id	sid	eid	entity	type
ma-156	1	1	2023	CARDINAL
ma-156	2	1	j. math	PERSON
ma-156	4	1	3 (	CARDINAL
ma-156	4	2	2023	CARDINAL
ma-156	4	3	16doi	CARDINAL
ma-156	4	4	10.28924	CARDINAL
ma-156	4	5	generalized geraghty	PERSON
ma-156	4	6	j. c. umudu1,∗	PERSON
ma-156	4	7	j. o. olaleru2	PERSON
ma-156	4	8	olaoluwa2	ORG
ma-156	4	9	a. a. mogbademu2 1department	PERSON
ma-156	4	10	university of jos	ORG
ma-156	4	11	2department	TIME
ma-156	4	12	university of lagos	ORG
ma-156	4	13	nigeria jolaleru@unilag.edu.ng	GPE
ma-156	4	14	holaoluwa@unilag.edu.ng	ORG
ma-156	5	1	α-φ-geraghty	ORG
ma-156	7	1	1	CARDINAL
ma-156	7	2	two	CARDINAL
ma-156	8	1	∈	ORG
ma-156	8	2	d(x	ORG
ma-156	10	1	1	CARDINAL
ma-156	11	1	3	CARDINAL
ma-156	11	2	4	CARDINAL
ma-156	11	3	5	CARDINAL
ma-156	12	1	2	CARDINAL
ma-156	12	2	6	CARDINAL
ma-156	12	3	8	CARDINAL
ma-156	12	4	16],[21	CARDINAL
ma-156	12	5	22	CARDINAL
ma-156	12	6	24	CARDINAL
ma-156	13	1	14	CARDINAL
ma-156	13	2	11	CARDINAL
ma-156	13	3	10	CARDINAL
ma-156	14	1	12	CARDINAL
ma-156	14	2	13	CARDINAL
ma-156	15	1	minx∈a d(x	PERSON
ma-156	15	2	applicationsin	PERSON
ma-156	16	1	7	CARDINAL
ma-156	17	1	9	CARDINAL
ma-156	17	2	10	CARDINAL
ma-156	18	1	11	CARDINAL
ma-156	18	2	12	CARDINAL
ma-156	19	1	13],[14	CARDINAL
ma-156	20	1	15	CARDINAL
ma-156	21	1	17	CARDINAL
ma-156	21	2	18	CARDINAL
ma-156	21	3	20	CARDINAL
ma-156	21	4	23	CARDINAL
ma-156	22	1	1.1	CARDINAL
ma-156	22	2	4	CARDINAL
ma-156	25	1	8 feb 2023	CARDINAL
ma-156	27	1	space.1	ORG
ma-156	28	1	j. math	PERSON
ma-156	30	1	10.28924	CARDINAL
ma-156	31	1	x.	GPE
ma-156	32	1	1.2	CARDINAL
ma-156	32	2	6	CARDINAL
ma-156	33	1	0 ≤	QUANTITY
ma-156	34	1	d(tx	NORP
ma-156	35	1	d(x	ORG
ma-156	35	2	d(x	ORG
ma-156	36	1	0,∞)→	CARDINAL
ma-156	36	2	0	CARDINAL
ma-156	36	3	1	CARDINAL
ma-156	36	4	lim	PERSON
ma-156	36	5	1	CARDINAL
ma-156	36	6	lim	PERSON
ma-156	36	7	n→∞	CARDINAL
ma-156	36	8	0	CARDINAL
ma-156	37	1	22	CARDINAL
ma-156	38	1	1.3	CARDINAL
ma-156	39	1	22	CARDINAL
ma-156	41	1	∈	ORG
ma-156	41	2	α(x	DATE
ma-156	42	1	1	CARDINAL
ma-156	42	2	mt	GPE
ma-156	43	1	d(x	ORG
ma-156	43	2	d(x	ORG
ma-156	44	1	φ	ORG
ma-156	44	2	φ	ORG
ma-156	45	1	0,∞)→	CARDINAL
ma-156	45	2	conditions:(i	ORG
ma-156	45	3	φ	GPE
ma-156	47	1	0	CARDINAL
ma-156	48	1	1.4	CARDINAL
ma-156	49	1	22	CARDINAL
ma-156	52	1	∈	ORG
