Citation: Ayub Samadi, M. Mosaee Avini, M. Mursaleen. Solutions of an infinite system of integral equations of Volterra-Stieltjes type in the sequence spaces ℓp(1<p<∞) and c0[J]. AIMS Mathematics, 2020, 5(4): 3791-3808. doi: 10.3934/math.2020246
[1] | Ateq Alsaadi, Manochehr Kazemi, Mohamed M. A. Metwali . On generalization of Petryshyn's fixed point theorem and its application to the product of n-nonlinear integral equations. AIMS Mathematics, 2023, 8(12): 30562-30573. doi: 10.3934/math.20231562 |
[2] | Lakshmi Narayan Mishra, Vijai Kumar Pathak, Dumitru Baleanu . Approximation of solutions for nonlinear functional integral equations. AIMS Mathematics, 2022, 7(9): 17486-17506. doi: 10.3934/math.2022964 |
[3] | Manalisha Bhujel, Bipan Hazarika, Sumati Kumari Panda, Dimplekumar Chalishajar . Analysis of the solvability and stability of the operator-valued Fredholm integral equation in Hölder space. AIMS Mathematics, 2023, 8(11): 26168-26187. doi: 10.3934/math.20231334 |
[4] | Mohamed M. A. Metwali, Shami A. M. Alsallami . Discontinuous solutions of delay fractional integral equation via measures of noncompactness. AIMS Mathematics, 2023, 8(9): 21055-21068. doi: 10.3934/math.20231072 |
[5] | Cheng-shi Huang, Zhi-jie Jiang, Yan-fu Xue . Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball. AIMS Mathematics, 2022, 7(10): 18194-18217. doi: 10.3934/math.20221001 |
[6] | Kanagaraj Muthuselvan, Baskar Sundaravadivoo, Kottakkaran Sooppy Nisar, Suliman Alsaeed . Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation. AIMS Mathematics, 2023, 8(7): 16846-16863. doi: 10.3934/math.2023861 |
[7] | Mouffak Benchohra, Zohra Bouteffal, Johnny Henderson, Sara Litimein . Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces. AIMS Mathematics, 2020, 5(1): 15-25. doi: 10.3934/math.2020002 |
[8] | Saud Fahad Aldosary, Mohamed M. A. Metwali, Manochehr Kazemi, Ateq Alsaadi . On integrable and approximate solutions for Hadamard fractional quadratic integral equations. AIMS Mathematics, 2024, 9(3): 5746-5762. doi: 10.3934/math.2024279 |
[9] | Rahul, Nihar Kumar Mahato, Sumati Kumari Panda, Manar A. Alqudah, Thabet Abdeljawad . An existence result involving both the generalized proportional Riemann-Liouville and Hadamard fractional integral equations through generalized Darbo's fixed point theorem. AIMS Mathematics, 2022, 7(8): 15484-15496. doi: 10.3934/math.2022848 |
[10] | Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad . On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686 |
The interest for studying the theory of infinite systems of integral equations is based on the fact that the theory of infinite systems of integral equations is a branch of nonlinear analysis which has been applied in various fields of science and numerous applications. In fact, most physical and engineering problems are formed by infinite systems of integral equations, see for example [1,2,3,4]. The problem of the existence of solutions for infinite systems of integral equations plays a significant role in the investigation of these types of equations and it is important to apply original studies in our investigations (cf.[5,6,7]). In some papers, integral equations of Volterra type have been converted in the form of integral equations of Volterra-Stieltjes type and numerous results have been obtained on the existence of solutions of nonlinear integral equations (cf.[8,9]). The aim of this paper is to present some results on the existence of solutions for an infinite system of integral equations of Volterra-Stieltjes type of the form
un(t,x)=Fn(t,s,f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds);u(t,x)={ui(t,x)}∞i=1,ui(t,x)∈BC(R+×R+,R), | (1.1) |
where BC(R+×R+,R) is the space of all real functions u(t,x)=u:R+×R+⟶R, which are defined, continuous and bounded on the set R+×R+ with a supremum norm ‖u‖=sup{|u(t,x)|:(t,x)∈R+×R+}. The obtained results extend and generalize the results of [6,8,9] in the Banach spaces c0 and ℓp. In our approach, this is done by applying the measure of noncompactness and Darbo fixed point theorem.
In future, we apply some notations, definitions and preliminary facts to obtain our main results.
For a bounded subset S of a metric space X, Kuratowski [10] defined the function α(S) by the formula
α(S)=inf{δ>0:S=n⋃i=1Si,diam(Si)≤δfor1≤i≤n<∞}, |
known as the Kuratowski measure of noncompactness. Another measure of noncompactness is the Hausdorff measure of noncompactness given by:
χ(S)=inf{ε>0:ShasfinitenetinX}. |
Let E be a real Banach space with norm ‖.‖ and zero element θ. Besides, we suppose ¯X and Conv(X) denote the closure and convex hull of X, respectively. Moreover, let us denote by ME the family of all nonempty and bounded subsets of E and by NE its subfamily consisting of all relatively compact sets.
Definition 1. [11] A mapping μ:ME⟶[0,∞) is called a measure of noncompactness if it satisfies the following conditions:
(1) The set Kerμ={X∈ME:μ(X)=0} is nonempty and Kerμ⊆NE.
(2) X⊆Y⟹μ(X)≤μ(Y).
(3) μ(¯X)=μ(X).
