m(t), t J, so v(), and T m(t) = Then, (4.2) yields (4.3) β(t) k(t, s)m(s)ds = β(t) k(t, s)e Ms v(s)ds. v (t) = Me Mt m(t) + e Mt m (t) e Mt [Nm(α(t)) + P T m(t)] = Ne M[t α(t)] v(α(t)) P β(t) k (t, s)v(s)ds with k (t, s) = e M(t s) k(t, s). We need to show that v(t) on J. Assume that it is not true, so there exists t (, b] such that v(t ) >. Let = A, A. If A =, min t [,t ] then v(t), t [, t ]. Hence, v (t), t [, t ], by (4.3). It shows that v(t), t [, t ], so v(t ). It is a contradiction. Let A >. Then there exists t [, t ) such that v(t ) = A. Moreover, there exists t 2 (t, t ) such that v(t 2 ) =. Now the mean value theorem gives so On the other hand we obtain v(t 2 ) v(t ) = v (t 3 )(t 2 t ), t 3 (t, t 2 ), v (t 3 ) = A t 2 t > A b. v (t 3 ) Ne M[t3 α(t3)] v(α(t 3 )) P β(t3) β(t3) k (t 3, s)v(s)ds NAe M[t3 α(t3)] + P A k (t 3, s)ds { ( NAe Mb + P Ak M e Mb ) if M >, NA + P Ak b if M = showing that < Nbe Mb + P kb ( M e Mb ) if M >, and < Nb + P k b 2 if M =. It is a contradiction because of Assumption H 6. Hence v(t) on J and therefore m(t), t J. Since g B is arbitrary, we get p(t) θ, t J. It ends the proof. Remark 4.2. If N = and β(t) = t, t J, then Lemma 4. becomes Lemma 3. of [6]. Lemma 4.3. Let Assumptions H to H 6 hold. Assume that u, v are lower and upper solutions of problem (.) and such that y (t) u(t) v(t)
DELAY INTEGRO DIFFERENTIAL EQUATIONS 325 z (t), t J. Let y (t) = f(t, u(t), u(α(t)), T u(t), Su(t)) + F (t, u(t), y(t)), t J, y() = x, (4.4) z (t) = f(t, v(t), v(α(t)), T v(t), Sv(t)) + F (t, v(t), z(t)), t J, z() = x, where F (t, u(t), y(t)) = M[y(t) u(t)] N[y(α(t)) u(α(t))] P [T y(t) T u(t)]. Then (i) u(t) y(t) z(t) v(t), t J, (ii) y and z are lower and upper solutions of (.), respectively. Proof. Lemma 3. shows that system (4.4) has a unique solution (y, z). First, we show (i). Put p = u y. Then p() θ, and p (t) f(t, u(t), u(α(t)), T u(t), Su(t)) f(t, u(t), u(α(t)), T u(t), Su(t)) F (t, u(t), y(t)) = Mp(t) Np(α(t)) P T p(t), t J since u is a lower solution of (.). This and Lemma 4. yield p(t) θ on J showing that u(t) y(t) on J. In the same way, we can show that z(t) v(t) on J. Now, we put p = y z. Then, using Assumptions H 4 and H 5, we obtain p (t) = f(t, u(t), u(α(t)), T u(t), Su(t)) f(t, v(t), v(α(t)), T v(t), Sv(t)) +F (t, u(t), y(t)) F (t, v(t), z(t)) M[v(t) u(t)] + N[v(α(t)) u(α(t))] + P [T v(t) T u(t)] +F (t, u(t), y(t)) F (t, v(t), z(t)) = Mp(t) Np(α(t)) P T p(t), t J, p() = θ. Hence, by Lemma 4., y(t) z(t) on J showing that property (i) holds. Now we need to show that y and z are lower and upper solutions of (.), respectively. Using Assumptions H 4 and H 5 we get y (t) = f(t, u(t), u(α(t)), T u(t), Su(t)) + F (t, u(t), y(t)) f(t, y(t), y(α(t)), T y(t), Sy(t)) + f(t, y(t), y(α(t))), T y(t), Sy(t)) f(t, y(t), y(α(t)), T y(t), Sy(t)) + M[y(t) u(t)] +N[y(α(t)) u(α(t))] + P [T y(t) T u(t)] + F (t, u(t), y(t)) = f(t, y(t), y(α(t)), T y(t), Sy(t)), t J,
326 T. JANKOWSKI and z (t) = f(t, v(t), v(α(t)), T v(t), Sv(t)) + F (t, v(t), z(t)) showing that (ii) holds. It ends the proof. f(t, z(t), z(α(t)), T z(t), Sz(t)) + f(t, z(t), z(α(t)), T z(t), Sz(t)) f(t, z(t), z(α(t)), T z(t), Sz(t)), t J 5. The next section gives sufficient conditions on existence of extremal solutions for problems of type (.). Theorem 5.. Let cone B be regular. Assume that Assumptions H to H 6 are satisfied. Then there exist monotone sequences {y n }, {z n } such that y n y, z n z as n uniformly and monotonically on J and y, z are minimal and maximal solutions of problem (.) on [y, z ], respectively. Proof. Let y n+ () = z n+ () = x, and { y n+ (t) = f(t, y n (t), y n (α(t)), T y n (t), Sy n (t)) + F (t, y n (t), y n+ (t)), t J, z n+(t) = f(t, z n (t), z n (α(t)), T z n (t), Sz n (t)) + F (t, z n (t), z n+ (t)), t J for n =,,..., where F is defined as in Lemma 4.3. Note that y, z are well defined, by Lemma 3.. Using Lemma 4.3, we obtain y (t) y (t) z (t) z (t), t J, and moreover y, z are lower and upper solutions of (.). Let us assume that y (t) y (t) y k (t) y k (t) z k (t) z k (t) z (t) z (t), for t J and let y k, z k be lower and upper solutions of problem (.) for some k. Then, by Lemma 3., the elements y k+, z k+ are well defined. Lemma 4.3 yields Hence, by induction, we have y k (t) y k+ (t) z k+ (t) z k (t), t J. y (t) y (t) y n (t) z n (t) z (t) z (t), t J for all n. The regularity of B and continuity of f imply that the sequences {y n }, {z n } converge uniformly to the limit functions y, z, so y n y, z n z, and y(t) z(t) on J. Indeed, y, z are solutions of problem (.). To prove that y, z are minimal and maximal solutions of (.) on the segment [y, z ], we need to show that if w is any solution of (.) such that y (t) w(t) z (t) on J, then y (t) y(t) w(t) z(t) z (t), t J.