ma-156	52	2	α(x	DATE
ma-156	52	3	y)φ(d(tx	ORG
ma-156	52	4	2	CARDINAL
ma-156	53	1	mt	ORG
ma-156	53	2	d(x	ORG
ma-156	53	3	d(x	ORG
ma-156	53	4	φ ∈ φ	ORG
ma-156	54	1	φ(t	PERSON
ma-156	54	2	2	CARDINAL
ma-156	54	3	1	CARDINAL
ma-156	55	1	mappingsin	PERSON
ma-156	57	1	j. math	PERSON
ma-156	59	1	10.28924	CARDINAL
ma-156	59	2	19	CARDINAL
ma-156	60	1	22	CARDINAL
ma-156	61	1	1.5	CARDINAL
ma-156	61	2	19	CARDINAL
ma-156	63	1	α(x	DATE
ma-156	63	2	1	CARDINAL
ma-156	63	3	2x	CARDINAL
ma-156	64	1	1	CARDINAL
ma-156	64	2	1.6	CARDINAL
ma-156	64	3	19	CARDINAL
ma-156	65	1	α(x	DATE
ma-156	65	2	1	CARDINAL
ma-156	65	3	α(x	DATE
ma-156	65	4	1	CARDINAL
ma-156	66	1	22	CARDINAL
ma-156	67	1	1.7	CARDINAL
ma-156	69	1	-φ-geraghty	ORG
ma-156	69	2	1	CARDINAL
ma-156	69	3	∈	ORG
ma-156	69	4	generalized α-φ-geraghty	PERSON
ma-156	71	1	2	CARDINAL
ma-156	73	1	∈	ORG
ma-156	74	1	d(x	ORG
ma-156	76	1	d(x	ORG
ma-156	78	1	2.1	CARDINAL
ma-156	78	2	14	CARDINAL
ma-156	79	1	d(x	ORG
ma-156	80	1	d(x	ORG
ma-156	82	1	11	CARDINAL
ma-156	84	1	j. math	PERSON
ma-156	86	1	10.28924	CARDINAL
ma-156	86	2	2.2	CARDINAL
ma-156	86	3	11	CARDINAL
ma-156	87	1	0 ≤	MONEY
ma-156	87	2	d(u	PERSON
ma-156	88	1	d(v	ORG
ma-156	90	1	d(x	ORG
ma-156	90	2	d(x	ORG
ma-156	90	3	d(y , u	ORG
ma-156	90	4	2.2	CARDINAL
ma-156	90	5	1.2	CARDINAL
ma-156	91	1	2.3	CARDINAL
ma-156	92	1	11	CARDINAL
ma-156	94	1	conditionshold:(i	ORG
ma-156	94	2	∅;(ii	GPE
ma-156	94	3	⊆	CARDINAL
ma-156	96	1	∀n ∈ n.any	CARDINAL
ma-156	96	2	2.3	CARDINAL
ma-156	96	3	∈	ORG
ma-156	96	4	a.	PERSON
ma-156	96	5	∈	ORG
ma-156	102	1	2.4	CARDINAL
ma-156	102	2	11	CARDINAL
ma-156	102	3	a0.if	GPE
ma-156	104	1	2.5	CARDINAL
ma-156	105	1	0,∞	DATE
ma-156	106	1	 α(x	ORG
ma-156	107	1	d(v	ORG
ma-156	108	1	1	CARDINAL
ma-156	109	1	j. math	PERSON
ma-156	111	1	10.28924	CARDINAL
ma-156	111	2	2.6	CARDINAL
ma-156	111	3	0,∞	DATE
ma-156	111	4	 α(x	ORG
ma-156	111	5	1	CARDINAL
ma-156	113	1	1	CARDINAL
ma-156	113	2	2.7	CARDINAL
ma-156	114	1	19	CARDINAL
ma-156	116	1	2.8	CARDINAL
ma-156	117	1	plane r2	PERSON
ma-156	117	2	two	CARDINAL
ma-156	117	3	0	CARDINAL
ma-156	117	4	0	DATE
ma-156	117	5	0	CARDINAL
ma-156	117	6	1	CARDINAL
ma-156	117	7	0	CARDINAL
ma-156	117	8	2	CARDINAL
ma-156	117	9	0	CARDINAL
ma-156	117	10	3	CARDINAL
ma-156	117	11	1	CARDINAL
ma-156	117	12	0	DATE
