Archivum Mathematicum

Archivum Mathematicum Deepak B. Pachpatte Some dynamic inequalities applicable to partial integrodifferential equations on time scales Archivum Mathematicum, Vol. 51 2015, No. 3, 143 152 Persistent URL: http://dml.cz/dmlcz/144425 Terms of use: Masaryk University, 2015 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz

ARCHIVUM MATHEMATICUM BRNO Tomus 51 2015, 143 152 SOME DYNAMIC INEQUALITIES APPLICABLE TO PARTIAL INTEGRODIFFERENTIAL EQUATIONS ON TIME SCALES Deepak B. Pachpatte Abstract. The main objective of the paper is to study explicit bounds of certain dynamic integral inequalities on time scales. Using these inequalities we prove the uniqueness of some partial integrodifferential equations on time scales. 1. Introduction The study time scale calculus was initiated in 1989 by Stefan Hilger 5 a German Mathematician in his Ph.D dissertation. Since then many authors have applied time scales calculus for various applications in Mathematics. Mathematical inequalities on time scales plays very important role. Recently many authors have studied various properties of dynamic inequalities on time scales 2, 6, 7, 8, 9, 10, 11, 12, 13, 14. Let R denotes the real number Z the set of integers and T denotes the arbitrary time scales. Let T 1 and T 2 be two time scales, and let Ω = T 1 T 2. The rd-continuous function is denoted by C rd. We denote the partial delta derivative of wx, y with respect to x, y and xy for x, y Ω by w 1 x, y, w 2 x, y, w 1 2 x, y = w 2 1 x, y. The basic information about time scales calculus can be found in 1, 3, 4, 5. 2. Main results In 7 the author has obtained some estimates of the some Gronwall like inequalities while in 6, 14 the authors have studied some nonlinear dynamic integral inequalities on time scales. Motivated by the above research work in this paper we obtain some explicit bounds of certain dynamic inequalities on time scales. Theorem 2.1. Let ut, ht C rd T, R, pτ, s C rd Ω, R defined for τ, s T, and c 0 be a constant. If τ 2.1 u t c h τ u τ p τ, s u s s τ, 2010 Mathematics Subject Classification: primary 26E70; secondary 34N05. Key words and phrases: explicit bounds, integral inequality, dynamic equations, time scales. Received December 21, 2014, revised June 2015. Editor R. Šimon Hilscher. DOI: 10.5817/AM2015-3-143

144 D.B. PACHPATTE for t T. Then 2.2 u t ce H1 t,, 2.3 H 1 t = h t p t, τ τ, Proof. Let c > 0 and define a function wt by right hand side of 2.1. Then wt > 0 and non decreasing for w = c, ut wt and 2.4 Integrating 2.4 we get w t = h t u t h t w t w t h t p t, τ u τ τ p t, τ w τ τ p t, τ τ w t t w t h t p t, τ τ. 2.5 w t ce H1 t,. Since ut wt we get 2.4. Theorem 2.2. Let ut, ht, pt, τ be as in Theorem 2.1 and ft, gt C rd T, R, pτ, s C rd Ω, R. If 2.6 u t f t g t h τ u τ p τ, s u s s t, for t T, then 2.7 u t f t g t K 2 t e H2 t,, 2.8 and 2.9 K 2 t =, H 2 t = h t g t p t, τg τ τ, h τ f τ p τ, s f s s τ. Proof. Define a function wt by right hand side of equation 2.6 2.10 wt = h τ u τ p τ, s u s s t, then 2.6 becomes 2.11 ut ft gtwt.

