Nonlocal problems for quasilinear functional partial differential equations of first order

Publicacions Matem`atiques, Vol 41 (1997), 507–517. NONLOCAL PROBLEMS FOR QUASILINEAR FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER Jan Turo Abstract Existence and uniqueness of almost everywhere solutions of nonlocal problems to functional partial differential systems in diagonal form are investigated. The proof is based on the characteristics and fixed point methods. 1. Introduction For any metric spaces Xand Ywe denote by C(X, Y ) the set of all continuous functions from Xto Y. Let a0>0 be a given constant and Ia0=[0,a 0]×Rm. Write D=[−τ,0] ×[−b, b], where τ∈ R+=[0,+∞) and b=(b1,... ,b m)∈Rm +.Forz:[−τ,a0]×Rm→Rn and (x, y)=(x, y1,... ,y m)∈Ia0, we define z(x,y):D→Rn by z(x,y)(s, t)=z(x+s, y +t), (s, t)∈D. Thus, we see that z(x,y) is a restriction of zto the rectangle [x−τ,x]×[y−b, y +b]. Put Ω=[0,a 0]×Rm×C(D,Rm) and I=[−τ,0] ×Rm. We assume that =[ij]:Ω→Rnm,i=1,... ,n,j=1,... ,m, f=(f1,... ,f n):Ω→Rn,hk=[hkij]:I→Rnn,i, j =1,... ,n, k=1,... ,r,ϕ=(ϕ1,... ,ϕ n):I→Rnare given functions. We consider quasilinear hyperbolic systems of functional partial differential equations (1) Dxzi(x, y)+ m  j=1 ij(x, y, z(x,y))Dyjzi(x, y)=fi(x, y, z(x,y)), i=1,... ,n,(x, y)∈Ia0, with nonlocal condition (2) z(0,y)+ r  k=1 (hk)(0,y)z(ak,y)=ϕ(0,y),y∈Rm, 508 J. Turo where ak,k=1,... ,r, are finite numbers such that 0 <a 1<a 2<···< ar≤a0. The nonlocal condition (2) may be also written in the form (3) z(x, y)+ r  k=1 hk(x, y)z(ak+x, y)=ϕ(x, y),(x, y)∈I. For r=n,τ= 0 and hkij =hijδki (δki is the Kronecker symbol) nonlocal boundary condition (2) reduces to the nonlocal condition “´a la Cesari” [8],[1].Ifhkij =δkiδij then (2) reduces to the Nicoletti condition [10],[12]. Furthermore, if all ak=0,k=1,... ,r then we get the usual Cauchy condition. Nonlocal condition was considered for parabolic problems in [4],[5], [8], and for hyperbolic problems in [2],[3],[6],[9]. Mixed problems for system (1) in two independent variables were investigated in [10]. System (1) contains as particular cases the system of differential equations with a retarded argument, differential-integral systems and differential-functional equations with operators of the Volterra type (see Section 4). In this paper, we consider the local existence and uniqueness of generalized solutions of nonlocal problem (1), (2). The method used in the paper is based on characteristics theory and the fixed point theorem. 