ma-156	117	13	2	CARDINAL
ma-156	117	14	1	CARDINAL
ma-156	117	15	2	CARDINAL
ma-156	117	16	2	CARDINAL
ma-156	117	17	1	CARDINAL
ma-156	117	18	0	CARDINAL
ma-156	117	19	1	CARDINAL
ma-156	117	20	0	DATE
ma-156	117	21	0	CARDINAL
ma-156	117	22	1	CARDINAL
ma-156	117	23	2	CARDINAL
ma-156	117	24	2	CARDINAL
ma-156	117	25	0	CARDINAL
ma-156	117	26	2	CARDINAL
ma-156	117	27	2	CARDINAL
ma-156	117	28	1	CARDINAL
ma-156	117	29	0	CARDINAL
ma-156	117	30	3	CARDINAL
ma-156	117	31	1	CARDINAL
ma-156	117	32	3).also	CARDINAL
ma-156	118	1	0,∞	TIME
ma-156	118	2	α(x	ORG
ma-156	118	3	0	CARDINAL
ma-156	118	4	0	DATE
ma-156	118	5	0	CARDINAL
ma-156	118	6	3	CARDINAL
ma-156	118	7	y ∈ a.	PERSON
ma-156	118	8	d(a	GPE
ma-156	118	9	1	CARDINAL
ma-156	118	10	1 d(v	CARDINAL
ma-156	119	1	1	CARDINAL
ma-156	120	1	∈	ORG
ma-156	120	2	0	CARDINAL
ma-156	120	3	0	DATE
ma-156	120	4	0	CARDINAL
ma-156	120	5	3	CARDINAL
ma-156	121	1	1	CARDINAL
ma-156	123	1	y ∈ a.	PERSON
ma-156	123	2	1	CARDINAL
ma-156	124	1	1	CARDINAL
ma-156	124	2	0	CARDINAL
ma-156	124	3	0	DATE
ma-156	124	4	0	CARDINAL
ma-156	124	5	3	CARDINAL
ma-156	125	1	1	CARDINAL
ma-156	126	1	j. math	PERSON
ma-156	128	1	10.28924	CARDINAL
ma-156	130	1	2.9	CARDINAL
ma-156	130	2	two	CARDINAL
ma-156	131	1	∈	ORG
ma-156	131	2	d(u	PERSON
ma-156	132	1	d(v	ORG
ma-156	134	1	y)φ(d(u	PERSON
ma-156	134	2	3	CARDINAL
ma-156	134	3	mt	ORG
ma-156	135	1	d(x	ORG
ma-156	135	2	d(x	ORG
ma-156	135	3	∈	ORG
ma-156	135	4	φ ∈ φ	ORG
ma-156	136	1	φ(t	PERSON
ma-156	137	1	2.9	CARDINAL
ma-156	138	1	2.10	CARDINAL
ma-156	138	2	two	CARDINAL
ma-156	139	1	∈	ORG
ma-156	139	2	d(u	PERSON
ma-156	141	1	d(v	ORG
ma-156	143	1	4	CARDINAL
ma-156	143	2	mt	GPE
ma-156	144	1	d(x	ORG
ma-156	144	2	d(x	ORG
ma-156	145	1	3.1	CARDINAL
ma-156	146	1	two	CARDINAL
ma-156	147	1	⊆	CARDINAL
ma-156	147	2	x0	ORG
ma-156	147	3	d(x1	GPE
ma-156	148	1	α(x0	CARDINAL
ma-156	148	2	1.then	CARDINAL
ma-156	150	1	α(x	DATE
ma-156	150	2	1	CARDINAL
ma-156	151	1	d(x1	GPE
ma-156	152	1	α(x0	CARDINAL
ma-156	152	2	1	CARDINAL
ma-156	154	1	j. math	PERSON
ma-156	156	1	10.28924	CARDINAL
ma-156	156	2	⊆	CARDINAL
ma-156	158	1	α(x0	CARDINAL
ma-156	158	2	1	CARDINAL
ma-156	161	1	1	CARDINAL
ma-156	163	1	1	CARDINAL
ma-156	164	1	⊆	CARDINAL
ma-156	166	1	α(xi	PERSON
ma-156	166	2	1	CARDINAL
ma-156	167	1	5	CARDINAL