INTEGRAL INEQUALITIES ON TIME SCALES 145 From 2.10 and 2.11 we get wt hτ{fτ gτwτ} τ pτ, s{fs gsws} s τ = hτfτ pτ, sfs s τ h τ g τ w τ p τ, s g s w s s τ 2.12 n t h τ g τ w τ p τ, s g s w s s τ, n t = ɛ h τ f τ p τ, s f s s τ, ɛ is arbitrarily small constant. From 2.12 we have w t t 2.13 n t 1 h τ g τ w τ n τ p τ, s g s w s n s s τ. Now applying Theorem 2.1 to 2.13 yields 2.14 wt n t e H2 t,. Using 2.14 in 2.11 and letting ɛ 0, we get 2.7. Theorem 2.3. Let a i C rd T, R, i = 1, 2 satisfying 2.15 0 a i t, u a i t, v b i t, v u v, for u v 0 b i t, v are non negative rd-continuous function. If 2.16 u t ft gt h τ a 1 τ, u τ p τ, s a 2 s, u s s τ, t T then 2.17 2.18 and 2.19 for t T. u t ft gtk 3 t e H3 t,, H 3 t = h t a 1 t, f t p t, τ a 2 t, f τ τ, K 3 t = h t a 1 t, f t g t p t, τ b 2 t, f τ g τ τ, Proof. Define a function wt by 2.20 w t = h τ a 1 τ, u τ p τ, s a 2 τ, u s τ,

146 D.B. PACHPATTE then from 2.16 we have 2.21 u t f t g t w t, From 2.20, 2.21 and 2.15 we have 2.22 wt hτ{a 1 τ, fτ gτwτ a 1 τ, fτ a 1 τ, fτ} τ pτ, s{a 2 s, fs gsws a2 s, fs a 2 s, fs} s τ hτb 1 τ, fτgτwτ Nt hτa 1 τ, fτ 2.23 Nt = ɛ hτb 1 τ, fτ gτwτ pτ, sb 2 s, fsgsws s τ, pτ, sb 2 s, fsgsws s τ pτ, sa 2 s, fs s τ hτa 1 τ, fτ pτ, sa 2 s, fs s τ, ɛ > 0 is an arbitrary small constant. Using the same steps as in Theorem 2.2 we get the result. Theorem 2.4. Let ux, y, hx, y C rd Ω, R be nonnegative functions defined for x, y, s, t T and px, y, s, t C rd Ω Ω, R. If 2.24 for x, y Ω then ux, y c hs, tus, t ps, t, ξ, ηus, t η ξ t s, 2.25 ux, y ce H1 x,, 2.26 H 1 x, y = hx, t x px, t, ξ, η η ξ.

INTEGRAL INEQUALITIES ON TIME SCALES 147 Proof. Let c > 0 and define a function wx, y by right hand side of 2.24 then wx, y > 0, w, y = wx, = c, ux, y wx, y and w x, y = wx, y hx, tux, t hx, twx, t hx, t ps, t, ξ, ηuξ, η η ξ t ps, t, ξ, ηwξ, η η ξ t ps, t, ξ, η η ξ t, i.e 2.27 w x, y wx, y hx, t ps, t, ξ, η η ξ t. Keeping y fixed in 2.27, set x = s and integrating it with respect to s from to x we get 2.28 w x, y ce H1 x,. Using 2.28 in ux, y wx, y we get the result in 2.25. Theorem 2.5. Let ux, y, hx, y, px, y, s, t, c as in Theorem 2.4 and fx, y, gx, y C rd Ω, R. If ux, y fx, y gx, y hs, tus, t 2.29 for x, y Ω then ps, t, ξ, ηuξ, η η ξ t s, 2.30 u x, y f x, y g x, y K 2 x, y e H2 x,, 2.31 2.32 H 2 x, y = K 2 x, y = hx, tgs, t hx, t Proof. Define a function wx, y by 2.33 wx, y = then 2.29 we have x hx, tus, t px, t, ξ, ηgξ, η η ξ s, ps, t, ξ, ηuξ, η η ξ t s. 2.34 u x, y f x, y g x, y w x, y. ps, t, ξ, ηuξ, η η ξ t s,