(4) μ(Conv(X))=μ(X).
(5) μ(λX+(1−λ)Y)≤λμ(X)+(1−λ)μ(Y) for λ∈[0,1].
(6) If {Xn} is a sequence of closed sets from ME such that Xn+1⊆Xn for n=1,2,… and limn→∞μ(Xn)=0, then ⋂∞n=1Xn is nonempty.
We will apply the following theorem as the main tool in our investigations.
Theorem 1. (Darbo[12]) Let C be a nonempty, bounded, closed and convex subset of a Banach space E and T:C→C be a continuous mapping. Assume that there exists a constant K∈[0,1) such that μ(TX)≤Kμ(X) for any nonempty subset X of C, where μ is a measure of noncompactness defined in E. Then T has at least a fixed point in C.
Samadi [13] extended Darbo's fixed point theorem as follows.
Theorem 2. Let C be a nonempty bounded, closed and convex subset of a Banach space E. Assume T:C⟶C be a continuous operator satisfying
θ(μ(X))+f(μ(T(X)))≤f(μ(X)) | (2.1) |
for all nonempty subsets X of C, where μ is an arbitrary measure of noncompactness defined in E and (θ,f)∈Δ=. Then T has a fixed point in C.
In Theorem 2, Δ is the set of all pairs (θ,f) satisfying the following:
(Δ1) θ(tn)↛0 for each strictly increasing sequence {tn};
(Δ2) f is strictly increasing function;
(Δ3) for each sequence {αn} of positive numbers, limn→∞αn=0 if and only if limn→∞f(αn)=−∞.
(Δ4) If {tn} is a decreasing sequence such that tn→0 and θ(tn)<f(tn)−f(tn+1), then we have ∑∞n=1tn<∞.
We know that the Hausdorff measure of noncompactness χ in the Banach space ℓp can be defined as follows:
χ(B)=limn⟶∞{supx∈B{Σk≥n∣xk∣p}1p}, | (2.2) |
where B∈Mℓp and x=(xk)∈ℓp. For the Banach space (c0,‖.‖c0), the Hausdorff measure of noncompactnes χ is given by (cf. Definition 1):
χ(B)=limn⟶∞{supu∈B{maxk≥n∣uk∣}}, | (2.3) |
where B∈Mc0 and u=(uk)∈c0.
Now, we recall some basic facts concerning the concept of the variation of a function (cf.[14,15]).
Assume that f is a real function defined on the interval [a,b]. The variation of the function f will be denoted by ⋁baf. If ⋁baf is finite, the function f has bounded variation on the interval [a,b]. Similarly, if g:[a,b]×[c,d]⟶R is a real function of two variables, then the variation of the function t⟶g(t,s) on the interval [p,q]⊆[a,b] will be denoted by ⋁qs=pg(t,s). Analogously, we can define ⋁qt=pg(t,s). Assume that f and g are two real functions defined on the interval [a,b], then under appropriate conditions we can define the Steiltjes integral ∫baf(t)dg(t) of the function f with respect to the function g. If the integral ∫baf(t)dg(t) is finite, then f is Stieltjes integrable on the interval [a,b].
The following lemmas will be applied in our investigations.
Lemma 1. If f is Stieltjes integrable on the interval [a,b] with respect to a function g of bounded variation, then
|∫baf(t)dg(t)|≤∫ba|f(t)|d(⋁tag). |
Lemma 2. Let f1 and f2 be Stieltjes integrable functions on the interval [a,b] with respect to a nondecreasing function g such that f1(t)≤f2(t) for t∈[a,b]. Then,
∫baf1(t)dg(t)≤∫baf2(t)dg(t). |
In this section, as an application of Theorem 2, the existence of solutions for the infinite system (1.1) is studied in the spaces ℓp and c0. First, we show that infinite system (1.1) has a solution that belongs to the space ℓp.
We consider the following conditions:
(H1)Fn:R+×R+×R×R⟶R is continuous and there exist positive real numbers τ>0 such that
|Fn(t,s,x1,y1)−Fn(t,s,x2,y2)|p≤e−τ(|x1−x2|p+|y1−y2|p), |
for all t,s∈R+ and x1,x2,y1,y2∈R. Moreover, we have
limi⟶∞Σ∞i=1|Fi(t,s,0,0)|p=0,N1=Σ∞i=1|Fi(t,s,0,0)|p. |
(H2)f1:R+×R∞⟶R is continuos with f0=supt∈R+|f(t,0)| and there exist positive real numbers τ>0 such that
|f1(t,u(t,x))−f1(t,v(t,x))|p≤e−τ‖u(t,x)−v(t,x)‖ℓp,|f1(t,u(t,x))|p≤e−τ‖u(t,x)‖ℓp. |
for all t,x∈R+ and u(t,x)={ui(t,x)}∞i=1,v(t,x)={vi(t,x)}∞i=1∈ℓp.
(H3)T:BC(R+×R+,ℓp)⟶BC(R+×R+,R) is a continuos operator such that
|(Tu)(t,x)−(Tv)(t,x)|≤‖u(t,x)−v(t,x)‖ℓp,|(Tu)(t,x)|≤1. |
for all u,v∈BC(R+×R+,ℓp) and t,x∈R+.
(H4) For any fixed t>0 the function s⟶gi(t,s) has a bounded variation on the interval [0,t] and the function t⟶⋁ts=0gi(t,s) is bounded over R+.