ma-156	167	2	α(xi , xi+1	DATE
ma-156	167	3	1 α(xi+1	CARDINAL
ma-156	167	4	1	CARDINAL
ma-156	169	1	1	CARDINAL
ma-156	170	1	α(xi	GPE
ma-156	170	2	xj	GPE
ma-156	170	3	1	CARDINAL
ma-156	170	4	0 ≤	MONEY
ma-156	171	1	xj−1	GPE
ma-156	173	1	xj−1	GPE
ma-156	174	1	1 ≤	QUANTITY
ma-156	175	1	5	CARDINAL
ma-156	175	2	d(xi	GPE
ma-156	175	3	xi+1	PERSON
ma-156	176	1	0	CARDINAL
ma-156	177	1	3	CARDINAL
ma-156	177	2	φ(d(xi	GPE
ma-156	177	3	xj	GPE
ma-156	177	4	xj−1)φ(d(xi	GPE
ma-156	177	5	xj	GPE
ma-156	178	1	xi−1	NORP
ma-156	178	2	xj−1	GPE
ma-156	178	3	6	CARDINAL
ma-156	179	1	xi−1	NORP
ma-156	179	2	xj−1	GPE
ma-156	179	3	xj−1	GPE
ma-156	179	4	xj	GPE
ma-156	179	5	xj	PERSON
ma-156	182	1	j. math	PERSON
ma-156	184	1	10.28924	CARDINAL
ma-156	184	2	xi−1	NORP
ma-156	184	3	xj−1	GPE
ma-156	185	1	φ(d(xi	GPE
ma-156	185	2	xj	GPE
ma-156	186	1	6	CARDINAL
ma-156	186	2	φ(d(xi	GPE
ma-156	186	3	xj	GPE
ma-156	187	1	xi−1	NORP
ma-156	187	2	xj−1	GPE
ma-156	187	3	xj	GPE
ma-156	187	4	φ(d(xi	GPE
ma-156	187	5	xj	GPE
ma-156	188	1	φ(d(xi	GPE
ma-156	188	2	xj	GPE
ma-156	189	1	xj−1	GPE
ma-156	190	1	xj	GPE
ma-156	191	1	lim	PERSON
ma-156	191	2	xj	GPE
ma-156	193	1	0	CARDINAL
ma-156	194	1	φ(d(xi	GPE
ma-156	194	2	xj	GPE
ma-156	194	3	xj−1	GPE
ma-156	194	4	xi−1	NORP
ma-156	194	5	xj−1	GPE
ma-156	195	1	∈	ORG
ma-156	195	2	lim	PERSON
ma-156	195	3	xi−1	NORP
ma-156	195	4	xj−1	GPE
ma-156	196	1	1	CARDINAL
ma-156	196	2	lim	PERSON
ma-156	196	3	xi−1	NORP
ma-156	196	4	xj−1	GPE
ma-156	197	1	0	CARDINAL
ma-156	197	2	7	CARDINAL
ma-156	197	3	6	CARDINAL
ma-156	198	1	xj	GPE
ma-156	199	1	0	CARDINAL
ma-156	200	1	φ	GPE
ma-156	200	2	d(xi	GPE
ma-156	200	3	xj	GPE
ma-156	203	1	0	CARDINAL
ma-156	204	1	lim	PERSON
ma-156	204	2	d(xi	GPE
ma-156	204	3	xj	GPE
ma-156	205	1	thereexists	NORP
ma-156	205	2	∈	ORG
ma-156	205	3	lim i→∞	PERSON
ma-156	205	4	xi = x∗.	PERSON
ma-156	206	1	⊆	CARDINAL
ma-156	206	2	d(y	GPE
ma-156	207	1	d(xi	GPE
ma-156	208	1	∀n ∈ n	GPE
ma-156	208	2	0	CARDINAL
ma-156	209	1	xi−1)φ(d(y	GPE
ma-156	211	1	j. math	PERSON
ma-156	213	1	10.28924	CARDINAL
ma-156	213	2	1	CARDINAL
ma-156	215	1	φ(d(y	ORG
ma-156	215	2	lim i→∞	PERSON
ma-156	215	3	1 ≤ lim i→∞	QUANTITY
ma-156	217	1	1	CARDINAL
ma-156	218	1	d(y	GPE
ma-156	219	1	0	CARDINAL
ma-156	224	1	y∗	DATE
ma-156	224	2	two	CARDINAL
ma-156	224	3	6=	CARDINAL