DELAY INTEGRO DIFFERENTIAL EQUATIONS 327 To do this, suppose that for some k, y k (t) w(t) z k (t) on J, and put p = y k+ w, q = w z k+. Then, Assumptions H 4 and H 5 yield p (t) = f(t, y k (t), y k (α(t)), T y k (t), Sy k (t)) F (t, w(t), w(α(t)), T w(t), Sw(t)) + F (t, y k (t), y k+ (t)) M[w(t) y k (t)] + N[w(α(t)) y k (α(t))] + P [T w(t) T y k (t)] +F (t, y k (t), y k+ (t)) Mp(t) Np(α(t)) P T p(t), t J, p() = θ, q (t) = F (t, w(t), w(α(t)), T w(t), Sw(t)) F (t, z k (t), z k (α(t)), T z k (t), Sz k (t)) F (t, z k (t), z k+ (t)) Mq(t) Nq(α(t)) P T q(t), t J, q() = θ. By Lemma 4., we obtain p(t) θ, q(t) θ on J showing that y k+ (t) w(t) z k+ (t), t J. Since y (t) w(t) z (t) it proves, by induction, that y n (t) w(t) z n (t) on J for all n. Taking the limit as n, we conclude that y(t) w(t) z(t), t J. The proof is complete. Remark 5.2. Note that Assumption H 4 holds if we assume that f(t, u, v, w, z) is nondecreasing in u, v, w for fixed t and z. Indeed, in this case, we have f(t, ū, v, w, z) f(t, u, v, w, z) M(ū u) N( v v) P ( w w) for some nonnegative M, N, P and ū u, v v, w w. Remark 5.3. If N =, β(t) = t, t J and f does not depend on the last argument, then Theorem 5. becomes Theorem 2 of [6]. Example 5.4. Consider the initial value problem of an infinite system for scalar delay integro differential equations of type x n (t) = [ ( [ 4n 2 2n x 2 n (t) x n+ 2 t)] ] 2(n+) 2 3 t 2 x n+ (s)ds [ (5.) + 3 2(n+) 3 x 2n(s) cos 4 (t s)ds], t J = [, ], x n () = for n =, 2,.... Let B = {u : (u,..., u n,...) : u n IR, u n < } with the norm u = n= u n and B = {u B : u n, n =, 2,...}. Then B n= is a normal cone in B. Since B is weakly complete, we know from Remarks 4.3. and.2.4 of [8] that B is regular. In this case f = (f,..., f n,...) with f n (t, x, y, z, w) = [ ] 4n 2 2n 2 x n y n+ 2(n + ) 2 z2 n+ + 2(n + ) 3 w3 2n.