(H5)gn:R+×R+×R+×R+×R∞⟶R is continuous and there exist continuous functions an:R+×R+⟶R+ such that
|gn(t,s,x,y,u(t,x))|≤an(t,s),limt⟶∞Σn≥1∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|dst⋁q=0g1(t,q)=0,φk=sup{Σn≥k[|∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s)|];t,s,x,y∈R+,u(t,x)∈R∞}. |
Moreover, assume that
A=sup{Σ∞n=1∫t0an(t,s)dss⋁p=0g1(t,p),t∈R+},G=sup{x⋁y=0g2(x,y);x∈R+},limk⟶∞φk=0. |
(H6)Vn:R+×R+×R∞⟶R is a continuous function and there exists continuous function k:R+×R+⟶R+ such that the function s⟶k(t,s) is integrable over R+ and the following conditions hold:
|Vn(t,s,u(t,x))|≤k(t,s)|un(t,x)|p,|Vn(t,s,u(t,x))−Vn(t,s,v(t,x)|≤|un(t,x)−vn(t,x)|pk(t,s). |
for all t,s,x∈R+ and u,v∈ℓp. Moreover, assume that
M=supt∈R+∫∞0k(t,s)ds. |
(H7) There exists a positive solution r0 such that
22pe−2τrp0(GA)p+22pe−τfp0(GA)p+2pe−τrp0Mp+2pN1≤rp0, |
Moreover, assume that 2pM<1.
Theorem 3. Under the assumptions (H1)−(H7), Eq (1.1) has at least one solution u(t,x)={ui(t,x)}∞i=1 in the space ℓp.
Proof. Let us define the operator G on BC(R+×R+,ℓp) by
(Gu)(t,x)={Fn(t,s,f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds)}. |
In view of our assumptions, for all t,x∈R+, we get
‖(Gu)(t,x)‖pℓp=Σ∞i=1|Fi(t,s,f1(t,u(t,x))∫t0∫x0gi(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds)|p≤2pΣ∞i=1|Fi(t,s,f1(t,u(t,x))∫t0∫x0gi(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds)−Fi(t,s,0,0)|p+2pΣ∞i=1|Fi(t,s,0,0)|p≤2pΣ∞i=1[e−τ|f1(t,u(t,x))∫t0∫x0gi(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s)|p+e−τ|(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds|p]+2pΣ∞i=1|Fi(t,s,0,0)|p≤22pe−τΣ∞i=1(|f1(t,u(t,x))−f1(t,0)|p)×(∫t0∫x0|gi(t,s,x,y,u(t,x))|dy⋁yq=0g2(x,q)⋁sp=0dsg1(t,p))p+22pe−τΣ∞i=1|f1(t,0)|p(∫t0∫x0|gi(t,s,x,y,u(t,x))|dy⋁yq=0g2(x,q)⋁sp=0dsg1(t,p))p+2pe−τ(∫∞0k(t,s)ds)pΣ∞i=1|ui(t,x)|p+2pΣ∞i=1|Fi(t,s,0,0)|p≤22pe−2τ‖u(t,x)‖pℓp(GA)p+22pe−τ(f0)p(GA)p+2pe−τMp‖u(t,x)‖pℓp+2pN1. | (3.1) |
Thus, by applying the last estimates and assumption (H7) one can easily seen that G maps ¯Br0 into itself, where
¯Br0={u∈BC(R+×R+,ℓp);‖u‖BC(R+×R+,ℓp)≤r0}. |
Next, we prove that the operator G is a continuous operator on the Ball ¯Br0. For this, take ε>0 arbitrarily and u(t,x)={ui(t,x)}∞i=1,v(t,x)={vi(t,x)}∞i=1∈¯Br0 with ‖u−v‖BC(R+×R+,ℓp)<ε. Acordingly, taking into account our assumptions, for (t,x)∈R+×R+ we have
‖(Gu)(t,x)−(Gv)(t,x)‖pℓp≤Σ∞i=1e−τ|f1(t,u(t,x))∫t0∫x0gi(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s)−f1(t,v(t,x))∫t0∫x0gi(t,s,x,y,v(t,x))dyg2(x,y)dsg1(t,s)|p+Σ∞i=1e−τ|(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds−(Tv)(t,x)∫∞0Vi(t,s,v(t,x))ds|p. | (3.2) |
On the other hand, we have