ma-156	224	4	1	CARDINAL
ma-156	225	1	d(y∗	PERSON
ma-156	226	1		ORG
ma-156	226	2	y∗	DATE
ma-156	226	3	y∗))φ(mt	ORG
ma-156	226	4	y∗	DATE
ma-156	226	5	mt	GPE
ma-156	226	6	y∗	DATE
ma-156	226	7	y∗	DATE
ma-156	226	8	d(y∗	PERSON
ma-156	230	1	3.2	CARDINAL
ma-156	231	1	two	CARDINAL
ma-156	232	1	⊆	CARDINAL
ma-156	232	2	x0	ORG
ma-156	232	3	d(x1	GPE
ma-156	233	1	α(x0	CARDINAL
ma-156	233	2	1.then	CARDINAL
ma-156	235	1	α(x	DATE
ma-156	235	2	1	CARDINAL
ma-156	237	1	j. math	PERSON
ma-156	239	1	10.28924	CARDINAL
ma-156	239	2	generalized α-φ-geraghty	PERSON
ma-156	240	1	22	CARDINAL
ma-156	245	1	1	CARDINAL
ma-156	245	2	a. abkar	PERSON
ma-156	245	3	m. gabeleh	PERSON
ma-156	245	4	j. optim	PERSON
ma-156	246	1	2011	DATE
ma-156	246	2	188	CARDINAL
ma-156	247	1	sur les operations dans les	ORG
ma-156	247	2	aux	ORG
ma-156	247	3	3 (1922	DATE
ma-156	247	4	133	CARDINAL
ma-156	247	5	s. sadiq basha	PERSON
ma-156	247	6	j. optim	PERSON
ma-156	248	1	151	CARDINAL
ma-156	248	2	2011	DATE
ma-156	248	3	210	CARDINAL
ma-156	248	4	//doi.org/10.1007	GPE
ma-156	248	5	e. karapınar	PERSON
ma-156	248	6	k. sadarangani	PERSON
ma-156	248	7	geraghty	ORG
ma-156	250	1	2013	DATE
ma-156	250	2	2013	DATE
ma-156	250	3	286	CARDINAL
ma-156	251	1	j. caballero	PERSON
ma-156	251	2	j. harjani	PERSON
ma-156	251	3	k. sadarangani	PERSON
ma-156	251	4	geraghty	ORG
ma-156	252	1	2012	DATE
ma-156	252	2	2012	DATE
ma-156	252	3	231	CARDINAL
ma-156	253	1	ciric	PERSON
ma-156	254	1	amer	PERSON
ma-156	256	1	45	CARDINAL
ma-156	256	2	1974	DATE
ma-156	256	3	267-273.[7	QUANTITY
ma-156	256	4	j. math	PERSON
ma-156	258	1	323	CARDINAL
ma-156	258	2	2006),1001-1006	DATE
ma-156	259	1	m. geraghty	PERSON
ma-156	260	1	amer	PERSON
ma-156	262	1	40 (1973	DATE
ma-156	262	2	604-608.[9	CARDINAL
ma-156	262	3	j. hamzehnejadi	PERSON
ma-156	262	4	r. lashkaripour	PERSON
ma-156	263	1	2016	CARDINAL
ma-156	263	2	2016	CARDINAL
ma-156	263	3	72	DATE
ma-156	264	1	https://doi.org/10.1186/ s13663-016-0561-0.[10	PERSON
ma-156	264	2	m. jleli	PERSON
ma-156	264	3	e. karapinar	PERSON
ma-156	264	4	b. samet	PERSON
ma-156	266	1	2013	DATE
ma-156	266	2	2013	DATE
ma-156	266	3	534127	DATE
ma-156	267	1	m. jleli	PERSON
ma-156	267	2	b. samet	PERSON
ma-156	268	1	2014	DATE
ma-156	268	2	2014	DATE
ma-156	268	3	141	CARDINAL