328 T. JANKOWSKI Indeed, f C(J B 4, B), α = 2 t, α C(J, J), α(t) t, β(t) = 3 t, β C(J, J), β(t) t, k(t, s) = for (t, s) J J, and l(t, s) = cos 4 (t s) for (t, s) J J. Let y (t) = (,...,,...), z (t) = Indeed, y (t) < z (t), t J. We see that and (,..., ) n 2,..., t J. f n (t, y (t), y (α(t)), T y (t), Sy (t)) = >, t J, n =, 2,..., 8n4 f n (t, z (t), z (α(t)), T z (t), Sz (t)) = [ 4n 2 2n 2 ] n 2 (n + ) 2 [ ] 2 3 t [ 2(n + ) 2 (n + ) 2 ds + 2(n + ) 3 4n 2 cos4 (t s)ds 3n2 + 2n 8n 4 (n + ) 2 t 2 <, t J, n =, 2,.... 8(n + ) 6 It proves that y, z are lower and upper solutions of problem (5.) respectively, so assumption H 3 holds. Let y (t) u ū z (t), y (α(t)) v v z (α(t)), T y (t) w w T z (t), Sy (t) z Sz (t) for all t J. Then f n (t, ū, v, w, z) f n (t, u, v, w, z) = = [ ] 4n 2 2n 2 ū n v n+ [ ] 4n 2 2n 2 u n v n+ + 2(n + ) 2 w2 n+ 2(n + ) 2 w2 n+ = 4n 2 [ū n u n ] 4n 2 [ v n+ v n+ ] 2(n + ) 2 [ w n+ + w n+ ][ w n+ w n+ ] 4 [ū n u n ] 4 [ v n+ v n+ ] 3 [ w n w n ]. It yields M = N = 4, P = 3 and therefore bne Mb + P k b M ( e Mb ).6997 <. It proves that assumption H 6 holds. Hence, problem (5.) has extremal solutions in the segment [y, z ], by Theorem 5.. ] 3
DELAY INTEGRO DIFFERENTIAL EQUATIONS 329 Example 5.5. Consider the following infinite problem of scalar equations [ x n(t) = t 4 2n x 2 n (t) ] + x ( [ n+ 2 t) ] 4 + t 2 x n+2 (s)ds [ (5.2) 4 +t x 2n(s) sin 2 (t s)ds], t J = [, ], x n () = for n =, 2,.... Indeed, α(t) = 2 t, β(t) = 4 t. Let B and B be defined as in Example. Note that f n (t, x, y, z, w) = [ ] t 4 2n 2 x n + y n+ + zn+2 2 + tw2n. 4 Let Then y (t) z (t), It yields and y (t) = (,...,,...), z (t) = t J, and y (t) = (,...,,...), z (t) = (,..., f n (t, y (t), y (α(t)), T y (t), Sy (t)) = ( ) t t,..., n 2,..., t J. ) n 2,.... t, t J, n =, 2,..., 8n2 f n (t, z (t), z (α(t)), T z (t), Sz (t)) = [ t 4 2n 2 t ] t n 2 + 2(n + ) 2 [ ] 2 4 t [ s ] 4 + (n + 2) 2 ds s + t 4n 2 sin2 (t s)ds [ t ] 2 8n 2 + t 2(n + ) 2 + 4 t (n + 2) 4 sds + t [ 256n 8 sds = t 8n 2 + t 2(n + ) 2 + t 4 24(n + 2) 4 + t 496n 8 < z n(t), t J, n =, 2,... It proves that y, z are lower and upper solutions of problem (5.2) respectively, so assumption H 3 holds. Let y (t) u ū z (t), y (α(t)) v v z (α(t)), T y (t) w w T z (t), Sy (t) z Sz (t) for all t J. Then f n (t, ū, v, w, z) f n (t, u, v, w, z) = = 4 [ ū n + u n ] + [ v n+ v n+ ] + w 2 n+2 w2 n+2 4 [ū n u n ], ]4
33 T. JANKOWSKI so M = 4, N = P =. Assumption H 6 is satisfied and problem (5.2) has extremal solutions in the segment [y, z ], by Theorem 5.. References [] R.P. Agarwal and D. O Regan, Integral and integrodifferential equations, Gordon and Breach Sci. Pub., Amsterdam, 2. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 8 (976), 62 79. [3] S.W. Du and V. Lakshmikantham, Monotone iterative technique for differential equations in a Banach space, J. Math. Anal. Appl. 87 (982), 454 459. [4] S.W. Du, Monotone iterative technique for delay differential equations in abstract cones, Appl. Math. Comput. 2 (983), 23 29. [5] L.H. Erbe and D. Guo, Periodic boundary value problems for second order integrodifferential equations of mixed type, Appl. Anal. 46 (992), 249 258. [6] D. Guo, Initial value problems for integro differential equations of Volterra type in Banach spaces, J. Appl. Math. Stoch. Anal. 7 (994), 3 23. [7] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 988. [8] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear integral equations in abstract spaces, Kluwer Academic Pub., Dordrecht, 996. [9] T. Jankowski, Differential equations in abstract cones, Serdica Math. J. 26 (2), 229 244. [] M.A. Krasnoselskii, G.M. Vainikko, P.P. Zabreiko, Ya.B. Ruticki and V.Ya. Stecenko, Approximate Solutions of Operator Equations (in Russian), Nauka, Moscow, 969 (English translation by Wolters Noorhoff Pub., Groningen, 972). [] G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, 985. [2] L. Liu, Iterative method for solutions of coupled quasi solutions of nonlinear integrodifferential equations of mixed type in Banach spaces, Nonlinear Anal. 42 (2), 583 598. [3] J.J. Nieto, Periodic boundary value problem for second order integro differential equations with general kernel, Internat. J. Math. & Math. Sci. 8 (995), 757 764. Technical University of Gdańsk Department of Differential Equations /2 G.Narutowicz Str., 8 952 Gdańsk, Poland. Received: 25.6.2. Revised: 7.2.22.