|f1(t,u(t,x))∫t0∫x0gi(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s)−f1(t,v(t,x))∫t0∫x0gi(t,s,x,y,v(t,x))dyg2(x,y)dsg1(t,s)|p≤2p|f1(t,u(t,x))−f1(t,v(t,x))|p×(∫t0∫x0|gn(t,s,x,y,u(t,x))|dy⋁yp=0g2(x,p)ds⋁tq=0g1(t,q))p+2p|f1(t,v(t,x))|p(∫t0∫x0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|dy⋁yp=0g2(x,p)ds⋁tq=0g1(t,q))p≤e−τ2p‖u(t,x)−v(t,x)‖ℓp(⋁xy=0g2(x,y)|∫t0an(t,s)ds⋁tq=0g1(t,q))p+2p|f1(t,v(t,x))|p(⋁xy=0g2(x,y)∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|ds⋁tq=0g1(t,q))p≤e−τ2p‖u(t,x)−v(t,x)‖ℓp(GAi)p+2pGp|f1(t,v(t,x))(∫t0|gi(t,s,x,y,u(t,x))−gi(t,s,x,y,v(t,x))|ds⋁tq=0g1(t,q))p. | (3.3) |
Further, by applying our assumptions, we arrive that
|(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds−(Tv)(t,x)∫∞0Vi(t,s,v(t,x))ds|p≤2p|(Tu)(t,x)∫∞0Vi(t,s,u(t,x))ds−(Tv)(t,x)∫∞0Vi(t,s,u(t,x))ds|p+2p|(Tv)(t,x)∫∞0Vi(t,s,u(t,x))ds−(Tv)(t,x)∫∞0Vi(t,s,v(t,x))ds|p≤2p‖u(t,x)−v(t,x)‖pℓp|ui(t,x)|pMp+Mp|ui(t,x)−vi(t,x)|p. | (3.4) |
Combining (3.2), (3.3) and (3.4), we conclude that
‖(Gu)(t,x)−(Gv)(t,x)‖pℓp≤Σ∞i=1e−2τ2p‖u(t,x)−v(t,x)‖pℓp(GAi)p+2pGpe−τ|f1(t,v(t,x))|p(Σ∞i=1∫t0|gi(t,s,x,y,u(t,x))−gi(t,s,x,y,v(t,x))|ds⋁tq=0g1(t,q))p+Σ∞i=1|ui(t,x)|pe−τ2pMp‖u(t,x)−v(t,x)|pℓp+e−τ2pMpΣ∞i=1|ui(t,x)−vi(t,x)|p. | (3.5) |
Using (H5), there exists T>0 such that for t>T, we get
Σ∞i=1∫t0|gi(t,s,x,y,u(t,x))−gi(t,s,x,y,v(t,x))|dst⋁q=0g1(t,q)<ε. |
Hence, by (3.5), we conclude that
‖(Gu)(t,x)−(Gv)(t,x)‖pℓp≤2pe−2τ‖u−v‖pBC(R+×R+,ℓp)(GA)p+2pGpεpe−τ‖v(t,x)‖pℓp+2pMpe−τ‖u−v‖pBC(R+×R+,ℓp)‖u(t,x)‖pℓp+e−τ2pMp‖u−v‖pBC(R+×R+,ℓp). | (3.6) |
For t∈[0,T] we have
‖(Gu)(t,x)−(Gv)(t,x)‖pℓp≤2pe−2τ‖u−v‖pBC(R+×R+,ℓp)(GA)p+‖v(t,x)‖pℓp2pGpω(g,ε)pe−τ+2pMpe−τ‖u−v‖pBC(R+×R+,ℓp)‖u(t,x)‖pℓp+e−τ2pMp‖u−v‖pBC(R+×R+,ℓp), | (3.7) |
where
ω(g,ε)=sup{Σ∞n=1|gn(t,s,x,,y,u)−gn(t,s,x,y,v)|;(t,s)∈Δ1,(x,y)∈Δ2,u,v∈ℓp,‖u−v‖BC(R+,R+,ℓp)<ε},Δ1={(t,s)∈R2;s≤t≤T},Δ2={(x,y)∈R2;y≤x≤T}. |
and ω(g,ε)⟶0 as ε⟶0. Consequently, G is continuous on the ball ¯Br0. To finish the proof, we prove that the condition (2.1) of Theorem 2 is fulfilled. Let X be a nonempty and bounded subset of the ball ¯Br0. Assume that
(Hn)(u)=f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Dn)(u)=(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds. |
Thus, by applying our assumptions, we infer that
χℓp(G(X))(t,x)=limn⟶∞[supu(t,x)∈X{Σk≥n|Fk(t,s,(Hk)(u),(Dk)(u))|p}1p]=limn⟶∞[supu(t,x)∈X{Σk≥n|Fk(t,s,(Hk)(u),(Dk)(u))−Fk(t,s,0,0)+Fk(t,s,0,0)|p}1p]≤2pe−τlimn⟶∞[supu(t,x)∈X{Σk≥n{|(Hk)(u)|p+|(Dk)(u)|p}}1p]≤2pe−τlimn⟶∞[supu(t,x)∈X{Σk≥n{e−τ‖u(t,x)‖pℓpφn+Mp|uk(t,x)|p}1p]=2pe−τMlimn⟶∞[supu(t,x)∈X{Σk≥n|uk(t,x)|p}1p]. | (3.8) |
Hence,
χℓp(G(X))(t,x))≤2pe−τMlimn⟶∞[supu(t,x)∈X{Σk≥n|uk(t,x)|p}1p]. | (3.9) |
Consequently,
sup(t,x)∈R+×R+χℓp(G(X))(t,x))=χBC(R+×R+,ℓp)(GX)≤sup(t,x)∈R+×R+2pe−τMlimn⟶∞[supu(t,x)∈X{Σk≥n|uk(t,x)|p}1p]. |
By passing to logarithms, we get
ln(χBC(R+×R+,ℓp))(GX))+τ≤ln(χBC(R+×R+,ℓp)(X)) | (3.10) |
Now applying Theorem 2 with f(t)=ln(t) and θ(t)=τ, we obtain that G has a fixed point and the proof is completed.
