GRONWALL BELLMAN OU-IANG TYPE INEQUALITIES Jaashree V. Patil Department of Mathematics Vasantrao Naik Mahavidalaa,Aurangabad - 43 3 (M.S.) INDIA ABSTRACT: In this paper we prove some retarded nonlinear inequalities which provide eplicit bounds on unknown function. Applications are also given to illustrate the usefulness of one of our results. Kewords: Integral Inequalities, Retarded Equation, Eplicit Bounds, Boundedness. INTRODUCTION In the development of the theor of differential and integral equations integral inequalities which provide eplicit bounds on unknown function take ver important place. The literature on such inequalities is vast see [6, 8] and the references there in. One of the most useful inequalities in the development of theor of differential equation is given in following Lemma b Ou-Iang see []. Lemma. Let u and f are non-negative functions on [, ) satisfing u (t) < k + t f (s)u(s)ds for all, t[, ), where k > is a constant then u(t) < k + for all t[, ). t f(s)ds In literature, the Ou-Iang tpe inequalities and their generalizations occurs which used effectivel in the stud of qualitative as well as quantitative properties of solutions of differential equations see [-5, 7,9 ].In this paper we prove some new retarded inequalities which is generalization of above lemma and used to stud the boundedness properties of solutions of integral and differential equations. 43
MAIN RESULTS Theorem. :If u(, ), m(, ) are real valued nonnegative continuous functions defined for >, >, f(,, s, t) be continuous non-decreasing in and for each t, s. < () <, < () <, ' (), '() > are real valued continuous functions defined for >, > satisfies u (, ) < m(, ) + f (,, s, t) u( s, t) dt ds ---- (.) then u(, ) < m(, ) f (,, s, t)dtds ---- (.) Proof. Let us assume that m(, ) >. Fied an numbers and with and Define a function z(, ) f (,,s, t)u(s, t)dtds ---- (.3) From (.) we get u (, ) < m, z, ---- (.4) and z(, ) = z (, ) =, clearl z (, ) is positive non-decreasing function in each variable. Hence for [, ] and [, ] b direct calculation, we get 44
z = f(,, (), ()) u( (), ()) '() '() '(),, (), t u (), t dt + + '(),,s, () u, () ds f (,,s, t) u(s, t) dt ds f,, (), () m(, ) z(, ) '() '() ( ) '() f,, (), t m (), t z (), t () '() f,, s, () m s, () z s, () ds () ( ) m s, t z s, t f,,s, t { dt ds () m, z, f,, (), () '() '() '() f,, (), tdt '() f,,s,, (ds z ( ) f,, s, t dt ds} Z, z, m, f,,s, t dt ds on the other hand z, f,,s, tdt ds m, z, ---- (.5) 45
Keeping fied in (.5), integrating with respect to from to, and using fact that z (, ) =, we deduce. Z, Z, m, f,, s, t dt ds ---- (.6) Now keeping fied in (.6) and integrating with respect to from to we get m, Z, m, f,,s, tdt ds for (, ) [, ] [, ] let = and = in above inequalit we get m, z, m, f,,s, tdt ds As and are arbitraril and u(, ) < m, z, we have obtain inequalit in (.). Corollar... Let m, n, g C (R+ R+, R+),, C (R+, R+) and let α, β is non-decreasing with () <, () < if u C (R+ R+, R+) Satisfies u(, ) < m, f,,s, tdt ds >, > Theorem.. Let m, n, g C (R+ R+, R+),, C (R+, R+) and let α, β is non-decreasing with () <, () < if u C (R+ R+, R+) Satisfies 46
(.7) International Journal of Mathematics and Statistics Studies u (, ) m(, ) n(, ) g(s, t) u(s, t)dt ds ---- then when >, > u(, ) < m(, ) n(, ) g s, t dt ds ---- (.8) Proof. Let us assume that m(, ) >. Fied an number and with < < and < < z(, ) = m (, ) + n (, ) g(s, t) u(s, t) dt ds Then z(, ) >, z(, ) = z (, ) = m (, ) and from (.7) we get u (, ) < z (, ) i.e. u (, ) < (, ) z ---- (.9) Clearl z(, ) is positive non-decreasing function in each variable. Hence we get z = n (, ) g (), (). u (), (). '(). '() n(, ) g( (), () ) z (), () '(). '() z n(, ) g (), () '() '() z, < i.e. 47
z International Journal of Mathematics and Statistics Studies z, n(, ) g (), () '() '() Keeping fied, integrating above inequalit with respect to from to and using fact that z (, ) =, we get z, z, n(, ) g (), (t) '() '(t) d t Now keeping fied in above inequalit and integrating with respect to from to, we deduce n, m, z, g (s), (t) '(s) '(t) dsdt B making change of variable on right-hand side of above inequalit, we get z, m, n, g s, tdt ds Taking =, = in the inequalit (.), since and are arbitrar and as u(, ) < z, we get the required inequalit in (.8). Theorem.3. Assume m, f, α, β be as in Theorem. ω(u) is positive continuous non-decreasing for u > with ω () = and ds s if u C(R+ R+, R+) satisfies u (, ) < m (, ) + f,,s, t u(s, t) dt ds,, ---- (.) then 48