148 D.B. PACHPATTE From 2.33 and 2.34 we have 2.35 wx, y x 0 x nx, y hs, t { fs, t gs, tws, t } ps, t, ξ, η { fs, tgs, tws, t } η ξ t s hs, tfs, t hs, tgs, tws, t ps, t, ξ, ηfs, t η ξ t s ps, t, ξ, ηgξ, ηwξ, η η ξ t s hs, tgs, tws, t ps, t, ξ, ηgξ, ηwξ, η η ξ t s, 2.36 nx, y = ε hs, tfs, t ps, t, ξ, ηfξ, η η ξ t s, and ɛ > 0 is an arbitrary small constant n x, y is positive, rd-continuous and nondecreasing. From 2.35 we have 2.37 wx, y nx, y 1 ws, t hs, tgs, t ns, t ws, t ps, t, ξ, ηgξ, η ns, t η ξ t s. Now an application of Theorem 2.4 to 2.37 we have 2.38 w x, y n x, y e H2 x,. Using 2.38, 2.34 and letting ɛ 0 we get 2.30. Theorem 2.6. Let ux, y, fx, y, gx, y, hx, y, px, y, s, t be as in Theorem 3.2. Let A i C rd Ω T, R, I = 1, 2 such that 2.39 0 A i x, y, u A i x, y, v B i x, y, v u v,

INTEGRAL INEQUALITIES ON TIME SCALES 149 for 0 v u B i nonnegative rd-continuous. If 2.40 ux, y fx, y gx, y s, ta 1 x, y, us, t ps, t, ξ, ηa 1 x, y, uξ, η η ξ t s, for x, y ω then 2.41 u x, y f x, y g x, y K 3 x, y e H3 x, 2.42 and 2.43 H 3 x, y = K 3 x, y = for x, y Ω. Proof. Define a function wx, y by hs, tb 1 s, t, fs, t gs, t ps, t, ξ, ηb 2 ξ, η, fξ, η gξ, η η ξ t s, hs, ta 1 s, t, fs, t gs, t ps, t, ξ, ηa 2 ξ, η, fξ, η gξ, η η ξ t s, 2.44 wx, y = hs, ta 1 s, t, us, t ps, t, ξ, ηa 2 ξ, η, u ξ, η η ξ t s, then we have 2.44 2.45 w x, y f x, y g x, y w x, y. From 2.43 and 2.44 we have w x, y hs, ta 1 s, t, fx, y gx, ywx, y ps, t, ξ, ηa 2 ξ, η, fξ, η gξ, ηwξ, η η ξ t s

150 D.B. PACHPATTE 2.46 2.47 = x 0 x hs, ta 1 s, t, fx, y ps, t, ξ, ηa 2 ξ, η, fξ, η η ξ t s hs, ta 1 s, t, gx, ywx, y ps, t, ξ, ηa 2 ξ, η, gξ, ηwξ, ηbig η ξ t s Nx, y hs, ta 1 s, t, gx, ywx, y Nx, y = ε ps, t, ξ, ηa 2 ξ, η, gξ, ηwξ, η η ξ t s, hs, ta 1 s, t, fx, y ps, t, ξ, ηa 2 ξ, η, fξ, η η ξ t s, in which ɛ arbitrary small constant. Clearly N x, y is positive, rd-continuous and nondecreasing. Remaining part of the proof can be done similarly as in Theorem 2.5. Remark. Theorem 2.2 and Theorem 2.5 are special cases of Theorem 2.3 and Theorem 2.6 respectively with at, u = u. 3. Applications In this section we give some applications of inequality proved in theorem to obtain the uniqueness and bound on the solution for dynamic partial integrodifferential equation of the form 3.1 3.2 V 2 1 x, y = F x, y, V x, y G x, y, ξ, η, V ξ, η η ξ, w x, = k 1 x, w, y = k 2 y, k 1 = k 2 = 0, F C rd Ω 2 R R, G Crd Ω 2 Ω 2 R R. Now we give the bounds on 3.1 and 3.2.