Example 1. Now, we investigate the following system of integral equations:
un(t,x)=(e−τ−t−n)1p2sin((e−t−τ)1psin(‖u(t,x)‖ℓp)2×∫t0∫x0arctan(12n×e−3t+s8+|x|+|y|+|un(t,x)|)ex1+y2e2xet1+t2dyds+cos(11+‖u(t,x)‖lp)∫∞0e−s1+t8sin(|un(t,x)|)ds); | (3.11) |
Observe that Eq (3.11) is a special case of the infinite system (1.1) if we put
Fn(t,s,x,y)=(e−τ−t−n)1p2sin(x+y),gn(t,s,x,y,u(t,x))=arctan(12n×e−3t+s8+|x|+|y|+|un(t,x)|),f1(t,u(t,x))=(e−t−τ)1psin(‖u(t,x)‖ℓp)2,an(t,s)=12ne−3t+s,g1(t,s)=set1+t2,g2(x,y)=arctan(yex),Vn(t,s,u(t,x))=e−s1+t8sin(|un(t,x)|),k(t,s)=e−s1+t8,(Tu)(t,x)=cos(11+‖u(t,x)‖lp). |
Thus, it is easily seen that Fn and f1 satisfy assumptions (H1) and (H2) with N1=0 and f0=0. Further, the operator T satisfies hypothesis (H3). To justify assumption (H5), let t,sx,y∈R+ and u,u∈ℓp. Then, we have
|gn(t,s,x,y,u(t,x))|≤12ne−3t+s=an(t,s). |
Since ∂g1∂s=et1+t2>0, then ⋁sq=0g1(t,q)=g1(t,s)−g1(t,0)=set1+t2. Consequently, we have
limt⟶∞∫t0an(t,s)dss⋁q=0g1(t,q)=limt⟶∞∫t012ne−3t+s(et1+t2)ds=limt⟶∞12ne−2t+s1+t2|t0=0 |
Inconsequence,
limt⟶∞Σn≥1∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|dst⋁q=0g1(t,q)=0,A=sup{Σ∞i=1∫t0an(t,s)dss⋁p=0g1(t,s),t∈R+},φk=sup{Σn≥k[∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s)];t,s,x,y∈R+,u(t,x)∈ℓp}≤G(e−2t1+t2−e−t1+t2)Σn≥k12n. |
So, φk⟶0. On the other hand the function Vn(t,s,u(t,x))=e−s1+t8sin(|un(t,x)|) verifies assumption (H6) with k(t,s)=e−s1+t8 and M=1. To show that the functions g1 and g2 satisfy assumption (H4), let first note that the functions g1 and g2 are increasing on every interval of the form [0,t] and g2 is bounded on the triangle △2. Consequently, the function y⟶g2(x,y) has bounded variation on the interval [0,x] and we have
x⋁y=0g2(x,y)=g2(x,y)−g2(x,0)=g2(x,y)≤π4. |
So, G≤π4. We can take G=π4. Consequently, all conditions of Theorem 3 are satisfied and Theorem 3 implies that the infinite system (3.11) has at least one solution which belongs to the space ℓp.
Now the existence of solutions of the system (1.1) is studied in the space c0. In this case, we need the following assumptions.
(D1)Fn:R+×R+×R×R⟶R is continuous and there exist positive real numbers τ>0 such that
|Fn(t,s,x1,y1)−Fn(t,s,x2,y2)|≤e−τ(|x1−x2|+|y1−y2|), |
for all t,s∈R+ and x1,x2,y1,y2∈R. Moreover, assume that
limi⟶∞|Fi(t,s,0,0)|=0,M1=sup{|Fi(t,s,0,0)|;t,s∈R+,i≥1}. |
(D2)f1:R+×R∞⟶R is continuous with f0=supt∈R+|f(t,0)| and there exist positive real numbers τ>0 such that
|f1(t,u(t,x))−f1(t,v(t,x))|≤e−τsupn≥1{|ui(t,x)−vi(t,x)|;i≥n},|f1(t,u(t,x))|≤e−τsupn≥1{|ui(t,x)|;i≥n} |
for all t,x∈R+ and u(t,x)={ui(t,x)},v(t,x)={vi(t,x)}∈c0
(D3)T:BC(R+×R+,c0)⟶BC(R+×R+,R) is a continuous operator such that
|(Tu)(t,x)−(Tv)(t,x)|≤supn≥1{|ui(t,x)−vi(t,x)|;i≥n},|(Tu)(t,x)|≤1. |
for all u,v∈BC(R+×R+,c0) and t,x∈R+.
(D4) For any fixed t>0 the function s⟶gi(t,s) has a bounded variation on the interval [0,t] and the functions t⟶⋁ts=0gi(t,s) are bounded on R+. Moreover, for arbitrarily fixed T>0 the function w⟶⋁wz=0gi(w,z) is continuous on the interval [0,T] for i=1,2.
(D5)gn:R+×R+×R+×R+×R∞⟶R is continuous and there exist continuous functions an:R+×R+⟶R+ such that
|gn(t,s,x,y,u(t,x))|≤an(t,s),limt⟶∞∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|dst⋁q=0g1(t,q)=0, |
for all t,s,x,y∈R+ and u,v∈R∞. Moreover, assume that
limn⟶∞∫t0an(t,s)dss⋁p=0g1(t,p)=0,A=sup{∫t0an(t,s)dss⋁p=0g1(t,p);n∈N},G=sup{x⋁y=0g2(x,y);x∈R+},G1=sup{w⋁z=0g1(w,z);w∈[0,T]}. |
where T>0 is arbitrarily fixed.