2. Assumptions and Lemma For η=(η1,... ,η k)∈Rkwe write |η|k= max{|ηi|:1≤i≤k}. For the matrix U=[uij ], i=1,... ,n,j=1,... ,m, we define ||U|| = max{m j=1 |uij|:1≤i≤n}. Let ||v|| denote the supremum norm of v∈C(D,Rn) and C(D, Rn;p)={v∈C(D,Rn):||v|| ≤ p},p∈R+. Let L([α, β],R) be the set of all integrable functions l:[α, β]→R.We denote by CL(D, Rn) the class of all functions v∈C(D, Rn) satisfying the condition (4) |v(s, t)−v(¯s, ¯ t)|n≤¯s s ω(ξ)dξ +q|t−¯ t|m,(s, t),(¯s, ¯ t)∈D, where ω∈L([−τ,0],R+), q∈R+(ωand qdepend on v). For v∈ CL(D,Rn) we define ||v||L=||v||∗+||v||, where ||v||∗= inf q+0 −τ ω(ξ)dξ :qand ωsatisfy (4). Functional partial differential equations 509 Let CL(D,Rn;p)={v∈CL(D,Rn):||v||L≤p},p∈R+. Denote by Λ the set of all functions λ:[0,a 0]×R+→R+such that λ(·,t)∈ L([0,a 0],R+) for each t∈R+and λ(s, ·) is nondecreasing on R+for almost every (a.e.) s∈[0,a 0]. Assumption H1.Suppose that 1. the matrix-valued function (·,y,v):[0,a 0]→Rnm is measurable for every (y, v)∈Rm×C(D, Rn) and (x, ·):Rm×C(D, Rn)→ Rnm is continuous for a.e. x∈[0,a 0]; 2. there exists d∈Λ such that ||(x, y, v)|| ≤ d(x, p) for all (y,v)∈Rm×C(D,Rn;p), a.e. x∈[0,a 0]; 3. there exists l∈Λ such that ||(x, y, v)−(x, ¯y, ¯v)|| ≤ l(x, p)[|y−¯y|m+||v−¯v||] for all (y,v),(¯y, ¯v)∈Rm×CL(D,Rn;p) and a.e. x∈[0,a 0]. Assumption H2.Suppose that 1. there exist constants ¯p∈0,1 2,¯q0∈R+and a function ¯ω0∈ L([−τ,0],R+) such that r  k=1 ||hk(x, y)|| ≤ ¯p, r  k=1 ||hk(x, y)−hk(¯x, ¯y)|| ≤ | ¯x x ¯ω0(s)ds|+¯q0|y−¯y|m; 2. there exist constants p0,q 0∈R+and a function ω0∈L([−τ,0],R+) such that |ϕ(x, y)|n≤p0, |ϕ(x, y)−ϕ(¯x, ¯y)|n≤¯x x ω0(s)ds +q0|y−¯y|m for all (x, y),(¯x, ¯y)∈I. 510 J. Turo For a∈(0,a 0] we denote by Cϕ,a[p, ω, q] the set of all functions v∈ C([−τ,a]×Rm,Rn) such that |z(x, y)|n≤p, |z(x, y)−z(¯x, ¯y)|n≤¯x x ω(s)ds +q|y−¯y|m for (x, y),(¯x, ¯y)∈Iaand vsatisfies condition (3) on I. We consider a Carath´eodory solution of (1), (2). More precisely, a function zis called a Carath´eodory solution of problem (1), (2) provided the following conditions hold: (i) z∈Cϕ,a[p, ω, q]; (ii) zsatisfies (1) almost everywhere in Iaand (2) everywhere in Rn. For z∈Cϕ,a[p, ω, q] we consider the following problem (5) η(t)=i(t, η(t),z (t,η(t))),η(x)=y, i =1,... ,n. Note that (5) is an ordinary differential equation. If Assumption H1is satisfied then for every z∈Cϕ,a[p, ω, q] the right hand side of (5) satisfies Carath´eodory assumptions and the following Lipschitz condition |i(t, ξ, z(t,ξ))−i(t, ¯ ξ,z(t,¯ ξ))|≤l(t, ra)(1 −q)|ξ−¯ ξ|m, holds, where ra=p+q+a −τω(s)ds. Thus, the existence and uniqueness of the solution gi[z](·;x, y):[0,a]→Rof (5) follows from classical theorems. Lemma 1. If Assumption H1is satisfied and z, ¯z∈Cϕ,a[p, ω, q]then (6) |gi[z](t;x, y)−gi[z](t;¯x, ¯y)|m ≤exp (1 + q)x t l(s, ra)ds¯x x d(s, p)ds +|y−¯y|m t∈[0,min(x, ¯x)],i=1,... ,n, and (7) |gi[z](t;x, y)−gi[¯z](t;x, y)|m ≤x t l(s, ra)ds exp (1 + q)x t l(s, ra)ds||z−¯z||a,t∈[0,x], Functional partial differential equations 511 where ||·|| adenotes the supremum norm in the space C(Ia,Rn). Proof: We will consider the case where x≤¯x. We have, by Assumption H1, |gi[z](t;x, y)−gi[z](t;¯x, ¯y)|m≤|y−¯y|m+¯x