INTEGRAL INEQUALITIES ON TIME SCALES 151 Theorem 3.1. Suppose that 3.3 F x, y, V x, y h x, y V x, y, 3.4 G x, y, ξ, η, V ξ, η p x, y, ξ, η V ξ, η, 3.5 k 1 x k 2 x c, h, p, c are as defined in Theorem 2.2. If wx, y is solution of 3.1 and 3.2 then 3.6 w x, y ce H1 x,, for x, y Ω H 1 is given by 2.26. Proof. The solution V x, y of 3.1 3.2 satisfy the equation V x, y = k 1 x k 2 y F s, t, V s, t 3.7 G x, y, ξ, η, V ξ, η η ξ t s. Now for 3.3 3.5 and 3.7 we get V x, y c h s, t V s, t 3.8 p x, y, ξ, η V ξ, η η ξ t s. Now we apply Theorem 2.4 to 3.8 gives 3.6. The right hand side of 3.8 gives the bounds of solution of 3.1 3.2. Now in the next theorem we give the uniqueness of solution of 3.1 3.2. Theorem 3.2. Suppose 3.9 F x, y, V x, y F x, y, V x, y hx, y V x, y V x, y, Gx, y, ξ, η, V ξ, η G x, y, ξ, η, V ξ, η 3.10 px, y, ξ, η V ξ, η V ξ, η, h, p are as defined in Theorem 2.4. Then 3.1 3.2 has at most one solution for x, y Ω. Proof. Let V x, y and V x, y be two solution of 3.1 3.2 then 3.11 V x, y V x, y {F } = s, t, V s, t F s, t, V s, t { } G s, t, ξ, η, V ξ, η G s, t, ξ, η, V ξ, η η ξ t s.

152 D.B. PACHPATTE From 3.9, 3.10 and 3.11 we have x V x, y V x, y 3.12 hs, t V s, t V s, t ps, t, ξ, η V ξ, η V ξ, η η ξ t s. Now applying Theorem 2.4 with k=0 gives V x, y V x, y 0 giving V x, y = V x, y for x, y Ω. Thus there is at most one solution of 3.1 3.2. References 1 Agarwal, R.P., O Regan, D., Saker, S.H., Dynamic inequalities on time scales, Springer, 2014. 2 Andras, S., Meszaros, A., Wendroff type inequalities on time scales via Picard operators, Math. Inequal. Appl. 17 1 2013, 159 174. 3 Bohner, M., Peterson, A., Dynamic equations on time scales, Birkhauser Boston Berlin, 2001. 4 Bohner, M., Peterson, A., Advances in dynamic equations on time scales, Birkhauser Boston Berlin, 2003. 5 Hilger, S., Analysis on measure chain A unified approch to continuous and discrete calculus, Results Math. 18 1990, 18 56. 6 Menga, F., Shaoa, J., Some new Volterra Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput. 223 2013, 444 451. 7 Pachpatte, D.B., Explicit estimates on integral inequalities with time scale, J. Inequal. Pure Appl. Math. 7 4 2006, Article 143. 8 Pachpatte, D.B., Properties of solutions to nonlinear dynamic integral equations on time scales, Electron. J. Differential Equations 2008 2008, no. 130, 1 8. 9 Pachpatte, D.B., Integral inequalities for partial dynamic equations on time scales, Electron. J. Differential Equations 2012 2012, no. 50, 1 7. 10 Pachpatte, D.B., Properties of some partial dynamic equations on time scales, Internat. J. Partial Differential Equations 2013 2013, 9pp., Art. ID 345697. 11 Saker, S. H., Some nonlinear dynamic inequalities on time scales and applications, J. Math. Inequalities 4 2010, 561 579. 12 Saker, S.H., Bounds of double integral dynamic inequalities in two independent variables on time scales, Discrete Dynamics in Nature and Society 2011, Art. 732164. 13 Saker, S.H., Some nonlinear dynamic inequalities on time scales, Math. Inequal. Appl. 14 2011, 633 645. 14 Sun, Y., Hassan, T., Some nonlinear dynamic integral inequalities on time scales, Appl. Math. Comput. 220 2013, 221 225. Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, Maharashtra 431004, India E-mail: pachpatte@gmail.com