(D6)Vn:R+×R+×R∞⟶R is a continuous function and there exists continuous function k:R+×R+⟶R+ such that the function s⟶k(t,s) is integrable over R+ and the following conditions hold:
|Vn(t,s,u(t,x))|≤k(t,s)supn≥1{|ui(t,x)|;i≥n},|Vn(t,s,u(t,x))−Vn(t,s,v(t,x)|≤supn≥1{|ui(t,x)−vi(t,x);i≥n}k(t,s). |
for all t,s,x∈R+ and u,v∈c0. Moreover, assume that
M=supt∈R+∫∞0k(t,s)ds<1,e−2τGA+f0GAe−τ+Me−τ+Me−τ<1. |
Theorem 4. Under assumptions (D1)−(D6), the infinite system (1.1) has at least one solution u(t)={ui(t,x)}∞i=1 belonging to the space c0.
Proof. Define the operator G on the space BC(R+×R+,c0) as
(Gu)(t,x)={Fn(t,s,f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds)} |
where t,x∈R+. We show that
¯Br0={u∈BC(R+×R+,c0);‖u‖BC(R+×R+,c0)≤r0} |
is G-invariant where i=1,2,... and t,x∈R+. Assume that
(Hn)(u)=∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Dn)(u)=(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds. |
For arbitrarily fixed (t,x)∈R+×R+, we have
‖(Gu)(t,x)‖c0=supn≥1|Fn(t,s,(Hn)(u),(Dn)(u))|≤supn≥1[|Fn(t,s,f1(t,u(t,x))(Hn)(u),(Dn)(u))−Fn(t,s,0,0)|+|Fn(t,s,0,0)|]≤supn≥1[e−τ|(f1(t,u(t,x))Hn)(u)|+e−τ|(Dn)(u)|]+supn≥1|Fn(t,s,0,0)|≤supn≥1[e−τ(|f1(t,u(t,x))−f1(t,0)|+|f1(t,0)|)(|Hn)(u)|+e−τ‖u(t,x)|c0M]≤supn≥1[e−2τ{|ui(t,x)|;i≥n}GA+f0GAe−τ+e−τ‖u(t,x)‖c0M≤(e−2τGA+e−τf0GA+Me−τ)‖u(t,x)‖c0. |
Consequently,
‖Gu‖≤‖u(t,x)‖c0 | (4.1) |
By applying (4.1), one can easily seen that G maps the ball ¯Br0 into itself. Next, the continuity property of the operator G will be proved on the ball ¯Br0. Let u,v∈Br0 and ε>0 such that ‖u−v‖BC(R+×R+,c0)<ε. Thus for all t,x∈R+, we have
‖(Gu)(t,x)−(Gv)(t,x)‖c0=supn≥1|Fn(t,s,f1(t,u(t,x))Hn(u),(Dnu))−Fn(t,s,f1(t,v(t,x))Hn)(v),(Dnv))|≤supn≥1{e−τ|f1(t,u(t,x))Hn)(u)−f1(t,v(t,x))Hn)(v)|+e−τ|(Dn)(u)−(Dn)(v)|}. | (4.2) |
Besides, we have
|f1(t,u(t,x))Hn)(u)−f1(t,v(t,x))Hn)(v)|≤2pGAe−τsupn≥1{|ui(t,x)−vi(t,x)|;i≥n}+2pe−τGsupn≥1{|vi(t,x)|;i≥n}×∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))ds(⋁tq=0g1(t,q). | (4.3) |
By assumption (D5), there exists T>0 such that for t>T, we have
∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|ds(t⋁q=0g1(t,q)<ε. |
Further, the assumptions (D3) and (D6) give us the following eastimates
|(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds−(Tv)(t,x)∫∞0Vn(t,s,v(t,x))ds|≤M‖u(t,x)−v(t,x)‖c0‖u(t,x)‖c0+|(Tv)(t,x)|∫∞0|Vn(t,s,u(t,x))−Vn(t,s,v(t,x))|ds≤M‖u(t,x)−v(t,x)‖c0‖u(t,x)‖c0+M‖u(t,x)−v(t,x)‖c0. | (4.4) |
Applying (4.2), (4.3) and (4.4), we have
‖(Gu)(t,x)−(Gv)(t,x)‖c0≤2pe−2τGAsupn≥1{|ui(t,x)−vi(t,x)|;i≥n}+2pe−2τGsupn≥1{|vi(t,x)|;i≥n}ε+M‖u−v‖BC(R+×R+,c0)+M(‖u(t,x)‖c0)‖u−v‖BC(R+×R+,c0)≤2pe−2τGAε+2pe−τG‖v(t,x)‖c0ε+e−τMε+Me−τ‖u(t,x)‖c0)ε. | (4.5) |
For t∈[0,T], we have
‖(Gu)(t,x)−(Gv)(t,x)‖c0≤2pe−2τGAsupn≥1{|ui(t,x)−vi(t,x)|;i≥n}+2pe−τGsupn≥1{|ui(t,x);i≥n}G1ω(gn,ε)+M‖u−v‖c0+M‖u‖BC(R+×R+,c0)‖u−v‖BC(R+×R+,c0)≤e−τGAε+e−τGG1‖v(t,x)‖c0ω(gn,ε)+Mε+M‖u‖BC(R+×R+,c0)ε, | (4.6) |
where
ω(gn,ε)=sup{|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|;(t,s)∈Δ1,(x,y)∈Δ2,u,v∈R∞;‖u−v‖BC(R+×R+,c0)<ε}. |