u(, ) < International Journal of Mathematics and Statistics Studies G G m, f (,,s, t) dt ds ---- (.) where G(r) = r ds s, r > Proof : Assume, be fied and let z(, ) = f (,,s, t) u(s, t) dt ds Clearl z (, ) is non-decreasing for and hence for [, ], [, ], we have z = f,, (), () u( (), () '() '() + '() f,,s, ( ) u(s, ( )) ds + + '() f,, (), t u( (), t) dt f,,s, t u(s, t) dt ds < f,, (), () m (), z(, ). '() '() + '() f,, (), t m (), t z (), t dt 49
'(),,s, () ds + f m s, () z () + f,,s, t m(s, t) z(s, t)dt ds m, z, f,, () (). '(). '() '() f,,,(), t dt '() f,,z, () ds f (,,s, t)dtds i.e. Z z, m, f (,,s, t)dt ds We have f (,,s, t) dt ds m, z, Z Integrating both sides of above inequailt with respect to from to we deduce Z z, m, f (,,s, t)dt ds Integrating above inequalit with respect to from to and using definition of G, we get 5
(.3) G International Journal of Mathematics and Statistics Studies G m, z, m, f (,,s, t) dt ds ---- for,,,. As ds s from (.3) we have m, z, G G m, f (,,s, t) dt ds let =, = in above inequalit, we get () m, z, G G m, f,,s, t dt ds As, are arbitraril and u(, ) < m(, ) z(, ) we obtain inequalit in (.). Theorem.4 : Suppose m, n, g C(R+ R+, R+),,C (R+, R+) and m, n, g, are nondecreasing function with () <, () < for >, > assume C(R+, R) be a nondecreasing function such that (t) > for t > and uc(r+ R+, R+) satisfies ds s if u(, ) < m (, ) + n (, ) g(t, s) (.4) u(s, t) dt ds >, > ---- 5
u (, ) < (.5) International Journal of Mathematics and Statistics Studies G G m(, ) n(, ) g(s, t) dt ds, >, > ---- where G(r) = r ds,r s Proof : Let us assume that m (, ) >, Fied an number and with < <, < <. Define a function z(, ) as z(, ) = m, + n, g s, t u(s, t) dt ds Then z (, ) >, z (, ) = z (, ) = m, and from (.4) we have u (, ) < z (, ) implies u (, ) < z, clearl z (, ) is positive non-decreasing function in each variable. Hence we get z = n, g (), () (u (), () '() '() n, g (), () z (), () '() '() z z(, ) n, g (), () '() '() i.e. 5
z n, g (), () '() '() z, Keeping fied, integrating above inequalit with respect to from to and using fact that Z (, ) = we get z, n, g (), () '() '(t) dt z, Keeping fied in above inequalit and integrating with respect to from to we get G G z(, ) m(, ) n, g (s), (t) '(s) '(t) dt ds B making change of variable on right-hand side of above inequalit, we get z(, ) G G m(, ) n, g (s, t), dtds for,,, let =, = in (.6) we have z, G G m, n, g s, t dt ds As, are arbitraril and u (, ) < z, we get required inequalit in (.5). 53
APPLICATION In this section we will show that our one of the result is used in proving the boundedness of solutions of nonlinear integral equation. Consider the nonlinear partial integral equation of the form u, h, G,,s, t, u(s, t) dt ds ---- (3.) where all the functions are continuous on their respective domains of their definitions and m, h, ---- (3.) F,, t,s, u f,, t,s u ----- (3.3) for >, > where m, f,, be as in Theorem (.). Using (3.), (3.3) in (3.) and using Theorem. we obtain the bound on the solution u(, ) of (3.). REFERENCES ) L Ou-Iang, The boundedness of solutions of linear differential equatioins, Ann. Mat. Pura. Appl., 68 (995) 37 49. ) C.M. Dafermos, The second law of thermodnamic and stabilit, Arch. Ration, Mech. Anal.,7(979) 67-79. 3) O. Lipovan, A retarded integral inequalit and its applications, J. math. Anal. Appl., 85 (3) 436-443. 4) O. Lipovan, Integral inequalities for retarded Volterra equations, J. Math. Anal. Appl., 3 (6) 349-358. 5) Lianzhong Li, Fanvei Meng, Leliang He, Some generalized integral inequalities and their applications, J. Math.Anal.Appl., 37 () 339-349. 54
6) B.G. Pachpatte, Inequalities for Differential and Integral Equations, Academic press, New York, 998. 7) B.G. Pachpatte, on some new inequalities related to certain inequalities in the theor of differential equations, J. Math. Anal.Appl.,89 (995) 8-44. 8) B.G. pachpatte, Integral and Finite Difference Inequalities and Applications, Elsevier, New York, 6. 9) H. Zhang, F. Meng, Integral inequalities in two independent variables for retarded Volterra equations, Appl.Math.Comp.,99 (8) 9-98. 55