ma-156	269	1	e. karapinar	PERSON
ma-156	269	2	i.m.	PERSON
ma-156	271	1	sci	ORG
ma-156	271	2	5 (2011),558-569.[13	CARDINAL
ma-156	271	3	e. karapinar	PERSON
ma-156	272	1	2013	DATE
ma-156	272	2	2013	DATE
ma-156	272	3	200	CARDINAL
ma-156	276	1	j. math	PERSON
ma-156	278	1	10.28924	CARDINAL
ma-156	279	1	14	CARDINAL
ma-156	279	2	kirk	PERSON
ma-156	279	3	p.s.	PERSON
ma-156	279	4	srinavasan	ORG
ma-156	280	1	4 (2003	DATE
ma-156	281	1	79-89.[15	DATE
ma-156	281	2	c. mongkolkeha	GPE
ma-156	281	3	y.j. cho	GPE
ma-156	281	4	p. kumam	PERSON
ma-156	281	5	geraghty	PERSON
ma-156	282	1	2013	DATE
ma-156	282	2	2013	DATE
ma-156	282	3	180	CARDINAL
ma-156	285	1	8 (	CARDINAL
ma-156	285	2	2007),87-95.[17	DATE
ma-156	286	1	m. abbas	PERSON
ma-156	287	1	j. math	PERSON
ma-156	289	1	optim	PERSON
ma-156	291	1	1 (	CARDINAL
ma-156	291	2	33-54.[18	CARDINAL
ma-156	291	3	j. olaleru	GPE
ma-156	291	4	h. akewe	PERSON
ma-156	292	1	2017	CARDINAL
ma-156	292	2	2017	CARDINAL
ma-156	292	3	6173468	DATE
ma-156	293	1	o. popescu	PERSON
ma-156	293	2	α-geraghty	ORG
ma-156	294	1	2014	DATE
ma-156	294	2	2014	DATE
ma-156	294	3	190	CARDINAL
ma-156	296	1	sankar raj	PERSON
ma-156	298	1	theorymeth	PERSON
ma-156	299	1	74 (2011	DATE
ma-156	299	2	4804-4808	DATE
ma-156	300	1	https://doi.org/10.1016/j.na.2011.04.052.[21] b.e. rhoades	ORG
ma-156	301	1	amer	PERSON
ma-156	303	1	226	CARDINAL
ma-156	303	2	1977	DATE
ma-156	303	3	257-290.[22	CARDINAL
ma-156	303	4	umudu	GPE
ma-156	304	1	olaleru	GPE
ma-156	304	2	a.a. mogbademu	GPE
ma-156	304	3	geraghty	ORG
ma-156	304	4	mappingsin	PERSON
ma-156	305	1	2020	DATE
ma-156	305	2	2020	DATE
ma-156	305	3	16	CARDINAL
ma-156	306	1	https://doi.org/10.1186/ s13663-020-00683-z.[23] j.c	PERSON
ma-156	306	2	umudu	GPE
ma-156	306	3	j.o.	GPE
ma-156	306	4	olaleru	GPE
ma-156	306	5	a.a. mogbademu	GPE
ma-156	306	6	geraghty	DATE
ma-156	306	7	21	CARDINAL
ma-156	306	8	2020	DATE
ma-156	306	9	21-31.[24	DATE
ma-156	306	10	j. umudu	PERSON
ma-156	306	11	a. mogbademu	PERSON
ma-156	306	12	j. olaleru	GPE
ma-156	306	13	geraghty	ORG
ma-156	306	14	caspian	NORP
ma-156	306	15	j. math	PERSON
ma-156	307	1	sci	ORG
ma-156	307	2	11	CARDINAL
ma-156	307	3	2022	CARDINAL
ma-156	307	4	191	CARDINAL
ma-156	309	1	1	CARDINAL
ma-156	309	2	2	CARDINAL
ma-156	309	3	3	CARDINAL
ma-156	309	4	4	CARDINAL