Moreover, in light of the continuity of V on △1×△2×R∞, we have ω(gn,ε)⟶0. Now, combining (4.5) and (4.6) implies that G is continuous on the Ball ¯Br0. In what follows let X be a nonempty subset of the ball ¯Br0, In view of the formula (2.3) and our assumptions, we have
χc0(GX)(t,x)=limn⟶∞{supu∈X(maxi≥n|Fi(t,s,(Hi)(u),(Di)(u)|)}≤limn⟶∞{supu∈X(maxi≥n|Fi(t,s,(Hi)(u),(Di)(u)|)−Fi(t,s,0,0)|+|Fi(t,s,0,0)|}≤limn⟶∞{supu∈X(maxi≥n(e−τ|(Hi)(u)|+e−τ|(Di)(u)|))}≤limn⟶∞{supu∈X(maxi≥n(e−τ|f1(t,u(t,x))−f1(t,0)|(Hi)(u)|+e−τ|f1(t,0)|(Hi)(u)|+e−τ|(Di)(u)|))}≤limn⟶∞{supu∈X(maxi≥n(e−2τsupn≥1{|ui(t,x);i≥n}GA+f0GA+e−τsupn≥1{|ui(t,x);i≥n}M)}. |
Consequently,
χBC(R+×R+,c0)(GX)≤Me−τsup(t,x)∈R+×R+limn⟶∞{supu∈X(maxi≥n|ui(t,x)|)}. |
As, M<1, by passing to logarithms, we have
τ+ln(χBC(R+×R+,c0)(GX))≤ln(χBC(R+×R+,c0)(X))). |
Thus all conditions of Theorem 2 hold true with f(t)=ln(t) and θ(t)=τ and by Theorem 2 there exists {ui(t,x)}∞i=1∈c0 such that
un(t,x)=Fn(t,s,f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds). | (4.7) |
Hence, the proof is completed.
Example 2.
Nowweinvestigateun(t,x)=e−t−s−τ−n3√5√arctan(e−τΣk≥n|uk(t,x)|1+k2)(Hn)(u)+7√(Dn)(u) | (4.8) |
on the space c0. Taking
(Dn)(u)=e−100Σk≥nsin(|uk(t,x)|)(1+k2)∫∞0e−t−s−nΣk≥n|uk(t,x)|10n(1+k2)ds,(Hn)(u)=∫t0∫x0arctan(es+t2−n8+|u(t,x)|)e−2t1+t2×ex1+y2e2xdyds,Fn(t,s,x,y)=e−τ−t−s−n3√5√x+7√y,f1(t,u(t,x))=arctan(e−τΣk≥n|uk(t,x)|1+k2),gn(t,s,x,y,u(t,x))=arctan(es+t2−n8+|u(t,x)|),g1(t,s)=se−2t1+t2,g2(x,y)=arctan(yex),Vn(t,s,u(t,x))=e−t−s−nΣk≥n|uk(t,x)|10n(1+k2),k(t,s)=e−t−s,(Tu)(t,x)=e−100Σk≥nsin(|uk(t,x)|)(1+k2)n∈N, |
in the system (1.1), the system of integral Eq (4.8) is obtained. Note that the functions Fn and f1 satisfy conditions (D1) and (D2). Indeed, we have
|Fn(t,x1,y1)−Fn(t,x2,y2)|=e−τ−n−t[|3√5√x1+7√y1−3√5√x1+7√y1|]≤e−τ[3√∣5√x1+7√y1−5√x2−7√y2∣]≤e−τ[3√5√∣x1−x2∣+7√∣y1−y2|]≤e−τ[|x1−x2|+|y1−y2|],M1=0,limn⟶∞Fn(t,s,0,0)=0,|f1(t,u(t,x))|≤supn≥1{|ui(t,x)|;i≥n},|f1(t,u(t,x))−f1(t,v(t,x))|≤supn≥1{|ui(t,x)|−|vi(t,x);i≥n} |
Also, it can easily be seen that the operator T satisfies assumption (D3) and
|(Tu)(t,x)|≤e−100π26supn≥1{|ui(t,x)|;i≥n},|(Tu)(t,x)−(Tv)(t,x)|≤e−τπ26supn≥1{|ui(t,x)−vi(t,x)|;i≥n}. |
Moreover, since ∂g1∂s=e−2t1+t2>0, so g1 is increasing and we have
s⋁q=0g1(t,q)=g1(t,s)−g1(t,0)=g1(t,s)=se−2t1+t2>0 |
Consequently,
|gn(t,s,x,y,u(t,x))|≤es+t2−n,limt⟶∞∫t0|gn(t,s,x,y,u(t,x))−gn(t,s,x,y,v(t,x))|dst⋁q=0g1(t,q)≤ 2limt⟶∞∫t0et+se−2t1+t2ds=0 |
Again, we have
y⋁q=0g2(x,y)=g2(x,y)−g2(x,0)=g2(x,y)≤π4,limn⟶∞∫t0an(t,s)dss⋁q=0g1(t,q)=limn⟶∞2−n(11+t2−e−t1+t2)=0. |
So, G=π4 and A<∞. On the other hand the function Vn(t,s,u(t,x))=e−t−s−nΣk≥n|uk(t,x)|10n(1+k2) verifies assumption (D6) with k(t,s)=e−t−s and M=1. By applying the continuity of the function h⟶⋁wz=0gi(h,z) on the interval [0,T] we can take G1=sup{⋁wz=0g1(w,z):w∈[0,T]} where T>0 is arbitrarily fixed. Thus all conditions of Theorem 4 are satisfied and by applying Theorem 4, infinite system (4) has at least one solution in the space c0
We studied the existence of solutions for an infinite system of integral equations of Volterra-Stieltjes type of the following form in the Banach sequence spaces ℓp and c0 via the techniques of measures of noncompactness and Darbo's fixed point theorem.