x d(s, p)ds +t x l(s, ra)(1 + q)|gi[z](s;x, y)−gi[z](s;¯x, ¯y)|mds ,t∈[0,x]. Hence,by the above inequality and by Gronwall’s inequality we get (6). We consider the case x>¯xanalogously to the case x≤¯x. It follows, from Assumption H1, that |gi[z](t;x, y)−gi[¯z](t;x, y)|m≤t x l(s, ra)[|gi[z](s;x, y)−gi[¯z](s;x, y)m +||z(s,gi[z](s;x,y)) −¯z(s,gi[¯z](s;x,y))||]ds|,t∈[0,x]. Since ||z(s,gi[z](s;x,y)) −¯z(s,gi[¯z](s;x,y))|| ≤||z−¯z||a+q|gi[z](s;x, y)−gi[¯z](s;x, y)|m, we get |gi[z](t;x, y)−gi[¯z](t;x, y)|m≤t x l(s, ra)ds ||z−¯z||a +t x l(s, ra)(1 + q)|gi[z](s;x, y)−gi[¯z](s;x, y)|ds . Utilizing the Gronwall’s inequality, we obtain (7). The proof of Lemma 1 is complete. Assumption H3.Suppose that 1. the vector-valued function f(·,y,v):[0,a 0]→Rnis measurable for every (y, v)∈Rm×C(D, Rn) and f(x, ·):Rm×C(D, Rn)→ Rnis continuous for a.e. x∈[0,a 0]; 2. there exists d1∈Λ such that |f(x, y, v)|n≤d1(x, p) for all (y,v)∈Rm×C(D,Rn;p), a.e. x∈[0,a 0]; 3. there exists l1∈Λ such that |f(x, y, v)−f(x, ¯y, ¯v)|| ≤ l1(x, p)[|y−¯y|m+||v−¯v||] for all (y,v),(¯y, ¯v)∈Rm×CL(D, Rn;p) and a.e. x∈[0,a 0]. 512 J. Turo 3. The main theorem Theorem 1. Suppose that Assumptions H1-H3are satisfied. Then there exist a∈(0,a 0],p, q ∈R+and ω∈L([−τ,a],R+)such that problem (1), (2) has a unique solution zin the class Cϕ,a[p, ω, q]. Proof: First, we take aso small that (8) a 0 d1(t, p)dt ≤1,K a= exp[La(1 + q)] ≤2, L1a(1 + q)≤1,R aLaKa+L1a<1 2, where La=a 0l(t, p)dt,L1a=a 0l1(t, p)dt and Ra=q0+¯q0p+¯pq + L1a(1 + q). Let us choose constants p, q with (9) p≥(1 −¯p)(1 + p0), q≥2(1 −2¯p)−1(q0+¯q0p+1) and function (10) ω(t) = max{qd(t, p)+d1(t, p),(1 −¯p)−1[ω0(t)+p¯ω(t)]}. We define the following operator (Tz)i(x, y)=ϕi(0,g i(0; x, y)) − r  k=1 hki(0,g i(0; x, y))zi(ak,g i(0; x, y)) +x 0 fi(t, gi(t;x, y),z (t,gi(t;x,y)))dt, (x, y)∈Ia, (Tz)i(x, y)=ϕi(x, y)− r  k=1 hki(x, y)zi(ak+x, y),(x, y)∈I. From Assumptions H2,H3and inequalities (8), (9), we obtain |(Tz)i(x, y)|≤|ϕi(0,g i(0; x, y))| + r  k=1 hki(0,g i(0; x, y))zi(ak,g i(0; x, y)) +x 0 fi(t, gi(t;x, y),z (t,gi(t;x,y)))dt ≤p0+¯pp +a 0 d1(t, p)dt ≤p, (x, y)∈Ia, Functional partial differential equations 513 and |(Tz)i(x, y)|≤|ϕi(x, y)|+ r  k=1 hki(x, y)zi(ak+x, y) ≤p0+¯pp ≤p, (x, y)∈I. For any (x, y),(¯x, ¯y)∈Ia, from Assumptions H1-H3and Lemma 1, we get |(Tz)i(x, y)−(Tz)i(¯x, ¯y)|≤Ra|gi(t;x, y)−gi(t;¯x, ¯y)|+¯x x d1(t, p)dt ≤RaKa¯x x d(t, p)dt +¯x x d1(t, p)dt +RaKa|y−¯y|m. For (x, y),(¯x, ¯y)∈I,wehave |(Tz)i(x, y)−(Tz)i(¯x, ¯y)|≤¯x x ω0(t)dt +¯p¯x x ω(t)dt +p¯x x ¯ω0(t)dt +(q0+¯pq +p¯q0)|y−¯y|m. Thus, from (8)-(10), we obtain |(Tz)i(x, y)−(Tz)i(¯x, ¯y)|≤¯x x ω(t)dt +q|y−¯y|m,(x, y)∈Ia. We see that Tz ∈Cϕ,a[p, ω, q]. Now, we prove that Tis a contraction. Indeed, for any z, ¯z∈Cϕ,a[p, ω, q], we have |(Tz)i(x, y)−(T¯z)i(x, y)|≤|ϕi(0,g[z](0; x, y)) −ϕi(0,g[¯z](0; x, y))| + r  