un(t,x)=Fn(t,s,f1(t,u(t,x))∫t0∫x0gn(t,s,x,y,u(t,x))dyg2(x,y)dsg1(t,s),(Tu)(t,x)∫∞0Vn(t,s,u(t,x))ds);u(t,x)={ui(t,x)}∞i=1,ui(t,x)∈BC(R+×R+,R), |
where BC(R+×R+,R) is the space of all real functions u(t,x)=u:R+×R+⟶R, which are defined, continuous and bounded on the set R+×R+ with a supremum norm ‖u‖=sup{|u(t,x)|:(t,x)∈R+×R+}. Some examples in the Banach sequence spaces ℓp and c0 are also given to ascertain the usefulness of our main result.
Research of the author M. Mursaleen was supported by SERB Core Research Grant, DST, New Delhi, under grant NO. EMR/2017/000340.
All authors declare no conflicts of interest in this paper.
[1] | K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. |
[2] | S. Chandrasekhar, Radiative Transfer, Oxford University Press, London, 1950. |
[3] | C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. |
[4] | A. Aghajani, M. Mursaleen, A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci., 35B (2015), 552-566. |
[5] | A. Das, B. Hazarika, M. Mursaleen, Application of measure of noncompactness for solvability of the infinite system of integral equations in two variables in ℓp(1<p<∞), Revista de la Real Academia de Ciencias Exactas, Fısicas y Naturales, 113 (2017), 31-40. |
[6] |
A. Das, B. Hazarika, R. Arab, et al. Solvability of the infinite system of integral equations in two variables in the sequence spaces c0 and ℓp, Jour. Comput. Appl. Math., 326 (2017), 183-192. doi: 10.1016/j.cam.2017.05.035
![]() |
[7] | M. Ghasemi, M. Khanehgir, R. Allahyari, On solutions of infinite systems of integral equations in navaribles in spaces of tempered sequences c.β0 and l.β1, J. Math. Anal., 6 (2018), 1-16. |
[8] | J. Banas, A. Dubiel, Solutions of a quadratic Volterra-Stieltjes integral equation in the class of functions converging at infinity, Electron. J. Qualitative Theory Differential Equations, 80 (2018), 1-17. |
[9] | B. Rzepka, Solvability of a nonlinear Volterra-Stieltjes integral equation in the class of bounded and continuous functions of two variables, Revista de la Real Academia de Ciencias Exactas, Fısicas y Naturales, 112 (2018), 311-329. |
[10] | K. Kuratowski, Sur les espaces completes, Fund. Math., 15 (1934), 301-335. |
[11] | J. Banas, K. Goebel, Measures of Noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, New York, 1980. |
[12] | G. Darbo, Punti uniti in transformazioni a condominio non compatto, Rend. Sem. Math. Uni. Padova., 24 (1955), 84-92. |
[13] |
A. Samadi, Applications of measure of noncompactness to coupled fixed points and systems of integral equations, Miskolc Math. Notes, 19 (2018), 537-553. doi: 10.18514/MMN.2018.2532
![]() |
[14] | J. Appell, J. Banas, M. Merentes, Bounded Variation and Around, Series in Nonlinear Analysis and Applications, 17 Walter de Gruyter, Berlin, 2014. |
[15] | D. O'Regan, M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht, 1998. |
1. | Vatan Karakaya, Derya Sekman, A new type of contraction via measure of non-compactness with an application to Volterra integral equation, 2022, 111, 0350-1302, 111, 10.2298/PIM2225111K | |
2. | Hemant Kumar Nashine, Rabha W. Ibrahim, 2021, Chapter 20, 978-981-16-2449-0, 447, 10.1007/978-981-16-2450-6_20 | |
3. | Moosa Gabeleh, Eberhard Malkowsky, Mohammad Mursaleen, Vladimir Rakočević, A New Survey of Measures of Noncompactness and Their Applications, 2022, 11, 2075-1680, 299, 10.3390/axioms11060299 | |
4. | Ayub Samadi, Jamshid Mohammadi, M. Mursaleen, Existence analysis on a coupled multiorder system of FBVPs involving integro-differential conditions, 2022, 2022, 1029-242X, 10.1186/s13660-022-02861-6 | |
5. | Anupam Das, Bhuban Chandra Deuri, Solution of Hammerstein type integral equation with two variables via a new fixed point theorem, 2022, 0971-3611, 10.1007/s41478-022-00537-4 | |
6. | Mesia Simbeye, Santosh Kumar, M. Mursaleen, Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β {c}_{0}^{\beta } and ℓ 1 β {\ell }_{1}^{\beta } , 2024, 57, 2391-4661, 10.1515/dema-2024-0025 |