k=1 [hki(0,g i[z](0; x, y)) −hki(0,g i[¯z](0; x, y))]zi(ak,g i[z](0; x, y)) + r  k=1 [hki(0,g i[¯z](0; x, y))[zi(ak,g i[z](0; x, y)) −¯zi(ak,g i[¯z](0; x, y))] +x 0 fi(t, gi[z](t;x, y),z (t,gi[z](t;x,y)))−fi(t, gi[¯z](t;x, y),¯z(t,gi[¯z](t;x,y)))] dt ≤(RaLaKa+L1a+¯p)||z−¯z||a,(x, y)∈Ia, 514 J. Turo and |(Tz)i(x, y)−(T¯z)i(x, y)|≤ r  k=1 hki(x, y)[zi(ak+x, y)−¯zi(ak+x, y) ≤¯p||z−¯z||a,(x, y)∈I. Thus, it follows from (8) that Tis a contraction. It remains to prove that the fixed point zof the operator Tis the Carath´eodory solution of (1), (2). By taking y=gi(x;0,η) in the equation zi(x, y)=(Tz)i(x, y), we get (11) zi(x, gi(x;0,η)) = ϕi(0,η)− r  k=1 hki(0,η)zi(ak,η) +x 0 fi(t, gi(t;0,η),z (t,gi(t;0,η)))dt, since the solution giof (5) satisfies the following group property gi(t;t, gi(t;x, y)) = gi(t;x, y). By differentiation of (11) with respect to x, using the chain rule differentiation Lemma (4.ii) of [7], and by putting again y=gi(x;0,η), we obtain that zsatisfies (1) almost everywhere in Ia. It follows immediately that zsatisfies (2). The proof of Theorem 1 is complete. 4. Special cases of system (1) We list below a few examples of problems which can be derived from (1) by specializing the functions and f. 1) Suppose that ¯:Ia0×Rn×Rn→Rnm and ¯ f:Ia0×Rn×Rn→Rn are given functions. Let (x, y, v)=¯x, y, v(0,0),D v(s, t)ds dt, f(x, y, v)= ¯ fx, y, v(0,0),D v(s, t)ds dt,(x, y, v)∈Ia×C(D,Rn). Then system (1) reduces to the differential-integral system Dxzi(x, y)+ m  j=1 ¯ij x, y, z(x, y),D z(x+s, y +t)ds dtDyjzi(x, y) =¯ fix, y, z(x, y),D z(x+s, y +t)ds dt,i=1,... ,n. Functional partial differential equations 515 2) Suppose that α=(α0,α ), β=(β0,β):Ia0×C(D,Rn)→R1+m, ¯:Ia0×Rn×Rn→Rnm and ¯ f:Ia0×Rn×Rn→Rnare given. Assume that (α0(x, y, v)−x, α(x, y, v)−y)∈D, (β0(x, y, v)−x, β(x, y, v)−y)∈D for (x, y, v)∈Ia0×C(D,Rn). Put (x, y, v)=¯(x, y, v(0,0),v(α0(x, y, v)−x, α(x, y, v)−y)), f(x, y, v)= ¯ f(x, y, v(0,0),v(β0(x, y, v)−x, β(x, y, v)−y)). Then system (1) reduces to the differential system with a retarded argument Dxzi(x, y)+ m  j=1 ¯ij (x, y, z(x, y),z(α0(x, y, z(x,y)),α(x, y, z(x,y))))Dyjzi(x, y) =¯ fi(x, y, z(x, y),z(β0(x, y, z(x,y)),β(x, y, z(x,y)))),i=1,... ,n. The functions αand βdepend on the functional argument. Therefore, we cannot apply existence theorems from [13],[14] to the above system. 3) If we take (x, y, v)=¯(x, y, v(0,0),(V(I(x,y)v))(x, y)), f(x, y, v)= ¯ f(x, y, v(0,0),(V(I(x,y)v))(x, y)), where (I(x,y)v)(s, t)=v(s−x, t −y) then system (1) reduces to the system of differential-functional equations ([13],[14]) Dxzi(x, y)+ m  j=1 ¯ij (x, y, z(x, y),(Vz)(x, y))Dyjzi(x, y) =¯ fi(x, y, z(x, y),(Vz)(x, y)),i=1,... ,n. References 1. P. Bassanini, The problem of Graffi-Cesari, Proc. Inter. Conf. Nonlinear Phenomena Math. Sci. Arlington, Acad. Press, USA (1982), 87–101.